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MathematicalProblems inEngineeringTheory, Methods, and Applications
Guest Editors: Cristian Toma, Carlo Cattani, Ezzat G. Bakhoum, and Ming Li
Special IssuePropagation Phenomena and Transitions in Complex Systems 2012
Hindawi Publishing Corporationhttp://www.hindawi.com
Propagation Phenomena and Transitionsin Complex Systems 2012
Mathematical Problems in Engineering
Propagation Phenomena and Transitionsin Complex Systems 2012
Guest Editors: Cristian Toma, Carlo Cattani,
Ezzat G. Bakhoum, and Ming Li
Copyright q 2012 Hindawi Publishing Corporation. All rights reserved.
This is a special issue published in “Mathematical Problems in Engineering.” All articles are open access articles distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.
Editorial BoardEihab Abdel-Rahman, Canada
Rashid K. Abu Al-Rub, USA
Salvatore Alfonzetti, Italy
Igor Andrianov, Germany
Sebastian Anita, Romania
W. Assawinchaichote, Thailand
Erwei Bai, USA
Ezzat G. Bakhoum, USA
Jose Manoel Balthazar, Brazil
Rasajit K. Bera, India
Jonathan N. Blakely, USA
Stefano Boccaletti, Spain
Daniela Boso, Italy
M. Boutayeb, France
Michael J. Brennan, UK
John Burns, USA
Salvatore Caddemi, Italy
Piermarco Cannarsa, Italy
Jose E. Capilla, Spain
Carlo Cattani, Italy
Marcelo M. Cavalcanti, Brazil
Diego J. Celentano, Chile
Mohammed Chadli, France
Arindam Chakraborty, USA
Yong-Kui Chang, China
Michael J. Chappell, UK
Xinkai Chen, Japan
Kui Fu Chen, China
Kue-Hong Chen, Taiwan
Jyh Horng Chou, Taiwan
Slim Choura, Tunisia
Swagatam Das, India
Filippo de Monte, Italy
M. do R. de Pinho, Portugal
Antonio Desimone, Italy
Yannis Dimakopoulos, Greece
Baocang Ding, China
Joao B. R. Do Val, Brazil
Daoyi Dong, Australia
Balram Dubey, India
Horst Ecker, Austria
M. Onder Efe, Turkey
Elmetwally Elabbasy, Egypt
Alex Elias-Zuniga, Mexico
Anders Eriksson, Sweden
Vedat Suat Erturk, Turkey
Qi Fan, USA
Moez Feki, Tunisia
Ricardo Femat, Mexico
Rolf Findeisen, Germany
R. A. Fontes Valente, Portugal
C. R. Fuerte-Esquivel, Mexico
Zoran Gajic, USA
Ugo Galvanetto, Italy
Xin-Lin Gao, USA
Furong Gao, Hong Kong
Behrouz Gatmiri, Iran
Oleg V. Gendelman, Israel
Didier Georges, France
Paulo Batista Goncalves, Brazil
Oded Gottlieb, Israel
Quang Phuc Ha, Australia
Muhammad R. Hajj, USA
Thomas Hanne, Switzerland
Katica R. Hedrih, Serbia
C. Cruz Hernandez, Mexico
M.I. Herreros, Spain
Wei-Chiang Hong, Taiwan
J. Horacek, Czech Republic
Chuangxia Huang, China
Gordon Huang, Canada
Yi Feng Hung, Taiwan
Hai-Feng Huo, China
Asier Ibeas, Spain
Anuar Ishak, Malaysia
Reza Jazar, Australia
Zhijian Ji, China
J. Jiang, China
J. J. Judice, Portugal
Tadeusz Kaczorek, Poland
Tamas Kalmar-Nagy, USA
Tomasz Kapitaniak, Poland
Hamid Reza Karimi, Norway
Metin O. Kaya, Turkey
Nikolaos Kazantzis, USA
Farzad Khani, Iran
Kristian Krabbenhoft, Australia
Jurgen Kurths, Germany
Claude Lamarque, France
F. Lamnabhi-Lagarrigue, France
Marek Lefik, Poland
Stefano Lenci, Italy
Jian Li, China
Shihua Li, China
Shanling Li, Canada
Tao Li, China
Ming Li, China
Teh-Lu Liao, Taiwan
P. Liatsis, UK
Shueei M. Lin, Taiwan
Jui-Sheng Lin, Taiwan
Yuji Liu, China
Wanquan Liu, Australia
Bin Liu, Australia
Fernando Lobo Pereira, Portugal
Paolo Lonetti, Italy
Vassilios C. Loukopoulos, Greece
Junguo Lu, China
Chien-Yu Lu, Taiwan
Alexei Mailybaev, Brazil
Manoranjan Maiti, India
Oluwole D. Makinde, South Africa
Rafael Martinez-Guerra, Mexico
Bohdan Maslowski, Czech Republic
Driss Mehdi, France
Roderick Melnik, Canada
Xinzhu Meng, China
Yuri Vladimirovich Mikhlin, Ukraine
Gradimir V. Milovanovic, Serbia
Ebrahim Momoniat, South Africa
Trung Nguyen Thoi, Vietnam
Hung Nguyen-Xuan, Vietnam
Ben T. Nohara, Japan
Anthony Nouy, France
Sotiris K. Ntouyas, Greece
Gerard Olivar, Colombia
Claudio Padra, Argentina
Francesco Pellicano, Italy
Vu Ngoc Phat, Vietnam
A. Pogromsky, The Netherlands
Seppo Pohjolainen, Finland
Stanislav Potapenko, Canada
Sergio Preidikman, USA
Carsten Proppe, Germany
Hector Puebla, Mexico
Justo Puerto, Spain
Dane Quinn, USA
Ruben R. Garcıa, Spain
K. R. Rajagopal, USA
Gianluca Ranzi, Australia
Sivaguru Ravindran, USA
G. Rega, Italy
Pedro Ribeiro, Portugal
J. Rodellar, Spain
R. Rodriguez-Lopez, Spain
A. J. Rodriguez-Luis, Spain
Ignacio Romero, Spain
Hamid Reza Ronagh, Australia
Carla Roque, Portugal
Mohammad Salehi, Iran
Miguel A. F. Sanjuan, Spain
Ilmar Ferreira Santos, Denmark
Nickolas S. Sapidis, Greece
Bozidar Sarler, Slovenia
Andrey V. Savkin, Australia
Massimo Scalia, Italy
Mohamed A. Seddeek, Egypt
Alexander P. Seyranian, Russia
Leonid Shaikhet, Ukraine
Cheng Shao, China
Daichao Sheng, Australia
Tony Sheu, Taiwan
Zhan Shu, UK
Dan Simon, USA
Luciano Simoni, Italy
Grigori M. Sisoev, UK
Christos H. Skiadas, Greece
Davide Spinello, Canada
Sri Sridharan, USA
Rolf Stenberg, Finland
Changyin Sun, China
Jitao Sun, China
Xi-Ming Sun, China
Andrzej Swierniak, Poland
Allen Tannenbaum, USA
Cristian Toma, Romania
Irina N. Trendafilova, UK
Alberto Trevisani, Italy
Jung-Fa Tsai, Taiwan
Kuppalapalle Vajravelu, USA
Victoria Vampa, Argentina
Josep Vehi, Spain
Stefano Vidoli, Italy
Xiaojun Wang, China
Dan Wang, China
Youqing Wang, China
Yongqi Wang, Germany
Moran Wang, USA
Yijing Wang, China
Cheng C. Wang, Taiwan
Gerhard-Wilhelm Weber, Turkey
Jeroen A.S. Witteveen, USA
Kwok-Wo Wong, Hong Kong
Zheng-Guang Wu, China
Ligang Wu, China
Wang Xing-yuan, China
X. Frank XU, USA
Xuping Xu, USA
Jun-Juh Yan, Taiwan
Xing-Gang Yan, UK
Suh-Yuh Yang, Taiwan
Mahmoud T. Yassen, Egypt
Mohammad I. Younis, USA
Huang Yuan, Germany
S. P. Yung, Hong Kong
Ion Zaballa, Spain
Arturo Zavala-Rio, Mexico
Ashraf M. Zenkour, Saudi Arabia
Yingwei Zhang, USA
Xu Zhang, China
Liancun Zheng, China
Jian Guo Zhou, UK
Zexuan Zhu, China
Mustapha Zidi, France
Contents
Propagation Phenomena and Transitions in Complex Systems 2012, Cristian Toma,Carlo Cattani, Ezzat G. Bakhoum, and Ming LiVolume 2012, Article ID 251791, 3 pages
Fast Detection of Weak Singularities in a Chaotic Signal Using Lorenz System andthe Bisection Algorithm, Tiezheng Song and Carlo CattaniVolume 2012, Article ID 102848, 10 pages
Fractional Calculus and Shannon Wavelet, Carlo CattaniVolume 2012, Article ID 502812, 26 pages
Parallel Motion Simulation of Large-Scale Real-Time Crowd in a Hierarchical EnvironmentalModel, Xin Wang, Jianhua Zhang, and Massimo ScaliaVolume 2012, Article ID 918497, 15 pages
Optimization of Resource Control for Transitions in Complex Systems, Florin PopVolume 2012, Article ID 625861, 12 pages
Mathematical Models of Dissipative Systems in Quantum Engineering, Andreea Sterian andPaul SterianVolume 2012, Article ID 347674, 12 pages
Power-Law Properties of Human View and Reply Behavior in Online Society, Ye Wu, Qihui Ye,Lixiang Li, and Jinghua XiaoVolume 2012, Article ID 969087, 7 pages
Sinogram Restoration for Low-Dosed X-Ray Computed Tomography Using Fractional-OrderPerona-Malik Diffusion, Shaoxiang Hu, Zhiwu Liao, and Wufan ChenVolume 2012, Article ID 391050, 13 pages
Difference-Equation-Based Digital Frequency Synthesizer, Lu-Ting Ko, Jwu-E. Chen,Yaw-Shih Shieh, Hsi-Chin Hsin, and Tze-Yun SungVolume 2012, Article ID 784270, 12 pages
Kernel Optimization for Blind Motion Deblurring with Image Edge Prior, Jing Wang, Ke Lu,Qian Wang, and Jie JiaVolume 2012, Article ID 639824, 10 pages
Dual-EKF-Based Real-Time Celestial Navigation for Lunar Rover, Li Xie, Peng Yang,Thomas Yang, and Ming LiVolume 2012, Article ID 578719, 16 pages
Hidden-Markov-Models-Based Dynamic Hand Gesture Recognition, Xiaoyan Wang, Ming Xia,Huiwen Cai, Yong Gao, and Carlo CattaniVolume 2012, Article ID 986134, 11 pages
Stable One-Dimensional Periodic Wave in Kerr-Type and Quadratic Nonlinear Media,Roxana Savastru, Simona Dontu, Dan Savastru, Marina Tautan, and Vasile BabinVolume 2012, Article ID 532610, 6 pages
Cutting Affine Moment Invariants, Jianwei Yang, Ming Li, Zirun Chen, and Yunjie ChenVolume 2012, Article ID 928161, 12 pages
Homotopy Perturbation Method and Variational Iteration Method for Harmonic WavesPropagation in Nonlinear Magneto-Thermoelasticity with Rotation, Khaled A. Gepreel,S. M. Abo-Dahab, and T. A. NofalVolume 2012, Article ID 827901, 30 pages
Simplicial Approach to Fractal Structures, Carlo Cattani, Ettore Laserra, and Ivana BochicchioVolume 2012, Article ID 958101, 21 pages
Gaussian Curvature in Propagation Problems in Physics and Engineering, Ezzat G. BakhoumVolume 2012, Article ID 371890, 10 pages
Solving Linear Coupled Fractional Differential Equations by Direct Operational Method andSome Applications, S. C. Lim, Chai Hok Eab, K. H. Mak, Ming Li, and S. Y. ChenVolume 2012, Article ID 653939, 28 pages
Study of the Fractal and Multifractal Scaling Intervening in the Description of FractureExperimental Data Reported by the Classical Work: Nature 308, 721722(1984),Liliana Violeta Constantin and Dan Alexandru IordacheVolume 2012, Article ID 706326, 10 pages
Multidimensional Wave Field Signal Theory: Transfer Function Relationships,Natalie BaddourVolume 2012, Article ID 478295, 27 pages
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 251791, 3 pagesdoi:10.1155/2012/251791
EditorialPropagation Phenomena and Transitions inComplex Systems 2012
Cristian Toma,1 Carlo Cattani,2 Ezzat G. Bakhoum,3 and Ming Li4
1 Faculty of Applied Sciences, University Politehnica of Bucharest, 70709 Bucharest, Romania2 Department of Mathematics, University of Salerno, 84084 Fisciano, Italy3 Department of Electrical and Computer Engineering, University of West Florida,Pensacola, FL 32514, USA
4 School of Information Science and Technology, East China Normal University,No. 500 Dong-Chuan Road, Shanghai 2002411, China
Correspondence should be addressed to Cristian Toma, [email protected]
Received 14 June 2012; Accepted 14 June 2012
Copyright q 2012 Cristian Toma et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
An increasing challenge in advanced engineering applications based on efficient math-
ematical models for propagation and transition phenomena can be noticed nowadays.
Fractal theory and special mathematical functions are used in modeling very small-scale
material properties (energy levels and induced transitions) for the design of nanostructures.
Differential geometry is adapted for solving nonlinear partial differential equations with very
great number of variables for modeling dynamics and transitions in complex optoelectronics
systems. Propagation aspects implying commutative and/or additive consequences of
quantum physics are used extensively in the design of long-range transmission systems. Time
series with extremely high-transmission rates are used for multiplexed transmission systems
for large communities, such as traffic in computer networks or transportations, financial time
series, and time series of fractional order in general. All these advanced engineering subjects
require efficient mathematical models in the development of classical tools for complex
systems. The objective in such applications is to take into consideration efficiency aspects
of mathematical and physical models required by basic phenomena of propagation and
transitions in complex systems, when specific limitations are involved (very long distance
propagation phenomena, fractal aspects and transitions in nanostructures, and complex
systems with great number of variables and infinite spatiotemporal extension of material
media). Using advanced mathematical tools for modeling propagation and transition
phenomena, this special issue presents high qualitative and innovative developments for
eficient mathematical approaches of propagation phenomena and transitions in complex
2 Mathematical Problems in Engineering
systems. Significant results were obtained in the research fields of low-scale physical
structures, propagation of waves in advanced materials, dynamics of complex systems,
and efficient signal and image analysis based on fundamental mathematical and physical
laws.
This special issue involves 19 original papers, selected by the editors so as to present
the most significant results in the previously mentioned topics. These papers are organised
as follows:
(a) Three papers on specific fractal approach for oscillation, propagation and diffusion
properties of low-scale structures: “Fractional calculus and Shannon wavelets” by C.
Cattani, “Simplicial approach to fractal structure” by I. Bochicchio, C. Cattani, and
E. Laserra, and “Sinogram restoration for low-dosed x-ray computed tomography usingfractional-order perona-malik diffusion” by S. Hu, Z. Liao, and W. Chen.
(b) Four papers on specific methods for analysis of complex movements: “Cutting afinemoment invariants” by J. Yang, M. Li, Z. Chen, and Y. Chen, “Dual-ekf based real-timecelestial navigation for lunar rover” by L. Xie, P. Yang, T. T. Yang, and M. Li, “Parallelmotion simulation of large-scale real-time crowd in a hierarchical environmental model” by
X. Wang, J. Zhang, and M. Scalia, and “Hidden Markov models based dynamic handgesture recognition” by X. Wang, M. Xia, H. Cai, Y. Gao, and C. Cattani.
(c) Five papers on accurate and efficient mathematical models for propagation
phenomena: “Gaussian curvature in propagation problems in physics and engineering”
by E. G. Bakhoum, “Multidimensional wave field signal theory: transfer functionrelationships” by N. Baddour, “Homotopy perturbation method and variational iterationmethod for harmonic waves propagation in nonlinear magneto-thermoelasticity withrotation” by S. M. Abo-Dahab, K. A. Gepreel, and T. A. Nofal, “Mathematical modelsof dissipative systems in quantum engineering” by A. Sterian and P. Sterian, and
“Stable one-dimensional periodic wave in kerr-type and quadratic nonlinear media” by R.
Savastru, S. Dontu, D. Savastru, M. Tautan, and V. Babin.
(d) Three papers on mathematical tools for analyzing the dynamics of complex
systems: “Solving linear coupled fractional differential equations by direct operationalmethod and some applications” by S. C. Lim, M. Li, C. H. Eab, K. H. Mak, and S. y.
Chen, “Difference-equation-based digital frequency synthesizer” by L. T. Ko, J. E. Chen,
Y. S. Shieh, H. C. Hsin, and T. Y. Sung, and ”Fast detection of weak singularities ina chaotic signal using Lorenz system and the bisection algorithm” by T. Song and C.
Cattani.
(e) Two papers on efficient image analysis based on fundamental mathematical and
physical laws: “Kernel optimization for blind motion deblurring with image edge prior”by J. Wang, K. Lu, Q. Wang, and J. Jia, and “Power-law properties of human view andreply behavior in online society” by Y. Wu, Q. Ye, J. Xiao, and L. Li.
(f) Two papers on scaling and optimization aspects: “Kernel optimization for blind motiondeblurring with image edge prior” by F. Pop, and “Study of the fractal and multifractalscaling intervening in the description of fracture experimental data reported by the classicalwork” by C. L. Violeta and D. Iordache.
Mathematical Problems in Engineering 3
Acknowledgment
Guest Editor Ming Li would like to acknowledge the supports in part by the 973 plan under
the project number 2011CB302802, and the National Natural Science Foundation of China
under the project Grant numbers 61070214 and 60873264.
Cristian TomaCarlo Cattani
Ezzat G. BakhoumMing Li
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 102848, 10 pagesdoi:10.1155/2012/102848
Research ArticleFast Detection of Weak Singularities ina Chaotic Signal Using Lorenz System andthe Bisection Algorithm
Tiezheng Song1 and Carlo Cattani2
1 School of Electrical Engineering and Automation, Hefei University of Technology, Anhui Province,Hefei City 230009, China
2 Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy
Correspondence should be addressed to Tiezheng Song, [email protected]
Received 1 March 2012; Accepted 1 May 2012
Academic Editor: Cristian Toma
Copyright q 2012 T. Song and C. Cattani. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.
Signals with weak singularities are important for condition monitoring, fault forecasting, andmedicine diagnosis. However, the weak singularity in a signal is usually hidden by strong noise. Anovel fast method is proposed for detecting a weak singularity in a noised signal by determininga critical threshold towards chaos for the Lorenz system. First, a rough critical threshold value iscalculated by local Lyapunov exponents with a step size 0.1. Second, the exact threshold valueis calculated by the bisection algorithm. The advantage of the method will not only reduce thecomputation costs, but also show the weak singular signal which can be accurately identifiedfrom strong noise. When the variance of an external signal method embeds into a Lorenz system,according to the parametric equivalent relation between the Lorenz system and the originalsystem, the critical threshold value of the parameter in a Lorenz system is determined.
1. Introduction
In engineering, most weak singular information often is submerged into strong signals, such
as the peaks, the discontinuities, and so forth. Moreover, when the some weak singular points
are magnified slowly with time, at the moment when the fault occurs, the output signals
usually contain jump points that are often singular points. Therefore, weak singular detection
has played an important role in condition monitoring, fault forecast and medicine diagnosis
[1, 2]. For example, some weak singular vibration signals in machine processes are important
for fault forecasting.
The weak-signal detection is a central problem in the general field of signal processing
and the use of chaos theory in weak-signal detection, and it is also a topic of interest in chaos
2 Mathematical Problems in Engineering
0
10
20
30
40
50
y(t)
0
10
20
30
40
50
y(t)
x(t) t
−20 −10 0 10 20 0 5 10 15 20
Figure 1: r = 24, phase plane of periodic state (sampling time 20 s).
control. At present, however, this research is mainly theory and simulation with MATLAB
in terms of the Duffing-Holmes oscillator [3–6]. Whether other chaos system had better to
characteristic than the Duffing-Holmes oscillator for detecting weak a singular signal. In this
paper, a weak singular signal embedded in the strong signal is detected by Lorenz system.
In 1963, an atmospheric scientist named E.N. Lorenz of M.I.T. proposed a simple model for
thermally induced fluid convection in the atmosphere [7]. In Lorenz’s mathematical model
of convection, three state variables are used (x, y, z). The variable x is proportional to the
amplitude of the fluid velocity circulation in the fluid ring, while y and z measure the
distribution of temperature around the ring. The so-called Lorenz systems may be derived
formally from the Navier-Stokes partial differential equations of fluid mechanics. The Lorenz
model reads in standard notation as follows:
x = a(x − y
),
y = −xz + rx − y,
z = xy − bz.
(1.1)
For a = 10 and b = 8/3 (a favorite set of parameters for experts in the field, integrated with
fourth-order Runge-Kutta method with a fixed step size, t = 0.01 s, there is an attractor for
r = 24 for which the origin is of periodic state (Figure 1). The r = 24.5 gives the phase plane
of chaotic state in which the other two attractors on the x-y plane become unstable spirals
which is called a strange attractor (sometimes called the “butterfly attractor”) and y(t) take
on a complex chaotic trajectory as shown in Figure 2.
Many researchers [8–10] analyze the Lorenz characteristic using r as a control variable.
Upwards, in terms of r = 24 and r = 25, a Lorenz system has proved that there is a huge
difference in the phase space trajectories between the chaotic state and the periodic state,
and this difference can be used for the detection of weak singular signals in strong noise.
Meanwhile, if Lyapunov exponents are adopted as the threshold value evaluated roughly
Mathematical Problems in Engineering 3
0
10
20
30
40
50
y(t)
0
10
20
30
40
50
y(t)
x(t) t
−20 −10 0 10 20 0 10 20 30 40 50
Figure 2: r = 24.5, phase plane of chaotic state (sampling time 50 s).
for a chaotic critical state, the bisection algorithm can fast approach any accurate threshold
value. Thus, when an external signal is embedded into parameter r, one chaotic threshold is
determined conveniently which can detect a weak singular signal in strong noise.
2. The Chaotic Behavior of the Detecting Lorenz byLyapunov Exponent
The Lyapunov exponent (LE) is frequently computed measure for characterizing of chaotic
dynamics. It describes a method for diagnosing whether or not a system is chaotic. For a
discrete mapping x(t+ 1) = f[x(t)], we calculate the local expansion of a flow by considering
the difference of two trajectories as follows:
x(t + 1) − y(t + 1) = f(x(t)) − f(y(t)
) ∂f
∂x[x(t)] ·
[x(t) − y(t)
]. (2.1)
If this grows like
∣∣x(t + 1) − y(t + 1)∣∣ eλ
∣∣x(t) − y(t)∣∣, (2.2)
then the exponent λ is called the Lyapunov exponent. If it is positive, bounded flows will
generally be chaotic. We can solve for this exponent, asymptotically,
λ ln
(∣∣xn+1 − yn+1
∣∣∣∣xn − yn
∣∣). (2.3)
4 Mathematical Problems in Engineering
Since the Lorenz system is in three dimensions, it has three Lyapunov exponents. How effi-
cient and reliable can algorithms to compute Lyapunov exponents be? For three-dimensional
mapping as Lorenz system
xn+1 = f1
(xn,yn, zn
)= a
(xn − yn
),
yn+1 = f2
(xn,yn, zn
)= −xnzn + rxn − yn,
zn+1 = f3
(xn,yn, zn
)= xnyn + bzn.
(2.4)
We get a Jacobian matrix for Lorenz flow
J(xn,yn, zn
)=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
∂f1
∂xn,∂f1
∂yn,∂f1
∂zn
∂f2
∂xn,∂f2
∂yn,∂f2
∂zn
∂f3
∂xn,∂f3
∂yn,∂f3
∂zn
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=
⎡⎣ a −a 0
r − zn −1 −xn
yn xn −b
⎤⎦. (2.5)
Algorithms to compute eigenvalues of matrices are remarkably efficient: supposing the point
successive mapping from the initial point P0(x0,y0, z0) to P1(x1,y1, z1), P2(x2,y2, z2),. . .,Pn(xn,yn, zn), the Jackobian matrix of the previous (n − 1)th point is J0 = J(x0,y0, z0),J1 = J(x1,y1, z1), J2 = J(x2,y2, z2), . . ., and Jn−1 = J(xn−1,yn−1, zn−1). Defining J(n) =Jn−1Jn−2 ···J1J0, the module of eigenvalue for J(n) is J1
(n), J2(n), and J3
(n), and J1(n) > J2
(n) >
J(n)3 , the Lyapunov exponents are defined as follows:
λ1 = limn
n
√J(n)1 , λ2 = lim
n
n
√J(n)2 , λ3 = lim
n
n
√J(n)3 . (2.6)
When (1.1) is in the chaotic state, at least one of the three Lyapunov exponents in (2.6) is
positive. The value is called maximum Lyapunov exponent. The chaotic behavior of the
detection (1.1) is established on the basis of maximum Lyapunov exponents. If the system
is not a point attractor, then the largest exponent cannot be negative. The Lyapunov exponent
links with self-similarity of fractal dimension [11, 12].
3. Threshold Calculated Based on Lyapunov Exponents
To confirm the existence of the weak singular signal, we need to define a proper index
for denoting the change in the states of Lorenz system. The index should be sensitive to a
weak singular signal, but insensitive to the random noise from the viewpoint of statistical
characteristics. Thus, the dynamic properties of Lorenz system are reflected statistically by
Lyapunov exponents which are described in the following as [13–15]:Let initial condition: [0.00001, 0.00001, 0.00001], with about typically 30 points in the
region r = [20, 30] chosen to calculate the Lyapunov exponents (LE), the computation’s
precision of r is two digits after the decimal point, shown in Table 1. The LE curve is plotted
in Figure 3. Obviously, when r = 24.05, (1.1) takes on the chaotic state, and when r = 24.10,
(1.1) takes on the periodic state.
Mathematical Problems in Engineering 5
Table 1: Lyapunov exponents in Lorenz.
No. r Max LE No. r Max LE
1 20 −0.15 16 24.4 0.782
2 20.4 −0.141 17 24.8 0.82
3 20.8 −0.127 18 25.2 0.833
4 21.2 −0.114 19 25.6 0.836
5 21.6 −0.099 20 26 0.844
6 22 −0.087 21 26.4 0.857
7 22.4 −0.074 22 26.8 0.873
8 22.8 −0.06 23 27.2 0.881
9 23.2 −0.047 24 27.6 0.892
10 23.6 −0.033 25 28 0.907
11 23.8 −0.027 26 28.4 0.909
12 24 −0.018 27 28.8 0.922
13 24.05 −0.014 28 29.2 0.923
14 24.1 0.736 29 29.6 0.939
15 24.2 0.758 30 30 0.945
1.2
1
0.8
0.6
0.4
0.2
0
−0.220 21 22 23 24 25 26 27 28 29 30
Ly
ap
un
ov
ex
po
nen
ts
Threshold r
Figure 3: The relational curve between max. LE and r.
The LE changes from positive to negative corresponding to the region r = [24.05, 24.1],and denotes the chaotic system’s extreme sensitivity to the changed parameters. If the
threshold r is equal to 24.05, and computation precision of r is only three effective digits after
decimal point, as the critical threshold between a chaotic and periodic state, the sensitivity
property is not precise enough.
4. Quickly Approaching Critical Threshold withthe Bisection Algorithm
First, the rough region of the system threshold r is estimated by Lyapunov exponents
with computation precision to be one digit after decimal point. Whatever the region of
r = [24.05, 24.1] is always sensitivity region changed from chaotic state to large periodic
state. Since the bisection algorithm can converge to an optimizing solution quickly [16],
6 Mathematical Problems in Engineering
Table 2: Threshold r based on the bisection algorithm in the region r = [24.05, 24.1].
Step Periodic, r1 Phase plane Chaos, r2 Phase plane (r1 + r2)/2 Phase plane
1 24.05 24.1 24.075
2 24.078 24.079 24.0785
3 24.0782 24.0783 24.07825
4 24.07820 24.07821
the threshold value is determined by the bisection algorithm in the region r = [24.05, 24.1].For the initial condition [0.00001, 0.00001, 0.00001], in order to improve the sensitivity of (1.1),the computation precision of r has risen from five digits after decimal point. The steps are as
follows:
(1) Because 24.1 corresponds to the chaotic state and 24.05 corresponds to the periodic
state, r = 24.075 is the midpoint value between 24.05 (chaotic) and 24.1 (periodic).
(2) Because r = 24.075 corresponds to periodic states, the region of r is [24.075, 24.1].Then r is accumulated from 24.075 to 24.1 with the step 0.001 up to 24.079 which
corresponds to the chaotic state and 24.078 which corresponds to the periodic state.
The 24.0785 is the middle value between 24.078 (periodic) and 24.079 (chaotic).
(3) Because r = 24.0785 corresponds to chaotic state, the region of r is taken [24.078,
24.0785]. Then r is accumulated from 24.078 to 24.0785 with the step 0.0001 up to
r = 24.0783 which corresponds to chaotic state and r = 24.0782 which corresponds
to periodic state. The 24.07825 is the middle value between 24.0782 (periodic) and
24.0783 (chaotic).
(4) Because r = 24.07825 corresponds to the chaotic state, the interval of r is [24.0782,
24.07825]. Then r is accumulated from 24.0782 to 24.07825 with the step 0.00001 up
to 24.07821 which corresponds to chaotic state.
(5) Finally, the threshold value calculated is 24.07820. When a weak noisy signal is
merged into (1.1), it takes on the large-scale chaotic state. The calculating process is
shown Table 2.
4.1. How Is an External Signal Merged into Lorenz System
The Lorenz system of differential equations contains the item of the power two, where a, b
are constants, parameter r can be motivated by exterior stimulations to generate a chaotic
trajectory or periodic trajectory. We can adjust the amplitude r of the reference signal to the
special value as in the chaotic critical state. The value is called the threshold value. How will
the external signal be embedded into the control variable r in (1.1)?The variance of a random signal is a measure of its statistical dispersion, indicating
how far from the expected value its values typically are. The variance of a real-valued random
Mathematical Problems in Engineering 7
0 200 400 600 800 1000
0
200
400
600
800
−200
−400
−600
−800
f(t)
t
Figure 4: Random signal with variance 7.59 × 104 (r = 24.0782, sampling time 20 s).
signal is its second central moment, and the variance is simply the square of the standard
deviation and also happens to be its second cumulant.
Let the time sequence of a random signal f(t) be x1,x2,x3, . . . ,xN.
The mean value is x = 1/N∑N
i=1 xi, and the variance of the time sequence is
σ =1
N − 1
N∑i=1
(xi − x)2, (4.1)
Since variance determines within what range values concentrated in a series fluctuate
around the series mean and provides a quantitative measure of these fluctuations [17], the
variance of an external signal f(t) is merged into r, that is as follows:
r = r0 + var(f(t)
). (4.2)
Var(f(t)) is variance function in MATLAB.
So long as the threshold is adjusted appropriately, the behavior of the Lorenz system
will be changed dramatically from chaotic states to periodic states. Then Lorenz equation
becomes
x = a(x − y
),
y = −xz +[r0 + var
(f(t)
)]x − y,
z = xy − bz.
(4.3)
8 Mathematical Problems in Engineering
0
10
20
30
40
50
y(t)
0
10
20
30
40
50
y(t)
x(t) t
−20 −10 0 10 20 0 5 10 15 20
Figure 5: x-y plane of f(t) merged into (4.3) (r = 24.0782, sampling time 20 s).
0 200 400 600 800 10000
50
100
150
200
250
300
350
400
s(t)
t
Figure 6: One-pulse signal (sampling time 50 s).
In (4.3), random signal f(t) is shown as Figure 4, its variance is var(f(t)) = 7.59 × 104,
then r0 = r − var(f(t)) = 24.07820 − 7.59 = 16.4882, (4.3) takes on periodic state (Figure 5).When one weak noisy signal s(t) (Figure 6) is merged into input f(t) (Figure 7), that is
f1(t) = S(m) + f(t), the variance of f1(t) is Var(f1(t)) = 8.04 × 104 = 8.04, r = r0 + var(S(m)) =16.4882 + 8.04 = 24.5282, then (4.3) takes on chaotic state (Figure 8).
5. Conclusion
Since the bisection algorithm can quickly converge to the critical threshold whose precision
can be changed freely, searching any precision grade of the critical threshold of a Lorenz
system will spent less time. If the variance of a random signal has a constant or a limited
range band, in case weak-singularities signal happens and arouses the variance of the random
signal to change infinitely small, the weak singularities signal can be detected.
Mathematical Problems in Engineering 9
0 200 400 600 800 1000
0
200
400
600
800
−200
−400
−600
−800
f1(t)
t
Figure 7: f1(t) = s(t) + f(t) (sampling time 50 s).
0
10
20
30
40
50
y(t)
x(t)
−20 −10 0 10 20
Figure 8: x-y plane of r = 24.5282 (sampling time 50 s).
Since the Runge-Kutta method of fourth-order is one kind of approximate solution
method for dynamic equations, a difference time step size will impact the computation’s pre-
cision for the threshold value.
References
[1] C. Toma, “Advanced signal processing and command synthesis for memory-limited complexsystems,” Mathematical Problems in Engineering, Article ID 927821, 13 pages, 2012.
[2] C. Cattani, “Wavelet based approach to fractals and fractal signal denoising,” Transactions onComputational Science VI, vol. 5730, pp. 143–162, 2009.
[3] N.-Q. Hu, X.-S. Wen, and M. Chen, “application of the Duffing chaotic oscillator for early faultdiagnosis-I. Basic theory,” International Journal of Plant and Management, vol. 7, no. 2, pp. 67–75, 2006.
10 Mathematical Problems in Engineering
[4] Y. Li and B. Yang, “Chaotic system for the detection of periodic signals under the background ofstrong noise,” Chinese Science Bulletin, vol. 48, no. 5, pp. 508–510, 2003.
[5] D. Liu, H. Ren, L. Song, and H. Li, “Weak signal detection based on chaotic oscillator,” in Proceedingsof the IEEE Industry Applications Conference, 40th IAS Annual Meeting, pp. 2054–2058, October 2005.
[6] B. Le, Z. Liu, and T. Gu, “Chaotic oscillator and other techniques for detection of weak signals,” IEICETransactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E88-A, no. 10,pp. 2699–2701, 2005.
[7] F. C. Moon, Chaotic and Fractal Dynamics, A Wiley-Interscience Publication, John Wiley & Sons, NewYork, NY, USA, 1992.
[8] C. K. Chen, J. J. Yan, T. L. Liao, and M. L. Hung, “Chaos suppression of generalized lorenz system:adaptive fuzzy sliding mode control approach,” in Proceedings of the IEEE Conference on Soft Computingon Industrial Applications (SMCia’08), pp. 318–321, June 2008.
[9] I. Pehlivan and Y. Uyaroglu, “A new chaotic attractor from general Lorenz system family andits electronic experimental implementation,” Turkish Journal of Electrical Engineering and ComputerSciences, vol. 18, no. 2, pp. 171–184, 2010.
[10] D. S. Lehrman, “A critique of Konrad Lorenz’s theory of instinctive behavior,” The Quarterly Reviewof Biology, vol. 28, no. 4, pp. 337–363, 1953.
[11] M. Li and W. Zhao, “Visiting power laws in cyber-physical networking systems,” MathematicalProblems in Engineering, vol. 2012, Article ID 302786, 13 pages, 2012.
[12] M. Li, C. Cattani, and S. Y. Chen, “Viewing sea level by a one-dimensional random function with longmemory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 13 pages, 2011.
[13] Q. H. Alsafasfeh and M. S. Al-Arni, “New chaotic behavior from lorenz and rossler systems and itselectronic circuit implementation,” Circuits and Systems, vol. 2, pp. 101–105, 2011.
[14] A. M. Al-Roumy, “The study of a new lorenz-like model,” Journal of Basrah Researches, vol. 37, no. 3 A,2011.
[15] M. Moghtadaei and M. R. H. Golpayegani, “Complex dynamic behaviors of the complex Lorenzsystem,” Scientia Iranica, vol. 19, no. 3, pp. 733–738, 2012.
[16] A. K. Kaw, E. E. Kalu, and D. Ngyen, Numerical Methods with Applications, 1st edition, 2008, http://numericalmethods.eng.usf.edu/topics/textbook index.html.
[17] R. G. Lyons, Understanding Digital Signal Processing, Prentice Hall PTR, 2004.
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 502812, 26 pagesdoi:10.1155/2012/502812
Research ArticleFractional Calculus and Shannon Wavelet
Carlo Cattani
Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy
Correspondence should be addressed to Carlo Cattani, [email protected]
Received 18 February 2012; Accepted 13 May 2012
Academic Editor: Cristian Toma
Copyright q 2012 Carlo Cattani. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
An explicit analytical formula for the any order fractional derivative of Shannon wavelet is givenas wavelet series based on connection coefficients. So that for any L2(R) function, reconstructedby Shannon wavelets, we can easily define its fractional derivative. The approximation error isexplicitly computed, and the wavelet series is compared with Grunwald fractional derivative byfocusing on the many advantages of the wavelet method, in terms of rate of convergence.
1. Introduction
Shannon wavelet theory [1, 2] is based on a family of orthogonal functions having many
interesting properties. They enjoy the many advantages of wavelets [3, 4]; moreover, being
analytical functions they are infinitely differentiable. Thus, enabling us to define the so-
called connection coefficients [5–7] for any order derivative. Connection coefficients are an
expedient tool for the projection of differential operators, useful for computing the wavelet
solution of integrodifferential equations [8–13].Wavelets are localized functions, in time and/or frequency, which are the basis for
energy-bounded functions and in particular for L2(R)-functions. So that localized pulse
problems [14, 15] can be easily approached and analyzed. Moreover, wavelet allows the
multiscale decomposition of problems, thus emphasizing the contribution of each scale. By
defining a suitable inner product on the orthogonal family of scaling/wavelet functions, any
L2(R)-function can be approximated at a fixed scale, by a truncated series having, as basis, the
scaling functions and the wavelet functions. The wavelet coefficients of these series represent
the contribution of each scale.
Shannon wavelets are related to the harmonic wavelets [3, 5, 8], being the real part
thereof, and to the well-known sinc function, which is the basic function in signal analysis.
It should be also noticed that, as compared with other wavelet families, the main advantage
2 Mathematical Problems in Engineering
of Shannon wavelets is that they are analytical functions, thus being infinitely differentiable.
Moreover, they are sharply bounded in the frequency domain, so that, by taking into account
the Parseval identity, any computation can be easily performed by their Fourier transforms.
The theory of connection coefficients was initially given [10, 13] for the compactly
supported wavelet families, such as the Daubechies wavelets [4]. The computation of these
connection coefficients was based on the recursive equations of the wavelet theory and the
explicit forms of these coefficients were given only up to the second order derivatives. The
connection coefficients are the wavelet coefficients of the derivatives of the wavelet basis.
These coefficients are a fundamental tool for the approximation of differential operators, with
respect to the wavelet basis.
In some recent papers, the connection coefficients for Shannon wavelets have been
explicitly computed up to any order derivative with a finite analytical form. This is due to
the analytical form of Shannon wavelets and the discovery by Cattani of a suitable series
expansion for the connection coefficients [2, 6, 7].In the following, we will define the wavelet representation of fractional derivative,
so that the fractional derivative of an L2(R)-function can be easily computed by knowing
the connection coefficients. The fractional derivatives of the Shannon scaling/wavelet basis
are defined and the error of the approximation will be explicitly computed. Moreover, a
comparison with the classical definition of Grunwald formula [16, 17] is given, by showing
the major performance of wavelets, in terms of rate of convergence.
In particular, Section 2 gives some preliminary remarks, definitions, and properties
about Shannon wavelets. Their corresponding connection coefficients are discussed in
Section 3. This Section deals with some properties of connection coefficients, functional
equalities, and error of approximation. Fractional derivatives of the Shannon scaling function
and wavelets are given in Section 4. In this section, it is also shown that the fractional
derivative is a semigroup. The error of the approximation is explicitly computed and
compared with classical definitions of the fractional derivative, and in particular with the
Grunwald formula.
2. Preliminary Remarks
In this section, some remarks on Shannon wavelets and connection coefficients are given (see
also [7]).Shannon wavelet theory (see e.g. [1, 2, 6, 7, 9]) is based on the scaling function ϕ(x),
also known as sinc function, and the wavelet function ψ(x), respectively, defined as
ϕ(x) = sincxdef=
sin πx
πx=
eπix − e−πix
2πix,
ψ(x) =sin 2π(x − (1/2)) − sin π(x − (1/2))
π(x − (1/2))
=e−2 i π x
(−i + ei π x + e3 i π x + i e4 i π x)
2π(x − (1/2)).
(2.1)
Mathematical Problems in Engineering 3
The corresponding families of translated and dilated instances wavelet [1, 2, 6, 7, 9], on which
is based the multiscale analysis [4], are
ϕnk(x) = 2n/2ϕ(2nx − k) = 2n/2 sin π(2nx − k)
π(2nx − k)
= 2n/2 eπi(2nx−k) − e−πi(2
nx−k)
2πi(2nx − k),
ψnk (x) = 2n/2 sin 2π(2nx − k − (1/2)) − sin π(2nx − k − (1/2))
π(2nx − k − (1/2))
=2n/2
2π(2nx − k − (1/2))
2∑s=1
i1+sesπi(2nx−k) − i1−se−sπi(2
nx−k) ,
(2.2)
being, in particular,
ϕ00(x) = ϕ(x), ψ0
0(x) = ψ(x), ϕ0k(x) = ϕk(x) = ϕ(x − k),
ψ0k(x) = ψk(x) = ψ(x − k).
(2.3)
Let
f(ω) = f(x) def=1
2π
∫∞
−∞f(x)e−iωxdx, f(x) =
∫∞
−∞f(ω)eiωxdω (2.4)
be the Fourier transform of the function f(x) ∈ L2(R), and its inverse transform, respectively.
The Fourier transform of (2.1) give us [2]
ϕ(ω) =1
2πχ(ω + 3π) =
⎧⎨⎩1
2π, −π ≤ ω < π
0, elsewhere
ψ(ω) =1
2πeiω/2
[χ(2ω) + χ(−2ω)
],
(2.5)
with
χ(ω) =
{1, 2π ≤ ω < 4π
0, elsewhere.(2.6)
Analogously for the dilated and translated instances of scaling/wavelet function, in
the frequency domain, it is
ϕnk(ω) =
2−n/2
2πeiωk/2nχ
(ω
2n+ 3π
)ψnk (ω) =
2−n/2
2πeiω(k+1/2)/2n
[χ
(ω
2n−1
)+ χ
(− ω
2n−1
)].
(2.7)
4 Mathematical Problems in Engineering
Both families of Shannon scaling and wavelet are L2(R)-functions therefore, for each
f(x) ∈ L2(R) and g(x) ∈ L2(R), the inner product is defined as
⟨f, g
⟩ def=∫∞
−∞f(x)g(x)dx = 2π
∫∞
−∞f(ω)g(ω)dω = 2π
⟨f , g
⟩, (2.8)
where the bar stands for the complex conjugate.
Shannon wavelets fulfill the following orthogonality properties (for the proof see e.g.,
[2, 7]):
⟨ψnk (x), ψ
mh (x)
⟩= δnmδhk,
⟨ϕ0k(x), ϕ
0h(x)
⟩= δkh ,
⟨ϕ0k(x), ψ
mh (x)
⟩= 0, m ≥ 0, (2.9)
δnm, δhk being the Kronecker symbols.
2.1. Properties of the Shannon Wavelet
According to (2.2), Shannon wavelets can be easily computed at some special points, being
in particular
ϕk(h) = ϕh(k) = ϕ(h − k) = ϕ(k − h) = δkh, (h, k ∈ Z), (2.10)
so that
ϕk(x) =
{0, x = h/= k, (h, k ∈ Z)1, x = h = k, (h, k ∈ Z).
(2.11)
It is also [7]
ψnk (h) = (−1)2nh−k 21+n/2(
2n+1h − 2k − 1)π,
(2n+1h − 2k − 1/= 0
)
ψnk (x) = 0, x = 2−n
(k +
1
2± 1
3
), (n ∈ N, k ∈ Z)
limx→ 2−n(h+(1/2))
ψnk (x) = −2n/2δhk.
(2.12)
In the following, we will be interested on the maximum values of these functions
which can be easily computed. The maximum value of the scaling function ϕk(x) can be
found at the integers x = k
max[ϕk(xM)
]= 1, xM = k, (2.13)
Mathematical Problems in Engineering 5
and the max values of ψnk(x) are
max[ψnk (xM)
]= 2n/2 3
√3
π, xM =
⎧⎪⎪⎨⎪⎪⎩−2−n
(k +
1
6
)2−n−1
3(18k + 7).
(2.14)
Both families of scaling and wavelet functions belong to L2(R), thus having a bounded
range and (slow) decay to zero
limx→±∞
ϕnk(x) = 0, lim
x→±∞ψnk (x) = 0. (2.15)
Let B ⊂ L2(R) the set of functions f(x) in L2(R) such that the integrals
αkdef=⟨f(x), ϕk(x)
⟩ (2.8)=
∫∞
−∞f(x)ϕ0
k(x)dx
βnkdef=⟨f(x), ψn
k (x)⟩ (2.8)
=∫∞
−∞f(x)ψn
k(x)dx
(2.16)
exist with finite values, then it can be shown [2–4, 7] that the series
f(x) =∞∑
h=−∞αh ϕh(x) +
∞∑n=0
∞∑k=−∞
βnkψnk (x) (2.17)
converges to f(x).According to (2.8), the coefficients can be also computed in the Fourier domain [7] so
that
αk =∫π
−πf(ω) eiωkdω,
βnk = 2−n/2
[∫2n+1π
2nπ
f(ω)eiω(k+1/2)/2ndω +∫−2nπ
−2n+1π
f(ω)eiω(k+1/2)/2ndω
].
(2.18)
In the frequency domain, (2.17) gives [7]
f(ω) =1
2πχ(ω + 3π)
∞∑h=−∞
αhei ωh
+1
2πχ
(ω
2n−1
) ∞∑n=0
∞∑k=−∞
2−n/2βnkeiω(k+1/2)/2n
+1
2πχ
(− ω
2n−1
) ∞∑n=0
∞∑k=−∞
2−n/2βnkeiω(k+1/2)/2n .
(2.19)
6 Mathematical Problems in Engineering
When the upper bound for the series of (2.17) is finite, then we have the approximation
f(x) ∼=K∑
h=−Kαhϕh(x) +
N∑n=0
S∑k=−S
βnkψnk (x). (2.20)
The error of the approximation has been estimated in [7].
2.2. Reconstruction of the Derivatives
In order to represent the differential operators in wavelet bases, we have to compute the
wavelet decomposition of the derivatives. It can be shown [2, 7] that the derivatives of the
Shannon wavelets are orthogonal functions:
d
dx ϕh(x) =
∞∑k=−∞
λ( )hk
ϕk(x),
d
dx ψmh (x) =
∞∑n=0
∞∑k=−∞
γ ( )mnhk ψn
k (x),
(2.21)
being
λ( )kh
def=
⟨d
dx ϕ0k(x), ϕ
0h(x)
⟩, γ ( )
mnkh
def=
⟨d
dx ψnk (x), ψ
mh (x)
⟩, (2.22)
the connection coefficients [2, 5, 6, 8–13].The computation of connection coefficients can be easily performed in the Fourier
domain, thanks to the equality (2.8)
λ( )kh
= 2π
⟨ d
dx ϕk(x), ϕh(x)
⟩, γ ( )
mnkh = 2π
⟨ d
dx ψnk(x), ψm
h(x)
⟩. (2.23)
In fact, in the Fourier domain, the -order derivative of the (scaling) wavelet functions
are simply
d
dx ϕnk(x) = (iω) ϕn
k(ω),d
dx ψnk(x) = (iω) ψn
k (ω), (2.24)
and, according to (2.7),
d
dx ϕnk(x) = (iω)
2−n/2
2πeiωk/2nχ
(ω
2n+ 3π
),
d
dx ψnk(x) = (iω)
2−n/2
2πeiω(k+(1/2))/2n
[χ
(ω
2n−1
)+ χ
(− ω
2n−1
)].
(2.25)
Mathematical Problems in Engineering 7
It has been shown [2, 6, 7] that the any order connection coefficients (2.22)1 of the
Shannon scaling functions ϕk(x) are
λ( )kh
=
⎧⎪⎪⎪⎨⎪⎪⎪⎩(−1)k−h
i
2π
∑s=1
!πs
s![i(k − h)] −s+1
[(−1)s − 1
], k /=h
i π +1
2π( + 1)
[1 + (−1)
], k = h,
(2.26)
or, by defining
μ(m) = sign(m) =
⎧⎪⎪⎨⎪⎪⎩1, m > 0
−1, m < 0
0, m = 0,
(2.27)
shortly as,
λ( )kh
=i π
2( + 1)
[1 + (−1)
](1 − ∣∣μ(k − h)
∣∣)+ (−1)k−h
∣∣μ(k − h)∣∣ i
2π
∑s=1
!πs
s![i(k − h)] −s+1
[(−1)s − 1
],
(2.28)
when ≥ 1, and for = 0,
λ(0)kh
= δkh. (2.29)
For the proof see [2].Analogously for the connection coefficients (2.22)2 we have that the any order
connection coefficients of the Shannon scaling wavelets ψnk(x) are
γ ( )nmkh = μ(h − k)δnm
{ +1∑s=1
(−1)[1+μ(h−k)](2 −s+1)/2 !i −s π −s
( − s + 1)!|h − k|s (−1)−s−2(h+k)2n −s−1
×{
2 +1[(−1)4h+s + (−1)4k+
]− 2s
[(−1)3k+h+ + (−1)3h+k+s
]}}, k /=h
γ ( )nmkh = δnm
[i π 2n −1
+ 1
(2 +1 − 1
)1 +
((−1)
)], k = h,
(2.30)
8 Mathematical Problems in Engineering
or, shortly
γ ( )nmkh = δnm
{i (1 − ∣∣μ(h − k)
∣∣)π 2n −1
+ 1
(2 +1 − 1
)(1 + (−1)
)
+ μ(h − k) +1∑s=1
(−1)[1+μ(h−k)](2 −s+1)/2 !i −s π −s
( − s + 1)! |h − k|s (−1)−s−2(h+k)2n −s−1
×{
2 +1[(−1)4h+s + (−1)4k+
]− 2s
[(−1)3k+h+ + (−1)3h+k+s
]}},
(2.31)
for ≥ 1, and
γ (0)nmkh = δkhδ
nm, (2.32)
= 0, respectively.
For the proof see [2].
3. Remarks on Connection Coefficients
3.1. Recursiveness
The connection coefficients fulfill some recursive formula as follows.
Theorem 3.1. The connection coefficients (2.26) are recursively given by
λ( +1)kh
=
⎧⎪⎪⎪⎨⎪⎪⎪⎩ + 1
k − hλ( )kh
− (−1)k−hi π +1
k − h
[(−1) + 1
], k /=h
iπ + 1
+ 2λ( )kh
+(−i) +1π +1
+ 2, k = h,
(3.1)
Proof. Let us show first when k = h. From the definition (2.26), it is
λ( +1)kk
=i +1π +2
2π( + 2)
[1 + (−1) +1
]= iπ
( + 1)( + 2)
i π +1
2π( + 1)
[1 + (−1) +1 + (−1) − (−1)
]= iπ
( + 1)( + 2)
i π +1
2π( + 1)
[1 + (−1) + 2(−1) +1
],
(3.2)
from where (3.1)2 follows. Analogously with simple computation we obtain (3.1)1.
Mathematical Problems in Engineering 9
Shorty and with some caution, (3.1) can be written as
λ( +1)kh
= (1 − δkh)
[ + 1
k − hλ( )kh
− (−1)k−hi π +1
k − h
[(−1) + 1
]]
+ δkh
[iπ
+ 1
+ 2λ( )kh
+(−i) +1π +1
+ 2
],
(3.3)
that is,
λ( +1)kh
=[(1 − δkh)
+ 1
k − h+ δkhiπ
+ 1
+ 2
]λ( )kh
− (1 − δkh)(−1)k−hi π +1
k − h
[(−1) + 1
]+ δkh
(−i) +1π +1
+ 2.
(3.4)
It is not so easy to find out a similar property also for the γ-coefficients as a function of
however, there is a simple rule for the recursiveness of the scale (upper) indexes, as follows.
Theorem 3.2. The connection coefficients (2.30) are recursively given by the matrix at the lowestscale level:
γ ( )nn
kh = 2 (n−1)γ ( )11kh . (3.5)
Proof. As can be seen from (2.30) parameter n appears only in the factor
2n −1, (3.6)
so that (3.5) follows from the identity
2n −1 = 2 (n−1)2 −1. (3.7)
Moreover, it can be shown also that
γ (2 +1)nnkh = −γ (2 +1)nnhk , γ (2 )nnhk = γ (2 )
nnhk . (3.8)
3.2. Taylor Series
By using the connection coefficients, it is easy to show the following theorem.
10 Mathematical Problems in Engineering
Theorem 3.3. If f(x) ∈ Bψ ⊂ L2(R) and f(x) ∈ CS the Taylor series of f(x) in x0 is
f(x) = f(x0)
+S∑r=1
[ ∞∑h,k=−∞
αh λ(r)hkϕk(x0) +
∞∑n=0
∞∑k,s=−∞
2r(n−1)βnkγ(r)11
skψns (x0)
](x − x0)r
r!+ RS(x,x0),
(3.9)
being αh and βnkgiven by (2.16), (2.18) and RS(x,x0) the error.
Proof. From (2.17), the -order derivative ( ≤ S) is
f ( )(x) =∞∑
h=−∞αh
d
dx ϕh(x) +
∞∑n=0
∞∑k=−∞
βnkd
dx ψnk (x),
(2.21)=
∞∑h=−∞
αh
∞∑k=−∞
λ( )hkϕk(x) +
∞∑n=0
∞∑k=−∞
βnk
∞∑m=−∞
∞∑s=−∞
γ ( )mnsk ψm
s (x),
=∞∑
h,k=−∞αh λ
( )hkϕk(x) +
∞∑n,m=0
∞∑k,s=−∞
βnkγ( )mn
sk ψms (x),
(3.10)
so that by taking into account (3.5) the proof follows.
In particular, by a suitable choice of the initial point x0, (3.9) can be simplified. For
instance, at the integers, x0 = h, (h ∈ Z), according to (2.10), (2.12) and (3.5), it is
f(x) ∼= f(h) +S∑r=1
[ ∞∑h=−∞
αh λ(r)hh
+∞∑n=0
∞∑k,s=−∞
(−1)2nh−s 2r(n−1)+1+n/2(2n+1h − 2s − 1
)πβnkγ
(r)11sk
](x − h)r
r!.
(3.11)
3.3. Functional Equations
The connection coefficients fulfill some identities as follows.
Theorem 3.4. For any k ∈ Z and ∈ N, it is
(iω) e−iωk =∞∑
h=−∞λ( )khe−iωh, −π ≤ ω ≤ π, (3.12)
or
(iω) =∞∑
h=−∞λ( )khe−iω(h−k), −π ≤ ω ≤ π, ∀k ∈ Z. (3.13)
Mathematical Problems in Engineering 11
Proof. From (2.21), by a Fourier transform of both sides and taking into account (2.24), we get
(iω) ϕk(ω) =∞∑
h=−∞λ( )khϕh(ω)
(iω) e−iωkχ(ω + 3π)(2.7)=
∞∑h=−∞
λ( )khe−iωhχ(ω + 3π),
(3.14)
from where the identity (3.12) follows.
In particular, by assuming, without restrictions, k = 0, we have the following (see
Figure 1).
Corollary 3.5. For any ∈ N it is
(iω) =∞∑
h=−∞λ( )0he−iωh, −π ≤ ω ≤ π, (3.15)
so that λ( )0h
are the Fourier coefficients of the power (iω) .
Analogously, from (2.21)2, we have the following.
Theorem 3.6. For any k ∈ Z and , n ∈ N it is
(iω) e−iω(k+1/2)/2n =∞∑
h=−∞γ ( )
nnkhe−iω(h+1/2)/2n , ω ∈
[−2n+1π,−2nπ
]∪[2nπ , 2n+1π
], (3.16)
or
(iω) =∞∑
h=−∞γ ( )
nnkhe−iω(h−k)/2n , ω ∈
[−2n+1π ,−2nπ
]∪[2nπ , 2n+1π
]. (3.17)
In particular, with k = 0, and taking into account (3.5), we have the following.
Corollary 3.7. For any , n ∈ N it is
(iω) = 2 (n−1)∞∑
h=−∞γ ( )
11
0he−iωh/2n , ω ∈
[−2n+1π ,−2nπ
]∪[2nπ , 2n+1π
]. (3.18)
As a consequence of the previous theorems we have the following.
Theorem 3.8. For any , n ∈ N it is
(iω) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩∞∑
h=−∞λ( )0he−iωh, −π ≤ ω ≤ π
2 (n−1)∞∑
h=−∞γ ( )
110he−iωh/2n , ω ∈ [−2n+1π,−2nπ
] ∪ [2nπ, 2n+1π].
(3.19)
12 Mathematical Problems in Engineering
−−1
1 x
(a)
−−1
1 x
(b)
−−1
x
(c)
−−1
x
(d)
Figure 1: Approximation of (iω) (plain) by the r.h.s of (3.15) at different scale: (a) �[(iω)3], k =5, |hmax| = 5; (b) �[(iω)3], k = 5, |hmax| = 10; (c) [(iω)2], k = 7, |hmax| = 5; (d) [(iω)2], k = 7, |hmax| =8.
There we have the following.
Corollary 3.9. The Fourier transform of the derivatives of a function is
d
dx f(x) =f(ω) ×
⎧⎪⎪⎪⎨⎪⎪⎪⎩∞∑
h=−∞λ( )0he−iωh, −π ≤ ω ≤ π
2 (n−1)∞∑
h=−∞γ ( )
11khe−iωh/2n , ω ∈ [−2n+1π,−2nπ
] ∪ [2nπ, 2n+1π].
(3.20)
Mathematical Problems in Engineering 13
If we express eiω as a Taylor series we have
eiω =∞∑ =0
(iω)
!, (3.21)
so that eiω with −π ≤ ω ≤ π is the solution of the functional equation
X =∞∑ =0
∞∑h=−∞
1
!λ( )0hX−h. (3.22)
Moreover, the theorem of moments
∫R
x f(x)dx = i df(ω)dω
(3.23)
can be written as
∫R
x f(x)dx = i f(ω) ×
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
∞∑h=−∞
λ( )khe−iωh, −π ≤ ω ≤ π
∞∑h=−∞
γ ( )nnkhe−iω(h−k)/2n , ω ∈ [−2n+1π,−2nπ
] ∪ [2nπ, 2n+1π].
(3.24)
3.4. Error of the Approximation by Connection Coefficients
For a fixed scale of approximation in (2.21), it is possible to estimate the error as follows. It
should be noticed that the approximation depends on a the upper bound of the limits in the
sums.
Theorem 3.10 (error of the approximation of scaling functions derivatives). The error of theapproximation in (2.21)1 is given by
∣∣∣∣∣ d
dx ϕh(x) −
N∑k=−N
λ( )hk
ϕk(x)
∣∣∣∣∣ ≤ ∣∣∣λ( )h(−N−1) + λ( )h(N+1)
∣∣∣. (3.25)
Proof. The error of the approximation (2.21)1 is defined as
d
dx ϕh(x) −
N∑k=−N
λ( )hkϕk(x) =
−N−1∑k=−∞
λ( )hkϕk(x) +
∞∑k=N+1
λ( )hkϕk(x). (3.26)
14 Mathematical Problems in Engineering
Concerning the r.h.s, and according to (2.13), it is
−N−1∑k=−∞
λ( )hk
ϕk(x) +∞∑
k=N+1
λ( )hk
ϕk(x)
≤ maxx∈R
[−N−1∑k=−∞
λ( )hk
ϕk(x) +∞∑
k=N+1
λ( )hk
ϕk(x)
]
= λ( )h(−N−1) ϕ−N−1(x) + λ
( )h(N+1) ϕN+1(x) ≤ λ
( )h(−N−1) + λ
( )h(N+1).
(3.27)
Theorem 3.11 (error of the approximation of wavelet functions derivatives). The error of theapproximation in (2.21)2 is given by
∣∣∣∣∣ d
dx ψmh (x) −
N∑n=0
S∑k=−S
γ ( )mnhk ψn
k (x)
∣∣∣∣∣ ≤∣∣∣∣∣2 (m−1)+m/2 3
√3
π
[γ ( )
11h(−S−1) + γ ( )
11h(S+1)
]∣∣∣∣∣. (3.28)
Proof. The error of the approximation is
d
dx ψmh (x) −
N∑n=0
S∑k=−S
γ ( )mnhk ψn
k (x) =∞∑
n=N+1
[ −S−1∑k=−∞
γ ( )mnhk ψn
k (x) +∞∑
k=S+1
γ ( )mnhk ψn
k (x)
]. (3.29)
If m < N, the r.h.s. according to (2.30) is zero; therefore, we assume that m > N so that the
last equation becomes
d
dx ψmh (x) −
N∑n=0
S∑k=−S
γ ( )mnhk ψn
k (x) =
[ −S−1∑k=−∞
γ ( )mmhk ψn
k (x) +∞∑
k=S+1
γ ( )mmhk ψn
k (x)
]
(3.5)= 2 (m−1)
[ −S−1∑k=−∞
γ ( )11hk +
∞∑k=S+1
γ ( )11hk
]ψmk (x)
≤ 2 (m−1) max
{[ −S−1∑k=−∞
γ ( )11hk +
∞∑k=S+1
γ ( )11hk
]ψmk (x)
}
= 2 (m−1)[γ ( )
11h(−S−1)ψm
(−S−1)(x) + γ ( )11h(S+1)ψm
(S+1)(x)]
≤ 2 (m−1)[γ ( )
11h(−S−1) maxψm
(−S−1)(x) + γ ( )11h(S+1) maxψm
(S+1)(x)]
(2.14)= 2 (m−1)2m/2 3
√3
π
[γ ( )
11h(−S−1) + γ ( )
11h(S+1)
].
(3.30)
Mathematical Problems in Engineering 15
4. Fractional Derivatives of the Wavelet Basis
The simplest way to define the fractional derivative is based on the assumption that the
noninteger derivative of the exponential function formally coincides with the derivative with
integer order so that
dν
dxνeax = aνeax ν ∈ Q. (4.1)
For negative values of ν, this formula still holds true and it represents the integration.
It is known that the fractional derivative cannot be analytically computed except for
some special functions, such as (see e.g., [16–18]) the following:
dν
dxνeax = aνeax ,
dν
dxνcosax = aν cos
(ax +
π
2ν),
dν
dxνsinax = aν sin
(ax +
π
2ν).
(4.2)
From these, classical examples, we can see that the fractional derivative can be also
interpreted as an interpolating function between derivatives with integer order, so that
dν
dxνf(x) = (1 − ν)f(x) + ν
d
dxf(x), 0 ≤ ν ≤ 1. (4.3)
More in general, let f(x) be a single-valued real function, then the Riemann-Liouville
fractional order derivative is defined as [16]
dν
dxνf(x) def=
1
Γ(1 − ν)d
dx
∫x
0
f(ξ)(x − ξ)ν
dξ, (0 < ν < 1, x > 0), (4.4)
Γ(ν) being the gamma function.
Other equivalent representations were given by Caputo (for a differentiable function)
dν
dxνf(x) def=
1
Γ(1 − ν)
∫x
0
f ′(ξ)(x − ξ)ν
dξ, 0 < ν < 1, (4.5)
and by Grunwald (see e.g., [17, 18])
dν
dxνf(x) = lim
N→∞1
Γ(−ν)( x
N
)−νN−1∑k=0
Γ(k − ν)Γ(k + 1)
f
[(1 − k
N
)x
], (0 < ν < 1, x > 0). (4.6)
However, a drawback in the Grunwald definition, as well as in the Riemann-Liouville, is that
it cannot be computed for negative values of the variable (x < 0).
16 Mathematical Problems in Engineering
4.1. Fractional Derivative of the Shannon Scaling Function
Let us assume that the fractional order derivative is defined by a linear interpolation of the
integer order derivatives, so that the fractional derivative of the scaling-wavelet basis
d +ν
dx +νϕh(x),
d +ν
dx +νψmh (x). (4.7)
with
0 ≤ ν ≤ 1, (4.8)
can be defined as
d +ν
dx +νϕh(x)
def= (1 − ν)d
dx ϕh(x) + ν
d +1
dx +1ϕh(x),
d +ν
dx +νψmh (x)
def= (1 − ν)d
dx ψmh (x) + ν
d +1
dx +1ψmh (x).
(4.9)
Let us show the following.
Theorem 4.1. The fractional derivative of the Shannon scaling functions is
d +ν
dx +νϕh(x)
def=∞∑
k=−∞λ( +ν)hk
ϕk(x) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩∞∑
k=−∞
[(1 − ν)λ( )
hk+ νλ
( +1)hk
]ϕk(x), > 0
∞∑k=−∞
[(1 − ν)δhk + νλ
(1)hk
]ϕk(x), = 0.
(4.10)
Proof. From (4.9), by taking into account (2.21), it is
d +ν
dx +νϕh(x)
def= (1 − ν)∞∑
k=−∞λ( )hkϕk(x) + ν
∞∑k=−∞
λ( +1)hk
ϕk(x)
(3.1)=
∞∑k=−∞
[(1 − ν)λ( )
hk+ νλ
( +1)hk
]ϕk(x),
(4.11)
and, when = 0,
dν
dxνϕh(x) =
∞∑k=−∞
[(1 − ν)δhk + νλ
(1)hk
]ϕk(x). (4.12)
With this definition, the fractional order derivative of the scaling functions is a
commutative operator according to the following.
Mathematical Problems in Engineering 17
Theorem 4.2. The operator (4.10) is a semigroup, so that
dμ
dxμ
dν
dxνϕh(x) =
dν
dxν
dμ
dxμϕh(x) =
dμ+ν
dxμ+ν ϕh(x). (4.13)
Proof. Without loss of generality, let us show that
dμ
dxμ
dν
dxνϕ(x) =
dν
dxν
dμ
dxμϕ(x). (4.14)
According to (4.10)2, it is
dν
dxνϕ0(x) =
∞∑k=−∞
[(1 − ν)δ0k + νλ
(1)0k
]ϕk(x), (4.15)
that is
dν
dxνϕ(x) = (1 − ν)ϕ(x) + ν
⎡⎢⎢⎣λ(1)00 ϕ(x) +∞∑
k /= 0k=−∞
λ(1)0kϕk(x)
⎤⎥⎥⎦, (4.16)
and, taking into account (2.26), by explicit computation we have
dν
dxνϕ(x) = (1 − ν)ϕ(x) + ν
∞∑k /= 0k=−∞
(−1)k
kϕk(x). (4.17)
By deriving, with respect to μ, we have
dμ
dxμ
dν
dxνϕ(x) = (1 − ν)
dμ
dxμϕ(x) + ν
∞∑k /= 0k=−∞
(−1)k
k
dμ
dxμϕk(x)
(4.17)= (1 − ν)
⎡⎢⎢⎣(1 − μ)ϕ(x) + μ
∞∑k /= 0k=−∞
(−1)k
kϕk(x)
⎤⎥⎥⎦+ ν
∞∑k /= 0k=−∞
(−1)k
k
dμ
dxμϕk(x),
(4.18)
18 Mathematical Problems in Engineering
that is, according to (2.26),
dμ
dxμ
dν
dxνϕ(x) = (1 − ν)
⎡⎢⎢⎣(1 − μ)ϕ(x) + μ
∞∑k /= 0k=−∞
(−1)k
kϕk(x)
⎤⎥⎥⎦+ ν
∞∑k /= 0k=−∞
(−1)k
k
∞∑s=−∞
[(1 − μ
)δsk + μλ
(1)sk
]ϕs(x)
= (1 − ν)
⎡⎢⎢⎣(1 − μ)ϕ(x) + μ
∞∑k /= 0k=−∞
(−1)k
kϕk(x)
⎤⎥⎥⎦+ ν
(1 − μ
) ∞∑k /= 0k=−∞
(−1)k
kϕk(x) + νμ
∞∑k /= 0k=−∞
(−1)k
k
∞∑s=−∞
λ(1)skϕs(x).
(4.19)
From where,
dμ
dxμ
dν
dxνϕ(x) = (1 − ν)
(1 − μ
)ϕ(x) +
[(1 − ν)μ + ν
(1 − μ
)] ∞∑k /= 0k=−∞
(−1)k
kϕk(x)
+ νμ∞∑
k /= 0k=−∞
(−1)k
k
∞∑s=−∞
λ(1)skϕs(x),
(4.20)
the proof follows due to the symmetry of the change μ → ν.
It can be easily seen that together with (4.17) also the following equations hold:
dν
dxνϕ1(x) = (1 − ν)ϕ1(x) + ν
∞∑k /= 0k=−∞
(−1)k
k − 1ϕk(x)
dν
dxνϕ−1(x) = (1 − ν)ϕ−1(x) + ν
∞∑k /= 0k=−∞
(−1)k
1 + kϕk(x),
(4.21)
and, in general,
dν
dxνϕh(x) = (1 − ν)ϕh(x) + ν
∞∑k /= 0k=−∞
(−1)k
k − hϕk(x). (4.22)
Moreover, when μ + ν = 1, then we can see that the definition (2.26) reduces to the
ordinary derivative, according to the following.
Mathematical Problems in Engineering 19
x
1
−1
= 0
= 1
A
A
Figure 2: Fractional derivative of the scaling functions (dν/dxν)ϕ(x) with upper limit N = 4 at differentvalues of ν = 0, 1/5, 2/5, 3/5, 4/5, 1.
Theorem 4.3. When μ + ν = 1, then
dμ
dxμ
dν
dxνϕh(x) =
dμ+ν
dxμ+ν ϕh(x) =d
dxϕh(x). (4.23)
Proof. If we restrict to ϕ(x), according to the definition (2.26), it is
dμ
dxμ
dν
dxνϕ(x) =
∞∑k=−∞
[(1 − (μ + ν
))δ0k +
(μ + ν
)λ(1)0k
]ϕk(x), (4.24)
and since (μ + ν) = 1 we have
dμ
dxμ
dν
dxνϕ(x) =
d
dxϕ(x) =
∞∑k=−∞
λ(1)0kϕk(x), (4.25)
According to the definition (4.10), the fractional derivative is an interpolation between
integer order derivative (see Figure 2).
4.2. Error of the Approximation of (4.10)
In the definition (4.10), the fractional derivative depends on a fixed bound N of the infinite
series. In this section, it will be shown that the rate of convergence of the series, on the r.h.s of
(4.10), is quite fast; already with low values of N, the approximation is quite good (Figure 3).
20 Mathematical Problems in Engineering
1 < N < 10
−x
(a)
10 < N < 50
−x
(b)
Figure 3: Fractional derivative of the scaling functions (d3/10/dx3/10)ϕ(x) with upper limit N = 1, . . . , 10(a) and N = 10, . . . , 50 (b).
4.2.1. Rate of Convergence
If we compare the fractional derivative (dν/dxν)ϕh(x) given by (4.10) with the Grunwald
definition (4.6), we can see that the approximation by connection coefficients is good (see
Figure 4), with a lower number of terms. Moreover, the definition based on connection
coefficients can be extended also to negative values of the variable.
Since we have defined the fractional derivative on an infinite series N → ∞, as well
as the Grunwald formula, we can explicitly compute the error of the approximation as the
difference between the approximated value at N + 1 and the corresponding value of the
infinite series at N. For instance, with respect to (4.10), it is
ενN = maxx∈R
∣∣∣∣∣∣N+1∑
k=−(N+1)
λ( +ν)hk
ϕk(x) −N∑
k=−Nλ( +ν)hk
ϕk(x)
∣∣∣∣∣∣, (4.26)
while for the Grunwald formula (4.6) we have
ενN = maxx>0
∣∣∣∣∣ 1
Γ(−ν)( x
N + 1
)−ν N∑k=0
Γ(k − ν)Γ(k + 1)
f
[(1 − k
N + 1
)x
]
− 1
Γ(−ν)( x
N
)−νN−1∑k=0
Γ(k − ν)Γ(k + 1)
f
[(1 − k
N
)x
]∣∣∣∣∣,(4.27)
Let us show the following.
Mathematical Problems in Engineering 21
1
−1
2x
(a)
1
−1
2x
(b)
1
−1
2x
(c)
1
−1
2x
(d)
Figure 4: Fractional derivative of the scaling functions (dν/dxν)ϕh(x) by Grunwald approximation (4.6)(shaded) and connection coefficients interpolation (4.10)2 (plain): (a) ν = 1/10, h = 0 with upper limitN = 1 (connection coefficients) and N = 4 (Grunwald); (b) ν = 1/10, h = 1 with upper limit N = 1(connection coefficients) and N = 1 (Grunwald); (c) ν = 1/20, h = 1 with upper limit N = 2 (connectioncoefficients) and N = 8 (Grunwald); (d) ν = 9/10, h = 1 with upper limit N = 10 (connection coefficients)and N = 50 (Grunwald).
Theorem 4.4. For = 0, the approximation error of (4.10)2 is given by
ενN = 2ν
∣∣∣∣∣ (−1)N+1h
(N + 1)2 − h2
∣∣∣∣∣. (4.28)
22 Mathematical Problems in Engineering
Proof. By taking into account (4.22), it is
N+1∑k=−(N+1)
λ( +ν)hk
ϕk(x) −N∑
k=−Nλ( +ν)hk
ϕk(x) = ν
[(−1)N+1
−(N + 1) − hϕ−(N+1)(x)
+(−1)N+1
(N + 1) − hϕ(N+1)(x)
]
(2.13)< ν
[(−1)N+1
−(N + 1) − h+
(−1)N+1
(N + 1) − h
]
=2ν(−1)N+1h
(N + 1)2 − h2.
(4.29)
Analogously, the following can be shown.
Theorem 4.5. For x > 0, the approximation error of (4.6)2 is given by
ενN =Nν
Γ(−ν)Γ(N − ν)Γ(N + 1)
. (4.30)
Proof. At the integer x = 1, it is
1
Γ(−ν)(
1
N + 1
)−ν N∑k=0
Γ(k − ν)Γ(k + 1)
f
[(1 − k
N + 1
)]− 1
Γ(−ν)(
1
N
)−νN−1∑k=0
Γ(k − ν)Γ(k + 1)
f
[(1 − k
N
)]
<1
Γ(−ν)Nν
N∑k=0
Γ(k − ν)Γ(k + 1)
f
[(1 − k
N + 1
)]− 1
Γ(−ν)NνN−1∑k=0
Γ(k − ν)Γ(k + 1)
f
[(1 − k
N
)](2.13)<
Nν
Γ(−ν)Γ(N − ν)Γ(N + 1)
.
(4.31)
4.3. Fractional Derivative of the Shannon Wavelet
Analogously to (4.10), the following can be proved.
Theorem 4.6. The fractional derivative of the Shannon wavelet functions is
d +ν
dx +νψmh (x)
def=∞∑
k=−∞γ ( +ν)
mmhk ψm
k (x)
=
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩2 (m−1)
[ ∞∑k=−∞
(1 − ν)γ ( )11hk + ν 2m−1γ ( +1)11
hk
]ψmk(x), > 0[ ∞∑
k=−∞(1 − ν)δhk + ν2m−1γ (1)
11hk
]ψmk(x), = 0.
(4.32)
Mathematical Problems in Engineering 23
Proof. From (4.9), by taking into account (2.21)
d +ν
dx +νψmh (x) = (1 − ν)
∞∑n=0
∞∑k=−∞
γ ( )mnhk ψn
k (x) + ν∞∑n=0
∞∑k=−∞
γ ( +1)mnhk ψn
k (x)
(3.5)=
[(1 − ν)
∞∑n=0
∞∑k=−∞
δmn2 (n−1)γ ( )11hk + ν
∞∑n=0
∞∑k=−∞
δmn2( +1)(n−1)γ ( +1)11hk
]ψnk (x)
=
[(1 − ν)
∞∑k=−∞
2 (m−1)γ ( )11hk + ν
∞∑k=−∞
2( +1)(m−1)γ ( +1)11hk
]ψmk (x)
= 2 (m−1)
[ ∞∑k=−∞
(1 − ν)γ ( )11hk + ν 2m−1γ ( +1)11
hk
]ψmk (x).
(4.33)
Analogously to the fractional derivative of the scaling function, also for the wavelet
function, the fractional order derivatives are enveloped by the integer order derivatives
(Figure 5).
4.4. Fractional Derivative of an L2(R) Function
Let f(x) ∈ B ⊂ L2(R) be a function such that (2.17) holds, then its fractional derivative can be
computed as
dν
dxνf(x) =
∞∑h=−∞
αhdν
dxνϕh(x) +
∞∑n=0
∞∑k=−∞
βnkdν
dxνψnk (x), (4.34)
where the fractional derivatives of the scaling functions ϕh(x) and wavelets ψnk(x) are given
by (4.10) and (4.32), respectively.
For instance, a good approximation of y = e−x2
is (Figure 6)
e−x2 ∼= 0.97ϕ(x) + 0.39
[ϕ−1(x) + ϕ1(x)
]. (4.35)
The fractional derivative is
dν
dxνe−x
2 ∼= 0.97dν
dxνϕ(x) + 0.39
dν
dxν
[ϕ−1(x) + ϕ1(x)
], (4.36)
24 Mathematical Problems in Engineering
= 1
= 0
x
−1
1
A
A
Figure 5: Fractional derivative of the wavelet functions (dν/dxν)ψ00(x) with upper limit N = 4 at different
values of ν = 0, 1/5, 2/5, 3/5, 4/5, 1.
= 1
= 0
x3−3
−1
1A
A
Figure 6: Fractional derivative of the function y = e−x2
with upper limit N = 4 at different values ofν = 0, 1/5, 2/5, 3/5, 4/5, 1.
Mathematical Problems in Engineering 25
so that by using (4.17) and (4.21) we have
dν
dxνe−x
2 ∼= 0.97
⎡⎢⎢⎣(1 − ν)ϕ(x) + ν∞∑
k /= 0k=−∞
(−1)k
kϕk(x)
⎤⎥⎥⎦+ 0.39(1 − ν)
[ϕ−1(x) + ϕ1(x)
]
+ 0.39ν
⎡⎢⎢⎣ ∞∑k /= 0k=−∞
(−1)k+1
kϕk(x) +
∞∑k /= 0k=−∞
(−1)k
k − 1ϕk(x)
⎤⎥⎥⎦,(4.37)
5. Conclusion
In this paper, fractional calculus has been revised by using Shannon wavelets. Fractional
derivatives of the Shannon scaling/wavelet functions, based on connection coefficients,
are explicitly computed and the approximation error is estimated. In the comparison with
the classical Grunwald formula of fractional derivative, Shannon wavelets and connection
coefficients make a better approximation and rate of convergence.
References
[1] C. Cattani, “Shannon wavelet analysis,” in Proceedings of the International Conference on ComputationalScience (ICCS ’07), Y. Shi, G. D. van Albada, J. Dongarra, and P. M. A. Sloot, Eds., Lecture Notes inComputer Science, LNCS 4488, Part II, pp. 982–989, Springer, Beijing, China, May 2007.
[2] C. Cattani, “Shannon wavelets theory,” Mathematical Problems in Engineering, vol. 2008, Article ID164808, 24 pages, 2008.
[3] C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as applied to Materials with Micro orNanostructure, vol. 74 of Series on Advances in Mathematics for Applied Sciences, World ScientificPublishing, Singapore, 2007.
[4] I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in AppliedMathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1992.
[5] C. Cattani, “Harmonic wavelet solutions of the Schrodinger equation,” International Journal of FluidMechanics Research, vol. 30, no. 5, pp. 463–472, 2003.
[6] C. Cattani, “Connection coefficients of Shannon wavelets,” Mathematical Modelling and Analysis, vol.11, no. 2, pp. 117–132, 2006.
[7] C. Cattani, “Shannon wavelets for the solution of integrodifferential equations,” Mathematical Problemsin Engineering, vol. 2010, Article ID 408418, 22 pages, 2010.
[8] C. Cattani, “Harmonic wavelets towards the solution of nonlinear PDE,” Computers & Mathematicswith Applications, vol. 50, no. 8-9, pp. 1191–1210, 2005.
[9] E. Deriaz, “Shannon wavelet approximation of linear differential operators,” Institute of Mathematicsof the Polish Academy of Sciences, no. 676, 2007.
[10] A. Latto, H. L. Resnikoff, and E. Tenenbaum, “The evaluation of connection coefficients of compactlysupported wavelets,” in Proceedings of the French-USA Workshop on Wavelets and Turbulence, Y. Maday,Ed., pp. 76–89, Springer, 1992.
[11] E. B. Lin and X. Zhou, “Connection coefficients on an interval and wavelet solutions of Burgersequation,” Journal of Computational and Applied Mathematics, vol. 135, no. 1, pp. 63–78, 2001.
[12] J. M. Restrepo and G. K. Leaf, “Wavelet-Galerkin discretization of hyperbolic equations,” Journal ofComputational Physics, vol. 122, no. 1, pp. 118–128, 1995.
[13] C. H. Romine and B. W. Peyton, “Computing connection coefficients of compactly supported waveletson bounded intervals,” Tech. Rep. ORNL/TM-13413, Computer Science and Mathematics Division,Mathematical Sciences Section, Oak Ridge National Laboratory, Oak Ridge, Tenn, USA, 1997.
26 Mathematical Problems in Engineering
[14] G. Toma, “Specific differential equations for generating pulse sequences,” Mathematical Problems inEngineering, vol. 2010, Article ID 324818, 11 pages, 2010.
[15] C. Toma, “Advanced signal processing and command synthesis for memory-limited complexsystems,” Mathematical Problems in Engineering, vol. 2012, Article ID 927821, 13 pages, 2012.
[16] K. B. Oldham and J. Spanier, The Fractional Calculus., Academic Press, London, UK, 1970.[17] B. Ross, A Brief History and Exposition of the Fundamental Theory of Fractional Calculus, Fractional Calculus
and Applications, vol. 457 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.[18] L. B. Eldred, W. P. Baker, and A. N. Palazotto, “Numerical application of fractional derivative model
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 918497, 15 pagesdoi:10.1155/2012/918497
Research ArticleParallel Motion Simulation ofLarge-Scale Real-Time Crowd in a HierarchicalEnvironmental Model
Xin Wang,1 Jianhua Zhang,2 and Massimo Scalia3
1 College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China2 TAMS Group, Department of Informatics, University of Hamburg, Vogt-Koelln-Straße 30,22527 Hamburg, Germany
3 Department of Mathematics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy
Correspondence should be addressed to Xin Wang, [email protected]
Received 17 February 2012; Accepted 28 March 2012
Academic Editor: Carlo Cattani
Copyright q 2012 Xin Wang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
This paper presents a parallel real-time crowd simulation method based on a hierarchicalenvironmental model. A dynamical model of the complex environment should be constructedto simulate the state transition and propagation of individual motions. By modeling of a virtualenvironment where virtual crowds reside, we employ different parallel methods on a topologicallayer, a path layer and a perceptual layer. We propose a parallel motion path matching methodbased on the path layer and a parallel crowd simulation method based on the perceptual layer.The large-scale real-time crowd simulation becomes possible with these methods. Numericalexperiments are carried out to demonstrate the methods and results.
1. Introduction
Real-time crowd simulation is one of the important research directions [1, 2] in computer
games, movies, and virtual reality. In the last decade, Shao et al. [3] proposed a
multilevel model for the virtual crowd simulation in the face of visual effects, perception,
routing, interactive and other issues directly and efficient support. Virtual environment
modeling consists of two parts, that is, geometric environment modeling and nongeometric
environment modeling [4]. Geometric environment model corresponds to the geometric layer
of the hierarchical environmental model, and nongeometric environment model corresponds
2 Mathematical Problems in Engineering
to the topology layer, the path layer, and the perception layer. There have existed some
parallel methods [5], which can fast compute a scene path map based on the topology layer.
However, they lacked parallel methods to accelerate real-time crowd simulation based on the
path layer and the perception layer.
Given a motion path, matching human movements with it is a common problem in the
field of character animation. Usually artist manually tuned the motion data. However, when
lots of human movements need to be matched with lots of paths, only using manual method
will be impractical and cannot meet the real-time requirements. Based on our analysis, we
find that a parallel matching algorithm is suitable for motion path matching. Because motion
matching between paths is independent of each other, the path segments among every two
points within the calculation of the discrete sampling are independent. Based on the above
investigations, this paper employs a parallel motion path matching algorithm based on the
path layer.
Researchers often use the agent-based simulation model in crowd simulation. We
found that single agent status updates only need to consider it within a limited range state
of the world. Based on this observation, we designed a parallel crowd simulation method,
employing a parallel update strategy and dividing the scene into bins (each bin is a square
area; agents are distributed among all bins); we then determined the scope of each agent. For
the nonadjacent bins, agents can employ parallel update strategy, which is divided into four
batches of parallel updates. Experimental results show that this strategy can greatly increase
update efficiency.
In the real world, each person makes appropriate movement decisions according to
the state of the environment around him [6, 7]. For example, when a car approaches, people
will maintain their status or change their walking direction according to the car’s driving
direction, speed, and distance. The agent model proposed by Reynolds [8] can simulate these
situations. The current research employed a similar but relatively simple agent model. Our
agent is controlled by three forces: [9] avoiding the force of obstacles; avoiding collision with
other agent forces; following path force [10, 11]. The three forces are in order of decreasing
priority.
This paper presents parallel real-time crowd simulation algorithms based on a
hierarchy environmental model and fully taps the multilevel environmental model in the
parallelism, making the simulation of large-scale real-time crowds possible.
2. Related Work
Given a motion path, matching human movements with it is a common problem in the
field of character animation. In the game industry, a relatively simple and efficient way is
to loop the motion clip while the character moves along a motion path. This method makes
the movements of characters appear mechanical, monotonous, and unrealistic in terms of
imitating body movements. To achieve realistic effects, some scholars have proposed complex
motion synthesis methods [12–14]. The general approach first uses motion data, which are
captured to construct a graph-like data structure. In searching this data structure for the
right movement sequences to match input path, these sequences meet specific constraints
that stitch up human posture, finally forming the results. When matching motion data with
motion path, there exist some difficulties [15] as follows: (1) in the construction of the motion
graph, if the graph is large and complex, determining what motion can be synthesized from
the graph is difficult, and the range of new movements is unknown [16]; (2) the structure
Mathematical Problems in Engineering 3
of the motion graph is complex, which takes a lot of time to search on the graph each
time; hence, this structure cannot improve system performance [17]; (3) for the presence
of path constraints, there are some low-level connection relationships in the motion graph,
so there will be some unnecessary local shaking when synthesizing motion [12]; (4) some
constraints need to be considered (e.g., obstacles); however, such methods are difficult to
estimate.
Lau and Kuffner [18] described a similar method for this paper. The method
organizes motion data into a finite state machine (FSM); each state represents a high-
level semantical movement, such as walking and running. If two states can be connected
together, there will be one connection in the state machine. Motion state machine is
used to find the character’s motion data when synthesizing motion, which can overcome
the difficulties mentioned above. For multiple paths, this paper divided long path into
short path segments. The length of these short path segments is almost equal to the
motion segments in a motion state machine. Thus, when performing matching work,
the efficiency of matching can be improved. This paper also used parallel computing
strategy to greatly reduce the time required for each planning step. In addition, when
using motion state machine to match, a benefit arises; that is, while the current matching
segment crosses paths with other motion path segments, a collision situation is possible;
if such a situation happens, there will be no displacement motion data (idle) for the
role to effectively avoid the collision. On the other hand, when more human movements
are required to match more paths, solely using such methods cannot meet the real-time
requirements.
The agent-based crowd simulation model is accorded the characteristics of crowd
movement in the real world and has a very flexible control strategy (e.g., controlling each
parameter of each agent). Therefore, agent-based crowd simulation model is widely used.
However, for large-scale real-time crowd simulation, each agent needs to update its own
status according to its surrounding environment, and each agent is a dynamic obstacle to
other agents. Computing cost increases rapidly as the number of agent increases [19] in the
face of time complexity O(n2). Thus, when n is large, the time required for each frame will
significantly increase, making the real-time status update of each agent impossible. Hence,
we need to find a new method to do large-scale crowd simulation.
3. Overview
The system consists of two parts: (1) hierarchy environment model and (2) multiple parallel
algorithms based on the hierarchy (see Figure 1).The hierarchy environmental model [20], as shown in Figure 1 (right panel), consists
of two parts: geometric and non-geometric environment model. The non-geometric environ-
ment model includes the topology layer, path layer, and perception layer.
Based on the multi-level environment, the topology layer, path layer, and perception
layer employ different parallel algorithms to speed up real-time crowd simulation, respec-
tively, as shown in Figure 1 (left panel). We employed the existing Voronoi diagram algorithm
[4] based on topology layer to segment the scene quickly and get the path layer map. We also
used parallel motion path matching based on path layer to quickly match human motion for
path, as well as parallel real-time crowd simulation based on perception layer to fully tap the
crowd behavior in parallelism.
4 Mathematical Problems in Engineering
Voronoi diagram
scene segmentation
Parallel motion
path matching
Parallel real-time
crowd simulation
Topology
layer
Path
layer
Perceptual
layer Room 5
Room 3
Room 4Room 3
Room 1
Room 5
Room 2
Room4Room3
Room1
Room5
Room2
Nongeometric environment model
A hierarchy environmental model
13
4
2
Geometric
layer
Geometric environment
model
Figure 1: System overview.
4. Parallel Motion Path Matching Based on Path Layer
This section explains parallel motion path matching based on path layer. The section contains
three parts: (1) motion state machine, (2) motion path parameterization, and (3) extraction of
motion segments.
4.1. Motion State Machine
The main differences between the motion state machine proposed by Kovar et al. [12] and
the motion state machine constructed in this paper are as follows: (1) trajectory arc length
of the node of motion segments used by each state is fixed and equal in this paper’s motion
state machine; (2) the length of most of the motion segments in this paper is longer than
that in Kovar et al. [12]. For example, there are 100 frame movements about walking used
in this paper, but the length of the movements in Da Silva et al. [21] is only 25 frames;
(3) the motion state machine in this paper employed group-based hierarchical status node,
which can ensure that “walk left” and “run left” are on the same level, and all directional
run and walk movements comprise a position move node on the up level. Three facts exist
in the motion synthesis process, which lead to the above differences: (1) this paper used
sectional matching; (2) the movement range of the motion path to be matched is large; (3) the
connection relationships among states are more complex.
The motion state machine (MSM) used in this paper is shown in Figure 2. The MSM
has a total of 12 base states, which have their own names and include 36 motion segments
(except start and end states). The motion segments’ arc length of “Crawl,” “Squat move,”
and “Stride over” is indeterminate, because these motion segments are mainly used to deal
with the path segments about obstacles. These are marked directly by users to explain which
Mathematical Problems in Engineering 5
Start
End
Crawl
Squat
move
Stride
over
Idle
Run
forward
Run leftRun
right
L
F
R
Position move
Walk
forward
Walk
left
Walk
right
Figure 2: Finite motion state machine.
motion segments are used, so that the system does not need to match in the rear. The function
of the “Idle” state’s motion segments is to make the virtual characters wait, so that the system
can avoid the collision between virtual characters. The states, which include “run forward,”
“run left,” “run right,” “walk forward,” “walk left,” and “walk right,” describe the ways in
which virtual characters can match according to the specific path situation. Furthermore, the
motion path’s arc length of motion segments is equal, which makes the matching of motion
path segments easy.
There are some directed edges among states in MSM. These directed edges express
whether two motion segments can be connected. For example, there is a directed edge
between “run right” and “walk right,” which means that the motion segments of “run right”
can stitch with the motion segments of “walk right.” Similarly, the motion segments of
“walk left” can stitch with the motion segments of “run left.” The motion machine in this
paper adopts a group-based hierarchical status node. For example, “walk forward” and “run
forward” constitute F, “run left” and “walk left” constitute L, and “run right” and “walk
right” constitute R. Finally, L, F, and R constitute locomotion. The child states can inherit
the father states’ connection. For example, the connection between F and R explains that
“walk forward” and “run right” can connect with each other, and the connection between the
“crawl” state and l locomotion can lead to the connection between “crawl” and “run left.”
This paper also prepares some transitional motion segments. These motion data are
used to connect the transitional fragments among motion segments. There are eight frames.
The transitional fragments exist at the beginning and the end of motion segments.
6 Mathematical Problems in Engineering
4.2. Motion Path Parameterization
In this paper, there is a path planner responsible for generating a parameterized original
movement path called “T” and decides the grid size of path planning pace. If the planning
space is planed in a lattice structure, which is composed of m ∗m square, the space between
2d point sequence is m, but in the specific application, the space of sample points cannot
keep the same with the sample degree of the original path. For example, in this research,
the human root node movement size between two frames is s in the MSM. However, the
motion clips in the MSM were prepared before motion synthesis and cannot be resampled.
The system initially needs to use some fitting methods to parameterize the original path
and then resamples the parametric path based on the desired sampling size. The resulting
path can be used to match the crowd animation better in the following phase. In detail,
our method constructs the cubic polynomial curve p(u) with C1 continuous based on
the original data point. If there is a curve formula called p(u), the system begins with a
starting point p0 and records those points p1, p2, . . . , pn, whose arc length spacing is s and
corresponding parameters u1, u2, . . . , un. The characteristics of cubic polynomial spline curve
with C1 continuous are simple and easy to control, and human motion path is generally not
so smooth that curve with C1 continuous is enough to make the motion path smooth through
fitting.
In addition, the algorithm needs the corresponding tangent vector of each sample
point. The system can calculate p(u) curve’s tangent vector in p0, p1, . . . , pn through
parameters u0, u1, . . . , un. Then, it can represent the motion path after normalizing with the
coordinates of parameters point and tangent vector, called T1:
T1 ={pi, �pi | i = 0, 1, . . . , n
}. (4.1)
The arc length in our motion segments in the MSM is almost equal. The arc length
of motion state i is represented by li. If li is n times as long as s, motion path T1 used in
experiment general is n times longer than s. Hence, it can cut up the whole path into many
small segments whose length is n ∗ s. For example, suppose there are n + 1 points from pj to
pj+n. The algorithm can find motion segments called motionk in the MSM, which is similar
to the path curve shape. If there are some obstacles in the path, direct matching cannot be
performed. The path is marked by specific segments of motion data through interaction.
Then, it can stitch directly in the matching phrase.
There are some crossings of motion paths and obstacles in T1. Thus, matching directly
to T1 is not enough. An artificial mark on this cross-path, such as “crawl” state, is needed.
In the matching, when meeting this path, motion data are directly found under the “crawl”
state and are associated with motion path T2:
T2 ={pi, �pi,
(motion
(j, t)= motionk
)r| i = 0, 1, . . . , n; 0 ≤ j < n; 0 < t ≤ n − j; r = 0, 1 . . . , u
}.
(4.2)
The path segments composed of continuous t points from pj have been associated with
motion data called motionk; r means that this mark is the r-section in all u+ 1 marks. Figure 3
shows the normalized conversion process from curve T to curve T2.
Mathematical Problems in Engineering 7
SS
S
Motionk
T
T
1
T1
−p0
−p1
−p2
−p3
−p4
p0
p1
p2
p3
−p4
p i−1 p i p i+1
p i−1 p i p i+1 −pn 1
−pn
−pn 1
pn
pi−1pi pi+1
−
−
p1
p0
p2
p3
p4
p0
p1
p2
p3
p4
pi−1 pi+1pipn
pn−1
pn−1
pn
m
m
Figure 3: Normalized conversion process from curve T to curve T2.
T2
di
−p0
−p1
−p2
−p3
p4
−p5
p5
−pn 1
p0
p1
p2
p3
p4
p5
p6
p5
p7
pi−
−−
1pi pi+1
pi+t
pn 1
pn
p5+k
d
Figure 4: Schematic diagram of T2 segmentation.
4.3. Extraction of Motion Segments
T1 changes into T2 through parametric resampling and marking the specific segment. Setting
the path consists of n + 1 points, and total arc length is n ∗ s. The basic motion segment of
the root node’s path curves arc length is certain, set as d, such as “walk” and “run.” d is also
integer times as s, set as k. Thus, T2 needs to be cut into path segments, whose arc length is d.
The set of this path segments is D. Consider
D = {di | i = 0, 1, . . . , Dsize − 1}, (4.3)
where di means the ith motion path segment. Dsize means the total number of segments after
cutting:
di ={pi | i = b + 0, b + 1, . . . , b + k, b = di starting coordinates
}. (4.4)
Figure 4 shows the T2 segmentation process.
8 Mathematical Problems in Engineering
Block 0 Block 1 Block 2 Block 3 Block m
Grid (m , 1)
······
Figure 5: Schematic diagram of Grid (m, 1).
The next work that needs to be done is using the distance formula D to find proper
motion segment motionk for each di.
This paper designs a one-dimensional Grid (motionNum, 1) for single curve.
motionNum means the number of motion segments in the database, and one Grid has
motionNum blocks. Each block is responsible for calculating its representative motion
segments. For example, block0 is only responsible for calculating the matching degree
between the current line and the 0th motion segment. Each block is one-dimensional. For
example, in block (PointNum, 1), PointNum means the number of discrete points currently
calculating the curve segment matching. Each thread in the block means the distance between
a discrete point in the curve and the corresponding point in the motion segment. For example,
thread0 is responsible for calculating the distance of 0th discrete point. The formula is
distancei =√
CurvePointi − MotionPointi. (4.5)
Each thread only needs to calculate its own data, and it does not need to interact with
other threads. After the final computation, thread0 accumulates matching value, calculating
by each thread, and acquires the total matching value between the curve and the motion
segment. The computation formula is
MatchDegree =n∑i=0
distancei. (4.6)
After calculating the matching value with all motion segments, the minimum
matching value is obtained through comparison. The final result is
Motion ={j | MatchDegreej = min
i∈(0,n)(MatchDegreei
)}. (4.7)
The matching effect of single curve is shown in Figure 6, and its internal thread
organization way is depicted in Figure 5. Before calculating the distance between a discrete
point in curve and the corresponding point in motion segment, every point in a motion
segment needs to do a rotating translation operation. This step is necessary because motion
segment only converts to a curve’s local coordinate. The matching value can be obtained
accurately. Now, the system selects thread0 to calculate rotation and offset. Meanwhile, the
other threads wait for the calculating result of thread. Then, they share the calculating result
of thread0.
We need to design a two-dimensional Grid (motionNum, CurveNum) to extend to
multiple curves, as shown in Figure 7. motionNum means the number of motion segments.
Mathematical Problems in Engineering 9
Figure 6: Matching effect of single curve.
Block(0,0)
Block(1,0)
Block(2,0)
Block(3,0)
Block(n,0)
Block(0,1)
Block(1,1)
Block(2,1)
Block(3,1)
Block(n,1)
Block(0,2)
Block(1,2)
Block(2,2)
Block(3,2)
Block(n,2)
Block(0,3)
Block(1,3)
Block(2,3)
Block(3,3)
Block(n,3)
Block(0,m )
Block(1,m )
Block(2,m )
Block(3,m )
Block(n,m )······
······
······
······
······
.
.
....
.
.
....
Grid (m ,n)
Figure 7: Schematic diagram of Grid (m,n).
CurveNum means the number of curves. Others are similar to the single curve. The matching
effect of multiple curves is illustrated in Figure 8.
This method can achieve better expansibility. If there are more motion segments that
need to match, then the dimension of Grid only increases.
5. Parallel Real-Time Crowd Simulation Based on Perception Layer
This section explains how to parallel real-time crowd simulation based on perception layer.
This section contains three parts: (1) scene segmentation; (2) nearest neighbor query; (3)parallel real-time crowd simulation.
5.1. Scene Segmentation
In agent-based crowd simulation, each agent has its own perceived range. In each update,
each agent must query the scope of its perception to obtain the range of obstacles or
information of other agents. Then, based on the information to calculate the forces acting
on the agents, their speed and direction can be adjusted, and the location can be updated.
Neighbor queries generally employ serial query that is executed when the current agent is
updated. Then, the next agent begins to update. In this algorithm, using parallel neighbor
10 Mathematical Problems in Engineering
Figure 8: Matching effect of multiple curves.
Agent
120 3
4 5
67
8
9 1011
12
13 14 15
16
17 18 19
2021 22 23
2425 26
Figure 9: Scene segmentation and agent distribution (circle represents the agent).
query, the main query is the bin, rather than each agent. Through these neighbor queries,
the bin can be obtained, in which agents need to consider all the potential neighbor agents.
Before the implementation of parallel neighbors’ query, the entire scene needs to be evenly
segmented to get all the information about the bin.
The segmentation needs to be completed in the initialization process. The size of the
entire scene is 1200 × 1200; setting the side length of each bin is 4 for the square, so that
the whole scene is to be divided into 300 × 300 bins. The square side length is 4, because the
sensing range of agent sets a circular area that is a query radius of 4. The formula is as follows:
queryRadius = predTime ∗ maxVelocity ∗ 2. (5.1)
The predTime is the forward predictive time, set as 1 s. maxVelocity is the agent’s
maximum move speed, set as 2 m/s. In the query process, only 8 bins need to be considered,
which are around the center bin.
As shown in Figure 9, the scene is evenly divided into a number of bins, the scene of
the agents located in each bin. There are multiple agents in a bin; however, a bin can also have
no agent.
Mathematical Problems in Engineering 11
Observation range ofgreen bin
0 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40 41 42 43
44 45 46 47 48 49 50 51 52 53 54
55 56 57 58 59 60 61 62 63 64 65
66 67 68 69 70 71 72 73 74 75 76
77 78 79 80 81 82 83 84 85 86 87
88 89 90 91 92 93 94 95 96 97 98
99 100 101 102 103 104 105 106 107 108 109
Figure 10: Observation range of bin (dashed circle).
5.2. Nearest Neighbor Query
After scene segmentation and statistical distribution among the agents in each bin, the nearest
neighbor query is employed. In the scene segmentation step, each Agent’s observation range
is a circular area of radius 4, and the bin side is also 4. Thus, the algorithm only needs to check
the bin and approximately 8 bins. Figure 10 shows that the number of green bins is 49 and the
observation range is the dashed circle. Hence, the bin numbers that need to be checked are
37, 38, 39, 48, 49, 50, 59, 60, and 61. As the neighbor queries between the bins only read data
with no write operation, this paper chooses the parallel query; the query results are saved in
the corresponding data structures.
5.3. Parallel Real-Time Crowd Simulation
After completing the nearest neighbor querying, crowd velocity, and position updating, the
next Agent begins to update. At the step of parallel nearest neighbor query, the algorithm
has queried all the neighbors of the bin. However, as the Agents of the adjacent bin would
interact, this step of the update operation cannot be completely parallel. For example, for bin
48 and bin 49, if the two parallels update, there would be a read/write conflict. The reason
is that when the Agents of bin 48 are updating, the Agents of bin 49 should be considered;
hence, the reading data may be outdated Agent data of bin 49, and vice versa. Therefore, this
paper chooses partial parallel update strategy; steps are described below in detail.
Referring to the behavior rules of pedestrians in the real world and traditional Agent
model, the Agent model used in this paper is acted upon by three kinds of forces, namely, (1)avoid the force of obstacles; (2) avoid collision with other Agent forces; (3) follow the path of
force. Figure 11 illustrates each kind of force.
Three kinds of forces decline in priority order. Avoiding the behavior of obstacles is the
highest priority; that is, when Agent k detects it will crash into obstacle d, it also may have
collided with Agent g, and then it would only consider avoiding obstacle d. Considering that
the obstacle is stationary, if the Agent does not adjust its speed, it will be hit and move. Even
if Agent k does not make an adjustment, Agent g itself can adjust the speed for Agent k.
Avoiding force is calculated as follows. Agent forward predictive distance with the observed
radius builds up a rectangular area; in the region of collision, we can find the nearest obstacle.
As shown in Figure 11 for obstacle d, avoiding force (Force k) for the current Agent is pull
12 Mathematical Problems in Engineering
Force j
Agent jP’
Pj’ i’
Agent i
PathObstacle e
Force h
Agenth
Obstacle dAgent g
Agent k
Force k
Figure 11: Schematic diagram of agent’s three kinds of forces.
force and the obstacle repulsive force direction. Another situation is when Agent j moves to
prediction j’ position, and Agent i will reach position i at the same time. When the distance
between position j and position i is less than the distance between the center of the radius of
their observations, a collision will be assumed; thus, they will not be influenced by the force
of path-following, but by the Agent force between the impact of avoidance, the avoidance of
force as a lateral tension (Force j), to deviate the current running direction. In Figure 11, there
is a lane in the path (dotted line parallel with the path and the distance between the path);the current updated Agent will forecast forward for some distance to predict the location of
P projected onto the path P’ point. If the distance between P’ and P is greater than the lane,
then the Agent’s travel direction deviates from the path; otherwise, there is no deviation. As
shown in Figure 11, Agent h would be acted upon by the force of path following (Force h), for
the projection point P’ of the direction of the Agent center of the connection to slowly move
closer to the path direction.
This paper makes parallel updates on the Agent rates four times. As shown in
Figure 10, the bin has a total of four colors. Each time parallel updates the bin that has the
same color. When all colors have been updated, a general update is completed. Agents within
the same bin have an impact on each other, but they cannot be updated in parallel; hence, the
serial update can only be used. In the implementation process, each block represents a bin,
following the Agent model above; meanwhile, according to the priority of various types of
forces, the research also carries out the appropriate adjustments.
6. Experimental Results
In general, the experiment platform uses a computer of 4 core (Q9950 CPU at 2.83 GHz
and 4 GB of RAM), equipped with a NVIDIA GTX 280 graphics card, which has 30
multiprocessors. Each multiprocessor has 8 cores; the total number of cores is 240.
The CPU-based algorithm achieves serial algorithms, and the GPU-based algorithm achieves
the proposed parallel algorithms in this paper.
Mathematical Problems in Engineering 13
0 20 40 60 80 100 120 140 200 400 600 800 1000
CurveNum
120000
100000
80000
60000
40000
20000
0
Parallel (GPU) time (ms)
Serial (CPU) time (ms)
Ex
ecu
tio
n t
ime(m
s)
Figure 12: Comparison between CPU-based and GPU-based algorithms.
Table 1: Comparison between CPU and GPU execution time.
10 100 200 500 1000
CPU execution time (ms) 1214.9124 11999.31 25312.8457 56183.2070 112101.531
GPU execution time (ms) 70.2647 549.388 1049.8966 2572.0297 5188.6655
Figure 12 clearly shows that in the path matching based on path layer, with the increase
in the number of curves that are required for matching, the CPU’s execution time increases
significantly, whereas the GPU’s execution time barely increases.
Table 1 lists the matching time based on the path matching on the path layer with the
CPU matching algorithm and GPU matching algorithm under some curve lines. The data in
the table indicate that GPU parallel algorithm would increase by about 20 times than the CPU
algorithm in execution speed.
Figure 13 shows the frame rate comparison between GPU parallel algorithm and CPU
serial algorithm in real-time crowd simulation based on the perceived layer.
As can be seen from Figure 13 GPU-based parallel computing speed is significantly
higher than the CPU-based parallel computing speed in crowd simulation. With the increase
in the number of Agents, CPU-based FPS drops very quickly. When the number reaches
more than 5,000, real-time simulation results are not achieved. Although the GPU program
number is above 10000 of the Agent, FPS can reach 24 or more, fully meeting the real-time
requirement. When the number is 1000, the FPS of the GPU is 60.
In addition, as the curves indicate, the FPS of GPU-based algorithms is significantly
higher than that of the CPU-based algorithms, but it still has not reached 10 or more times.
The main reason is the speed of Agent updating; there are a lot of conditional executions,
such as statements reducing the parallel computing efficiency of the GPU-based algorithm.
7. Discussion
Through the proposed parallel algorithms in this paper, we achieve a completely parallel
real-time crowd simulation based on a hierarchy environmental model, making topology
14 Mathematical Problems in Engineering
1000 3000 5000 7000 9000 11000
60
55
50
45
40
35
30
25
20
15
10
5
0
Number of agents
FPS
FP
S
GPU’
CPU’ FPS
Figure 13: Performance comparison between CPU and GPU in crowd simulation.
layer, path layer, and perception layer have corresponding parallel algorithms to speed up
the calculation.
This paper describes the detailed process with the path layer-based motion path
matching, through the structure of motion state machine, setting up the transfer relationship
among movement segments. Specifying the sequence of key points, this paper chooses cubic
spine curve fitting to get a continuous path and then samples the sequence of points needed.
We use the distance function to calculate the matching degree between the motion segment
and the path, then accumulate the value of discrete points, and get the total matching degree
of the corresponding segment. Moreover, this paper explains the use of parallel computing
algorithms to accelerate and verify the algorithm through data analysis.
This paper introduces parallel computing algorithm based on the perception layer to
achieve real-time simulation of large crowds. Using scene segmentation evenly, according to
his own location, each Agent is assigned to the appropriate bin where the original calculations
must be serialized into parallel computing. Each bin is a separate update unit.
Experimental results suggest that the proposed method in this paper is consistent with
the serial method in effect, but efficiency has been greatly improved. Given that the Agent
model used in this paper is relatively simple, the focus of future work is how to use more
complex models to achieve a more realistic real-time crowd simulation, reduce the occurrence
times of logic-based computing in the parallel algorithm framework, and further improve the
algorithm’s parallelism.
Acknowledgments
This work was supported by Natural Science Foundation of Zhejiang Province (Y1110882,
Y1110688, R1110679), Department of Education of Zhejiang Province (Y200907765,
Y201122434), and Doctoral Fund of Ministry of Education of China (20113317110001).
Mathematical Problems in Engineering 15
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[12] L. Kovar, M. Gleicher, and F. Pighin, “Motion graphs,” in Proceedings of the ACM Transactions onGraphics (ACM SIGGRAPH ’02), pp. 473–482, July 2002.
[13] O. Arikan and D. A. Forsyth, “Interactive motion generation from examples,” in Proceedings of theACM Transactions on Graphics (ACM SIGGRAPH ’02), pp. 483–490, July 2002.
[14] J. Lee, J. Chai, P. S. A. Reitsma, J. K. Hodgins, and N. S. Pollard, “Interactive control of avatarsanimated with human motion data,” in Proceedings of the ACM Transactions on Graphics (ACMSIGGRAPH ’02), pp. 491–500, July 2002.
[15] S. Y. Chen, H. Tong, and C. Cattani, “Markov models for image labeling,” Mathematical Problems inEngineering, vol. 2012, Article ID 814356, 18 pages, 2012.
[16] P. S. A. Reitsma and N. S. Pollard, “Evaluating motion graphs for character navigation,” in Proceedingsof the ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 89–98, Grenoble, France,2004.
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 625861, 12 pagesdoi:10.1155/2012/625861
Research ArticleOptimization of Resource Control forTransitions in Complex Systems
Florin Pop
Faculty of Automatic Control and Computers, University Politehnica of Bucharest,Splaiul Independentei 313, 060042 Bucharest, Romania
Correspondence should be addressed to Florin Pop, [email protected]
Received 7 March 2012; Accepted 2 April 2012
Academic Editor: Cristian Toma
Copyright q 2012 Florin Pop. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
In complex systems like Large-Scale Distributed Systems (LSDSs) the optimization of resource con-trol is an open issue. The large number of resources and multicriteria optimization requirementsmake the optimization problem a complex one. The importance of resource control increases withthe need of use for industrial process and manufacturing, being a key solution for QoS assuring.This paper presents different solutions for multiobjective decentralized control models for tasksassignment in LSDS. The transaction in real-time complex system is modeled in simulation bytasks which will be scheduled and executed in a distributed system, so a set of specificationsand requirements are known. The paper presents a critical analysis of existing solutions andfocuses on a genetic-based algorithm for optimization. The contribution of the algorithm is thefitness function that includes multiobjective criteria for optimization in different way. Severalexperimental scenarios, modeled using simulation, were considered to offer a support for analysisof near-optimal solution for resource selection.
1. Introduction
The resources control in complex systems requires information about resources and tasks. A
near-optimal assignment could be made based on some criterion function, such as minimum
execution time or load balancing. There has been a steadily increasing interest, supported
by advanced technological and economic developments, into dealing with very complex
(dynamical) systems describing natural phenomena or manufacturing processes [1]. The
particular interest in the study of complex systems is tipping points where one observes a
sudden change in the dynamics, sometimes referred to as critical transitions, modeled as
tasks to be submitted for execution on physical resources. For instance, medical conditions
such as asthma attacks and epileptic seizures can change quickly from regular to irregular
2 Mathematical Problems in Engineering
behavior, the financial markets are known to suddenly break trends in a crisis, and climate
conditions and ecological environments can change rather abruptly. The understanding of
the dynamical behavior near tipping points would enable human interaction to attenuate or
control the consequences of critical transitions [2].The scheduling process is considered to be the core of resources control in complex
systems. Due to the NP-complete nature of scheduling algorithms, current research directions
are focused on finding suboptimal (near-optimal) solutions, which can be further divided
into the following two general categories: approximate and heuristic algorithms. At global
level, two-phase scheduling solution comprised of a set of heuristic subalgorithms to achieve
optimized scheduling performance over the scope of overall resources is a new research
subject in the present [3, 4].Today, engineers face an increasing challenge in advanced applications with different
requirements and constrains. Innovative developments for efficient mathematical approaches
focused on approximate algorithms, heuristics-based methods, and bio-inspired models. The
approximate algorithms use formal computational models, but instead of searching the entire
solution space for an optimal solution, they are satisfied when a solution that is sufficiently
good is found. In the case where a metric is available for evaluating a solution, this technique
can be used to decrease the time taken to find an acceptable schedule. The factors which
determine whether this approach is worthy of pursuit include [5, 6] availability of a function
to evaluate a solution, the time required to evaluate a solution, the ability to judge the
value of an optimal solution according to some metric, and availability of a mechanism for
intelligently pruning the solution space. The paper proposes a mathematical approach for
resources control based on a multicriteria optimization genetic algorithm. One of the well-
known problems of genetic algorithms is that, for large solution space, the convergence time
is high.
The rest of the paper is structured as follows. Section 2 describes the optimization
methods. Section 3 approaches the multidimensional optimization methods for real-time
system. The proposed genetic algorithm is described in Section 4. The tests conducted on the
proposed algorithm (Section 5) highlight the improvement provided by this new approach
not only in terms of convergence time, but also in terms of solution quality.
2. Optimization Methods for Decentralized Control
Optimization methods for resource control use heuristic (multiobjective) approaches. The
allocation problem considers a set of n tasks, T = {T1, T2, . . . , Tn}, for some finite integer n,
that models a set o transitions that will use a multiple processor system (e.g., cluster system)in which each transition can be characterized by multiple parameters: Ti = {ai, ti, Ci, ri, ωi . . .},
where ai is the arrival time (the time when the transition is produce), ti is the execution time(it can be estimated or calculated), Ci is the completion time (with the following condition:
ai + ti ≤ Ci), ri is a rate 0 < ri ≤ 1 (can be a normalized priority with∑n
i=1 ri = 1), 0 ≤ ωi ≤ 1 is
a weight (with the special normalization condition∑n
i=1 ωi = 1), and we can have some other
parameters which characterize the task.
In this model, a complex system has a number of m resources R = {R1, R2, . . . , Rm}.
Each resource Rj has specific characteristics like capacity, latency, memory type and space,
CPU processing characteristics, and storage limitation. The most important characteristic
used in the proposed model is the utilization rate (uj) that measures the processing capacity.
The workj done by Rj in order to process a task Ti is defined as its running time multiplied
Mathematical Problems in Engineering 3
by the resource utilization rate, workj(Ti) = tiuj ≤ (Ci − ai)uj . A valid schedule is defined
as Sched = {(Ti, Rj) | Ti ∈ T, Rj ∈ R}. Similarly, the work of set of scheduled tasks is
work(T, R) =∑
(Ti,Rj )∈ Sched workj(Ti). Usually it is assumed that in the case of malleable tasks
the work of a task cannot be decreased by spending more processors on it (preservation of
work). Similarly the work of a task cannot be decreased by using virtualization.
To evaluate the efficiency o resource utilization a resource is considered to be Active(working) or Idle (waiting for new tasks). Efficiency E(t) at time t is E(t) = Resource
Active(t)/(Resource Active(t) + Resource Idle(t)). In general we are looking for a feasible
solution to scheduling problem. This is a schedule which meets all the requirements and
constrains.
For optimization, there are bottleneck objectives and sum objectives. The scheduling
problem considers the following objective for optimization: maximum completion time (Cmax =maxi{Ci}), weighted completion time (Cw =
∑ni=1 wiCi), or maximum lateness (Lmax = maxi{|(Ci−
ai) − ti|}).Another important aspect of scheduling optimization considers real-time systems.
These type of systems are defined as those systems in which the correctness of the system
depends not only on the logical result of computation, but also on the time at which the results
are produced. If the timing constraints of the system are not met, system failure is said to have
occurred. Hence, it is essential that the timing constraints of the system are guaranteed to be
met.
2.1. Heuristics for Resources Control
Opportunistic Load Balancing (OLB)
The Opportunistic Load Balancing heuristic selects the task Ti arbitrarily from the group of
tasks and assigns it to the next resource that is expected to be available [7, 8]. It does not
consider the workj(Ti), which may lead to very high Cmax. If all tasks are scheduled with
respect to condition a1 < a2 < · · · < an the heuristic is called First Come First Served (FCFS) [9].
Minimum Execution Time (MET)
The heuristic assigns each task selected arbitrarily to the machine with the least expected
execution time for that task [10].
Minimum Completion Time (MCT)
The heuristic assigns each task selected in arbitrary order to the machine with the minimum
expected completion time for that task [10]. The MCT combines the benefits of OLB and MET
and tries to avoid the circumstances in which OLB and MET perform poorly.
Min-Min
The heuristic begins with the set T of all task to be unscheduled. Then, the set C of minimum
possible completion times of all tasks on any of the machines is computed: Cmin = mini{Ci}.
The task with the Cmin is then assigned on a processor with minimum expected work.
4 Mathematical Problems in Engineering
Max-Min
The heuristic is very similar to min-min, but considers Cmax and then assigns the task on a
processor with minimum expected work [10, 11].
Duplex
The heuristic is a combination of the min-min and max-min heuristics. The heuristic performs
both of the min-min and max-min heuristics and used the better solution [8, 10, 11].
Genetic Algorithms (GAs)
They are used for searching large solution spaces with multiple possible schedule of tasks.
Each possible schedule is modeled by a chromosome that has a fitness value, which is the
result of an objective function designed in accordance with the performance criteria of the
problem (Cmax or Lmax) [12].
Simulated Annealing (SA)
It is an iterative technique that considers only one possible solution (mapping) for each task at
a time [13]. This solution is modeled like in a GA. SA uses a procedure that probabilistically
allows poorer solutions to be accepted to attempt to obtain a better search of the solution
space.
A∗
Heuristic is a search technique that has been applied in various task allocation problems. The
A∗ heuristic begins at a root node that is a null solution. As the tree grows, nodes represent
partial schedule. A pruning process is performed to limit the maximum number of active
resources at any one time. A cost function f(Sched) is associated with a partial solution (e.g.,
f(Sched) = Cmax).Guaranteeing timing behavior requires that the system could be predicted. Predictabil-
ity means that when a task is activated it should be possible to determine its execution time
with certainty. It is also desirable that the system attains a high degree of utilization while
satisfying the timing constraints of the system [14–16].
2.2. Resource Control in Real-Time Complex System
A complex system is said to be real-time if there exists at least one task Ti ∈ T , which falls into
one of the following categories.
(1) Task Ti is a hard real-time task. The execution of the task Ti should be completed by
a given deadline, ai + ti ≤ Ci.
(2) Task Ti is a soft real-time task. If a task Ti finishes the work after a given deadline Ci, thepenalty is pays. A penalty function P(Ti) is defined for the task. If ai + ti ≤ Ci, the
penalty function P(Ti) = 0. Otherwise P(Ti) = (ai + ti) − Ci > 0.
Mathematical Problems in Engineering 5
(3) Task Ti is a firm real-time task. If a task Ti finishes the work before a given deadline
Ci, the more rewards it gains. A reward function R(Ti) is defined for the task. If
ai+ ti ≥ Ci, the reward function R(Ti) = 0 is zero. Otherwise R(Ti) = Ci− (ai+ ti) > 0.
The set of real-time tasks T can be a combination of hard, firm, and soft real-time
tasks. Let TS be the set of all soft real-time tasks in T . The penalty function of the system is
P(T) =∑|TS|
i=1 P(TS,i). Let TF be the set of all soft real-time tasks in T . Similarly, the reward
function of the system is R(T) =∑|TF |
i=1 R(TF,i).The following goals should be considered in scheduling a real-time system: (i) meeting
the timing constraints of the system; (ii) preventing simultaneous access to shared resources
and devices; (iii) attaining a high degree of utilization while satisfying the timing constraints
of the system; (iv) reducing the cost of context switches caused by preemption; (v) reducing
the communication cost in real-time distributed systems. In addition, the following criteria
are considered in advanced real-time systems: (vi) considering a combination of hard, firm,
and soft real-time activities, which implies the possibility of applying dynamic scheduling
policies that respect the optimality criteria; (vii) task scheduling for a real-time system whose
behavior is dynamically adaptive, reconfigurable, reflexive, and intelligent; (viii) covering
reliability, security, and safety. Basically, the scheduling problem is to determine a schedule
for the execution of the task so that they are all completed before the overall deadline [14, 15].
3. Multidimensional Optimization Methods: Applications
Multidimensional optimization methods are useful when the search space is likely to have
many local optima, making it hard to locate the global optimum. In low-dimensional or
constrained problems it may be enough to apply a local optimizer starting at a set of possible
start points, generated either randomly or systematically (for instance, at systems locations),and choose the best result. However this approach is less likely to locate the true optimum as
the ratio of volume of the search region to number of starting points increases. The application
of different multidimensional optimization method proves that finding the global optimum
is a hard problem.
Application of Simplex Method
Scheduling of vehicles in the container terminal is often studied as a static problem in the
literature, where all information, including the number of task, their arrival time, and so
forth, is known beforehand. The objective is generally minimizing the total traveling and/or
waiting times of the vehicles. When the situation changes, for example, new jobs arrive or a
section of the terminal is blocked, new solutions are generated from scratch.
Application of Simulated Annealing Method
A parallel approach of a modular simulated annealing (MSA) algorithm, a shortened
SA algorithm, applied to classical job-shop scheduling (JSS) problems is presented. The
JSS problems tackled are very well-known difficult benchmarks, which are considered to
measure the quality of such systems.
6 Mathematical Problems in Engineering
Step 1. User requests
Step 3. Task preparation
for scheduling
Step 4. Monitoring thesystem receive theavailable resources
Step 6. Migrate to the
best solution
Step 7. Get a localoptimum and end the GA
Step 8. Aggregate the
results for a near-globaloptimum
Step 9. Keep the schedule
in a dedicated repository
Step 5. Run thescheduling algorithm(GA)
Step 2. Create the “batch
of tasks“
Figure 1: Main actions of proposed algorithm.
Application of Genetic Algorithms
GAs are developed for solving the machine-component grouping problem required for
example a cellular manufacturing systems. GA provides a collection of satisfactory solutions
for a two-objective environment (minimizing cell load variation and minimizing volume of
inter cell movement), allowing the decision maker to then select the best alternative.
4. Genetic Algorithm for Resources Control in LDSD
In [17] Iordache et al. present a genetic algorithm for decentralized scheduling. The
description of the scheduling algorithm in a logical flow of activities is described in the
following steps. The important contribution of this algorithm is the fitness function that
considers multiobjective criteria for optimization (see Figure 1).
Step 1. A user requests that one or more tasks are scheduled.
Step 2. The input is processed as a “batch of tasks” (group of tasks). The batch of tasks is
broadcast to all the resources in the cluster.
Step 3. The resources receive the group of tasks to be scheduled. The tasks are inserted-sorted
in a queue according to a sorting criteria like arriving time (ai) or scheduling priority (ri).If the number of tasks in the queue is less than a predefined length of the chromosome,
they wait for τ units of time before starting the genetic algorithm. If the chromosome is still
not complete at the end of the waiting period, a noninfluential padding is added. On the
contrary, if the length of an arriving group of tasks exceeds the predefined dimension of the
chromosome, some tasks are saved in the waiting queue and will be scheduled at the next
time.
Mathematical Problems in Engineering 7
Step 4. On each resource, a tool keeps an up-to-date status of the computers in the LSDS on
which tasks are sent for execution, by constantly interrogating a monitoring system.
Step 5. The resources in the cluster run the GA. Each resource starts with a different, specific
initialization of the genetic algorithm. The subsequent steps of the GA are similar for all the
nodes in the cluster, and so is the fitness formula. The clients will compute different optimum
from which the best one will be chosen.
Step 6. The migration of the best current solutions is performed after each step of the GA,
thus ensuring that the population finds a better optimum. The resources exchange the fittest
individuals and insert them into the next generation.
Step 7. The reproduction process stops after a finite, predefined number of steps. Each
resource in the cluster computes its optimal individual.
Step 8. Each resource sends its optimum to all the other nodes in the cluster and the final
optimal individual is decided.
Step 9. The scheduling obtained is saved in a history file on each resource in the cluster of
resources.
The fitness function is an essential element of proposed GA. It gives an appreciation
of the quality of a potential solution according to the problem’s specification. For the
scheduling problem, the goal is to obtain task assignments that ensure minimum execution
time, maximum processor utilization, a well-balanced load across all machines, and last
but not least to ensure that the precedence of the task’s is not violated. According to the
chromosome encoding and genetic operators presented previously all individuals respect
the task dependencies, so the focus should be on the other goals of the problem. The fitness
function has the following representations: (1) F =∑
i cifi or (2) F =∏
ifi, where fi encode
a criterion in fitness function and ci is a weight for a criterion (∑
i ci = 1). In both cases, if
0 ≤ fi ≤ 1, then 0 ≤ F ≤ 1. For the proposed genetic scheduling algorithm three criteria are
considered: load balancing over the resources, f1 = tmin/tmax = mini{ti}/maxi{ti}, average idletime of the resources, f2 = (1/n)
∑ni=1 ti/tmax, and the schedule penalty, f3 = |Sched|/|T |, where
|Sched| represents the length of a schedule (the number of tasks that respect deadlines and
resource restrictions) and |T | is the total number of tasks.
5. Experimental Methodology and Results
5.1. Simulation Environment
Due to the complexity of the LSDS, involving many resources and many jobs being
concurrently executed in heterogeneous environments, there are not many simulation tools
to address the general problem of LSDS computing. The simulation instruments tend to
narrow the range of simulation scenarios to specific subjects, such as scheduling or data
replication. The simulation model provided by MONARC is more generic than others, as
demonstrated in [18]. It is able to describe various actual distributed system technologies
and provides the mechanisms to describe concurrent network traffic, to evaluate different
strategies in data replication, and to analyze job scheduling procedures. In order to provide a
realistic simulation, all the components of the system and their interactions were abstracted.
8 Mathematical Problems in Engineering
DB serverDB server
CPU CPU CPU
Task scheduler
Activity Task Transition
Regional centerRegional
center
Regionalcenter
LAN
WAN
Figure 2: MONARC simulation tool: the Regional center model for LSDS control.
The chosen model is equivalent to the simulated system in all the important aspects. A first
set of components was created for describing the physical resources of the distributed system
under simulation. The largest one is the regional center (see Figure 2), which contains a site
of processing nodes (CPU units), database servers and mass storage units, as well as one or
more local and wide area networks.
The maturity of the simulation model was demonstrated in previous work. For
example, a number of data replications experiments were conducted in [17], presenting
important results for the future LHC experiments, which will produce more than 1 PB of data
per experiment and year, data that needs to be then processed. In [19] the simulation model
was used to conduct a series of simulation experiments to compare a number of different
scheduling algorithms.
5.2. Evaluation Criteria for LSDS Control
It is quite difficult to make a comparison among different control systems for LSDS,
since each of them is suitable for different situations. For different control systems, the
class of targeted applications and LSDS resource configurations may differ significantly.
The adequate evaluation criteria for LSDS control systems are as follows. (i) ApplicationPerformance Promotion involves reviewing how well the applications can benefit from the
deployment of the control system (ii) System Performance Promotion concerns how well the
whole system can benefit (iii) Control and Efficient Allocation it is desired so that the LSDS
control system can always produce good allocation. However, it is also required that the
scheduling system should introduce additional overhead as low as possible. (iv) Reliability—
a reliable LSDS control system should provide some level of fault tolerance. An LSDS
Mathematical Problems in Engineering 9
80
75
70
65
60
55
50
45
40
35
30
GA convergence
0 10 20 30 40 50 60 70 80 90 100
Generation number
Mak
esp
an(C
max)
Figure 3: Makespan comparison for the scheduling of 38 tasks on 8 processors.
is a large collection of loosely coupled resources, and therefore it is inevitable that some
of the resources may fail due to diverse reasons. The control system should handle such
frequent resource failures. For example, in case of resource failure, the control system should
guarantee an applications completion. (v) Scalability—since an LSDS environment is in nature
heterogeneous and dynamic, a scalable scheduling infrastructure should maintain good
performance with not only increasing number of applications, but also increasing number
of participating resources with diverse heterogeneity.
When designing the control infrastructure for LSDSs, these criteria are expected to
receive careful consideration. Emphasis may be laid on different concerns among these
evaluation criteria according to practical needs in real situations. The performance of
scheduling algorithms for LSDS control is usually estimated using a certain number of
standard parameters, like total time or schedule length. In the tests performed we used the
following evaluation parameters [20, 21]:
(i) total schedule length (SL)—Cmax;
(ii) convergences time—the number of generations needed to obtain performances
better then a certain threshold;
(iii) load balancing (where umed, denotes the average utilization for all processors in the
system):
L = 1 − 1
umed
√√√√ 1
n
n∑i=1
(ui − umed)2, 0 < L ≤ 1. (5.1)
The load balancing of system, for a given schedule, converges to 1 when all resources
have approximately the same utilization rate, equal to makespan. In these conditions the
square deviation Δ → 0.
5.3. Experimental Results
The test case considered task dependencies containing 38 tasks. The processors’ topology
contained 8 processors connected in a full mesh. The results presented in Figure 3 show that
10 Mathematical Problems in Engineering
Mak
esp
an(s)
1000900800700600500400300200100
0
200
805
211
816
219
896
203
810
GA Duplex SA Min-min
90 tasks + 8 resources
506 tasks 38 resources−
Figure 4: Makespan comparison for the following scenarios: 90 tasks + 8 resources, 506 tasks − 38 resources.
Min-min
SA
Duplex
GA
Proc4
Proc3
Proc2
Proc1
0 50 100 150 200 250 300 350 400 450
Execution time (s)
Figure 5: Processor utilization overview for 90 tasks.
the genetic algorithm has a very good convergence (after 50 generations there is no significant
improvement, so the algorithm could be stopped).In order to analyze the schedule length of different dependent task scheduling
algorithms, it has been used a processor topology containing 8 processors connected in a
full mesh. Two tests have been run for DAGs containing 90 and 506 tasks (see Figure 4).For the first test (90 tasks—left side of the figure), the best result, 263 time units, was
provided by the proposed GA. On the second place came the GA without the initialization
phase with the value of makespan equal to 266. From the classic scheduling algorithm,
Duplex provided the best solution equal to 281, while the result of SA was the worst equal
to 292. The test containing 506 tasks was an extreme test. The best result, 1073 time units,
was offered by GA, proving once more the importance of the proposed algorithm. The worst
solution was given by SA. The other compared algorithm is min-min [22] (see Figure 5).Memory is another important factor since it is the characteristic that controls most
of the allocation algorithms and also since it cannot be oversubscribed. As can be seen, the
memory allocator gets to the maximum memory value slower and thus allows for better
Mathematical Problems in Engineering 11
UsedNo Share
Total
Low (<50%)
High ( 50%)
9080706050403020100
Time (s)
0 250 500 750 1000 1250
Memory type: Load:
Mem
ory
(GB)
Figure 6: Memory usage for Best Fit allocation.
performance. Also this allocator is the first to leave the maximum value barrier when the
load is decreased (see Figure 6).
6. Conclusions
We present in this paper an algorithm for controling the resources allocation for special tasks
type (transitions) in LSDS (considered to be a complex one). The novelty of the proposed
algorithm is represented by the multicriteria optimization fitness function for special tasks
with specific requirements and constrains. The process was modulated using a genetic
scheduling algorithm. The paper analyzed the existing methods for control optimization
in LSDS. The multidimensional optimization criteria were considered with the real-time
behavior introducing two measures for evaluation: penalty and reward. In accordance
with this behavior, the convergence of proposed control method is very good in terms of
convergence, solution cost, and memory usage.
The most important contribution of this paper is the innovative method for the
optimization of dependent task scheduling control in LSDS. Inspired from the natural
models, this algorithm evolves an initial population of chromosomes in order to achieve
a good average fitness for the population. The experimental results have proven that the
proposed algorithm offers the best solutions in most cases. For comparison were used several
classical algorithms such as SA, Duplex, and min-min.
Acknowledgments
The research presented in this paper is supported by the Romanian Project: SORMSYS-Resource Management Optimization in Self-Organizing Large-Scale Distributes Systems (Contract
no. 5/28.07.2010, Project CNCSIS-PN-II-RU-PD ID: 201). The work has been cofunded
by the Sectorial Operational Program Human Resources Development 2007–2013 of the
Romanian Ministry of Labor, Family and Social Protection through the Financial Agreement
POSDRU/89/1.5/S/62557.
12 Mathematical Problems in Engineering
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 347674, 12 pagesdoi:10.1155/2012/347674
Review ArticleMathematical Models of Dissipative Systems inQuantum Engineering
Andreea Sterian and Paul Sterian
Academic Center of Optical Engineering and Photonics, Polytechnic University of Bucharest,313 Spl. Independentei, 060042 Bucharest, Romania
Correspondence should be addressed to Paul Sterian, [email protected]
Received 9 February 2012; Accepted 18 March 2012
Academic Editor: Ezzat G. Bakhoum
Copyright q 2012 A. Sterian and P. Sterian. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.
The paper shows the results of theoretical research concerning the modeling and characterizationof the dissipative structures generally, the dissipation being an essential property of the systemwith self-organization which include the laser-type systems also. The most important resultspresented are new formulae which relate the coupling parameters ain from Lindblad equation withenvironment operators Γi; microscopic quantitative expressions for the dissipative coefficients ofthe master equations; explicit expressions which describe the changes of the environment densityoperator during the system evolution for fermion systems coupled with free electromagnetic field;the generalized Bloch-Feynman equations for N-level systems with microscopic coefficients inagreement with generally accepted physical interpretations. Based on Maxwell-Bloch equationswith consideration of the interactions between nearing atomic dipoles, for the dense optical mediawe have shown that in the presence of the short optical pulses, the population inversion oscillatesbetween two extreme values, depending on the strength of the interaction and the optical pulseenergy.
1. Introduction
An essential problem of the quantum information systems is the controllability and observ-
ability of the quantum systems. In this context, Fermi systems are essential for several
important physical effects in quantum engineering as the dynamics of semiconductor na-
nostructures and high temperature superconductivity, nuclear resonances, fusion-fission
reactions, and analysis of optical quantum systems. These effects are essentially determined
by the dissipative coupling of the system.
Dissipation in quantum systems is a complex phenomenon which raises important
theoretical investigations. A dissipative system is a system of interest, coupled with another
system usually considered as being of much larger-environment. Fundamental and difficult
2 Mathematical Problems in Engineering
problem of dissipative quantum theory is to design the total system (system of interest +environment) on the space system of interest. In this way obtain a quantum master equation
describing the evolution of the system using two terms: (1) a hamiltonian term for processes
with energy conservation and (2) a nonhamiltonian term with coefficients that depend on the
dissipative coupling. A master equation is based on approximations that consist in mediating
rapid oscillations of reduced density matrix describing the interaction.
Such an approximation is the assumption that the evolution operators of a dissipative
system forms a semigroup, not a group like for isolated systems. In this framework was
derived a quantum master equation with dissipative terms which is consistent with all
principles of quantum mechanics. Considering two operators, coordinate q and momentum
p, master equation was used to describe the harmonic oscillator. In this theoretical framework,
dissipation is described by the friction and diffusion coefficients that satisfy certain conditions
called basic restrictions and Heisenberg’s uncertainty relations are observed during the whole
evolution of the system.
A rigorous method for deducting the master equation with microscopic expressions of
the dissipative coefficients is developed in the literature.
For a weak dissipative coupling one obtains a master equation of Lindblad form [1],but with the microscopic expressions of the dissipative coefficients.
In the development of quantum theory of dissipative systems an important step was
the connection between Lindblad’s generator and the previous phenomenological descrip-
tions, realized by Sandulescu and Scutaru [2]. Besides, we must mention Isar et al.’s con-
tributions [3]. This school developed by the above-mentioned researchers in the field are
well recognized in the scientific world [4–8].Firstly, in the paper general expressions which relate the coupling parameters ain in
Lindblad equation with environment operators Γi have been established [9–11]. In this way,
became possible deeper causality understanding of processes of friction and diffusion and of
related quantum effects: broadening and shift of spectral lines, tunneling rates, bifurcations
and instability [12, 13].Secondly, for a system of fermions, coupled with a dissipative environment quanti-
tative microscopic expressions for the coefficients of the dissipative master equation depend-
ing on the potential matrix elements, the densities of states of the environment and the occu-
pation probabilities of these states are presented [14–19].The study continue with the systems of fermions coupled by electric dipole inter-
actions of free electromagnetic field for which has established general explicit expressions
which describe the changes of the environment density operator during the system evolution.
This description is not restricted to the Born approximation, taking into account the envi-
ronment time evolution as a function of the system evolution. The results of the dissipative
dynamics of the system of fermions in the presence of laser field are applicable to the dis-
sipative structures [14, 20–29].Next, generalized Bloch-Feynman equations for N-level systems with microscopic
coefficients in agreement with generally accepted physical interpretations are presented.
In the last part, we study the dynamics of dense media under the action of ultrafast
optical pulses using Maxwell-Bloch formalism to include interaction between close atomic
dipoles [30–34]. It is shown that, in a system initially without inversion, in the presence
of optical pulses, the final population has two extreme values, results which contribute to
understanding the specific mechanisms of switching for applications, with specific examples
concerning the coherent radiation generation and amplification [35–43]. A computational
specific software, to verify the experimental and numerical existing models and in the same
Mathematical Problems in Engineering 3
time to discover new important situations for operative systems design and implementation,
was developed [44–48].
2. Relationship between Coupling Coefficients in Lindblad MasterEquation and Environment Observables
Research on dissipative processes has led to evidence for the first time concerning the
relationship between coupling coefficients ain in the Lindblad equation:
ρ ≡ − i
ħ
[H,ρ
]+
1
2ħ
∑n
{[Xnρ,X
+n
]+[Xn, ρX
+n
]}(2.1)
depending on the system Hamiltonian H and the operators of opening Xn:
Xn ≡∑i
ainsi, (2.2)
where si are system operators, ain are complex coupling coefficients or amplitudes and Γioperators of environment defined using the interaction Hamiltonian as
HSE = ħ∑i
siΓi. (2.3)
These relationships have been established under the form [10]
∑n
aina∗jn = 2ħ
⟨ΓiΓj
⟩, (2.4)
and allow an understanding of the physical causes of quantum processes of friction and
diffusion, with their known effects: broadening and shift of spectral lines [11], increased rates
of tunneling, nonlinear characteristics, leading to bifurcation, instability, and chaos.
3. Microscopic Quantitative Expressions ofthe Dissipative Coefficients in Master Equations
A general quantum master equation for a many-level many-particle system, with microscopic
coefficients, that preserves the quantum-mechanical properties of the density matrix was
obtained [12]:
d
dtρ(t) = − i
ħ
[H,ρ(t)
]+∑i,j
λij{[
c+i cjρ(t), c+j ci]+[c+i cj , ρ(t)c
+j ci]}
(3.1)
with dissipative coefficients:
λij = λFij + λBij (3.2)
4 Mathematical Problems in Engineering
including a component λFij for a dissipative environment of fermions and a component λBij for
a dissipative environment of bosons.
Equation (3.1) is of Lindblad’s form, with dissipative operators depending on the
transition/population operators c+i cj . For a system with N levels, the total number of these
operators is N2 − 1 the number of the independent operators defined.
If we denote by V F and VB the interaction dissipative potentials of the environment
containing YF fermions and YB bosons, respectively, it is possible to write the expressions
of the coefficients λFij si λBij for the resonant transition |j〉 → |i〉 of the system coupled with
|β〉 → |α〉 environmental transition, with fermionic state having densitiesaly gFα , g
Fβ
and
populations fFα (εα), f
Fβ(εβ), and bosonic states with densities gB
α , gBβ
and population fBα (εα) si
fBβ(εβ). The probability that the final state |α〉 of the environment to be free is 1 − f(εα) while
the probability the initial state of the environment to be occupied is f(εβ).General expressions of the dissipative coefficients are written for this type of
interaction in the form:
λFij =π
ħYF
∫ ∣∣∣〈αi|V F∣∣βj⟩∣∣∣2[
1 − fFα (εα)
]fFβ
(εβ)gFα (εα)g
Fβ
(εβ)dεβ, εα − εβ = εj − εi,
λBij =π
ħYB
∫ ∣∣〈αi|VB∣∣βj⟩∣∣2[
1 + fBα (εα)
]fBβ
(εβ)gBα (εα)g
Bβ
(εβ)dεβ, εα − εβ = εj − εi.
(3.3)
4. The Environment Dynamics Correlated with that ofa Fermion Systems Coupled with Free Electromagnetic Field
We consider a system of Z charged fermions with the coordinates rn and momenta pn (n =1, 2, . . . , Z) in a single-particle potential U(1)(rn), while U(2)(rn, rm) represents the two-
particle residual potential. This system is coupled to the modes ν of the free electromagnetic
field. In order to describe the dynamics of this system, for simplicity, we neglect the particle
spin and its dimensions with respect to the electromagnetic field wavelength (the electric
dipole approximation). In this case, the total hamiltonian is of the form [14]
HT =Z∑n=1
(pn − eA
B)2
2m+
Z∑n=1
U(1)(rn) +1
2
Z∑n,m=1
U(2)(rn, rm) +HB. (4.1)
In the total hamiltonian (4.1),
V = − e
m
Z∑n=1
pnAB
(4.2)
is the system-field interaction potential, while
HS =Z∑n=1
p2n
2m+
Z∑n=1
U(1)(rn) +1
2
Z∑n,m=1
U(2)(rn, rm) (4.3)
Mathematical Problems in Engineering 5
is the fermion system hamiltonian, and
HB =∑ν
H(ν) (4.4)
is the field hamiltonian, where
H(ν) = ħων
(a+νaν +
1
2
)(4.5)
is the field mode ν hamiltonian.
Let us take the density operator χ(t) of the total system with hamitonian (4.1) and the
reduced density matrix
ρ(t) = TrB{χ(t)
}(4.6)
over the environment states.
The total density operator χ(t) satisfies the equation of motion:
dχ
dt= − i
ħ
[εV R(t) + εV (t), χ(t)
], (4.7)
where the sign above χ designs operators within the framework of interaction picture of the
system and environment
χ(t) = e(i/ħ)(HB+HS
0 )tχ(t)e−(i/ħ)(HS0 +H
B)t, (4.8)
while ε is an intensity parameter used to show the orders of the series expansion of this
density. Considering the radiation field of the black body in the initial state R, the total density
operator of the system can be taken under the form:
χ(t) = R ⊗ ρ(t) + εχ(1)(t) + ε2χ(2)(t) + · · · , (4.9)
where χ(1)(t), χ(2)(t) represent modifications of the field during the system evolution. The first
term of this expression corresponds to the Born approximation when the environment state
is a constant state R, while the higher-order terms, which satisfy the normalization relations
TrB{χ(1)(t)
}= TrB
{χ(2)(t)
}= · · · = 0, (4.10)
describe the environment dynamics that is correlated to the system dynamics. For an equation
of motion of the form
dρ
dt= εB(1)[ρ(t), t] + ε2B(2)[ρ(t), t] (4.11)
6 Mathematical Problems in Engineering
From (4.7), (4.9), and (4.11) we get a system of coupled equations:
R ⊗ B(1)[ρ(t), t] + dχ(1)
dt= − i
ħ
[V R(t) + V (t), R ⊗ ρ(t)
],
R ⊗ B(2)[ρ(t), t] + dχ(2)
dt= − i
ħ
[V R(t) + V (t), χ(1)(t)
].
(4.12)
By calculating the partial traces over the environment states and using the normalization
conditions (4.10), from these equations we get successively the terms of the equation of
motion (4.11):
B(1)[ρ(t), t] = − i
ħTrB
[V R(t) + V (t), R ⊗ ρ(t)
],
B(2)[ρ(t), t] = − i
ħTrB
[V R(t) + V (t), χ(1)(t)
],
(4.13)
while, integrating by time, we get ”excitation” terms of the total density operator (4.9):
χ(1)(t) =∫ t
0
{− i
ħ
[V R
(t′)+ V
(t′), R ⊗ ρ
(t′)] − R ⊗ B(1)[ρ(t′), t′]}dt′,
χ(2)(t) =∫ t
0
{− i
ħ
[V R
(t′)+ V
(t′), χ(1)(t′)] − R ⊗ B(2)[ρ(t′), t′]}dt′.
(4.14)
The first-order equation (4.13) represents the system evolution when the environment is
considered as being in a constant state R, while for the higher-order term (28), we take
into consideration some changes of the environment matrix (4.14). Further on, we will show
that the first-order terms (4.13) describe the hamiltonian dynamics of the system, while the
second-order term (28) describes system one-particle transitions related to environment.
5. The Generalized Bloch-Feynman Equations
An alternative description of dissipative system dynamics is given by Bloch-Feynman
equations for systems of fermions obtained by defining the pseudo-spin operators [14].In particular, for a system with two-level known Bloch-Feynman, equations are
obtained, where, Q12 is the field operator, P12 is the polarization operator, and N2 is
population operator:
d
dt〈Q12〉 = −γ⊥〈Q12〉 +ω21〈P12〉,
d
dt〈P12〉 = −ω21〈Q12〉 − γ⊥〈P12〉,
d
dt〈N2〉 = −γ||
[〈N2〉 −N
(0)2
],
(5.1)
Mathematical Problems in Engineering 7
with microscopic coefficients γ⊥ si γ|| expressed by dissipative coefficients λij of the master
equation:
γ⊥ = λ12 + λ21 + λ11 + λ22, (5.2)
γ|| = 2(λ12 + λ21), (5.3)
N(0)2 =
λ21
λ12 + λ21. (5.4)
The condition 2γ⊥ ≥ γ|| is a confirmation of master equation (4.7) which led to the estab-
lishment of Bloch equations-Feynman, because this condition is verified experimentally.
6. Dynamics of Dense Media under the Action of Short Optical Pulses
Maxwell-Bloch equations of a two-level atomic medium generalized to include interactions
between the dipoles approach [15, 16, 32] have been used to describe the system dynamics
under the action of ultrafast optical pulses. These equations, for systems with homogeneous
broadening of spectral lines in about semiclassical treating, were established using the density
matrix formalism as
dw
dt= −γL(w + 1) +
μ
ħ
(E∗Rab + ER∗
ab
), (6.1)
dRab
dt= −[γT + i(Δ + εw)
]Rab −
μ
2ħEw. (6.2)
In the above equations, w is the inversion of population, Rab nondiagonal elements of density
matrix slow variable, indices a and b refer to lower and higher energy states, with the gap
ħω0, EL is slowly varying local field Δ = ω0 − ω is the frequency deviation in relation to the
center frequency of the field resonance frequency, μ is the transition matrix element of the
electric dipole, and γ||, γ⊥ are longitudinal and transverse relaxation rates.
Contributions of the dipole-dipole interactions occur in (6.2) by term iεwRab, where
ε = nμ2/3ħεd � ω0 is the strength parameter of dipole-dipole interactions having
a dimension of a frequency. Equations (6.1) and (6.2) for the atomic variables and for
field variables realise the description Maxwell-Bloch of optically dense environment. These
equations were generalized and used to study intrinsic optical bistability, propagation effects
in nonlinear media, and so forth.
For numerical simulation, we considered the case resonant (Δ = 0), a characteristic
distance between dipoles much smaller than the wavelength of the central field (propagation
effects are negligible) and ultrafast pulses (pulses much shorter than γ−1|| ; this enables us to
8 Mathematical Problems in Engineering
−1
−0.5
0
0.5
1
w(t
)
0.8 0.9 1 1.1 1.2 1.3
Ω0/
20
30
Figure 1: Final state of population inversion, depending on the hyperbolic secant pulse maximum value
E(t) = E0sech(t/τp) (solid line), ετp = 20 (continuous line) and ετp = 30.
neglect dissipation processes). In these conditions, the matrix element Rab is decomposed
into its real and imaginary parts Rab = 1/2(ν + iu), resulting in system
du
dt′= −(ετp)νw,
dν
dt′=(ετp
)(u +
Ωε
)w,
dw
dt′= −(ετp)(Ω
ε
)ν
(6.3)
whose outcome is possible only numerically.
In the above equations t′ = t/τp is the normalized time, τp is the measured width pulse,
Ω(t) = μE(t)/h is the instantaneous Rabi frequency, and E(t) is the intensty of electrical pulse.
In Figure 1, we present the final population inversion function of maximum Rabi
frequency for hyperbolic secant pulses E(t) = E0sech(t/τp). As long as the Rabi frequency
has a value so that Ω0/ε < 1, the final population inversion is w = −1. In the region Ω0/ε > 1,
the final population inversion has an oscillatory behavior, almost rectangular wave. As the
parameter ετp value is greater, the oscillation period decreases, the transitions become abrupt,
and the first half cycle of the rectangular wave becomes more centered to Ω0/ε = 1.
In Figure 2, temporal evolution of the system is presented for a hyperbolic secant pulse with
a peak higher than one (when t → ∞, the population inversion performs a number of
oscillations before reaching a value 1; under certain conditions when t → ∞, after a number
of oscillations, the system remains in the ground state).
Mathematical Problems in Engineering 9
−1
0
0.5
1.5
1
−0.5
w,Ω/
w
Ω/
−8 −6 −4 −2 0 2 4 6 8
t
Figure 2: Temporal evolution of the system for a hyperbolic secant pulse, E(t) = E0sech(t/τp).
7. Conclusions
General expressions which relate the coupling parameters ain in Lindblad equation with envi-
ronment operators Γi have been established. These expressions allow deeper understanding
of causal processes of friction- and diffusion-related quantum effects: broadening and shift of
spectral lines, tunneling rates, bifurcations, and instability.
For a system of fermions coupled with a dissipative environment quantitative micro-
scopic expressions for the coefficients of the dissipative master equation are presented.
These coefficients depend on the potential matrix elements, the densities of states of
the environment, and the occupation probabilities of these states.
Expressions of the dependence of the particle distributions on temperature are taken
into account. It can be shown that a system of fermions located in a dissipative environment
of bosons tends to a Bose-Einstein distribution.
Studying the systems of fermions coupled by electric dipole interactions of free elec-
tromagnetic field, has established general explicit expressions which describe the changes of
the environment density operator during the system evolution for fermion systems coupled
with free electromagnetic field. This description is not restricted to the Born approximation,
taking into account the environment time evolution as a function of the system evolution. The
study can be continued with the calculation of the higher-order term of the reduced matrix
equation in order to describe the correlated transition of the system particles. The results of
the dissipative dynamics of the system of fermions in the presence of laser field are applicable
to the dissipative structures.
Generalized Bloch-Feynman equations for N-level systems with microscopic coef-
ficients in agreement with generally accepted physical interpretations are presented. On
this basis, the problem of a quantum system control is explicitly formulated in terms of
microscopic quantities: matrix elements of the dissipative two-body potential, densities of
the environment states, and occupation probabilities of these states.
10 Mathematical Problems in Engineering
Studying the dynamics of dense media under the action of ultrafast optical pulses
using Maxwell-Bloch formalism to include interaction between close atomic dipoles showed
that, in a system initially without inversion, in the presence of optical pulses, the final
population has two extreme values, the ratio of Rabi frequency and the parameter that des-
cribes the interactions between close dipoles, which contribute to understanding the specific
mechanisms of switching.
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 969087, 7 pagesdoi:10.1155/2012/969087
Research ArticlePower-Law Properties of Human View andReply Behavior in Online Society
Ye Wu,1, 2 Qihui Ye,2 Lixiang Li,3 and Jinghua Xiao1, 2
1 State Key Lab of Information Photonics and Optical Communications,Beijing University of Posts and Telecommunications, Beijing 100876, China
2 School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China3 Information Security Center, Beijing University of Posts and Telecommunications, Beijing 100876, China
Correspondence should be addressed to Ye Wu, [email protected]
Received 11 February 2012; Accepted 26 March 2012
Academic Editor: Ming Li
Copyright q 2012 Ye Wu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
Statistical properties of the human comment behavior are studied using data from “Tianya” and“Tieba” which are very popular online social systems (or forums) in China. We find that both thereply number R and the view number V of a thread in a subforum obey the power-law distributionsP(R) = Rα and P(V ) ∝ V β, respectively, which indicates that there exists a kind of highly populartopics. These topics should be specially paid much attention, because they play an important role inthe public opinion formation and the public opinion control. In addition, the relationship betweenR and V also obeys the power-law function R ∝ V γ . Based on the human comment habit, a modelis introduced to explain the human view and reply behaviors in the forum. Numerical simulationsof the model fit well with the empirical results. Our findings are helpful for discovering collectivepatterns of human behaviors and the evolution of public opinions on the virtual society as well asthe real one.
1. Introduction
Statistical properties and models of human behaviors have received much attention in dif-
ferent scientific fields, such as sociology, psychology, and economics. However, most of the
existing findings are only qualitative analyses for the lack of real data about the complexity
of human behaviors. Usually, it is assumed that the human behavior is a Poisson progress
[1, 2] which is a kind of the Markov progress. However, some researchers found that the in-
terevent time distribution of some human behaviors is power-law which means that it is a
non-Markov [3–14] one. More and more researchers are interested in it for its importance in
the theory and potential applications.
As an important part of modern life and human dynamics, the human behavior on the
Internet also attracts more and more attention. Chmiel et al. investigated the flows of visitors
2 Mathematical Problems in Engineering
migrating between different portal subpages. A model of portal surfing was developed where
a browsing process corresponds to a self-attracting walk on weighted networks with a short
memory [15]. Grabowski found that the distribution of human activity has the form of a
power-law [16] distribution. Based on the data from “Tianya”, Wu et al. found that the
dynamics of human comment in the online society is non-Markov. Further, they proposed
a model to explain it [17]. All these researches indicated that some kinds of human behavior
in on-line systems were non-Markov. They have some common statistic properties. More and
more researchers considered the forum as a virtual society to study the property and the
evolution of complex friendship networks [18, 19].A forum is very important for the information and the spreading of public opinions.
Many public opinions were also formatted and then spread in the forum. Analyzing the
user behavior in the forum is not only helpful for understanding the human behavior
and enhancing the information spreading, but also for designing a better website which is
important for the information spreading. Recently in China, the news about controlling public
opinions on purpose by news have attracted more and more attention. There was a report
that at least a half of public opinions in the Internet were proposed by some companies on
purpose. So it is very important to study the human comment behavior in the forum. Yu et
al. analyzed the view and reply data in the forum which was the beginning of researches on
the human comment behavior in the forums. They found that the view and reply numbers
of a thread in the sub-forum were power-law. However, they mainly considered statistic
properties of the behavior and did not present a model to explain the basic mechanism [20].In this paper, we consider the data collected from “Tianya” and “Tieba” which are very
popular on-line social system in China and different from those in [17]. We show that both
the view number (V ) and the reply number (R) of a thread in the sub-forum obey power-
law distributions which confirmed Yu et al.’s finding [20]. The relationship between V and
R is also power-law. These present that a lot of topics are important in the formation and
evolution of public opinions. Furthermore, based on the human habit, a model is proposed to
explain these phenomena. Numerical simulations are given to explain the human comment
behavior in the forum. We hope it is useful for understanding complex human behaviors in
the forums.
This paper is organized as follows: in Section 2, the origin of the data is introduced.
The statistical results are presented in Section 3. The model and numerical simulations are
presented in Section 4. Finally, our conclusion is given in Section 5.
2. Description of the Original Data
Our data are obtained from “Tianya” (http://www.tianya.cn) and “Tieba” (http://tieba.bai-
du.com), which are two most popular on-line social systems in China. Our data are collected
from the sub-forums of “Tianya” and “Tieba.” Each user is assigned a different identity name
(ID) in the forums. A topic in the sub-forum is called a thread. A thread is a minimal unit, and
it can be divided into a root thread and the reply threads. A root thread is a new topic, and
the reply threads are related to a root one. The users discuss the public opinion in both the
root and reply threads. Until 2010/02/11, there were 33,296,350 IDs in “Tianya,” and about
200,000 IDs on average were on-line at the same time. The topics and the public opinion in
“Tianya” and “Tieba” reflect part of the public opinions of the real society in China. Our data
sets are collected from the threads in four sub-forums. The types of these topics are different
from public news to personal stories which indicate that our results are general for different
Mathematical Problems in Engineering 3
Table 1: Detailed format of a subforum.
Topic Last update ID Reply/View Last update time
A ID 1 123/23124 2009/02/12, 12:11:05
B ID 3 3243/323532 2009/02/12, 12:10:05
C ID 32 323/42421 2009/02/12, 12:03:05
· · · · · ·Y ID 31 43/232 2009/02/12, 12:01:05
Table 2: Detailed informations about four randomly selected sub-forums.
Subforum A B C D
Total threads 19,492 19,479 27,359 5,302
Total clicks 1.05 × 109 1.62 × 109 2.33 × 108 5.19 × 106
Total replies 1.11 × 108 1.43 × 108 7.89 × 105 1.77 × 105
Duration (day) 247 2569 3 15
contents. The format of the data is shown in Table 1, where the first column is the title of a
thread, the second one gives the author’s name of a root thread, the third one shows R and
V , and the last one is the last update time of a thread.
3. Statistical Results
In the forum, the view and reply times of a thread reflect the influencing ability of a topic.
Further, more reply times mean more discussions and more communications. These two
parameters play an important role in the public opinion formation and the web design.
Hence, we study statistical properties of V and R in the thread of each sub-forum. Four sub-
forums are randomly selected as our data sets. The topics and some prosperities are listed in
Table 2.
The distributions of V and R in each sub-forum are shown in Figure 1 from which
we can clearly see that all the distributions are power-law, although the threads differ in
their contents. Their exponents vary with different sub-forums. These results show that the
process of human comments is non-Markov which is the same as the human dynamics of
the letter and e-mail communications, the web browsing, online movie watching, and broker
trades. The heavy tail of the distribution allows for much more numbers of threads which
have larger amounts of V and R than the Poisson progress. The thread which has more V and
R has much more influences on the public opinion. The number of these kind of threads is
so large that they cannot be ignored. A large population will read the thread by which their
opinions may be influenced. So we must pay much attention to them.
As is known to all, the more the view, the more the reply. However, the quantity
relationship between V and R is not very easy to know and it is the basic property of a
thread. Hence, next we mainly focus on the relationship between the human’s view and reply
behaviors in Figure 2. We found that it can be illustrated as a straight line in a log-log plot,
which means R ∝ V γ . It is easy to understand that the more the view, the more the reply.
Moreover, the nonlinear relationship here also means that the reply number increases slower
than the view one when the view number is large enough. It also indicates that human’s
interest in reply decreases as the increment of V .
4 Mathematical Problems in Engineering
100
10−2
10−4
102 104 106
P(R
,V)
R,V
(a)
100
10−2
10−4
102 104 106100
P(R
,V)
R,V
(b)
100
10−2
10−4
100 105 101010−6
R,V
P(R
,V)
(c)
100
10−5
100
R,V
P(R
,V)
102 104 106
(d)
Figure 1: Distributions of V (∗) and R (•) in each sub-forum. The solid line and the dashed one showthe slopes of fitting function for distributions of V and R, respectively, where (a) sub-forum A, the slopesα = 1.40 ± 0.01, β = 1.44 ± 0.02, (b) sub-forum B, the slopes α = 1.35 ± 0.02, β = 1.12 ± 0.01, (c) sub-forum C,the slopes α = 1.51 ± 0.01, β = 1.68 ± 0.01, (d) sub-forum D, the slopes α = 1.12 ± 0.02, β = 1.76 ± 0.02.
4. The Model and the Simulations
In order to get a better understanding of our empirical observations in Section 3, we propose
a model based on our intuitive experience about the human comment habit. We see that the
view number of each sub-forum increases more quickly as the time evolves. There are many
threads on each sub-forum. Each thread will be viewed based on its content and its previous
view time. Hence, our model is defined by the following scheme.
Step 1 (growing). At time t = 0, there are a few threads on the sub-forum, and each thread
has a random small V and R. At each step, a new thread is created, and there are c ∗ tθ views
on the old thread. All the old threads have the probabilities to be viewed.
Step 2 (view habit). The probability that an old thread is viewed at each step is based on
its attraction Π(i) = Ai(t)/ΣAi(t), where Ai(t) is the attraction of a thread i at time t and it is
reflected by the previous view number Vi(t), that is, Ai(t) = A(0)+Vi(t). Here A(0) represents
the initial attraction which is different due to different topics.
Step 3 (reply habit). At each step, when the user views a thread, he has a probability P(i) =L ∗ (R(i)/V (i))η to reply the thread.
Mathematical Problems in Engineering 5
104
100
108
R
V
100 104
(a)
104
100
108
R
V
100 104
(b)
104
100
108
R
V
100 104
(c)
104
100
R
V
100 104
(d)
Figure 2: The power-law relationship between V and R, where (a) sub-forum A, the slope γ = 0.77, (b) sub-forum B, the slope γ = 0.89, (c) sub-forum C, the slope γ = 0.85, (d) sub-forum D, the slope γ = 0.90.
Mathematically, the model is similar to the growing networks in [21]. Based on the
analysis of the growing network in this paper, we obtain that the distribution of Vi is a power-
law one, that is, P(Vi) ∝ (Vi)−α at a large enough time t where the exponent α is 1+ 1/(1+ θ) .
To compare our model with empirical observation results, let us take the sub-forum C
in our data sets as an example. Here we use the parameters θ = 0.9, L = 0.1, η = 0.5 in the
simulation. The results are shown in Figure 3. Figure 3(a) presents that the distribution of the
view is indeed a power-law one with a similar exponent as that from the data. Figure 3(b)shows that the reply number also obeys a power-law distribution. The nonlinear relationship
between the view and reply times is shown in Figure 3(c) which is the same as that from
the data. In Figure 3(d), we further study the relationship between the parameter η and the
slope γ . We see that γ decreases as η increases. From the analyses above, we can see that
the proposed model can well-describe most important features in the human view and reply
behaviors in online social systems.
5. Conclusion
In this paper, we analyze the statistical properties of the view and reply behaviors in on-line
social systems. We find that they are different types of interactive human dynamics which
are non-Markov. The view and the reply behaviors follow power-law distributions, and the
relationship between them also follows a power-law one. A model based on the personal
6 Mathematical Problems in Engineering
100
10−2
10−4
102 104
V
100
P(V
)
(a)
100
10−2
10−4
R
100
P(R
)
102 104 106
(b)
100
102 104100
102
R
V
106
(c)
1
0.8
0.6
0.2 0.4 0.6
γ
η
(d)
Figure 3: Simulation results of the model whose parameters are selected as θ = 0.9, L = 0.1, η = 0.5, wherethe dashed line shows the slope of the fitting function. (a) The distribution of V of a thread. The slopeof fitting function is α = 1.51 ± 0.02. (b) The distribution of R of a thread. The slope of fitting function isα = 1.41± 0.04. (c) The relationship between V and R. The slope of fitting function is γ = 0.9± 0.03. (d) Therelationship between the slope γ and the parameter η.
attraction is introduced to explain the human complex behavior. Numerical simulations of the
model fit well with empirical results. Our work is useful to understand the human complex
behavior in realistic society, for example, the human discussion behavior in a meeting
or group communications in trunked mobile telephony [22]. We expect that quantitative
understanding of human view and reply behaviors, when combined with additional content
analyses, will open a new perspective on distinguishing fraud public opinions from realistic
opinions.
Acknowledgment
This paper is supported by the National Natural Science Foundation of China (Grants nos.
61104152, 60804046), the Fundamental Research Funds for the Central Universities (Grant
no. 2011R01), the Foundation for the Author of National Excellent Doctoral Dissertation of
China (Grant no. 200951), and the Asia Foresight Program under NSFC Grant (Grant no.
61161140320).
References
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[3] A. L. Barabasi, “The origin of bursts and heavy tails in human dynamics,” Nature, vol. 435, no. 7039,pp. 207–211, 2005.
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[12] H. Sheng, H. Sun, Y. Chen, and T. Qiu, “Synthesis of multifractional Gaussian noises based on varia-ble-order fractional operators,” Signal Processing, vol. 91, no. 7, pp. 1636–1644, 2011.
[13] H. Sheng, Y. Q. Chen, and T. Qiu, “Heavy-tailed distribution and local long memory in time series ofmolecular motion on the cell membrane,” Fluctuation and Noise Letters, vol. 10, no. 1, pp. 93–119, 2011.
[14] H. Sheng, Y. Q. Chen, and T. S. Qiu, Fractional Processes and Fractional Order Signal Processing, Springer,New York, NY, USA, 2012.
[15] A. Chmiel, K. Kowalska, and J. A. Holyst, “Scaling of human behavior during portal browsing,” Phys-ical Review E, vol. 80, no. 6, Article ID 066122, 7 pages, 2009.
[16] A. Grabowski, “Human behavior in online social systems,” European Physical Journal B, vol. 69, no. 4,pp. 605–611, 2009.
[17] Y. Wu, C. Zhou, M. Chen, J. Xiao, and J. Kurths, “Human comment dynamics in on-line social sys-tems,” Physica A, vol. 389, no. 24, pp. 5832–5837, 2010.
[18] A. Grabowski, N. Kruszewska, and R. A. Kosinski, “Properties of on-line social systems,” EuropeanPhysical Journal B, vol. 66, no. 1, pp. 107–113, 2008.
[19] G. Csanyi and B. Szendroi, “Structure of a large social network,” Physical Review E, vol. 69, no. 3,Article ID 036131, 5 pages, 2004.
[20] J. Yu, Y. Hu, M. Yu, and Z. Di, “Analyzing netizens’ view and reply behaviors on the forum,” PhysicaA, vol. 389, no. 16, pp. 3267–3273, 2010.
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 391050, 13 pagesdoi:10.1155/2012/391050
Research ArticleSinogram Restoration for Low-Dosed X-RayComputed Tomography Using Fractional-OrderPerona-Malik Diffusion
Shaoxiang Hu,1 Zhiwu Liao,2 and Wufan Chen3
1 School of Automation Engineering, University of Electronic Science and Technology of China,Chengdu 611731, China
2 School of Computer Science, Sichuan Normal University, Chengdu 610101, China3 Institute of Medical Information and Technology, School of Biomedical Engineering,Southern Medical University, Guangzhou 510515, China
Correspondence should be addressed to Zhiwu Liao, [email protected]
Received 18 January 2012; Accepted 16 March 2012
Academic Editor: Ming Li
Copyright q 2012 Shaoxiang Hu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
Existing integer-order Nonlinear Anisotropic Diffusion (NAD) used in noise suppressing willproduce undesirable staircase effect or speckle effect. In this paper, we propose a new scheme,named Fractal-order Perona-Malik Diffusion (FPMD), which replaces the integer-order derivativeof the Perona-Malik (PM) Diffusion with the fractional-order derivative using G-L fractionalderivative. FPMD, which is a interpolation between integer-order Nonlinear Anisotropic Diffusion(NAD) and fourth-order partial differential equations, provides a more flexible way to balancethe noise reducing and anatomical details preserving. Smoothing results for phantoms and realsinograms show that FPMD with suitable parameters can suppress the staircase effects and speckleeffects efficiently. In addition, FPMD also has a good performance in visual quality and root meansquare errors (RMSE).
1. Introduction
Radiation exposure and associated risk of cancer for patients receiving CT examination have
been an increasing concern in recent years. Thus minimizing the radiation exposure to
patients has been one of the major efforts in modern clinical X-ray CT radiology [1–8].A simple and cost-effective means to achieve low-dose CT applications is to lower X-
ray tube current (mA) as low as achievable [6, 7]. However, the presentation of strong noise
degrades the quality of low-dose CT images dramatically and decreases the accuracy of the
diagnosis dose.Filtering noise from clinical scans is a challenging task, since these scans contain arti-
facts and consist of many structures with different shape, size, and contrast, which should be
2 Mathematical Problems in Engineering
preserved for making correct diagnosis. Many strategies have been proposed to reduce
the noise, for example, by nonlinear noise filters [8–20] and statistics-based iterative image
reconstructions (SIIRs) [21–29].The SIIRs utilize the statistical information of the measured data to obtain good denois-
ing results but are limited for their excessive computational demands for the large CT image
size. Although the nonlinear filters show effectiveness in reducing noise both in sinogram
space and image space, they cannot handle the noise-induced streak artifacts. Some nonlinear
filters, such as Nonlinear Anisotropic Diffusion (NAD), even produce new artifacts in L-CT
denoising [30–37].To eliminate the undesirable staircase effect, high-order PDEs (typically fourth-order
PDEs) for image restoration have been introduced in [38–43]. Though these methods can
eliminate the staircase effect efficiently, they often lead to a speckle effect [44].Recently, fractional-order PDEs have been studied in many fields [38–49]. The frac-
tional derivative can be seen as the generalization of the integer-order derivative. It has been
studied by many mathematicians (e.g., Euler, Hardy, Littlewood, and Liouville) [47]. Not
until Mandelbrot found fractals and applied the G-L fractional derivative to the Brownian
motion did the fractional derivative cause great attention. There are many methods that can
define the fractional derivative. The usual definitions among them involve G-L fractional
derivative, Cauchy-integral fractional derivative, frequency-domain (Fourier-domain) frac-
tional derivative.
Li and Zhao investigate relation between the data of cyber-physical networking systems
and power laws and then suggest that power-law-type data may be governed by stochasti-
cally differential equations of fractional order [45]. They also propose that one-dimensional
random functions with long-range dependence (LRD) based on a specific class of processes
called the Cauchy-class (CC) process maybe a possible model of sea level data [46].You and Kaveh develop a class of fractional-order multiscale variational model using
G-L definition of fractional-order derivative and propose an efficient condition of the conver-
gence for the model [38]. The experiments show that the model can improve the peak signal-
to-noise ratio, preserve texture, and eliminate the stair effect efficiently.
Bai and Feng proposed a class of fractional-order anisotropic diffusion equations based
on PM equation for image denoising using Fourier-domain fractional derivative in [49].The numerical results showed that both of the staircase effect and the speckle effect can be
eliminated effectively by using the fractional-order derivative.
Inspired from previous works and in order to eliminate the staircase effects and pre-
serve anatomical details, we propose to replace the first-order and the second-order deriva-
tive of the PM Diffusion with the fractional-order derivative using G-L fractional derivative.
It should be indicated that the method proposed in this paper, which is carried on the
sinogram space directly, is different to the method proposed in [49], which is carried on the
Fourier space.
The arrangement of this paper is as follows. In Section 2, the noise model of Low-dosed
CT (L-CT) is introduced; and then the PM diffusion is given in Section 3, new fractional-order
PM method is developed using G-L fractional definition in Section 4; the experiment results
are shown and discussed in Section 5; the final part is the conclusions and acknowledgement.
2. Noise ModelsBased on repeated phantom experiments, low-mA (or low-dose) CT-calibrated projection
data after logarithm transform were found to follow approximately a Gaussian distribution
Mathematical Problems in Engineering 3
with an analytical formula between the sample mean and sample variance, that is, the noise
is a signal-dependent Gaussian distribution [20].In this section, we will introduce signal-independent Gaussian noise (SIGN), Poisson
noise, and signal-dependent Gaussian noise.
2.1. Signal-Independent Gaussian Noise (SIGN)
SIGN is a common noise for imaging system. Let the original projection data be {xi}, i = 1,
. . . , m, where i is the index of the ith bin. The signal has been corrupted by additive noise
{ni}, i = 1, . . . , m and one noisy observation
yi = xi + ni, (2.1)
where yi, xi, ni are observations for the random variables Yi, Xi, and Ni where the uppercase
letters denote the random variables and the lower-case letters denote the observations for
respective variables. Xi is normal N(0, σ2X); Ni is normal N(0, σ2
N) and independent to the
Gaussian random variable Xi. Thus Yi is normal N(0, σ2X + σ2
N).
2.2. Poisson Model and Signal-Dependent Gaussian Model
The photon noise is due to the limited number of photons collected by the detector [36]. For
a given attenuating path in the imaged subject, N0(i, α) and N(i, α) denote the incident and
the penetrated photon numbers, respectively. Here, i denote the index of detector channel or
bin, and α is the index of projection angle. In the presence of noises, the sinogram should be
considered as a random process and the attenuating path is given by
ri = − ln
[N(i, α)N0(i, α)
], (2.2)
where N0(i, α) is a constant and N(i, α) is Poisson distribution with mean N.
Thus we have
N(i, α) = N0(i, α) exp(−ri). (2.3)
Both its mean value and variance are N.
Gaussian distributions of ploy-energetic systems were assumed based on limited
theorem for high-flux levels and followed many repeated experiments in [20]. We have
σ2i
(μi
)= fi exp
(μi
γ
), (2.4)
where μi is the mean and σ2i is the variance of the projection data at detector channel or bin i,
γ is a scaling parameter, and fi is a parameter adaptive to different detector bins.
The most common conclusion for the relation between Poisson distribution and
Gaussian distribution is that the photon count will obey Gaussian distribution for the case
4 Mathematical Problems in Engineering
with large incident intensity and Poisson distribution with feeble intensity [20]. In addition,
in [36], the authors deduce the equivalency between Poisson model and Gaussian model.
Therefore, both theories indicate that these two noises have similar statistical properties and
can be unified into a whole framework.
3. Perona-Malik Diffusion
In image smoothing, Nonlinear Anisotropic Diffusion (NAD), also called Perona-Malik dif-
fusion (PMD), is a technique aiming at reducing image details without removing significant
parts of the image contents, typically edges, lines, or textures, which are important for the
image [50].With a constant diffusion coefficient, the anisotropic diffusion equations reduce to the
heat equation, which is equivalent to Gaussian blurring. This is ideal for smoothing details
but also blurs edges. When the diffusion coefficient is chosen as an edge seeking function, the
resulting equations encourage diffusion (hence smoothing) within regions and stop it near
strong edges. Hence the edges can be preserved while smoothing from the image [50].Formally, NAD is defined as
∂u(x,y, t
)∂t
= div(g(x,y, t
)∇u(x,y, t
)), (3.1)
where u(x,y, 0) is the initial gray scale image, u(x,y, t) is the smooth gray scale image at time
t, ∇ denotes the gradient, div(·) is the divergence operator, and g(x,y, t) is the diffusion
coefficient. g(x,y, t) controls the rate of diffusion and is usually chosen as a monotonically
decreasing function of the module of the image gradient. Two functions proposed in [50] are
g(∥∥∇u
(x,y, t
)∥∥) = e−(‖∇u(x,y,t)‖/σ)2
, (3.2)
g(∥∥∇u
(x,y, t
)∥∥) = 1
1 +(∥∥∇u
(x,y, t
)∥∥/σ)2, (3.3)
where ‖ · ‖ is the module of the vector and the constant σ controls the sensitivity to edges.
Perona and Malik propose a simple method to approach the modules of gradients,
which is called PM diffusion [50]. Its discretization for Laplacian operator is
u(i, j, t + 1
)= u
(i, j, t
)+
1
4
[cN · ∇2
Nu(i, j, t
)+cS · ∇2
Su(i, j, t
)+ cE · ∇2
Eu(i, j, t
)+ cW · ∇2
Wu(i, j, t
)],
(3.4)
where
∇2Nu
(i, j, t
)= u
(i − 1, j, t
) − u(i, j, t
),
∇2Su(i, j, t
)= u
(i + 1, j, t
) − u(i, j, t
),
Mathematical Problems in Engineering 5
∇2Eu(i, j, t
)= u
(i, j + 1, t
) − u(i, j, t
),
∇2Wu
(i, j, t
)= u
(i, j − 1, t
) − u(i, j, t
). (3.5)
According to (3.2)-(3.3), the diffusion coefficient is defined as a function of module of the
gradient. However, computing a gradient accurately in discrete data is very complex and the
module of the gradient is simplified as the absolute values of four directions and diffusion
coefficients are
cN(i, j, t
)= g
(∣∣∣∇2Nu
(i, j, t
)∣∣∣),cS(i, j, t
)= g
(∣∣∣∇2Su(i, j, t
)∣∣∣),cE(i, j, t
)= g
(∣∣∣∇2Eu(i, j, t
)∣∣∣),cW(i, j, t
)= g
(∣∣∣∇2Wu
(i, j, t
)∣∣∣),(3.6)
where | · | is the absolute value of the number and g(·) is defined in (3.2) or (3.3).The main default for PM diffusion is that it will lead to staircase effect or sometimes
details oversmoothing. In order to eliminate the staircase effects and preserve anatomical
details, we propose to replace the first-order and the second-order derivative of the PM Dif-
fusion with the fractional-order derivative using G-L fractional derivative. The new diffusion
model will be introduced in the next section.
4. The Fractional-Order PM Diffusion (FPMD)
The FPMD is developed using G-L fractional-order derivative, which is defined as [38]
Dαg(x) = limh→ 0+
∑k≥0 (−1)kCα
kg(x − kh)
hα, α > 0, (4.1)
where g(x) is a real function, α > 0 is a real number, Cαk= Γ(α + 1)/[Γ(k + 1)Γ(α − k + 1)] is
the generalized binomial coefficient and Γ(·) denotes the Gamma function. If h = 1, the finite
fractional difference is
�αg(x) =K−1∑k=0
(−1)kCαkg(x − k). (4.2)
An image U will be a 2-dimensional matrix of size N × N and its discrete fractional-
order gradient ∇αu is an 8-dimensional vector:
∇αu(i, j)
=(∇α
0u(i, j),∇α
1u(i, j),∇α
2u(i, j),∇α
3u(i, j),∇α
4u(i, j),∇α
5u(i, j),∇α
6u(i, j),∇α
7u(i, j))T
,(4.3)
where T represents the transpose of the vector and ∇αuk(i, j), k = 0, . . . , 7 are defined as
∇α0u(i, j)=
K−1∑k=0
(−1)kCαku(i, j + k
), ∇α
1u(i, j)=
K−1∑k=0
(−1)kCαku(i − k, j + k
),
6 Mathematical Problems in Engineering
∇α2u(i, j)=
K−1∑k=0
(−1)kCαku(i − k, j
), ∇α
3u(i, j)=
K−1∑k=0
(−1)kCαku(i − k, j − k
),
∇α4u(i, j)=
K−1∑k=0
(−1)kCαku(i, j − k
), ∇α
5u(i, j)=
K−1∑k=0
(−1)kCαku(i + k, j − k
),
∇α6u(i, j)=
K−1∑k=0
(−1)kCαku(i + k, j
), ∇α
7u(i, j)=
K−1∑k=0
(−1)kCαku(i + k, j + k
).
(4.4)
Thus
∇2αu(i, j)
=(∇2α
0 u(i, j),∇2α
1 u(i, j),∇2α
2 u(i, j),∇2α
3 u(i, j),∇2α
4 u(i, j),∇2α
5 u(i, j),∇2α
6 u(i, j),∇2α
7 u(i, j))T
,
(4.5)
where T represents the transpose of the vector. From (4.3), we have
∇2α0 u
(i, j)=
K−1∑k=0
(−1)kCαk∇α
0u(i, j + k
), ∇2α
1 u(i, j)=
K−1∑k=0
(−1)kCαk∇α
1u(i − k, j + k
),
∇2α2 u
(i, j)=
K−1∑k=0
(−1)kCαk∇α
2u(i − k, j
), ∇2α
3 u(i, j)=
K−1∑k=0
(−1)kCαk∇α
3u(i − k, j − k
),
∇2α4 u
(i, j)=
K−1∑k=0
(−1)kCαk∇α
4u(i, j − k
), ∇2α
5 u(i, j)=
K−1∑k=0
(−1)kCαk∇α
5u(i + k, j − k
),
∇2α6 u
(i, j)=
K−1∑k=0
(−1)kCαk∇α
6u(i + k, j
), ∇2α
7 u(i, j)=
K−1∑k=0
(−1)kCαk∇α
7u(i + k, j + k
).
(4.6)
Let
g =(g0, g1, g2, g3, g4, g5, g6, g7
)T,
(4.7)
where T represents the transpose of the vector and gk, k = 0, . . . , 7 is defined as
gk =g(∣∣∇α
ku(i, j)∣∣)∑7
n=0 g(∣∣∇α
nu(i, j)∣∣) , k = 0, 1, . . . , 7, (4.8)
where ∇αku(i, j), k = 0, . . . , 7, defined in (4.3) are the components of vector ∇αu(i, j) and∑7
n=0 g(|∇αnu(i, j)|) is the normalized constant, g is the decreasing function of absolute value
Mathematical Problems in Engineering 7
of ∇αku(i, j), k = 0, . . . , 7. Following (2.2) and (2.3), g(|∇uα
k(x,y, t)|) can be defined as
g(∣∣∇uα
k
(x,y, t
)∣∣) = e−(|∇uαk(x,y,t)|/σ)2
, k = 0, . . . , 7 (4.9)
or
g(∣∣∇uα
k
(x,y, t
)∣∣) = 1
1 +(∣∣∇uα
k
(x,y, t
)∣∣/σ)2, k = 0, . . . , 7, (4.10)
where | · | is the absolute value of the number and the constant σ controls the sensitivity to
edges.
The new FPMD based on G-L fractional-order derivative is defined as
∂u(i, j, t
)∂t
= div
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
g0∇α0u(i, j, t
)g1∇α
1u(i, j, t
)g2∇α
2u(i, j, t
)g3∇α
3u(i, j, t
)g4∇α
4u(i, j, t
)g5∇α
5u(i, j, t
)g6∇α
6u(i, j, t
)g7∇α
7u(i, j, t
)
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
, (4.11)
where the ∇αku(i, j, t), k = 0, . . . , 7, are the components of vector ∇αu(i, j, t) in (4.3) and gk,
k = 0, . . . , 7, defined in (4.8) are the components of g in (4.7).The above equation can be represented as
∂u(i, j, t
)∂t
=7∑
k=0
gk∇2αk u
(i, j, t
), (4.12)
where∑7
k=0 gk = 1 and ∇2αku(i, j, t) can be computed according to (4.5).
Thus the explicit form for solving (4.12) is
u(i, j, t + 1
)= u
(i, j, t
)+
7∑k=0
gk∇2αk u
(i, j, t
), (4.13)
where u(i, j, t + 1) is the gray level of (i, j) at time t + 1 and gk, ∇2αku(i, j, t) are the same as in
(4.12).
5. Experiments and Discussion
The main objective for smoothing L-CT images is to delete the noise while to preserve
anatomy details for the images.
8 Mathematical Problems in Engineering
Table 1: RMSE of different smoothing methods.
Noisy Median Wlener GaussianPMD
FPMD FPMD FPMD
image Filter Filter Filter α = 0.2 α = 0.5 α = 1.5
RMSE 0.0962 0.0804 0.0634 0.0963 0.0774 0.0603 0.0735 0.0752
In order to show the performance of FPMD, a 2-dimensional 256 × 256 Shepp-Logan
head phantom developed in MatLab. The number of bins per view is 888 with 984 views
evenly spanned on a circular orbit of 360◦. The detector arrays are on an arc concentric to
the X-ray source with a distance of 949.075 mm. The distance from the rotation center to the
X-ray source is 541 mm. The detector cell spacing is 1.0239 mm. The L-CT projection data
(sinogram) is simulated by adding Gaussian-dependent noise (GDN) whose analytic form
between its mean and variance has been shown in (2.4). In this paper, set fi = 4.0 and T =2e + 4. The projection data is reconstructed by standard Filtered Back Projection (FBP). Since
both the original projection data and sinogram have been provided, the evaluation based on
root-mean-square error (RMSE) between the ideal reconstructed image is and reconstructed
images defined as
√√√√ 1
256 × 256
256∑i=1
256∑j=1
(frecon
(i, j) − fPh
(i, j))2
, (5.1)
where frecon(i, j) denotes the reconstructed value on position (i, j) while fPh(i, j) denotes the
ideal reconstructed value on position (i, j).Two abdominal CT images of a 62-year-old woman with different doses were scanned
from a 16 multidetector row CT unit (Somatom Sensation 16; Siemens Medical Solutions)using 120 kVp and 5 mm slice thickness. Other remaining scanning parameters are gantry
rotation time, 0.5 second; detector configuration (number of detector rows section thickness),16 × 1.5 mm; table feed per gantry rotation, 24 mm; pitch, 1 : 1 and reconstruction method,
Filtered Back Projection (FBP) algorithm with the soft-tissue convolution kernel “B30f”.
Different CT doses were controlled by using two different fixed tube current 30 mAs and
150 mAs ((60 mA or 300 mAs) for L-CT and standard-dose CT (SDCT) protocols, resp.).The CT dose index volume (CTDIvol) for LDCT images and SDCT images are in positive
linear correlation to the tube current and are calculated to be approximately ranged between
15.32 mGy to 3.16 mGy [51] (see Figures 2(a) and 2(b)).On sinogram space, FPMD with α = 0.2, α = 0.5, and α = 1.5 is carried on two image
collections. Other compared methods include median filter with 5 × 5 window; wiener filter
with 5 × 5 window; Gaussian filter whose mean is 0 and its standard deviation is 1.8. The
diffusion coefficient for PMD and FPMDs is selected as a Gaussian function whose standard
deviation is 2. All smoothed projection data will be reconstructed by standard FBP.
Table 1 summarized RMSE between the ideal reconstructed image and filtered
reconstructed image. The FPMD with α = 0.2 has the best performance in RMSE, while other
FPMD with α = 0.5 and α = 1.5 also has better performance than almost other comparing
methods except for 5 × 5 wiener. In summary, the FPMD has a very good performance in
RMSE. Since FPMD provides a more flexible way for diffusion than PMD, FPMD has much
good performance in denoising while preserving structures.
Mathematical Problems in Engineering 9
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Figure 1: Shepp-Logan phantoms. (a) Original ideal reconstructed phantom. (b) Simulated LDCTreconstructed phantom. (c) LDCT reconstructed phantom processed by 5 × 5 median filter. (d) LDCTreconstructed phantom processed by 5 × 5 wiener filter. (e) LDCT reconstructed phantom processedby Gaussian smoothing with [σ = 1.8, μ = 0]. (f) LDCT reconstructed phantom processed by PMDwith [σ = 2]. (g) LDCT reconstructed phantom processed by FPMD with [σ = 2, α = 0.2]. (h) LDCTreconstructed phantom processed by FPMD with [σ = 2, α = 0.5]. (i) LDCT reconstructed phantomprocessed by FPMD with [σ = 2, α = 1.5].
Comparing all the original SDCT images and L-CT images in Figures 1 and 2, we found
that the L-CT images were severely degraded by nonstationary noise and streak artifacts.
In Figures 2(g)–2(i), for the proposed FPMD approach, experiments with fractional-order
α gradually increased will obtain more smooth images. Both in Figure 1 and 2, we can
observe better noise/artifacts suppression and edge preservation when α = 0.2. Especially,
compared to their corresponding original SDCT images, the fine features representing the
intrahepatic bile duct dilatation and the hepatic cyst were well restored by using the
10 Mathematical Problems in Engineering
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Figure 2: Abdominal CT images of a 62-year-old woman. (a) Original SDCT image with tube current timeproduct 150 mAs. (b) Original LDCT image with tube current time product 60 mAs. (c) LDCT imageprocessed by 5 × 5 median filter. (d) LDCT image processed by 5 × 5 wiener filter. (e) LDCT imageprocessed by Gaussian smoothing with [σ = 1.8, μ = 0]. (f) LDCT image processed by PMD with [σ = 2].(g) LDCT image processed by FPMD with [σ = 2, α = 0.2]. (h) LDCT image processed by FPMD with[σ = 2, α = 0.5]. (i) LDCT image processed by FPMD with [σ = 2, α = 1.5].
proposed FPMD. We can observe that, the noise grains and artifacts were significantly
reduced for the FPMD processed L-CT images with suitable α both in Figures 1 and 2. The
fine anatomical/pathological features can be well preserved compared to the original SDCT
images (Figures 1(a) and 2(a)) under standard dose conditions.
6. Conclusions
In this paper, we propose a new fractional-order PMD (FPMD) for L-CT sinogram imaging
based on G-L fractional-order derivative definition. Since FPMD is a interpolation between
Mathematical Problems in Engineering 11
integer-order Nonlinear Anisotropic Diffusion (NAD) and fourth-order partial differential
equations, it provides a more flexible way to balance the noise reducing and anatomical
details preserving. Smoothing results for phantoms and real sinograms show that FPMD
with suitable parameters can suppress the staircase effects and speckle effects efficiently. In
addition, FPMD also has good performance in visual quality and root mean square errors
(RMSE).
Acknowledgments
This paper is supported by the National Natural Science Foundation of China (no. 60873102),Major State Basic Research Development Program (no. 2010CB732501), and Open Foundation
of Visual Computing and Virtual Reality Key Laboratory Of Sichuan Province (no. J2010N03).This paper was supported by a Grant from the National High Technology Research and
Development Program of China (no. 2009AA12Z140) and Open foundation of Key Labora-
tory of Land Resources Evaluation and Monitoring of Southwest Sichuan Normal University,
Ministry of Education.
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 784270, 12 pagesdoi:10.1155/2012/784270
Research ArticleDifference-Equation-BasedDigital Frequency Synthesizer
Lu-Ting Ko,1 Jwu-E. Chen,1 Yaw-Shih Shieh,2Hsi-Chin Hsin,3 and Tze-Yun Sung2
1 Department of Electrical Engineering, National Central University, Chungli 320-01, Taiwan2 Department of Electronics Engineering, Chung Hua University, Hsinchu 300-12, Taiwan3 Department of Computer Science and Information Enginnering, National United University,Miaoli 360-03, Taiwan
Correspondence should be addressed to Tze-Yun Sung, [email protected]
Received 9 February 2012; Accepted 2 March 2012
Academic Editor: Ming Li
Copyright q 2012 Lu-Ting Ko et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
This paper presents a novel algorithm and architecture for digital frequency synthesis (DFS). It isbased on a simple difference equation. Simulation results show that the proposed DFS algorithmis preferable to the conventional phase-locked-loop frequency synthesizer and the direct digitalfrequency synthesizer in terms of the spurious-free dynamic range (SFDR) and the peak-signal-to-noise ratio (PSNR). Specifically, the results of SFDR and PSNR are more than 186.91 dBc and127.74 dB, respectively. Moreover, an efficient DFS architecture for VLSI implementation is alsoproposed, which has the advantage of saving hardware cost and power consumption.
1. Introduction
Many modern devices, for example, radio receivers, ADSL (Asymmetric Digital Subscriber
Line), XDSL (X Digital Subscriber Line), 3G/4G mobile phones, walkie-talkies, CB radios,
satellite receivers, and GPS systems, require frequency synthesizers with fine resolutions,
fast channel switching, and large bandwidths. There are two types of frequency synthesizer
available: phase-locked loop (PLL) and direct digital frequency synthesis (DDFS).PLL is a control system, which generates an output signal with phase matched that
of the input reference signal. Figure 1 shows the conventional PLL frequency synthesizer
consisting of a phase detector, a charge bump, a lowpass filter, a voltage control oscillator, and
a frequency divider [1–8]. The lower frequency signal, Fdiv, obtained by dividing the output
signal via the frequency divider, is compared with the reference signal, Fref, in the phase
2 Mathematical Problems in Engineering
Phase detector Charge bump Low-pass filter
Voltage
controlled
oscillator
Frequency
divider
OutputFref
Fdiv
Figure 1: Block diagram of the conventional PLL system.
vv
Fclk
Phase
accumulator
Sine/cosine
generator
Digital to
analog
converter
Low pass filter
θ cos θ
sin θFCW
Figure 2: Block diagram of the conventional DDFS architecture.
detector to generate an error signal, which is proportional to the phase difference. The charge
bump converts the error signal pulse into analog current pulses, which are then integrated
by using the lowpass filter, and drives the voltage-controlled oscillator to obtain the desired
frequency.
The commonly used architecture of DDFS [9] shown in Figure 2 consists of a phase
accumulator, a sine/cosine generator, a digital-to-analog converter (DAC), and a lowpass
filter (LPF). It takes two inputs: a reference clock and a v-bit frequency control word (FCW).In each clock cycle, the phase accumulator integrates FCW with periodical overflow to
produce an angle in the range of [0, 2π), the sine/cosine generator computes its sinusoidal
value, which in practice is implemented digitally and, therefore, follows by DAC and LPF
[10–23]. Various fractional-order ideal filters and fractional oscillators were proposed in [24–
29].Instead of using the conventional methods above, we propose a novel digital
frequency synthesis (DFS) algorithm based on a simple difference equation. The rest of the
paper is organized as follows. In Section 2, a novel DFS algorithm is proposed. In Section 3,
the VLSI (very large-scale integration) digital frequency synthesizer is presented. In Section 4,
the FPGA implementation and the performance evaluation are given. Conclusion can be
found in Section 5.
2. The Proposed DFS Algorithm
The difference equation of DFS is as follows:
y[n − 2] − 2a1y[n − 1] + a2y[n] = x[n]. (2.1)
Mathematical Problems in Engineering 3
Thus, we have the following characteristic equation:
z−2 − 2a1z−1 + a2 = 0. (2.2)
The eigen-functions of (2.2) are represented as
z1,2 = re±jθ. (2.3)
The ZIR (zero-input response) of DFS can be written as
ZIR = rn(B1 cos(nθ) + B2 sin(nθ)), (2.4)
where B1 and B2 are determined by initial conditions, and n = 0, 1, 2, . . . .
For DFS with sine wave generator, we have
r = 1, B1 = 0, B2 = 1, ZIR = sin(nθ). (2.5)
The eigen-functions of DFS are therefore as follows:
z1,2 = e±jθ = cos θ ± j sin θ = c ± jd. (2.6)
Thus, the characteristic equation can be expressed as
z−2 − 2cz−1 + c2 + d2 = 0, (2.7)
where c2 + d2 = 1.
Equation (2.7) could be rewritten as
z−2 − 2cz−1 + 1 = 0, (2.8)
and the transfer function of DFS can be derived as
H(z) =1
z−2 − 2cz−1 + 1. (2.9)
According to (2.9), the corresponding difference equation could be derived as
y[n] = x[n] − y[n − 2] + 2c · y[n − 1], (2.10)
where
y[0] = B1 = 0,
y[1] = −B1 cos θ + B2 sin θ = d.(2.11)
4 Mathematical Problems in Engineering
As one can see, a rotation of angle φ in the circular coordinate system can be obtained
by performing a sequence of microrotations in an iterative manner. In particular, a vector
can be successively rotated through the use of a sequence of predetermined step angles:
α(i) = tan−1(2−i). This technique can be applied to generate many elementary functions,
in which only simple adders and shifters are required. Thus, the well-known coordinate
rotation digital computer (CORDIC) algorithm can be used for the DFS applications. The
conventional CORDIC in the circular coordinate system is as follows [36–39]:
u(i + 1) = u(i) − σ(i)2−iv(i),
v(i + 1) = v(i) + σ(i)2−iu(i),
w(i + 1) = w(i) − σ(i)α(i),
α(i) = tan−12−i,
(2.12)
where σ(i) ∈ {−1,+1} denotes the direction of the ith microrotation, σi = sign(w(i)) with
w(i) → 0 in the vector rotation mode, σi = − sign(u(i)) · sign(v(i)) with v(i) → 0 in the
angle accumulated mode, the corresponding scale factor k(i) is equal to√
1 + σ2(i)2−2i, and
i = 0, 1, . . . , l − 1. The product of all scale factors after l microrotations is given by
K1 =n−1∏i=0
k(i) =n−1∏i=0
√1 + 2−2i. (2.13)
In the vector rotation mode, sinφ and cosφ can be obtained, where the initial value (u(0),v(0)) = (1/K1, 0). In principle, uout and vout can be computed from the initial value (uin,vin) =(u(0),v(0)) by using the following equation:
[uout
vout
]= K1
[cosφ − sinφ
sinφ cosφ
][uin
vin
]. (2.14)
In order to evaluate the sinusoidal parameters: c and d for the proposed digital fre-
quency synthesizer, the inputs of the CORDIC processor are uin = 1/K1, vin = 0, and win = θ
as shown in Figure 3.
3. Proposed Architecture for Digital Frequency Synthesizer
In this section, the architecture and the terminology associated with the proposed digital
frequency synthesizer are presented. Our scheme is based on the proposed DFS algorithm
combined with a CORDIC processor. It consists mainly of the radian converter, the CORDIC
processor, and the sine generator as shown in Figure 4.
Figure 5 shows the radian converter. It is a constant multiplier, which converts the
input signal into radians. Figure 6 shows the CORDIC processor, which evaluates the sinus-
oidal value and consists of three adders and two shifters.
Figure 7 shows the architecture of sine generator, which is the core of the proposed
digital frequency synthesizer. It consists of one multiplier, one adder, and two latches only.
Mathematical Problems in Engineering 5
CORDIC
Angle accumulated mode
uin =
vin = 0
win = θvout = sin θ
uout = cos θ
1
K1
Figure 3: The CORDIC arithmetic for the proposed digital frequency synthesizer.
sin θ
cos θ
θRadian
converter
CORDIC
processor
Sine
generator
Fo
FS
Figure 4: The proposed digital frequency synthesizer.
The key terminologies associated with the proposed digital frequency synthesizer are
as follows.
3.1. Output Frequency
The output frequency of the proposed digital frequency synthesizer is determined by the
coefficients d and c, since
θ = ωTs = tan−1
(d
c
),
Fo =1
2π· tan−1
(d
c
)· Fs.
(3.1)
3.2. Frequency Resolution
For m-bit digital frequency synthesizer, the minimum change of the output frequency ΔFo,min
is expressed as
ΔFo,min =1
2π· tan−1
(2−(m−1)
)· Fs. (3.2)
6 Mathematical Problems in Engineering
Input
2 1 2
CSA(3,2) CSA(3,2)
CSA(4,2)
+
Output
5 9 15
Figure 5: The radian converter.
3.3. Bandwidth
The bandwidth of digital frequency synthesizer is defined as the difference between the
highest and lowest attainable output frequencies, which are expressed as follows:
Fo,max =1
2π· tan−1(1) · Fs,
Fo,min =1
2π· tan−1
(2−(m−1)
)· Fs.
(3.3)
3.4. Peak Signal-to-Noise Ratio (PSNR)
A good direct digital frequency synthesizer should have an output signal with low noise,
which can be evaluated by using the following signal-to-noise-ratio (PSNR) measured in dB:
PSNR = 20 log
(255√MSE
), (3.4)
where MSE is the mean square error.
3.5. Spurious-Free Dynamic Range (SFDR)
The spurious-free dynamic range (SFDR) is defined as the ratio of the amplitude of the
desired frequency component to that of the largest undesired frequency component at the
output of a DDFS. It is expressed in decibels (dBc) as follows:
SFDR = 20 log
(Ap
As
)= 20 log
(Ap
) − 20 log(As), (3.5)
Mathematical Problems in Engineering 7
Latch
10
10
+
+
+
Sign
detector
0
1LUT
Barrel
shifter
Barrel
shifter
clk
u(i)
i
i
α(i)
v(i + 1)
w(i + 1)
−1
−1
−1
u(i + 1)
v(i)
w(i)
i
Figure 6: The CORDIC processor (LUT: Lookup table).
Latch
Latch
2c = 2 cos θ
Output
Initial value: sin θ
clk
−1
+ ×
Figure 7: The sine generator.
where Ap is the amplitude of the desired frequency component, As is the amplitude of the
largest undesired frequency component, and the higher the better.
4. FPGA Implementation of Digital Frequency Synthesizer
In this section, the proposed high-performance architecture of digital frequency synthesizer
is presented. Figure 8 depicts the system block diagram. The PSNR and SFDR of the proposed
8 Mathematical Problems in Engineering
sin θ
cos θθRadian
converter
CORDIC
processor
Output
Control
CLA
Latch Latch
1 0 1-bit
shifter
Multiplier
Fo
FS
Figure 8: The proposed digital frequency synthesizer.
0 10 14 18 22 26 30 3460
80
100
120
140
160
180
200
220
240
Word length
PS
NR
(d
B)
Figure 9: The PSNR of the proposed digital frequency synthesizer at various word lengths (100 MHzsampling rate and the maximum output frequency 12.5 MHz).
digital frequency synthesizer at various word lengths at 100 MHz sampling rate and the
maximum output frequency 12.5 MHz are shown in Figures 9 and 10, respectively.
The platform for architecture development and verification has also been designed
and implemented to evaluate the development cost. The proposed architecture of digital
frequency synthesizer has been implemented on the field programmable gate array (FPGA)emulation board [40]. The FPGA has been integrated with the microcontroller (MCU)and I/O interface circuit (USB 2.0) to form the architecture development and verification
platform.
Figure 11 depicts the block diagram and circuit board of the architecture development
and evaluation platform. In which, the microcontroller reads data and commands from PC
and writes the results back to PC via USB 2.0 bus; the FPGA implements the proposed
Mathematical Problems in Engineering 9
0 10 14 18 22 26 30 340
100
200
300
400
500
600
700
Word length
SF
DR
(d
Bc)
Figure 10: The SFDR of the proposed digital frequency synthesizer at various word lengths (100 MHzsampling rate and the maximum output frequency 12.5 MHz).
MCU FPGA
Architecture evaluation
boardPC
USB 2.0
Figure 11: Block diagram and circuit board of the architecture development and verification platform.
architecture of digital frequency synthesizer. The hardware code in portable hardware
description language runs on PC with the logic circuit simulator [41] and FPGA compiler
[42]. It is noted that the throughput can be improved by using the proposed pipelined
architecture while the computation accuracy is the same as that obtained by using the
conventional architecture with the same word length. Thus, the proposed digital frequency
synthesizer improves the power consumption and performance significantly. Moreover,
all the control signals are internally generated onchip. The proposed digital frequency
synthesizer provides a high-performance sinusoid waveform.
5. Conclusion
In this paper, we present a novel digital frequency synthesizer based on a simple difference
equation with pipelined data path. Circuit emulation shows that the proposed high-
performance architecture has the advantages of high precision, high data rate, and simple
hardware. For 16-bit digital frequency synthesizer, the PSNR and SFDR obtained by using
the proposed architecture at the maximum output frequency are 127.74 dB and 186.91 dBc,
respectively. As shown in Table 1, the proposed digital frequency synthesizer is superior to
the previous works in terms of SFDR, PSNR, and hardware [18, 30–35]. The proposed digital
10 Mathematical Problems in Engineering
Table 1: Comparisons between the proposed DFS and other related works.
AuthorsItems
Outputresolution (bits) SFDR (dBc) PSNR (dB) ROM
(words) Adders Multipliers
Strollo et al. [18] 13 90.2 — 1344 8 0
Song and Kim [30] 16 100 — 270 6 6
Langlois and Al-Khalili[31] 14 96.2 — 1152 8 0
De Caro et al. [32] 12 80 — 0 2 2
De Caro and Strollo [33] 12 83.6 — 896 2 2
Yang et al. [34] 12 80 — 2176 6 0
Curticapean andNiittylahti [35] 14 85 — 832 2 2
Ko et al. (This work) 14 133 113 14 8 1
16 187 128 16 8 1
frequency synthesizer designed by portable hardware description language is a reusable IP,
which can be implemented in various VLSI processes with trade-offs of performance, area
and power consumption.
Acknowledgment
The National Science Council of Taiwan, under Grants NSC100-2628-E-239-002-MY2, sup-
ported this work.
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 639824, 10 pagesdoi:10.1155/2012/639824
Research ArticleKernel Optimization for Blind Motion Deblurringwith Image Edge Prior
Jing Wang, Ke Lu, Qian Wang, and Jie Jia
College of Computing & Communication Engineering, Graduate University of Chinese Academy of Science,Beijing 100049, China
Correspondence should be addressed to Ke Lu, [email protected]
Received 10 January 2012; Accepted 20 February 2012
Academic Editor: Ming Li
Copyright q 2012 Jing Wang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
Image motion deblurring with unknown blur kernel is an ill-posed problem. This paper proposesa blind motion deblurring approach that solves blur kernel and the latent image robustly. Forkernel optimization, an edge mask is used as an image prior to improve kernel update, then anedge selection mask is adopted to improve image update. In addition, an alternative iterativemethod is introduced to perform kernel optimization under a multiscale scheme. Moreover, forimage restoration, a total-variation-(TV-) based algorithm is proposed to recover the latent imagevia nonblind deconvolution. Experimental results demonstrate that our method obtains accurateblur kernel and achieves better deblurring results than previous works.
1. Introduction
Motion deblurring is a type of image restoration problems [1, 2]. Commonly, image motion
blur is caused by camera sensor motion, where the track of the sensor motion is represented
by a blur kernel [3]. Theoretically, the motion blur process is modeled as the convolution
of the latent image and a blur kernel with additive noise (Figure 1). Therefore, motion
deblurring not only solves for blur kernel but also recovers latent image. As a blind de-
convolution process, motion deblurring is always split into two stages: kernel estimation
and nonblind image deconvolution. Note that motion deblurring with single-input image is
more complicated than that with two-or-more-input images because multiple blurred images
always provide more information in solving the problem [4–6]. In this paper, we mainly focus
on the single-image-based motion deblurring.
To address such challenging problem, various theories and methods have been pro-
posed. In early days, blind deconvolution recovers sharp images by simple motion and
Gaussian blur based on frequency-domain constraints or assumptions [7]. Recently, many
2 Mathematical Problems in Engineering
Noise
∗ +
Figure 1: Image motion blur process.
researchers believe that the more accurate the obtained kernel is, the more clear the recovered
image will be. For this reason, kernel estimation becomes a principal task in motion
deblurring research. Several novel ideas based on image spatial domain priors are put
forward to solve for the blur kernel. For example, Fergus et al. [8] introduced a statistics
research of natural image as image gradient prior. Their method uses a Bayesian approach to
solve the kernel and then adopts the Richardson-Lucy deconvolution algorithm to reconstruct
the image with the estimated kernel. Jia [9] suggested that the blur kernel can be determined
by the transparency on the object boundary, and a maximum a posteriori (MAP) model
was implemented to estimate the blur kernel. Cai et al. [10] handled a joint optimization
problem with the linearized Bregman iteration method, which maximizes the sparsity of the
blur kernel and the latent image under curvelet system and framelet system, respectively.
Moreover, Joshi et al. [11] solved the simple motion blur and defocus blur kernel by a
predicted sharp edge of the blurry image. Xu and Jia [12] proposed a two-phase kernel es-
timation scheme, which uses a gradient selection process to measure the usefulness of image
edges [13].Even with the estimated kernel, the restoration of the latent image is still a tough
problem. In the process of motion blur, the latent image loses much high-frequency infor-
mation. The traditional methods (inverse filter, wiener filter, etc.) always give undesirable
restoration results because of the effect of the additive noise [14, 15]. To overcome such
difficulty, novel image restoration method with total variation regularization term was
proposed recently which removes the image noise and preserves edge details simultaneously
[16, 17]. To solve the total variation deconvolution problem, it is common to transform it into
a partial difference equation first. Rudin and Osher [16] proposed a time marching scheme to
solve the TV model. Vogel and Oman [18] used the fix point iteration method to optimize the
TV image deconvolution. According to the variable split method and half quadratic penalty
function method, Wang et al. [19] presented a fast total variation deconvolution (FTVd)algorithm to compute TV image deconvolution. Afonso et al. [20] proposed a split augmented
Lagrangian shrinkage algorithm (SALSA), where the augmented Lagrangian method is used
in computing. Similarly, Chan et al. [21] adopted the alternating direction method (ADM)which considered another variant of the augmented Lagrangian method.
In this paper, a complete blind deblurring algorithm is proposed to handle image
motion blur with image edge prior. In kernel optimization, an edge mask is used as im-
age prior to improve kernel update and an edge selection mask is adopted to improve im-
age update. Moreover, an alternative iterative method is introduced to implement kernel
optimization under a multiscale scheme. For image restoration, a total-variation-based image
nonblind deconvolution algorithm is proposed to restore latent image. The rest of the paper
Mathematical Problems in Engineering 3
is organized as follows. In Section 2, the edge character in the blurry image is analyzed and
an edge detection process is introduced. In Section 3, given the image edge prior, a complete
blind deblurring algorithm is described including a kernel estimation model and an image
restoration model. Then, the algorithm is implemented with motion blurred images and
the restored results are shown and discussed in Section 4. Finally, Section 5 concludes our
work.
2. Image Edge Prior
Motion deblurring is an ill-posed problem where the number of unknowns is more than the
number of observed measurements. Generally, the motion blur process can be modeled as
B = K ∗ I +N, (2.1)
where B is the blurry image, K is the motion blur kernel, I is the latent image, N is the
noise effect, and ∗ is the convolution operator. This model suggests that the blurry image is
the convolution of a blur kernel and the latent image. Accordingly, our goal is to solve the
kernel and the latent image inversely from a single blurred image, which is obviously an
ill-posed problem. Therefore, such inverse problem can be solved only with other necessary
prior knowledge being provided.
In image processing system, an edge is defined as the continuous boundary pixels that
connect two separate regions with changing image amplitude attributes [22–24]. It offers
information including magnitude and orientation, which has been widely used as image
prior knowledge in solving many image processing and computer vision problems, such as
image restoration [11, 13] and image superresolution [25, 26]. After edge detection, there
always exist some particular edges caused by blur or noise in edge map. Such edges are
called false edges, which can be further removed by other image processing techniques.
In motion deblurring problem, it can be seen that the edge in the blurry image usually
appears fuzzy or unsharp, as shown in Figures 2(c) and 2(d), while the latent image has
clear edges, as shown in Figures 2(a) and 2(b). If an edge map can be found from the motion
blurred image, which is also assumed to be closed enough to the edge map of the latent image,
then it might be used to refine the kernel estimation. In this paper, it is assumed that the fuzzy
edges in the blurry image are viewed as false edges, which are removed through an edge
detection process. In other words, sharp edges can be found through certain edge detection
process. This sharp edge map is taken as an edge prior to improve kernel estimation.
The edge detection process finds the presence and locations of the intensity transitions.
To find an ideal edge map from the blurry image, a modified edge detection process is used
and described as follows. First the blurry image is convolved by the derivatives of Gaussian.
Then, the magnitude and orientation of its gradient are computed. Thirdly, a nonmaxima
suppression method is used to get the thinned gradient magnitudes. Finally, hysteresis is
used to get the sharp edge map by doing threshold operation on the gradient magnitude.
Especially, two adaptive thresholds are used to suppress the false edges. As shown in Figures
2(e) and 2(f), the detected edge map and its close-up are sufficiently clear and sharp in detail.
According to this edge map, the edge locations are labeled in a mask, which is used to solve
for the blur kernel as described in the Section 3.
4 Mathematical Problems in Engineering
(a) (b)
(c) (d)
(e) (f)
Figure 2: Edge features analysis in both motion blurred image and latent image. (a) Edge map beforemotion blur, (b) close-up of (a), (c) edge map after motion blur, (d) close-up of (c), (e) edge map detectedfrom the blurry image by our edge detection process, and (f) close-up of (e).
3. Blind Deblurring Algorithm
3.1. Kernel Estimation Model
Before kernel estimation, the blurry image and the initial kernel, which are later used as the
inputs of our algorithm, need to be preprocessed. More specifically, the bilateral filter and the
Mathematical Problems in Engineering 5
shock filter are used to smooth the noise and keep the edge details, respectively [27]. The blur
kernel is defined as a smooth convolution mask with nonnegativity values and normalized.
So the initial kernel is set as a unit matrix with unit value at its central position.
For kernel optimization, an iterative optimization problem model is constructed, and
its task is to optimize the blur kernel and the latent image alternately under a multiscale
scheme. The edge map mentioned above is introduced as image prior, which adds mask on
both latent image and blurry image. On the other hand, the L1 norm of the kernel is used as
a regularization term to suppress the noise in the blur kernel. According to the motion blur
model, the minimization energy function for motion blur kernel is as follows:
minK
{‖K ∗ME(∇I) −ME(∇B)‖2 + α‖K‖1
}, (3.1)
where ‖K ∗ME(∇I) −ME(∇B)‖2 is the data fitting term and ‖K‖1 is the L1 regularization
term of K. ME is the edge location mask mentioned in Section 2, ∇I is the gradient of latent
image, ∇B is the gradient of blurry image, and parameter α controls the relative strength
of the data fitting and kernel regularization terms. Before adding the edge mask ME, lateral
filter is used to suppress noise in blurry image B. Here an unconstrained iterative reweighted
least squares (IRLS) system [28, 29] is adopted to solve this minimization problem, and the
conjugate gradient (CG) method is used to solve the inner IRLS system.
For latent image optimization, an edge selection mask mentioned in [13] is used to
recover a coarse latent image. Image edges do not always profit kernel estimation, so we
need an edge selection process to choose useful ones. The energy function is then modeled as
follows:
‖K ∗ I − B‖2 + β‖∇I −MS(∇I)‖2, (3.2)
where ‖K ∗ I − B‖2 is the data fitting term, β‖∇I −MS(∇I)‖2 is the regularization term, β
is playing the same role as α, and MS is the edge selection mask. According to Parseval’s
theorem, this equation has a closed-form solution by using FFTs:
I = F−1
⎛⎜⎝F(K)F(B) + β(F(∂x)F
(IxMs
)+ F
(∂y)F(Iy
Ms
))F(K)F(K) + β
(F(∂x)F(∂x) + F
(∂y)F(∂y))
⎞⎟⎠, (3.3)
where F denotes the FFT and F−1 denotes the inverse FFT.
To solve for the kernel accurately, a multiscale scheme is introduced to implement
the whole blind deblurring algorithm. Under this scheme, blur kernel and latent image are
estimated by using a coarse-to-fine pyramid of image resolutions. The number of scale levels
is computed by the size of blur kernel and the scale level step is√
2. The blurry image is
downsampled as the algorithm input. In each scale, the latent image is updated by solving
(3.2). Then the updated latent image is used to update the blur kernel by solving (3.1). Finally,
the updated kernel is used in the next scale.
6 Mathematical Problems in Engineering
3.2. Image Restoration Model
Given the estimated blur kernel, the latent image can be restored by using a nonblind image
deconvolution algorithm. As mentioned above, a clear image should have sharp edge details.
For this reason, an image restoration model with TV regularization term is built to recover
the latent image. This model is a minimum optimization problem:
minI
{λ
2‖K ∗ I − B‖2 + ‖DI‖1
}, (3.4)
where ‖K ∗ I − B‖2 is the data fitting term, ‖DI‖1 is the TV regularization term, D is the finite
difference operator, and λ is weight factor. When (3.4) is used to restore the latent image, the
TV regularization term can keep the image edge details satisfyingly.
It can be seen that (3.4) is an L1 norm regularization optimization problem. In this
paper, the split Bregman method [30, 31] is introduced to solve the problem. The split Breg-
man method, proposed by Goldstein and Osher, is a fast scheme to solve a type of optimiza-
tion problem with the form
minu
{‖l(u)‖1 + f(u)}, (3.5)
where l(u) and f(u) are both convex functions. According to the variable split method, the
split Bregman method transforms (3.5) into an unconstraint optimization problem with an
auxiliary variable and quadratic penalty function. Then this unconstraint optimization model
is divided into two or three optimization subproblems and solved alternatively with the
Bregman iteration.
At first, an auxiliary variable G takes the place of DI, and (3.4) is transformed into a
unconstrained optimization equation related to I and G as follows:
minG,I
{λ
2‖K ∗ I − B‖2 +
γ
2‖G −DI‖2 + ‖G‖1
}. (3.6)
Then, (3.6) is divided into two subproblems related to I and G, respectively. According to the
Bregman iteration, these two optimization subproblems are modeled as
minI
{λ
2‖K ∗ I − B‖2 +
γ
2‖G −DI − b‖2
}, (3.7)
minG
{λ
2‖G −DI − b‖2 + ‖G‖1
}, (3.8)
where b is an iteration parameter and b = b + (G −DI).To solve these two sub-problems, an alternative minimization method (AMD) is used
to optimize them. In each iteration, (3.7) is transformed into the equation as follows:
(DTD +
γ
λKTK
)I =
γ
λKTB +DT (G − b), (3.9)
Mathematical Problems in Engineering 7
(a)
(b)
(c)
Figure 3: Testing our algorithm with the synthetic images. (a)–(c) In order from left to right, the imagesare the original synthetic images, the synthetic images after motion blur, restored results and estimatedkernels by using our algorithm, and the close-ups of them (the red rectangle in the blurry image shows thelocation of close-ups).
where K and D are both block circulant matrices. So (3.9) is computed by FFTs. On the
other hand, (3.8) is optimized by the shrinkage technique, and it can be solved by using
the following equation:
G = max
{∥∥g∥∥2− 1
λ, 0
}g∥∥g∥∥
2
, (3.10)
where g = DI + b. Interested readers can refer to [32] for more details.
4. Experiments
In this Section, the proposed blind deblurring algorithm was tested with both synthetic
motion blurred images and real-life motion blurred images. In the kernel estimation process,
8 Mathematical Problems in Engineering
(a)
(b)
(c)
Figure 4: Testing our algorithm with real-life motion blurry images. (a)–(c) In order from left to right, theimages are original blurry images, restored results and estimated kernels by our algorithm, restored resultsand estimated kernels by using the algorithm in [28] and the close-ups of them (the red rectangle in theoriginal blurry image shows the location of close-up).
the parameters α and β were set as 1e − 4, 2e − 3, the initial minimum kernel size was 3 × 3,
and the initial maximum kernel size was 35 × 35. According to the multiscale scheme, the
outer iteration was controlled by the maximum kernel size and the inner iteration was set as
8 uniformly. In the image recovery process, the parameter λ was set as 2e + 3. Our algorithm
was implemented on Matlab experimental platform.
To verify the validity of our algorithm, the synthetic blurry images were generated by
convolving the synthetic images with a 15 × 15 synthetic kernel. The Gaussian white noise
was added whose standard deviation was 0.001. Figure 3 shows the experimental results of
several synthetic images. The deblurring results are extremely close to the original synthetic
images, and they manifest the significance of the proposed algorithm.
On the other hand, the proposed algorithm was compared with the approach
described in [28] by deblurring the real-life motion blur images. In Figure 4, the restored
results of some real-life images are given. As shown in recovery results, our method is
robust to restored sharp images and accurate kernels. In contrast to the approach in [28],the deblurring images and the close-ups show that our algorithm could obtain clearer image
detail information.
Mathematical Problems in Engineering 9
5. Conclusion
In this paper, a novel blind deblurring algorithm is presented for motion blur occurring in
photography. The approach consists of two stages: kernel estimation and image reconstruc-
tion. The edge information in blurry images is explored as an image prior for obtaining
accurate blur kernel and the use of total variation regularization keeps image details during
image recovery. The proposed algorithm was tested with synthetic and really captured
motion blur images. The experimental results demonstrated the efficacy of our algorithm
in image motion deblurring. On the other hand, there still exist some defects (cartoon effect
and unclear texture detail) in the restored images. Our future work is to extend the current
research by considering more complex blurs (such as blur with rotation and shift-variant
blur) and other image analysis problems [33–35].
Acknowledgments
This work was supported by the China Special Fund for Meteorological-scientific Research
in the Public Interest (GYHY201106044), NSFC (Grant nos. 61103130, 61070120, 61141014);National Program on Key basic research Project (973 Programs) (Grant nos. 2010CB731804-1,
2011CB706901-4).
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 578719, 16 pagesdoi:10.1155/2012/578719
Research ArticleDual-EKF-Based Real-Time Celestial Navigationfor Lunar Rover
Li Xie,1, 2 Peng Yang,1, 2 Thomas Yang,3 and Ming Li4
1 Department of Information Science and Electronic Engineering, Zhejiang University,Hangzhou 310027, China
2 Zhejiang Provincial Key Laboratory of Information Network Technology, Hangzhou 310027, China3 The Department of Electrical, Computer, Software, and Systems Engineering, Embry-RiddleAeronautical University, Daytona Beach, FL 32114, USA
4 School of Information Science and Technology, East China Normal University,Shanghai 200241, China
Correspondence should be addressed to Li Xie, [email protected]
Received 27 December 2011; Accepted 14 February 2012
Academic Editor: Carlo Cattani
Copyright q 2012 Li Xie et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
A key requirement of lunar rover autonomous navigation is to acquire state information accuratelyin real-time during its motion and set up a gradual parameter-based nonlinear kinematics modelfor the rover. In this paper, we propose a dual-extended-Kalman-filter- (dual-EKF-) based real-time celestial navigation (RCN) method. The proposed method considers the rover position andvelocity on the lunar surface as the system parameters and establishes a constant velocity (CV)model. In addition, the attitude quaternion is considered as the system state, and the quaterniondifferential equation is established as the state equation, which incorporates the output of angularrate gyroscope. Therefore, the measurement equation can be established with sun direction vectorfrom the sun sensor and speed observation from the speedometer. The gyro continuous outputensures the algorithm real-time operation. Finally, we use the dual-EKF method to solve the systemequations. Simulation results show that the proposed method can acquire the rover positionand heading information in real time and greatly improve the navigation accuracy. Our methodovercomes the disadvantage of the cumulative error in inertial navigation.
1. Introduction
In order to conduct scientific exploration on the lunar surface, lunar rover must have the
ability to execute tasks in unstructured environment. Its navigation system must have a
high degree of autonomy and the capabilities of high-accuracy real-time positioning and
orientation. On lunar surface, some commonly used navigation methods on the earth are not
2 Mathematical Problems in Engineering
applicable. There is no GPS system on the moon. If we use radio navigation, the rover control
may fail because of the two-way communication delay. The moon rotation is very slow, so we
cannot use north seeking gyro. Also, lunar magnetic field is very weak, so magnetic sensor-
based methods are ineffective.
Lunar rover navigation techniques mainly include absolute positioning and relative
positioning. For absolute positioning, such as autonomous celestial navigation [1], position
and heading errors are bounded and do not accumulate over time, and the output is discrete.
The initial positioning is generally absolute positioning, and its accuracy directly affects
relative positioning accuracy. Relative positioning, such as inertial navigation, achieves high
accuracy of position and heading in short time, but the errors accumulate over time (which
may lead to divergence), and the output is continuous. The current trend for lunar rover
navigation is integrated navigation, which combines the advantages of celestial navigation
and inertial navigation.
The earliest researchers [2–4] carried out celestial navigation by the altitude difference
method through observing the sun, earth, and fixed stars. Kuroda et al. [5] utilized celestial
navigation and dead-reckoning-based integrated navigation method to obtain lunar rover’s
absolute position and heading, which is achieved by observing the altitude and azimuth of
the sun and the earth. However, on the moon, the time period during which the sun and
the earth appear simultaneously is very short. Therefore, the application of this method is
limited. Altitude difference method is very sensitive to measurement noise, and positioning
accuracy [6] is low. Vision-based navigation is often used in robotics (Chen 2012, see [7, 8]),but it has difficulty in determining the absolute location and attitude.
Recent researchers use vector-observations-based quaternion estimation (QUEST) to
get the rover heading angle [9–12]. Ashitey proposed an absolute heading detection method
for the field integrated, design and operation (FIDO) rover [9]. When stopped, it uses sun
sensor and accelerometer to sense the sun orientation and the local gravity orientation, supply
absolute heading for rover with QUEST, and correct the gyroscope cumulative error. Ali
described that the US Mars rovers (the “Hope” and the “Spirit”) utilized this method to self-
correct the heading information [10]. Some recent technologies used in robotics can be found
in the works of Chen et al. [13]. Methods in Chinese literature are similar to the method
by Ashitey and they also calculate the heading through QUEST [11, 12]. Thein analyzed
the relationship between lunar rover positioning accuracy and astronomical instrument
measurement noise [14]. If we want to limit the position error within 50 m, measurement
noise should be less than 5.93 arcsec.
The above celestial navigation methods (except [5]) do not combine celestial
positioning with orientation and cannot get the absolute heading and location information
in real time. Ning established a position and attitude determination method based on
celestial observations [15], but its reference frame is moon fixed coordinate system rather
than local level coordinate system, and it does not consider the impact of the position and
speed changes on the gyro angular rate output. Ning proposed a lunar rover kinematics
model-based augmented unscented particle filter (ASUPF) as a new autonomous celestial
navigation method for dealing with systematic errors and measurement noise [16]. However,
the altitude angle measurements in this method are based on the local level provided by
the inertial measurement unit (IMU), assuming the rover is keeping static or in constant
motion. When the rover is moving, it needs the support of attitude update algorithm in
inertial navigation, because the gyro accumulates error and the local level precision is low. Pei
proposed a strapdown inertial navigation and celestial-navigation-based integrated method
for lunar rovers [17].
Mathematical Problems in Engineering 3
Since lunar rover position change on the lunar surface is very slow, in order to reduce
the dimension of the system, we can set position as a gradual system parameter to estimate
the rover state, that is, heading and attitude. To correct the position of the lunar rover, the
velocity observation is introduced. Meanwhile, in order to obtain real-time navigation output,
the output of the gyro is needed in integrated navigation. In this paper, a method of real-time
celestial navigation is proposed, in which positioning and orientation are simultaneous. Also,
the error bounded sun sensor output and high accuracy rate gyro output are fused, which
ensures the navigation output to be both real-time and of higher accuracy when the rover
is moving. Because there is no accelerometer in the system, the impact of the accelerometer
error and gravity anomaly on navigation is avoided.
The organization of the paper is as follows. Section 1 describes the principles of
celestial navigation and attitude quaternion kinematics. Section 2 describes the dual EKF-
based real-time celestial navigation method. Section 3 presents the results of computer
simulations and compares the accuracy of the results obtained with and without velocity
observation. Section 4 presents conclusions and discussions.
2. Celestial Navigation Principle and Attitude Kinematics
2.1. Principle of Celestial Navigation
Set the selenocenter celestial coordinate system as the inertial coordinate system (i), the moon
fixed coordinate system as m, geographic coordinate system (NED) as n, the lunar rover body
coordinate system as b, the local level coordinate system as l, and the sun sensor coordinate
system as c. After installation, the sensor coordinate systems b and c coincide with each other.
Celestial navigation system can detect the rover geographical position and heading
provided that the local gravity datum (level posture) is known. The outputs are lunar rover
position (latitude and longitude) on the moon and the attitude, including the heading, pitch,
and roll. State vector of the system x = [λ, L,An] is used to describe the lunar rover position
and heading information, in which (λ, L) is the lunar rover longitude and astronomical
latitude and An is just heading relative to North Pole of the moon.
In Figure 1, the moon fixed coordinate system, after rotating λ (east longitude is
positive) around the Z axis, becomes coordinate system O − x1y1z1. After further rotating
by −L − π/2 (north latitude is positive) around the Oy1 axis, the navigation coordinate w is
obtained. The attitude matrix about the latitude and longitude is as follows [18]:
nm = y
(−L − π
2
) z(λ). (2.1)
The attitude matrix about the navigation coordinate system and the lunar body coordinate
system is:
bn = x
(ϕ) y
(ψ) z(θ). (2.2)
Here, θ is the heading, ψ is the pitch angle, and ϕ is the roll. To prevent the risk of rollover,
lunar rover pitch and roll should between ±45◦. The attitude matrix about the moon fixed
coordinate system and the local level coordinate system is A = lm (called target matrix
4 Mathematical Problems in Engineering
Moon-fixedframe
The moon
The sun
North-Earth-downframe
Zm
O m
Ym
X n
Yn
O nZn
X m
L
Figure 1: Moon-fixed (m) and navigation (n) coordinate system.
here). Substituting z(θ), y(−L − π/2), z(λ) into the above formula, we get the following
equation:
lm = z(θ) y
(−L − π
2
) z(λ). (2.3)
2.2. Quaternion Attitude Kinematics
Attitude can be expressed in several mathematical parameters: quaternion, attitude matrix,
Euler angles, Rodrigues parameters, and so on. The attitude matrix contains a total of nine
parameters, but because it is orthogonal matrix, only three components are independent. One
of the most useful parameters is the attitude quaternion, which is a four-dimensional vector,
defined as q = [ρT q4]T , where ρ = [q1 q2 q3]
T = e sin(ϑ/2) and q4 = cos(ϑ/2). Here, e is the
rotation axis and ϑ is the rotation angle. When using a four-dimensional vector to describe
the three-dimensional rotation, the four parameters of quaternion are not independent, and
they are subject to the constraint qTq = 1. The relationship between the attitude matrix and
the quaternion from the inertial coordinate system i to the body coordinate system b is
biA(q) = ΞT (q)Ψ(q), (2.4)
where
Ξ(q) ≡[q4I3×3 + [ρ×]
−ρT
], Ψ(q) ≡
[q4I3×3 − [ρ×]
−ρT
]· Ξ(q) ≡
[q4I3×3 + [ρ×]
−ρT
],
Ψ(q) ≡[q4I3×3 − [ρ×]
−ρT
].
(2.5)
Mathematical Problems in Engineering 5
Here [ρ×] is the cross-product matrix, defined as [ρ×] =[
0 −q3 q2
q3 0 −q1
−q2 q1 0
]. One advantage of using
quaternion is that the attitude matrix is quadratic equation of the parameter and thus does
not include any transcendental function. For small angles, the vector part of the quaternion
is approximately half of the rotation angle, and therefore ρ ≈ α/2, q4 ≈ 1, where the 3-
dimensional vector α includes roll, pitch, and heading. Therefore, the attitude matrix can be
approximated as biA ≈ I3×3 − [α×], which is effective in the first-order approximation.
The attitude kinematics equation is
bi A = −
[ωb
ib×]biA . (2.6)
Here, ωbib
is the angular velocity of the b frame relative to i frame expressed in b coordinates.
The quaternion differential equation is
q =1
2Ξ(q)ωb
ib =1
2Ω(ωb
ib
)q, (2.7)
where
Ω(ωb
ib
)≡
⎡⎢⎣−[ωb
ib×]
ωbib
−(ωb
ib
)T0
⎤⎥⎦. (2.8)
The main advantage of using the quaternion is that the kinematics equation is linear
and there is no singularity. Another advantage is that continuous rotation of coordinate
frames can be expressed as the quaternion multiplication. Suppose a continuous rotation can
be expressed as
A(q′)A(q) = A
(q′ ⊗ q
). (2.9)
The composition of the quaternion is bilinear, with
q′ ⊗ q =[Ψ(q′) q′]q = [Ξ(q) q]q′, (2.10)
and the inverse quaternion is defined by q−1 =[ −ρq4
]. Note that q ⊗ q−1 = [ 0 0 0 1 ]T is the
identity quaternion.
3. Dual-EKF-Based RCPO Method
Assume the state vector of the navigation system is xs, the system parameter vector is xp, and
the observation vector is yk. According to the problem, a continuous-discrete nonlinear state
space model can be derived:
xs(t) = f{t, xs(t),u(t), xp(t)
}+w(t),
yk = hk
(xs,k, xp,k
)+ vk,
(3.1)
6 Mathematical Problems in Engineering
where f(·), h(·) are implicit vector functions, w(t) is the continuous process noise, and vkis the discrete measurement noise. In the state vector xs = [qT , βT ]T , q is the heading and
attitude quaternion in the navigation frame (w) for the lunar rover and β is the constant bias
for gyro. In parameter vector xp = [pT , vT ]T , p = [L λ]T is the rover position, which is the
latitude and longitude; V = [vL vλ]T is the north speed and east speed on the lunar surface.
3.1. System Parameter and State Equations
The lunar rover position and velocity equations constitute the system parameter equations:
xp = Fpxp +wp (3.2)
with the parameter vector xp = [pT VT ]T , the state transition matrix Fp =[
0 0 1 00 0 0 10 0 0 00 0 0 0
], the
parameters process noise wp =[
00wLwλ
], and the noise covariance Qp = diag[0 0 σ2
L σ2λ].
The lunar rover attitude constitutes the system state equations, and the quaternion
differential equation is expressed by
q2 =1
2Ω(ωb
nb
)q2. (3.3)
Here, ωbnb is the angular velocity of the b frame relative to n frame expressed in b coordinates.
The gyro measurement model is
ωbib = ωb
ib + β + ηv,
β = ηu.(3.4)
Here, ωbib
is the angular velocity of the b frame relative to i frame expressed in b coordinates.
β is the constant bias of the gyro, ηv and ηu are zero mean Gaussian white noise processs, and
their spectral density functions are σ2vI3×3 and σ2
uI3×3, respectively.
Because the selenocenter celestial coordinate system is the inertial coordinate system
here, so
ωbnb = ωb
ib − bnA(q2)ωn
in. (3.5)
Also, ωnin is the angular velocity of the n frame relative to i frame expressed in n coordinates
ωnin = ωim
⎡⎣ cosL
0
− sinL
⎤⎦ +
⎡⎢⎢⎢⎢⎢⎢⎢⎣
VE
R
−VN
R
−VE tanL
R
⎤⎥⎥⎥⎥⎥⎥⎥⎦, VE = vλR cosL, VN = vLR. (3.6)
Mathematical Problems in Engineering 7
In (3.6), ωim is the angular velocity of the m frame relative to i frame, and the second
expression on the right side is the angular velocity of the n frame relative to m frame. The
angular velocity of the m frame relative to i frame ωim is
ωim = ωgz + mi Aωzz . (3.7)
In (3.7), ωgz is the revolution angular velocity of the moon around the earth, ωzz is the moon
spin velocity, and mi A is the attitude matrix from the inertial reference frame i to the moon
fixed frame m, which can be calculated after querying ephemeris [18].
3.2. Celestial and Speed Observation Equations
The measurement principle of vector observation attitude sensor can be expressed as bi =A(q)ri + vi, i = 1, . . . , n. If n celestial bodies are observable simultaneously, we can get n
vector pairs, so the measurement equation at time k is
bk =
⎡⎢⎢⎢⎣A(q2)A(p1)r1
A(q2)A(p1)r2
...
A(q2)A(p1)rn
⎤⎥⎥⎥⎦∣∣∣∣∣∣∣∣ tk
+
⎡⎢⎢⎢⎣v1
v2
...
vn
⎤⎥⎥⎥⎦∣∣∣∣∣∣∣∣ tk
, (3.8)
where A(q2) = bnA, A(p1) = n
mA = y(−L − π/2) z(λ).
Set vk =
⎡⎣ v1v2
...vn
⎤⎦∣∣∣∣ tk , its variance is R = diag[σ21I3×3, σ
22I3×3, . . . , σ
2nI3×3], where diag[· · ·]
is the diagonal matrix. In this paper, n = 1, r = is, b = bs, where is is the sun unit vector in
inertial frame and bs is the sun unit vector in the body frame.
The speed observation equation of the speedometer is
Vk = Vk + uk, (3.9)
where Vk is speed measurement at time k, uk is the measurement noise, and its covariance
matrix is Ru = σ2uI2×2.
3.3. Dual Continuous-Discrete EKF
Dual-EKF algorithm uses two mutual coupling extended Kalman filters working in parallel
and a state estimator working between the system parameter time update process and the
measurement update process [19]. Dual-EKF can estimate the system state and parameter
online. Using the above model, a continuous-discrete extended Kalman filter can be derived
(Chen 2012, [20]). The process equation about the system parameter is a continuous linear
equation, which can be discretized directly. The process equations about the system state
8 Mathematical Problems in Engineering
are nonlinear equations, and the Jacobian matrix needs to be calculated. Finally, we get the
discrete linear state space model (without considering the control input uk):
xs,k+1 = f{xs,k, xp,k
}+wk,
yk = hk
(xs,k, xp,k
)+ vk.
(3.10)
3.3.1. Linearization of State Process Equations
In order to maintain the quaternion normalization constraint, we use the multiplicative error
quaternion in the body frame to express the attitude error:
δq = q ⊗ q−1, (3.11)
where q−1 is the inverse of the quaternion estimate and δq ≡ [δρT δq4]T . If the error
quaternion δq is very small, we can use the small angle approximation. After a series of
derivation, the linear kinematic model of the attitude error [21] is obtained:
δα = −[ωb
nb×]δα + δωb
ib −A(q2)δωnin, δq4 = 0, (3.12)
where δωbib= ωb
ib− ωb
iband δωn
in = ωnin − ωn
in = 0. Also, δωbib= −(Δβ + ηv) is available by the
above gyro model, in which Δβ ≡ β − β. So the above formula becomes
δα = −[ωb
nb×]δα − (Δβ + ηv
). (3.13)
The remaining error equation can be obtained by similar methods. The state vector,
the state error vector, and the process noise vector and covariance in this EKF are defined as
xs ≡[qβ
], Δxs ≡
[δα
Δβ
], ws ≡
[ηvηu
], Qs =
[σ2vI3×3 03×3
03×3 σ2uI3×3
]. (3.14)
The error dynamics of time update in the EKF is Δx = FΔx + Gw. Here, the state
transition matrix F and the noise coefficient matrix G are
F ≡[−[ωb
nb×]
−I3×3
03×3 03×3
], G ≡
[−I3×3 03×3
03×3 I3×3
]S. (3.15)
3.3.2. Linearization of Measurement Equations
Next we determine the sensitive matrix Hs(x−s ) of the system state observation equation. The
true value and the estimate of the celestial bodies vector in the body coordinate system are
b = A(q2)A(p−
1
)r, b− = A
(q−
2
)A(p−
1
)r. (3.16)
Mathematical Problems in Engineering 9
According to (2.6),
A(q2) = A(δq2)A(q−
2
)= (I3×3 − [δα2×])A
(q−
2
). (3.17)
From (3.16), we have
Δb = b − b− =[A(q−
2
)A(p−
1
)r×]δα2. (3.18)
Note that Hsq = [A(q−2 )A(p−
1 )r×], so the sensitivity matrix for all measurements is
Hs
(x−s)=
⎡⎢⎢⎢⎣Hsq1 03×3
Hsq2 03×3
......
Hsqn 03×3
⎤⎥⎥⎥⎦∣∣∣∣∣∣∣∣ tk
. (3.19)
Next we determine the sensitive matrix Hp(x−p) of the system parameter observation
equation.
The true value and the estimate of the celestial bodies vector in the body coordinate
system are
b = A(q−
2
)A(p)r, b− = A
(q−
2
)A(p−)r. (3.20)
Function A(p) is expanded as a Taylor series, which is
A(p) ≈ A(p−) + 2∑
j=1
A−j Δpj , (3.21)
where A−1 = ∂A/∂L|L− , A
−2 = ∂A/∂L|λ− .
Finally, we have
Δb = b − b− =2∑
j=1
A(q−
2
)A−
j rΔpj . (3.22)
Note that Hp =[A(q−
2 )A−1 r A(q−
2 )A−2 r]. Combined with the speed observations, the
sensitivity matrix of all measurements is
Hp
(x−p)=
⎡⎢⎢⎢⎢⎢⎢⎣Hp1 03×2
Hp2 03×2
......
Hpn 03×2
02×2 I2×2
⎤⎥⎥⎥⎥⎥⎥⎦
∣∣∣∣∣∣∣∣∣∣∣ tk. (3.23)
10 Mathematical Problems in Engineering
Table 1: Dual-EKF algorithm.
InitializationParameter: xp(t0) = xp,0, Pp(t0) = Pp,0
State: xs(t0) = xs,0, Ps(t0) = Ps,0
State measurement update
Ks,k = P−s,kHT
s,k[Hs,kP
−s,kHT
s,k+ R]−1
εk = bs,k − hk(x−s,k, x−
p,k)
Δx+s,k
= Ks,kεk
q+2,k
= q−2,k
+1
2Ξ(q−
2,k)δα+
2,k, normalization
β+k = β
−k + Δβ
+k
P+s,k
= [I −Ks,kHs,k]P−s,k
Parameter measurement update
Kp,k = P−p,k
HTp,k
[Hp,kP−p,k
HTp,k
+ R′]−1
R′ = diag([R,Ru])x+p,k
= x−p,k
+Kp,k[εk ; (Vk − V−k)]
P+p,k
= [I −Kp,kHp,k]P−p,k+1
Parameter time updatex−p,k+1
= Φpx+p,k
P−p,k+1
= ΦpP+p,k
ΦTp +Qp
State time update
ωbnb
= (ωbib− β+
k) −A(q+
2,k)ωn
in(x−p,k+1
)
˙q2 =1
2Ω(ωb
nb)q2
˙β = 0
Ps = FsPs + PsFsT +GQsG
T
3.3.3. Dual-EKF Algorithm
Finally the proposed algorithm of dual-EKF is shown in Table 1.
4. Simulations and Discussions
4.1. Simulation Conditions
Specific simulation parameters are shown in Table 2.
4.2. Simulation of Moving Lunar Rover
In this paper, we carried out lunar rover simulation under various moving conditions
described in Table 3, and navigation accuracy with and without the speed observation is
compared. The lunar rover movement includes rotational and translational movements,
where the former can be sensed by the gyro angular velocity and the latter can be measured
by the speedometer line speed.
The simulation results of the lunar rover are shown in Figure 2, with the left diagram
on each figure representing the simulation result without speed observation and the right
diagram representing the simulation result with speed observation.
Figure 2 shows the position error and its 3σ boundary, and we see the latitude and
longitude errors in the left diagram diverge at last. After the uniform motion error expands,
we mainly have the lunar rover speed changes, so the constant velocity (CV) model is no
Mathematical Problems in Engineering 11
0 200 400 600
0
50
100
−100
−50
Time (s)
Lati
tud
e(a
rcse
c)
(a)
0 200 400 600
0
50
100
−100
−50Lati
tud
e(a
rcse
c)
Time (s)
(b)
0 200 400 600
0
50
100
Time (s)
−100
−50Lo
ng
itu
de(a
rcse
c)
(c)
0 200 400 600
0
50
100
Lo
ng
itu
de(a
rcse
c)
Time (s)
−100
−50
(d)
Figure 2: Position error and 3σ boundary.
Table 2: Simulation parameters.
Beginning time 2011-01-01 00:00:00
Sampling interval Δt = 1 s
Initial origin λ(t0) = 0◦, L(t0) = 0◦
Initial velocity vL = −0.1m/(s · R),vλ = 0.1m/(s · R)Initial attitude q(t0) = [ 0 0 0 1 ]T
Gyro biases β(t0) = 0.1[ 1 1 1 ]T deg/hr
Initial covariance
Pp
0 = 0.052 deg2
PV0 = 0.12(m/s)2
Pα0 = 0.12 deg2
Pβ
0 = 0.22(deg/hr)2
Gyro noise (Qs)σgv =
√10 × 10−7 rad/s1/2
σgu =√
10 × 10−10 rad/s3/2
CV model (Qp)σL = σL = 0.0001m/(s · R)(R: moon radius, the same below)
Sun sensor (R) 1′(3σs)
Velocity sensor (Ru) σu = 0.001m/(s · R)
12 Mathematical Problems in Engineering
0 200 400 600
0
0.2
0.4
−0.4
−0.2
Time (s)
Vel
oci
ty l
ati
tud
e(m
/s)
(a)
Time (s)
Vel
oci
ty l
ati
tud
e(m
/s)
0 200 400 600−0.04
−0.02
0
0.02
0.04
(b)
0 200 400 600
0
0.2
0.4
Vel
oci
ty l
on
git
ud
e(m
/s)
−0.4
−0.2
Time (s)
(c)
0 200 400 600
0
0.02
0.04
Vel
oci
ty l
on
git
ud
e(m
/s)
−0.04
−0.02
Time (s)
(d)
Figure 3: The speed error and 3σ boundary.
Table 3: Lunar rover motion.
Motion Time (s) Angular velocity( ◦/s)
Linear velocity(m/(s · R))
(1) Static 1∼100 0 0
(2) Rotation 101∼200 ωz = 1 0
(3) Uniform motion 201∼300 0vL = 0.2
vλ = −0.25
(4) Rotation and uniform motion 301∼500 ωz = 1vL = 0.2
vλ = −0.25
(5) Static again 501∼600 0 0
longer applicable. The navigation error in the right diagram is kept within the 3σ boundary,
and it does not diverge. Because of the speedometer line speeds information, the absolute
position of the rover can be adjusted in real time. The mean of the latitude error is 3.97′′,
and the standard deviation is 0.83′′; the mean of longitude error is 1.07
′′, and the standard
deviation is 1.42′′. Converted into the line error according to the lunar radius, the error is
35.51 m.
Mathematical Problems in Engineering 13
0 200 400 600
0
100
−100
Ro
ll(a
rcse
c)
Time (s)
(a)
0 200 400 600
0
100
−100
Ro
ll(a
rcse
c)
Time (s)
(b)
0 200 400 600
0
100
Pit
ch (
arc
sec)
−100
Time (s)
(c)
0 200 400 600
0
100
Pit
ch (
arc
sec)
Time (s)
−100
(d)
0 200 400 600
0
100
Yaw
(arc
sec)
−100
Time (s)
(e)
0 200 400 600
0
100
Yaw
(arc
sec)
Time (s)
−100
(f)
Figure 4: Attitude, heading error, and 3σ boundary.
Figure 3 shows the speed error and its 3σ boundary. The initial velocity is not accurate.
In the left diagram, when uniform motion speed changes cannot be sensed any longer, the
error shape exhibits phase steps. While the speed observation is available, the navigation
system can sense it after the speed change. We see the two speed changes in the lunar rover
movement are in zigzag fashions on the speed error figure and then quickly disappear.
Figure 4 shows the attitude, heading error, and its 3σ boundary, but the heading
information is of main interest in the navigation. The mean of the heading error in the left
diagram is −5.55” with a standard deviation of 3.03”. The mean of the heading error in the
right diagram is 1.71”, with a standard deviation of 3.53”.
Figure 5 shows the constant gyro bias error and its 3σ boundary. As can be seen from
the graph, the 3-channel constant bias basically converges in the Motion 1 stage, that is, static,
and completes the initial alignment of the gyroscope.
14 Mathematical Problems in Engineering
0 200 400 600
0
0.1
0.2
−0.1
Time (s)
x(d
eg/
hr)
(a)
−0.1
x(d
eg/
hr)
0 200 400 600
0
0.1
0.2
Time (s)
(b)
0 200 400 600
0
0.1
0.2
−0.1
Time (s)
y(d
eg/
hr)
(c)
0
0.1
0.2
0 200 400 600−0.1
Time (s)y(d
eg/
hr)
(d)
−0.1
Time (s)
0 200 400 600
0
0.1
0.2
z(d
eg/
hr)
(e)
0 200 400 600
0
0.1
0.2
−0.1
Time (s)
z(d
eg/
hr)
(f)
Figure 5: The constant gyro bias error and 3σ boundary.
4.3. Discussions and Remarks
From the above analysis and simulation, it can be seen that the significance of this work is to
combine celestial and inertial sensor data to obtain the attitude and heading information for
the real-time navigation of the lunar rover. The simulation results indicate that the dual-EKF
method is valid in this field. To obtain better results, the following two properties are worth
of being further investigated in the future work on navigation.
Computational accuracy: the technology of imaging processing plays a role in the
celestial navigation. The performance of noise filtering and feature extraction for
the astronomical images will affect the navigation precision directly (Liao et al.,
see [22, 23]; Yang et al., see [24, 25]). In addition, the nonlinear properties, such as
fractals [26, 27], in the astronomical images can affect the navigation effect also.
Computational complexity: though the Kalman filter is the most widely used attitude
estimation algorithm for navigation and it offers the optimal recursive solution to
the nonlinear estimation problem, the implementation efficiency of the recursive
Mathematical Problems in Engineering 15
Kalman estimator has been an issue. Correlation is a useful technique in the field.
Real-time navigation may use it to help in Kalman filtering [28, 29].
5. Discussion and Conclusions
In this paper, a sun-orientation-and-speed-observations-based lunar rover real-time celestial
navigation method is proposed, using dual-EKF to estimate system parameters and state.
The method treats the position and velocity as system parameters and establishes a position,
velocity differential equation. Further, the rover attitude quaternion is treated as the system
state, and the quaternion differential equation is established as the state equation. To establish
the measurement equation, the sun direction vector is obtained from the sun sensor and the
speed observation is obtained from the speedometer. Finally, the rover position and heading
information is obtained in real-time through the dual-extended Kalman filter (Dual-EKF).The proposed system does not use accelerometers and thus avoids the acceleration noises.
Also, the system uses a high-precision gyro to improve the navigation accuracy.
Simulation results show that the proposed technique is able to obtain the rover
navigation information in real time, and it overcomes the two shortcomings of more
traditional navigation methods: the discrete output (of pure celestial navigation) and
cumulative error (of inertial navigation).
Acknowledgments
L. Xie and P. Yang were supported by the National Natural Science Foundation of China
(NSFC) under Grant no. 60534070, Zhejiang Provincial Program of Science and Technology
under Grant no. 2009C33085, Wenzhou Program of Science and Technology under Grant no.
S20100029. M. Li would like to acknowledge the support from the 973 plan under the project
no. 2011CB302802 and from the National Natural Science Foundation of China under Project
Grant no. 61070214 and 60873264.
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[14] M. W. L. Thein, D. A. Quinn, and D. C. Folta, “Celestial navigation (CelNav): lunar surfacenavigation,” in Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu,Hawaii, USA, August 2008.
[15] X. L. Ning and J. C. Fang, “Position and pose estimation by celestial observation for lunar rovers,”Journal of Beijing University of Aeronautics and Astronautics, vol. 32, no. 7, pp. 756–759, 2006 (Chinese).
[16] X. L. Ning and J. C. Fang, “A new autonomous celestial navigation method for the lunar rover,”Robotics and Autonomous Systems, vol. 57, no. 1, pp. 48–54, 2009.
[17] F. J. Pei, H. H. Ju, and P. Y. Cui, “A long-range autonomous navigation method for lunar rovers,” HighTechnology Letters, vol. 19, no. 10, pp. 1072–1077, 2009 (Chinese).
[18] X. N. Xi, Lunar Probe Orbit Design, National Defense Industry, Beijing, China, 2001.[19] E. A. Wan and A. T. Nelson, “Dual extended kalman filter methods,” in Kalman Filtering and Neural
Networks, John Wiley & Sons, New York, NY, USA, 2001.[20] S. Y. Chen, “Kalman filter for robot vision: a survey,” IEEE Transactions on Industrial Electronics, vol.
59, no. 99, 2012.[21] S. G. Kim, J. L. Crassidis, Y. Cheng, A. M. Fosbury, and J. L. Junkins, “Kalman filtering for relative
spacecraft attitude and position estimation,” in Proceedings of the AIAA Guidance, Navigation, andControl Conference, pp. 2518–2535, San Francisco, Calif, USA, August 2005.
[22] Z. W. Liao, S. X. Hu, D. Sun, and W. Chen, “Enclosed laplacian operator of nonlinear anisotropicdiffusion to preserve singularities and delete isolated points in image smoothing,” MathematicalProblems in Engineering, vol. 2011, Article ID 749456, 15 pages, 2011.
[23] Z. W. Liao, S. X. Hu, M. Li et al., “Noise estimation for single-slice sinogram of low-dose x-raycomputed tomography using homogenous patch,” Mathematical Problems in Engineering, vol. 2012,Article ID 696212, 16 pages, 2012.
[24] J. W. Yang, Z. Chen, W. S. Chen, and Y. Chen, “Robust affine invariant descriptors,” MathematicalProblems in Engineering, vol. 2011, Article ID 185303, 2011.
[25] J. W. Yang, M. Li, Z. Chen et al., “Cutting affine invariant moments,” Mathematical Problems inEngineering. In press.
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[29] E. Pardo-Iguzquiza, K. V. Mardia, and M. Chica-Olmo, “MLMATERN: a computer program formaximum likelihood inference with the spatial Maern covariance model,” Computers and Geosciences,vol. 35, no. 6, pp. 1139–1150, 2009.
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 986134, 11 pagesdoi:10.1155/2012/986134
Research ArticleHidden-Markov-Models-Based DynamicHand Gesture Recognition
Xiaoyan Wang,1 Ming Xia,1 Huiwen Cai,2Yong Gao,3 and Carlo Cattani4
1 College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China2 Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China3 Zhejiang Jieshang Vision Science and Technology Cooperation, Hangzhou 310013, China4 Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy
Correspondence should be addressed to Xiaoyan Wang, [email protected]
Received 12 January 2012; Accepted 3 February 2012
Academic Editor: Ming Li
Copyright q 2012 Xiaoyan Wang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
This paper is concerned with the recognition of dynamic hand gestures. A method based onHidden Markov Models (HMMs) is presented for dynamic gesture trajectory modeling andrecognition. Adaboost algorithm is used to detect the user’s hand and a contour-based handtracker is formed combining condensation and partitioned sampling. Cubic B-spline is adopted toapproximately fit the trajectory points into a curve. Invariant curve moments as global features andorientation as local features are computed to represent the trajectory of hand gesture. The proposedmethod can achieve automatic hand gesture online recognition and can successfully reject atypicalgestures. The experimental results show that the proposed algorithm can reach better recognitionresults than the traditional hand recognition method.
1. Introduction
The goal of Human Computer Interaction (HCI) is to bring the performance of human
machine interaction similar to human-human interaction [1]. Gestures play an important part
in our daily life, and they can help people convey information and express their feelings.
Among different body parts, the hand is the most effective, general-purpose interaction
tool. Therefore, hand gesture tracking and recognition becomes an active area of research
in human computer interaction and digital entertainment industry [2–4]. A gesture can be
static or dynamic or both. According to this, there are three types of gesture recognition: static
hand posture recognition, dynamic hand gesture recognition, and complicated hand gesture
recognition. Our work in this paper concentrates on dynamic gesture recognition, which
2 Mathematical Problems in Engineering
characterizes the hand movements. Tracking frameworks have been used to handle dynamic
gestures. Isard and Blake [5] established a hand tracking approach based on 2D deformable
contour model and Kalman filter [6]. However, it is inefficient to track an articulated object
which has a high dimension state space using condensation alone. MacCormick and Blake
[7] introduced a partition sampling method to track more than one object. MacCormick and
Isard [8] implemented a vision-based articulated hand tracker using this technique after that.
Their tracker is able to track position, rotation, and scale of the user’s hand while maintaining
a pointing gesture. Based on Blake’s work, Tosas [9] makes some technique extensions and
implements a full articulated hand tracker.
Several methods on hand gesture recognition have been proposed [10–13], which
differ from one another in their models, just like Neural Network, Fuzzy Systems and
Hidden Markov Models (HMMs) [14]. The most challenging problem of dynamic gesture
recognition is its spatial-temporal variability, when the same gesture can differ in velocity,
shape, duration, and integrality. These characteristics make it more difficult to recognize
dynamic hand gestures than to recognize static ones. HMM is a statistical model widely
used in hand writing, speech, and character recognition [13, 15] because of its capability of
modeling spatial-temporal time series. HMM has also been successfully used in hand gesture
recognition [13, 16–18], in respect that it can preserve the spatial-temporal identity of hand
gesture and have an ability to do the segmentation automatically. Motion features of each
time point have been modeled in most of the dynamic hand gesture recognition methods
using HMM, nevertheless, the whole trajectory shape characters are not considered at the
same time. The recognition based on local features is very sensitive to sampling period and
velocity, and the continuous local process of gesture will cause false recognition.
Researches on psychology indicate that human brains lean to perceive object from a
whole, and then apprehend its details, which illustrates that an object can only be described
perfectly when the local and global information are integrated. In this paper, we propose
a dynamic gesture trajectory modeling and recognition method based on HMM. Cubic B-
spline is adopted to approximately fit the trajectory points into a curve, and invariant curve
moments as global features and orientation as local features are computed to represent the
trajectory of hand gesture. Threshold model is used to model all the atypical gesture patterns,
and automatically segment and recognize the dynamic gesture trajectory. The proposed
method can achieve automatic hand gesture online recognition and can successfully reject
atypical gestures. Meanwhile, the experiment results show that the recognition performance
of the proposed algorithm can be greatly improved by combining the global invariant curve
features with local orientation features.
The rest of the paper is organized as follows: Section 2 describes the dynamic gesture
representation and the global and local features we used. Section 3 gives the continuous
hand gesture recognition procedure, which contains hand detection, tracking, and gesture
recognition based HMM. The experimental results are shown in Section 4. Finally, Section 5
and ends the paper with a summary.
2. Dynamic Gesture Representation
A dynamic hand gesture is a spatial-temporal pattern and has four basic features: velocity,
shape, location, and orientation. The motion of the hand can be described as a temporal
sequence of points with respect to the hand centroid of the person performing the gesture.
Mathematical Problems in Engineering 3
50
40
30
20
10
0400
300
200
100 150200
250300
XY
Tim
e450
400
350
300
250
200
150
100
50
00 100 200 300 400 500 600
X
Y
Figure 1: A dynamic hand gesture instance.
In this paper, the hand shape is not considered and each dynamic hand gesture instance is
represented by a time series of the hand’s location:
pt =(xt,yt
), (t = 1, 2, . . . , T), (2.1)
where T represents the length of gesture path and varies across different gesture instances.
Consequently, a gesture containing an ordered set of points can be regarded as a mapping
from time to location. Figure 1 shows a dynamic hand gesture instance and gives its
projection along the time axis onto the image plane.
2.1. Local Feature Representation
There is no doubt that selecting good features plays significant role in hand gesture
recognition performance. The orientation feature is proved to be the best local representation
in terms of accuracy results [19–21] and it is considered as the most important feature in
dynamic gesture recognition using HMM [22, 23]. Therefore, we will rely upon it as a main
local feature in our system. The orientation of hand movement is computed between two
consecutive points of the hand gesture trajectory:
θt = arctan
(yt+1 − yt
xt+1 − xt
), (t = 1, 2, . . . , T). (2.2)
A feature vector will be determined by converting the orientation to directional
codewords by a vector quantizer. For example, in Figure 2 the orientation is quantized to
generate the codewords from 1 to 20 by dividing it by 20 degree. Thereby, the discrete feature
vector will be used as an input to discrete HMM.
2.2. Global Feature Presentation
The human brain is inclined to sense object from a whole, and people also try to understand
a gesture as integrity. Accordingly, we try to connect all the discrete points of gesture using
4 Mathematical Problems in Engineering
1
2
3
456
78
9
10
11
12
1314
15 1617
18
19
20
90
180
270
dy
dx
(xt+1, yt+1)
(xt, yt)
θt
Figure 2: The orientation and its codewords.
a slippery line. Cubic B-spline function is adopted to approximately fit the trajectory points
into a curve:
p(t) =3∑
m=0
Bm(t)CPm, (2.3)
where B0(t) = (1 − t)3, B1(t) = 3t(1 − t)2, B2(t) = 3t2(1 − t), B3(t) = t3, CPm are control
points. After the curve is shaped, an issue to be addressed is the variation of speed of the
same gesture. To overcome this problem, all curves are scaled such that they lie within the
same range. Those curves for faster moves are relatively expanded by interpolation and those
of slower moves are contracted.
The trajectories of a same gesture vary in size and shape. We use invariant curve
moments as global features to represent the trajectory [24]. The advantage of moment
methods is that they are mathematically concise and invariant to translation, rotation, and
scale. Furthermore, they reflect not only the shape but also the density distribution within
the curve.
The (p + q)th-order moments of plane curve l are defined as
mpq =∫xpyq ds,
(p, q = 0, 1, 2, . . .
), (2.4)
where ds is the arc differentiation of curve l. The (p + q)th-order central moments are defined
as:
μpq =∫(x − x)p
(y − y
)qds,
(p, q = 0, 1, 2, . . .
), (2.5)
where x = m10/m00, y = m01/m00.
Mathematical Problems in Engineering 5
For a digital image f(x,y),
mpq =∑x,y
xpyqf(x,y
),
μpq =∑x,y
(x − x)p(y − y
)qf(x,y
).
(2.6)
This paper defines f(x,y) as
f(x,y
)=
{1,
(x,y
) ∈ l,
0,(x,y
)/∈ l.
(2.7)
Thus, the global descriptors of hand gestures have been calculated using the central
moments of the curve. As we use discrete HMM, all the features extracted need to be
represented as an integer. The statistical distributions of the central moments are calculated
and then a feature is denoted as one or two digits.
3. The Continuous Hand Gesture Recognition Scheme
In this paper, we consider online-continuous-handed dynamic gestures based on discrete
HMM. The hand gesture recognition system consists of three major parts: palm detection,
hand tracking, and trajectory recognition. Figure 3 shows the whole process. The hand
tracking function is trigged when the system detects an opened hand before the camera; the
hand gesture classification based on HMM is activated when the user finishing the gesture.
The basic algorithmic framework for our recognition process is the following.
(1) Detect the palm from video and initialize the tracker with the template of hand
shape.
(2) Track the hand motion using a contour-based tracker and record the trajectory of
palm center.
(3) Extract the discrete vector feature from gesture path by the global and local feature
quantization.
(4) Classify the gesture using HMM which gives maximum probability of occurrence
of observation sequence.
3.1. Hand Detection and Tracking
We use Adaboost algorithm with (histograms of gradient) HOG feature to detect the user’s
hand. The shape information of an opening hand is relatively unique in the scene. We
calculate the HOG features of a new observed image to detect the opened hand at different
scales and location. When the hand is detected, we update the hand color model which will
be used in hand tracking. The system requires user to keep his palm opened vertically and
statically before the palm is captured by the detection algorithm. In this paper, we have
considered single handed dynamic gestures. A gesture is composed of a sequence of epochs.
Each epoch is characterized by the motion of distinct hand shapes.
6 Mathematical Problems in Engineering
Image input
Tracking Detection
2D mappingHand
trajectory
B-spline fit
Global
featureLocal
feature
Quantization the features
discrete vector
Feature extraction
using one integer
Combine the features to
Initialize the HMM
Gesture Nongesture
Satisfy the
gesture
ending
condition?
Satisfy the
gesture
ending
condition?
Gesture
model
Nongesture
model
Fit which model?
HMM classification
endend
Figure 3: Overview of the hand gesture recognition process.
Figure 4: Hand contour.
We have implemented a contour-based hand tracker, which combines two techniques
called condensation and partitioned sampling. During tracking, we record the trajectory of
the hand which will be used in the hand recognition stage. The hand contour is represented
with B-Splines, as shown in Figure 4. A fourteen-dimension state vector is used to describe
the dynamics of the hand contour:
χ =(tx, ty, α, s, θL, lL, θR, lR, θM, lM, θI , lI , θTh1, θTh2
), (3.1)
where the subvector (tx, ty, α, s) is a nonlinear representation of a Euclidean similarity
transform applied to the whole hand contour template, (tx, ty) is the palm center. (θL, lL)represents the nonrigid movement of the little finger, θL means the little finger’s angle with
respect to the palm, and lL means the little finger’s length relative to its original length in
Mathematical Problems in Engineering 7
the hand template. (θR, lR), (θM, lM), and (θI, lI) have the same meaning as the subvector
(θL, lL), but for different fingers. θTh1 represents the angle of the first segment of the thumb
with respect to the palm, and the last part θTh2 represents the angle of the second segment of
the thumb with respect to the first segment of the thumb.
We use a second-order autoregressive processes to predict the motion of the hand
contour:
xt = A1xt−1 +A2xt−2 + Bωt, (3.2)
where A1 and A2 are fixed matrices representing the deterministic components of the
dynamics, B is another fixed matrix representing the stochastic component of the dynamics,
and ωt is a vector of independent random normal N(0, 1) variants.
In prediction, lots of candidate contours will be produced. We choose the one which
matches the image feature (edges, boundaries of regions in skin color) best. Usually,
more dimensions of the state space are required to make the condensation filter achieve
considerable performance. However, this will increase computation complexity. In order to
alleviate the problem, partitioned sampling is used, which divides the hand contour tracking
into two steps: first, track the rigid movement of the whole hand, which is represented by
(tx, ty, α, s); second, track the nonrigid movement of the each finger, which is represented by
angle and length of each finger. The above operations can reduce the amount of candidate
contours and improve the efficiency of tracking.
3.2. Recognition Based on HMM
After the trajectory is obtained from the tracking algorithm, features are abstracted and used
to compute the probability of each gesture type with HMM. We use a vector to describe those
features and as the input of the HMM.
There are three main problems for HMM: evaluation, decoding, and training, which
are solved by using Forward algorithm, Viterbi algorithm, and Baum-Welch algorithm,
respectively [25]. The gesture models are trained using BW re-estimation algorithm and the
numbers of states are set depending on the complexity of the gesture shape.
We choose left-right banded model (Figure 5(a)) as the HMM topology, because the
left-right banded model is good for modeling-order-constrained time-series whose properties
sequentially change over time [26]. Since the model has no backward path, the state index
either increases or stays unchanged as time increases. After finishing the training process
by computing the HMM parameters for each type of gesture, a given gesture is recognized
corresponding to the maximal likelihood of seven HMM models by using viterbi algorithm.
Although the HMM recognizer chooses a model with the best likelihood, we cannot
guarantee that the pattern is really similar to the reference gesture unless the likelihood is
high enough. A simple threshold for the likelihood often does not work well. Therefore, we
produce a threshold model [22] that yields the likelihood value to be used as a threshold.
The threshold model is a weak model for all trained gestures in the sense that its likelihood
is smaller than that of the dedicated gesture model for a given gesture and is constructed
by collecting the states of all gesture models in the system using an ergodic topology shown
in Figure 5(b). A gesture is then recognized only if the likelihood of the best gesture model
is higher than that of the threshold model; otherwise, it is recognized as nongesture type.
Therefore, we can segment the online gestures using the threshold model.
8 Mathematical Problems in Engineering
S1 S2 S3 S4
(a) Left-right banded topology
S1 S2
S3 S4
(b) Ergodic topolgy
Figure 5: HMM topologies.
4. Experiments
For experimentation, we develop a human machine interaction interface based on hand
gesture. It can work with regular webcams that is connected to PC, which is used to capture
live images of the users’ hand movement. The minimum requirements of webcams are (1)frame rate up to 25 frames per second and (2) capture capability up to 640 × 480 pixels. The
interface can be deployed in indoors environment, which generally has static background
and less light changes. Hand gestures are those articulated with poses and movement with
hands. The interface is able to track and recognize the following predefined hand gestures:
(1) user drawing three circles continuously in a line horizontally with hand movement
in the air,
(2) user drawing a question mark (?) with hand movement in the air,
(3) user drawing three circles continuously in a line vertically with hand movement in
the air,
(4) hand being vertically lifted upwards,
(5) hand waving from left to right,
(6) hand waving from right to left,
(7) user drawing an exclamation mark (!) with hand movement in the air.
For the quantification of local oriental features, we pick 18 as the codeword number
from experience. Figure 6 shows the distribution histogram of central moment μ11 of the
seven gestures as our global feature, where all the sample amounts are 450. We can set
the number of the vector quantizer of global features to 20 according to the distribution.
It can also be seen that the central moment feature can express the shape characteristic of
trajectories. For example, gesture 1 and gesture 3, gesture 4 and gesture 7, gesture 5 and
gesture 6 are close in their integral form, respectively, and it can be separated easily using the
global feature.
We choose the state number of HMM for each gesture according to the experiment
results and find that the recognition rate cannot be promoted when the state numbers of
gestures 1 and gesture 3 are 10, and the other state numbers are set to 8. Therefore, we use
this setting in the following experiments.
We collected more than 800 trajectory samples of each isolated gesture from seven
people for training and more than 330 trajectory samples of each isolated gesture from eight
different users for testing. The recognition results are listed in Table 1. It can be seen that the
proposed method can greatly improve the recognition process, especially for those relatively
complicated gestures such as predefined gesture 1 and gesture 3. It is difficult to separate
Mathematical Problems in Engineering 9
180
160
140
120
100
80
60
40
20
00 5 10 15 20 25 30 35
Sam
ple
nu
mb
er
Gesture 1
Gesture 2
Gesture 3
Gesture 4
Gesture 5
Gesture 6
Gesture 7
μ11
Figure 6: The distribution of μ11.
Table 1: Recognition results comparison.
Gestures Test sets’ numbers Our method (%) Traditional method (%)1 339 84.1 94.7
2 407 95.1 98.2
3 372 73.4 89.7
4 454 95.9 98
5 424 98.1 100
6 476 95.8 99.8
7 474 98.9 99.6
gesture 1 and gesture 3 only using local features, because their motions resemble temporally.
Our algorithm can resolve this problem effectively.
5. Conlusion
We have implemented an automatic dynamic hand gesture recognition system in this paper.
The user’s hand is detected using Adaboost algorithm with HOG features and tracked using
condensation and partitioned sampling. The trajectory of hand gesture is represented by both
local and global features. Then, we take a discrete HMM method to recognize the gestures.
The experimental results show that the proposed algorithm can reach better recognition
results than the traditional hand recognition method. However, the tracking algorithm is still
very sensitive to light and the system can only report the detection until a gesture reaches its
end. Therefore, our future work will focus on improving the tracking algorithm and making
the recognition more natural.
10 Mathematical Problems in Engineering
Acknowledgments
This work was supported by the Research Project of Department of Education of Zhe-
jiang Province (Y201018160), and the Natural Science Foundation of Zhejiang Province
(Y1110649).
References
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[3] X. Zhang, X. Chen, Y. Li, V. Lantz, K. Wang, and J. Yang, “A framework for hand gesture recognitionbased on accelerometer and EMG sensors,” IEEE Transactions on Systems, Man, and Cybernetics Part A,vol. 41, no. 6, pp. 1064–1076, 2011.
[4] I. N. Junejo, E. Dexter, I. Laptev, and P. Perez, “View-independent action recognition from temporalself-similarities,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 33, no. 1, pp. 172–185, 2011.
[5] M. Isard and A. Blake, “Condensation—conditional density propagation for visual tracking,”International Journal of Computer Vision, vol. 29, no. 1, pp. 5–8, 1998.
[6] S. Chen, “Kalman filter for robot vision: a survey,” IEEE Transactions on Industrial Electronics, vol. 59,Article ID 814356, 18 pages, 2012.
[7] J. MacCormick and A. Blake, “Probabilistic exclusion principle for tracking multiple objects,” inProceedings of the 7th IEEE International Conference on Computer Vision (ICCV ’99), pp. 572–578,September 1999.
[8] J. MacCormick and M. Isard, “Partitioned sampling, articulated objects, and interface-quality handtracking,” in Proceedings of the European Conferene Computer Vision, 2000.
[9] M. Tosas, Visual articulated hand tracking for interactive surfaces, Ph.D. thesis, University of Nottingham,2006.
[10] X. Deyou, “A neural network approach for hand gesture recognition in virtual reality driving trainingsystem of SPG,” in Proceedings of the 18th International Conference on Pattern Recognition (ICPR ’06), pp.519–522, August 2006.
[11] D. B. Nguyen, S. Enokida, and E. Toshiaki, “Real-time hand tracking and gesturerecognition system,”in Proceedings of the International Conference on on Graphics, Vision and Image Processing (IGVIP ’05 ), pp.362–368, CICC, 2005.
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[17] N. D. Binh and T. Ejima, “Real-time hand gesture recognition using pseudo 3-d Hidden MarkovModel,” in Proceedings of the 5th IEEE International Conference on Cognitive Informatics (ICCI ’06), pp.820–824, July 2006.
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Mathematical Problems in Engineering 11
[20] N. Liu, B. C. Lovell, P. J. Kootsookos, and R. I. A. Davis, “Model structure selection & trainingalgorithms for an HMM gesture recognition system,” in Proceedings of the 9th International Workshopon Frontiers in Handwriting Recognition (IWFHR-9 ’04), pp. 100–105, October 2004.
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 532610, 6 pagesdoi:10.1155/2012/532610
Research ArticleStable One-Dimensional Periodic Wave inKerr-Type and Quadratic Nonlinear Media
Roxana Savastru, Simona Dontu, Dan Savastru,Marina Tautan, and Vasile Babin
Department of Constructive and Technological Engineering—Lasers and Fibre Optic Communications,National Institute of R&D for Optoelectronics INOE 2000, 409 Atomistilor Street, P.O. Box MG-5,077125 Magurele, Ilfov, Romania
Correspondence should be addressed to Simona Dontu, [email protected]
Received 6 December 2011; Revised 9 February 2012; Accepted 13 February 2012
Academic Editor: Cristian Toma
Copyright q 2012 Roxana Savastru et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.
We present the propagation of optical beams and the properties of one-dimensional (1D) spatialsolitons (“bright” and “dark”) in saturated Kerr-type and quadratic nonlinear media. Specialattention is paid to the recent advances of the theory of soliton stability. We show that thestabilization of bright periodic waves occurs above a certain threshold power level and thedark periodic waves can be destabilized by the saturation of the nonlinear response, while thedark quadratic waves turn out to be metastable in the broad range of material parameters. Thepropagation of (1+1) a dimension-optical field on saturated Kerr media using nonlinearSchrodinger equations is described. A model for the envelope one-dimensional evolution equationis built up using the Laplace transform.
1. Introduction
The discrete spatial optical solitons have been introduced and studied theoretically as
spatially localized modes of periodic optical structures [1]. A standard theoretical approach
in the study of the discrete spatial optical solitons is based on the derivation of an effective
discrete nonlinear Schrodinger equation and the analysis of its stationary localized solitons-
discrete localized modes [1, 2].The spatial solitons may exist in a broad branch of nonlinear materials, such as cubic
Kerr, saturable, thermal, reorientation, photorefractive, and quadratic media, and periodic
systems. Furthermore, the existence of solitons varies in topologies and dimensions [3].The theory of spatial optical solitons has been based on the nonlinear Schrodinger
(NLS) equation with a cubic nonlinearity, which is exactly integrable by means of the inverse
2 Mathematical Problems in Engineering
scattering (IST) technique. From the physical point of view, the integrable NLS equation
describes the (1+1)-dimensional beams in a Kerr (cubic) nonlinear medium in the framework
of the so-called paraxial approximation [4].Bright solitons are formed due to the diffraction or dispersion compensated by self-
focusing nonlinearity and appear as an intensity hump in a zero background. Solitons, which
appear as intensity dips with a CW background, are called dark soliton [3].Kerr solitons rely primarily on a physical effect, which produces an intensity-
dependent change in refractive index [3].The periodic wave structures play an important role in the nonlinear wave domain
so that they are core of instability modulation development and optics chaos on continuous
nonlinear media, modes of quasidiscrete systems or discrete system on mechanic and electric
domain. Thus, periodic wave structures are unstable in the propagation process. For example,
photorefractive crystals accept relatively high nonlinearity of saturated character at an
already known intensity for He-Ne laser in continuous regime.
2. Methodology
The propagation of the optical radiation in (1+1) dimensions in saturable Kerr-type medium
is described by the nonlinear Schrodinger equation for the varying field amplitude Φ(ς, ρ)[5]:
2i∂Φ
(ς, ρ
)∂ς
+∂2Φ
(ς, ρ
)∂ρ2
− 2Φ(ς, ρ
)∣∣Φ(ς, ρ)∣∣2
1 + S∣∣Φ(ς, ρ)∣∣2
= 0. (2.1)
The transverse ς and the longitudinal ρ coordinates are scaled in terms of the
characteristic pulse (beam) width and dispersion (diffraction) length, respectively; S is the
saturation parameter; σ = −1 (+1) stands for focusing (defocusing) media [5]
ς = σKZ,
ρ =√σK
√X2 + Y 2,
ϕ = arctg
(Y
X
),
η = ρ sinϕ,
ξ = ρ cosϕ.
(2.2)
The simplest periodic stationary solutions of (2.1) have the following form:
Φ(ς, ρ
)= U
(ρ)e+2ihς, (2.3)
where h is the propagation constant.
By replacing the field in such a form into (2.1), one gets
∂2U(ρ)
∂ρ2− 2hU
(ρ) − 2U3
(ρ)
1 + SU2(ρ) = 0. (2.4)
Mathematical Problems in Engineering 3
To perform the linear stability analysis of periodic waves in the saturable medium,
we use the mathematical formalism initially developed for periodic waves in cubic nonlinear
media [5].We consider an analytic model, which used the Laplace transform of (2.4):
(α(U(ρ))
=∫+∞
0
U(ρ)e−pρdρ = U
(p))
p = u1 + iv1
−(
p2
2− 2h
)U(p)+
⎡⎣p
2U(0) +
1
2
(∂U
(ρ)
∂ρ
)ρ=0
⎤⎦ +∫∞
0
U3(ρ)
1 + SU2(ρ)e−pρdρ = 0.
(2.5)
With the boundary conditions,
U(ρ)∣∣
ρ=0= U(0) = U0,
∂U(ρ)
∂ρ
∣∣∣∣∣ρ=0
= 0.(2.6)
From (2.5) we get the Laplace transform of the field:
(i) direct form:
U(p)=
pUo + 2∫+∞
0
((U3(ρ))/(1 + SU2
(ρ)))
e−pρdρ((p2/2
) − 2h) (2.7)
(ii) inverse transformation form:
U(ρ)=
1
2πi
∫u+i∞
u−i∞
pU0 + 2∫+∞
0
(U3(ρ)/1 + SU2
(ρ))e−pρdρ(
p2 − 4h) e+pρdp, (2.8)
where u is a finite number.
For the integration on real (h > 0) and imaginary (h < 0) poles, we calculated the
complex amplitude of nonlinear equation such as
U(ρ)= U0ch
(2√hρ)− 4
∫+∞
0
d
⎛⎜⎝sh2√h(ρ − ρ′
)(2√h)2
⎞⎟⎠(U3(ρ′)
1 + SU2(ρ′)),
U(ρ)= U0 cos
(2√hρ)+ 4
∫+∞
0
dρ′(
U3(ρ′)
1 + SU2(ρ′))(cos 2
√h(ρ − ρ′
)2√h
).
(2.9)
4 Mathematical Problems in Engineering
For the harmonic case (h < 0) integration form of the complex amplitude is
U(ρ)= U0 cos
(2√hρ)+
1
hcos
(2√hρ)∫+∞
0
U3(ρ′)
1 + SU2(ρ′)d(sin 2
√hρ′
)− 1
hsin
(2√hρ)∫+∞
0
U3(ρ′)
1 + SU2(ρ′)d(cos 2
√hρ′
).
(2.10)
By using the integration, we get
U(ρ)= U0 cos
(2√hρ)+
1
h
U3(ρ)
1 + SU2(ρ) sin
(2√hρ)
(2.11)
or
U(ρ)=
√√√√U2
0 +
(1
h
U3(ρ)
1 + SU2(ρ))2
sin(
2√hρ + ϕ1
),
ϕ1 = arctg
(U0
(1/h)(U3(ρ)/1 + SU2
(ρ))).
(2.12)
The total phase of the optical field envelope is as follows:
ϕT = 2√hρ + arctg
(U0
(1/h)(U3(ρ)/1 + SU2
(ρ))). (2.13)
We assume a frequency (ω) as a speed variation of total phase such as
ωDef=
dϕT
dρ=(
2√h)+
d
dρ
{arctg
(hU0
1 + SU2(ρ)
U3(ρ) )}
. (2.14)
We have the complex amplitude of envelope field with the following form:
U(ρ)= A
(ω, ρ
)cos
(2√hρ)+ B
(ω, ρ
)sin
(2√hρ). (2.15)
Mathematical Problems in Engineering 5
un2, un21, un22
0
0.5
1
1.5
2
2.5
un2un21un22
nh
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
(a)
nh
−90
−75
−60
−45
−30
−15
0
15
30
45
60
75
90ϕn2, ϕn21
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
jn2
jn21
(b)
Figure 1: Numerical simulations of complex amplitude and phase.
The hyperbolic secant plays this equation resulting in a conservative effect. The longi-
tudinal component is
A(ω, ρ
)= U0 − 5ω
2(Sh)3/2(1 + (ω3/4h))ch(ωρ
) ,B(ω, ρ
)=
(1
h
U30
1 + SU2(0)
)−
(5ω(ω2/2
√h))
+ h(ωρ
)2(Sh)3/2(1 + (ω3/4h))ch
(ωρ
) .(2.16)
Some numerical simulations of the complex amplitude of the nonlinear equation and
the total phase of the optical field depending on the propagation constant and an integer
number n are illustrated in Figure 1.
6 Mathematical Problems in Engineering
Figure 1 represents the model amplitude and the phase functions of the complex total
number, which explained the theoretical model presented. Thanks to the complex model, the
initial solution includes the hyperbolic secant and the conjugate complex part
Φ(ξ, ρ
)=
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
[U0 − 5ω
2(Sh)3/2(1 + (ω3/4h))ch(ωρ
)] cos(
2√hρ)
+
⎡⎢⎣( 1
h
U30
1 + SU20
)−
5ω(ω2/2
√h)+ h
(ωρ
)2(Sh)3/2(1 + (ω3/4h))ch
(ωρ
)⎤⎥⎦ sin
(2√h)ρ
⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭e2ihξ,
ω0 = 2√h,
ω =dϕT
dρ,
ω(ρ)= ω0 +
e−4ω0ρ
4((ω2
0/4)U0S3/2
)2
1
ch2(ω0ρ) ,
ω(0) −ω0 =16
U20S
3
1
ω40
.
(2.17)
3. Conclusions
We have described the propagation in quadratic nonlinear media of the periodic waves in sat-
urated Kerr type. The analytic solution for one-dimensional, bright and dark spatial solitons
was found. To describe the spatial optical solitons in saturated Kerr type and the quadratic
nonlinear media, we propose an analytical model based on Laplace transform. The theoretical
model consists in solving analytically the Schrodinger equation with photonic network
using Laplace transform. The propagation properties were found by using different forms
of saturable nonlinearity. However, an exact analytic solution of the propagation problem
presented herein creates possibilities for further theoretical investigation. As a result, it is a
useful structure, which obtains one-dimensional “bright” and “dark” solitons with transver-
sal structure and transversal one-dimensional periodic waves.
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[2] F. Lederer, S. Darmanyan, and A. Kobyakov, Spatial Solitons, springer, Berlin, Germany, 2001.[3] X. u. Zhiyong, All-optical Soliton Control in Photonic Lattices, Master work, Universitat Politecnica de
Catalunya, Barcelona, Spain, 2007.[4] Y. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Optical and Quantum Electronics,
vol. 30, no. 7–10, pp. 571–614, 1998.[5] Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Stable one-dimensional periodic waves
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 928161, 12 pagesdoi:10.1155/2012/928161
Research ArticleCutting Affine Moment Invariants
Jianwei Yang,1 Ming Li,2 Zirun Chen,1 and Yunjie Chen1
1 School of Math and Statistics, Nanjing University of Information Science and Technology,Nanjing 210044, China
2 School of Information Science and Technology, East China Normal University, no. 500 Dong-Chuan Road,Shanghai 200241, China
Correspondence should be addressed to Jianwei Yang, [email protected]
Received 18 December 2011; Accepted 26 January 2012
Academic Editor: Carlo Cattani
Copyright q 2012 Jianwei Yang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
The extraction of affine invariant features plays an important role in many fields of imageprocessing. In this paper, the original image is transformed into new images to extract more affineinvariant features. To construct new images, the original image is cut in two areas by a closed curve,which is called general contour (GC). GC is obtained by performing projections along lines withdifferent polar angles. New image is obtained by changing gray value of pixels in inside area. Thetraditional affine moment invariants (AMIs) method is applied to the new image. Consequently,cutting affine moment invariants (CAMIs) are derived. Several experiments have been conductedto evaluate the proposed method. Experimental results show that CAMIs can be used in objectclassification tasks.
1. Introduction
The extraction of affine invariant features plays a very important role in object recognition
and has been found applicable in many fields such as shape recognition and retrieval [1, 2],watermarking [3], identification of aircrafts [4, 5], texture classification [6], image registration
[7], and contour matching [8].Many algorithms have been developed for affine invariant features extraction. Based
on whether the features are extracted from the contour only or from the whole shape region,
the approaches can be classified into two main categories: region-based methods and contour-
based methods [9]. For good overviews of the various techniques, refer to [9–12].Contour-based methods [4, 5, 13–18] provide better data reduction and the contour
usually offers more shape information than interior content [9]. But these methods are unap-
plicable to objects with several separable components (like some Chinese characters).
2 Mathematical Problems in Engineering
In contrast to contour-based methods, region-based techniques take all pixels within
a shape region into account to obtain the shape representation. Moment invariant methods
are the most widely used techniques. The commonly used affine moment invariants (AMIs)[19–21] are extensions of the classical moment invariants firstly developed by Hu [22].Although the moment-based methods can be applicable to binary or gray-scale images with
low computational demands, they would be sensitive to noise. Hence, only a few low-order
moment invariants can be used and limit the ability of object classification with a large-sized
database [18].A number of new region-based methods have also been introduced, such as Ben-
Arie’s frequency domain technique [23, 24], cross-weighted moment (CWM) [25], and Trace
transform [26]. A novel approach called multi scale autoconvolution (MSA) was derived by
Rahtu et al. [27]. These new methods give high accuracy, but usually at the expense of high
complexity and computational demands [27]. It is reported in [27] that one needs O(N4)and O(N2log2N) operations for computing CWM and MSA, respectively. It can be shown
that some of these methods are sensitive to noise in the background. To derive robust affine
invariant features, in [28], we cut the object into slices by division curves which are derived
from the object based on the obtained general contour (GC). The affine invariant descriptors
are constructed by summing up the gray value associated with every pixels in each slice.
However, the maximum of the division quantity τ is hard to be determined. To cut object into
small slices, the computational complexity is very large.
Recently, structure moment invariants have been introduced in [29, 30]. These invari-
ants are very efficient in object classification tasks for gray level images or color images, but
they are unapplicable to binary images. The density of binary images can not be changed
only by squaring.
All in all, contour-based methods can only be used to objects with single boundary;
whereas some region-based methods can achieve high accuracy but usually at the expense
of high computational demands, and some region-based methods are unapplicable to binary
images.
To extract affine invariant features more efficiency, we transform the original image
into new images in this paper. Affine invariants are extracted from new images. In order
to construct new images, the original image is cut in two areas: the inside area and the
outside area. To establish correspondence between areas of an image and those of its
affine transformed image, as in [28], general contour (GC) of the image is constructed by
performing projection along lines with different polar angles. A nonnegative constant is
added to the gray value associated with every pixel of inside area. As a result, new images
are obtained. Consequently, affine invariant features can be derived from these new images.
In this paper, AMIs method is applied to the obtained new images. More affine invariant
features, cutting affine moment invariants (CAMIs), are extracted. Furthermore, we combine
CAMIs with the original AMIs (we call the obtained affine invariants as CCAMIs). To test
and evaluate the proposed method, several experiments have been conducted. Experimental
results show that CAMIs and CCAMIs can be used in object classification tasks.
The rest of the paper is organized as follows: in Section 2, the GC of an image is
introduced. Consequently, the image is cut in two areas by putting GC on the image. New
image is formed by changing gray value of the inside area. We apply AMIs method to the new
image in Section 3. The performance of the proposed method is evaluated experimentally in
Section 4. Finally, some conclusion remarks are provided in Section 5.
Mathematical Problems in Engineering 3
2. The Construction of New Images
To derive affine invariant features, we construct new images by cutting the original image in
two areas. New images can be obtained by changing the gray value associated with pixels in
one of these areas.
2.1. GC of an Image
Suppose that an image is represented by I(x,y) in the 2D plane. Firstly, the origin of the
reference system is transformed to the centroid of the image. To derive general contour of
an image, the Cartesian coordinate system should be converted to polar coordinate system.
Hence, the shape can be represented by a function f of r and θ, namely,
I(x,y
)= f(r, θ), (2.1)
where r ∈ [0,∞), and θ ∈ [0, 2π). Take projection along lines from the centroid with different
angles by computing the following integral:
g(θ) =∫∞
0
f(r, θ)dr, (2.2)
where θ ∈ [0, 2π).
Definition 2.1. For an angle θ ∈ R, if g(θ) is given in (2.2), then (θ, g(θ)) denotes a point in the
plane of R2. Let θ go from 0 to 2π , then {(θ, g(θ)) | θ ∈ [0, 2π)} forms a closed curve. We call
this closed curve the general contour (GC) of the image.
By (2.2), a single value is correspond to an angle θ ∈ R. Consequently, a single closed
curve can be derived from any image. For an image I, we denote the GC extracted from
it as ∂I. Equation (2.2) is called central projection transform in [31–33]. It has been used in
those papers to extract rotation invariant signature by combining wavelet analysis and fractal
theory. Satisfying classification rates have been achieved in the recognition of rotated English
letters, Chinese characters, handwritten signatures, and so forth. As aforementioned, in [28],by employing GC, we derive division curves to cut object into slices. The affine invariant
descriptors are constructed by summing up the gray value associated with every pixels in
each slice. However, the maximum of the division quantity is hard to be determined. In this
paper, we use GC to construct new images. Affine invariant features are extracted from these
new images.
2.2. The Affine Property of GC
An affine transformation A of coordinates x ∈ R2 is defined as
x′ = Ax + b, (2.3)
where b =(
b1
b2
)∈ R2, and A = ( a11 a12
a21 a22) is a 2-by-2 nonsingular matrix with real entries.
4 Mathematical Problems in Engineering
Affine maps parallel lines onto parallel lines, intersecting lines into intersecting lines.
Based on these facts, it can be shown that the GC extracted from the affine transformed image
is also the same affine transformed version of GC extracted from the original image. In other
words, if two images I and I ′ are related by an affine transformation A,
�′ ={x′ | x′ = Ax + b, x ∈ �
}, (2.4)
where � and �′ are supports of I and I ′, respectively. Then ∂I and ∂I ′, GCs of I, and I ′ are
related by the same affine transformation A too:
∂I ′ ={x′ | x′ = Ax + b, x ∈ ∂I
}. (2.5)
2.3. The Construction of New Images
To construct new images, we put the GC on the original image. The image is cut in two areas:
the inside area (denoted as Dinside) and the outside area (denoted as Doutside). In Figure 1(b),we put the GC of Figure 1(a) on the image. Figure 1(c) is the inside area of the image, and
Figure 1(d) is the outside area of the image.
As aforementioned, GC preserves the affine transformation signature. As a result, the
inside area preserves affine transformation too. If two images I and I ′ are related by an affine
transformation A as in (2.4), then DIinside
and DI ′inside
, inside areas of I and I ′, are related by the
same affine transformation A too:
DI ′inside =
{x′ | x′ = Ax + b, x ∈ DI
inside
}. (2.6)
For example, Figure 2(a) is an affine transform version of Figure 1(a). Put the GC of
Figure 2(a) on the image (as shown in Figure 2(b)). Figure 2(c) is the inside area of the image
of Figure 2(b). Figure 2(d) is the outside area of the image of Figure 2(b). We observe that
Figures 2(c) and 2(d) are affine transformed versions of Figures 1(c) and 1(d). The affine
transformation is the same as that of Figure 2(a) to Figure 1(a).Consequently, new images can be constructed by changing gray value associated with
pixels in Dinside. For an image, a constant d (d ≥ 0) is added to the gray value associated with
every pixels in Dinside. The obtained new image is denoted as I(d)(x,y):
I(d)(x,y
)=
⎧⎨⎩I(x,y
),
(x,y
) ∈ Doutside,
I(x,y
)+ d,
(x,y
) ∈ Dinside.(2.7)
For different d, various new images can be derived. It is obvious that I(d)(x,y) is the original
image if d = 0.
Suppose that I(x,y) is an affine transformed image of the original image I(x,y).
I(d)(x,y) is the new image constructed from I(x,y) by (2.7). ˜I(d)(x,y) is the new image
constructed from I(x,y) by (2.7). Then ˜I(d)(x,y) is the same version affine transformed image
of I(d)(x,y). For example, we add 0.1 to the inside area of Figure 1(a); the obtained new image
is shown in Figure 3(a). The gray value of the inside area of Figure 2(a) is also added 0.1;
Mathematical Problems in Engineering 5
(a) (b)
(c) (d)
Figure 1: (a) Chinese character “Fu”. (b) Put the GC of Figure 1(a) on the image. (c) The inside area ofFigure 1(a). (d) The outside area of Figure 1(a).
the obtained image is shown in Figure 3(b). We observed that Figure 3(b) is the same affine
transform version of Figure 3(a) as that of Figure 2(a) to Figure 1(a).Some well-developed methods can be applied to the derived new images. More affine
invariant features can be constructed. As aforementioned, only a few low-order moment
invariants can be used for object classification. We can apply AMIs method to the derived
new images. More low-order moment invariants can be extracted. We construct new affine
moment invariants in the next section.
3. Cutting Affine Moment Invariants
By applying various region-based methods to the derived new image, some affine invariant
features can be extracted. As aforementioned, AMIs method is region-based method with low
computational demands. We apply AMIs to the constructed new image.
Geometric moment m(d)pq of the new image I(d)(x,y) is defined as
m(d)pq =
∫xpyqI(d)
(x,y
)dx dy, (3.1)
6 Mathematical Problems in Engineering
(a) (b)
(c) (d)
Figure 2: (a) An affine transformation version of Figure 1(a). (b) Put the GC of Figure 2(a) on the image.(c) The inside area of Figure 2(a). (d) The outside area of Figure 2(a).
(a) (b)
Figure 3: (a) New image constructed from Figure 1(a). (b) New image constructed from Figure 2(a).
where p, q are nonnegative integers. μ(d)pq is the central moments:
μ(d)pq =
∫(x − x0)p
(y − y0
)qI(d)
(x,y
)dx dy, (3.2)
where x0, y0 are the coordinates of the centroid of the image.
Mathematical Problems in Engineering 7
For two points x1 = (x1,y1)T , x2 = (x2,y2)
T ∈ R2, we denote the cross product C12 as
C12 = x1y2 − x2y1. (3.3)
After an affine transform, the following equation holds:
C12 = JC12, (3.4)
where J denotes the Jacobian of affine transformation: J = det(A).For N points (N ≥ 2): Ui = (xi,yi)
T ∈ R2, i = 1, 2, . . .N, and non-negative integers
nkj(1 ≤ k < j ≤ N), we define RCAMI(d) of the form:
RCAMI(d) =∫ ∏
1≤k<j≤NC
nkj
kj
N∏i=1
I(d)(xi,yi
)dxi dyi. (3.5)
Denote w =∑
1≤k<j≤N nkj . We normalized RCAMI(d) as follows:
CAMI(d) =RCAMI(d)(μ(d)00
)w+N. (3.6)
Using similar argument with that of affine moment invariants (see [20], etc.), it can be shown
that CAMI(d) is affine invariant. We call these invariants as cutting affine moment invariants(CAMIs). If d = 0, these invariants are the same as moment invariants given in [20].
By expanding Ckj in (3.5), RCAMI(d) becomes a polynomial of moments given in
(3.2). Consequently, we can compute CAMIs from moments given in (3.2). Invariants can
be derived by replacing moments in AMIs with the moments given in (3.2). Here, we use the
well-developed theory for the AMIs as described in [19]. The following form invariants are
used in this paper:
F(d)1 =
(μ(d)20 μ
(d)02 −
(μ(d)11
)2)
(μ(d)00
)4,
F(d)2 =
((μ(d)30
)2(μ(d)03
)2 − 6μ(d)30 μ
(d)21 μ
(d)12 μ
(d)03 + 4μ
(d)30
(μ(d)12
)3+ 4
(μ(d)21
)3μ(d)03 − 3
(μ(d)21
)2(μ(d)12
)2)
(μ(d)00
)10
F(d)3 =
(μ(d)20
(μ(d)21 μ
(d)03 −
(μ(d)12
)2)
− μ(d)11
(μ(d)30 μ
(d)03 − μ
(d)21 μ
(d)12
)+ μ
(d)02
(μ(d)31 μ
(d)12 −
(μ(d)21
)2))
(μ(d)00
)7.
,
(3.7)
8 Mathematical Problems in Engineering
If we set d = 0, (3.7) results in AMIs used in [19]. By changing the constant d, different
invariants can be constructed. Consequently, more low-order moment invariants can be
extracted. We will show that the obtained CAMIs can be used in object classification.
Furthermore, we will combine the obtained CAMIs with the traditional AMIs. The obtained
features (we call them CCAMIs) are also used in object classification.
4. Experiments
In this section, we evaluate the proposed method in object classification tasks. We will show
that the derived affine invariants (CAMIs) can be used in object classification. Furthermore,
CAMIs can be combined with the original AMIs (we call the obtained affine invariants as
CCAMIs). We denote AMIs used in [19] as: f1, f2, f3 (d = 0 in (3.7)).In the first experiment, some binary images of Chinese characters are used. The CAMIs
used in this experiment are obtained by setting d equal to 10% of the maximum gray
value in the image. Hence, d is set to 0.1. These CAMIs are denoted as: F(0.1)1 , F
(0.1)2 , F
(0.1)3 .
Figure 4(a) shows the original six Chinese characters. Some of these characters are very
similar. Figure 4(b) shows the same set of characters deformed by affine transforms. The
values of invariants AMIs: f1, f2, f3 and CAMIs: F(0.1)1 , F
(0.1)2 , F
(0.1)3 are given in Table 1. It can
be seen clearly that CAMIs really are invariant under affine transform. Furthermore, CAMIs
are different with the original AMIs.
In the second experiment, we test the combined invariants (CCAMIs): f1, f2, f3,
F(0.1)1 , F
(0.1)2 , F
(0.1)3 . Two groups of Chinese characters, shown in Figures 5(a) and 5(b), are
chosen as databases. Each group include 40 Chinese characters with regular script font. The
images in Figure 5(a) have size of 128 × 128, and those in Figure 5(b) have size of 256 × 256.
Some characters in these databases have the same structures, but the number of stokes or the
shape of specific stokes may be a little different. The affine transformations are generated by
the following transformation matrix [4]:
T = l
(cos θ − sin θ
sin θ cos θ
)⎛⎜⎝a b
01
a
⎞⎟⎠, (4.1)
where a ∈ {1, 2}, b ∈ {−1.5, −1, −0.5, 0, −0.5, 1, 1.5}, θ ∈ {0◦, 72◦, 144◦, 216◦, 288◦}, and
l ∈ {0.8, 1.2}. l, θ denote the scaling, rotation transformation, respectively, and a, b denote
the skewing transformation.
Each character will be transformed 140 times as described above. With these affine
transformations and the database, 5600 tests run using the proposed method for each group.
In our experiments, the classification accuracy is defined as
η =γ
η× 100%, (4.2)
where γ denotes the number of correctly classified images, and η denotes the total number of
images applied in the test.
The AMIs, CAMIs, and the combined invariants CCAMIs are applied to databases in
Figures 5(a) and 5(b). Classification is performed by the method used in [19]. Table 2 shows
Mathematical Problems in Engineering 9
Da Quan Tai Tian Yao Fu
(a)
Test1
Test2
Test3
Test4
Test5Test6
(b)
Figure 4: (a) The original six model Chinese characters. (b) Deformed Chinese characters to be recognized.
Table 1: AMIs and CAMIs for some similar Chinese characters.
f1 · 104 f2 · 108 f3 · 106 F(0.1)1 · 104 F
(0.1)2 · 108 F
(0.1)3 · 106
Da 1033 −29586 −6040 889 −20096 −4579
Test1 1029 −29457 −6013 894 −20689 −4645
Quan 1132 −27041 −6406 980 −18884 −4915
Test2 1131 −26560 −6336 1000 −19504 −5051
Tai 1080 −50285 −8123 904 −31880 −5861
Test3 1081 −51263 −8201 915 −33214 −6019
Tian 939 −22855 −5612 805 −15556 −4220
Test4 939 −23053 −5627 816 −16216 −4341
Yao 1037 −29694 −6538 907 −21196 −5118
Test5 1031 −28759 −6415 915 −20846 −5153
Fu 850 −23926 −5847 726 −15462 −4351
Test6 846 −23686 −5795 726 −15324 −4354
the results. For the first group of Chinese characters, we observe that the performance of
CAMIs is a little better than that of AMIs, and the combined invariants CCAMIs have better
performance than the original AMIs and CAMIs. For the other group of Chinese characters,
we observe that the performance of the traditional AMIs is better than that of CAMIs, and
the combined invariants CCAMIs have also better performance than the original AMIs and
CAMIs. Hence, the original AMIs can be combined with CAMIs, more shape information
may be extracted.
5. Conclusions
In this paper, an approach is developed for the extraction of affine invariant features
by cutting image into areas: the inside area and the outside area. In order to establish
correspondence between areas of an image and those of its affine transformed version,
10 Mathematical Problems in Engineering
(a)
(b)
Figure 5: (a) First group of 40 characters. (b) Second group of 40 characters.
Table 2: Classification accuracies of AMIs, CAMIs, and CCAMIs in case of different affine transformations.
AMIs CAMIs CCAMIs
Group one 86.46% 87.62% 90.14%
Group two 95.55% 89.64% 96.13%
general contour (GC) of the object is employed. A nonnegative constant is added to the
gray value associated with every pixel of inside area. Consequently, new image is obtained,
and CAMIs are constructed from the new image. To test and evaluate the proposed method,
several experiments have been conducted. Experimental results show that CAMIs can be
used in object classification tasks.
Acknowledgments
This work was supported in part by the National Science Foundation under Grant
60973157, 61003209 in part by the Natural Science Foundation of Jiangsu Province Education
Mathematical Problems in Engineering 11
Department under Grant 08KJB520004. Ming Li acknowledges the 973 plan under the Project
no. 2011CB302802 and the NSFC under the Project Grant nos. 61070214 and 60873264.
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 827901, 30 pagesdoi:10.1155/2012/827901
Research ArticleHomotopy Perturbation Method and VariationalIteration Method for Harmonic Waves Propagationin Nonlinear Magneto-Thermoelasticitywith Rotation
Khaled A. Gepreel,1, 2 S. M. Abo-Dahab,2, 3 and T. A. Nofal2, 4
1 Math. Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt2 Math. Department, Faculty of Science, Taif University, Saudi Arabia3 Math. Department, Faculty of Science, SVU, Qena 83523, Egypt4 Math. Department, Faculty of Science, El-Minia University, Egypt
Correspondence should be addressed to Khaled A. Gepreel, [email protected] and
S. M. Abo-Dahab, [email protected]
Received 17 August 2011; Accepted 3 October 2011
Academic Editor: Cristian Toma
Copyright q 2012 Khaled A. Gepreel et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.
The homotopy perturbation method and variational iteration method are applied to obtain theapproximate solution of the harmonic waves propagation in a nonlinear magneto-thermoelasticityunder influence of rotation. The problem is solved in one-dimensional elastic half-space model sub-jected initially to a prescribed harmonic displacement and the temperature of the medium. The dis-placement and temperature are calculated for the methods with the variations of the magnetic fieldand the rotation. The results obtained are displayed graphically to show the influences of the newparameters and the difference between the methods’ technique. It is obvious that the homotopyperturbation method is more effective and powerful than the variational iteration method.
1. Introduction
In the past recent years, much attentions have been devoted to simulate some real-life pro-
blems which can be described by nonlinear coupled differential equations using reliable
and more efficient methods. The nonlinear coupled system of partial differential equations
often appear in the study of circled fuel reactor, high-temperature hydrodynamics, and
thermoelasticity problems, see [1–4]. From the analytical point of view, lots of work have been
done for such systems. With the rapid development of nanotechnology, there appears an ever-
increasing interest of scientists and researchers in this field of science. Nanomaterials, because
of their exceptional mechanical, physical, and chemical properties, have been the main topic
2 Mathematical Problems in Engineering
of research in many scientific publications. Wave generation in nonlinear thermoelasticity
problems has gained a considerable interest for its utilitarian aspects in understanding the
nature of interaction between the elastic and thermal fields as well as for its applications.
A lot of applications was paid on existence, uniqueness, and stability of the solution of the
problem, see [5–7].Much attention has been devoted to numerical methods, which do not require dis-
cretization of space-time variables or linearization of the nonlinear equations, among which
the variational iteration method (VIM) suggested in [8–20] shows its remarkable merits
over others. The method was successfully applied to a nonlinear one dimensional coupled
equations in thermoelasticity [21], revealing that the method is very convenient, efficient, and
accurate. The basic idea of variational iteration method is to construct a correction functional
with a general Lagrange multiplier which can be identified optimally via variational theory.
The homotopy perturbation method [8, 22] has the merits of simplicity and easy
execution. Unlike the traditional numerical methods, the HPM does not need discretization
and linearization. Most perturbation methods assume that a small parameter exists, but
most nonlinear problems have no small parameter at all. Many new methods have been
proposed to eliminate the small parameter. Recently, the applications of homotopy theory
among scientists appeared, and the homotopy theory becomes a powerful mathematical
tool, when it is successfully coupled with perturbation theory. Sweilam and Khader [1]investigated variational iteration method for one dimensional nonlinear thermoelasticity.
Applying He’s variational iteration method for solving differential-difference equation is
discussed by Yildirim [23]. Noor and Mohyud-Din [24], Mohyud-Din et al. [25–27] used
He’s polynomials or Pade approximants to solve solving higher-order nonlinear boundary
value problems, second-order singular problems, and nonlinear boundary value problems.
Mohyud-Din et al. [28] applied the modified variational iteration method for free-convective
boundary-layer equation using Pade approximation. Mohyud-Din and Noor [29, 30] used
Homotopy perturbation method for solving some new boundary value problems. Mohyud-
Din et al. [31] investigated some relatively new techniques for nonlinear problems.
In this paper, the homotopy perturbation method and variational iteration method
are used to solve the coupled harmonic waves nonlinear magneto-thermoelasticity equations
under influence of rotation. The Maple and Mathematica software packages are used to
obtain the approximate solutions in one-dimensional half-space. The displacement and tem-
perature which obtained have been calculated numerically and presented graphically.
2. Basic Idea of He’s Homotopy Perturbation Method
We illustrate the following nonlinear differential equation [8, 22]:
A(u) − f(r) = 0, r ∈ Λ, (2.1)
with the boundary conditions:
B
(u,
∂u
∂n
)= 0, r ∈ Γ, (2.2)
Mathematical Problems in Engineering 3
where A is a general differential operator, B is a boundary operator, f(r) is an analytic
function, and Γ is the boundary of the domain Λ. Generally speaking, the operator A can
be divided into two parts which are L and N, where L is linear operator but N is nonlinear
operator. Equation (2.1) can therefore be rewritten as follows:
L(u) +N(u) − f(r) = 0. (2.3)
By the homotopy technique, we construct a homotopy V (r, p): Λ × [0, 1] → R which satisfies
H(V, p
)=(1 − p
)[L(V ) − L(u0)] + p
[A(V ) − f(r)
]= 0, r ∈ Λ, (2.4)
or
H(V, p
)= L(V ) − L(u0) + pL(u0) + p
(N(V ) − f(r)
)= 0, r ∈ Λ, (2.5)
where p ∈ [0, 1] is an embedding parameter and u0 is an initial approximation of (2.1) which
satisfies the boundary conditions (2.2). Obviously, from (2.4) and (2.5) we have
H(V, 0) = L(V ) − L(u0) = 0,
H(V, 1) = A(V ) − f(r) = 0.(2.6)
The changing process of p from zero to unity is just that of V (r, p) from u0(r) to u(r). In
topology, this is called deformation, and L(V ) − L(u0) and A(V ) − f(r) are called homotopy.
According to the homotopy perturbation method, we can first use the embedding parameter
“p” as a small parameter and assume that the solution of (2.4) and (2.5) can be written as a
power series in “p” as follows:
V = V0 + pV1 + p2V2 + · · · . (2.7)
On setting p = 1 results in the approximate solution of (2.3), we have
u = limp→ 1
V = V0 + V1 + V2 + · · · . (2.8)
The combination of the perturbation method and the homotopy method is called the homo-
topy perturbation method, which has eliminated the limitations of the traditional per-
turbation methods. On the other hand, this technique can have full advantage of the
traditional perturbation techniques. The series (2.8) is convergent to most cases. However,
the convergent rate depends on the nonlinear operator A(V ).
(1) The second derivative of N(V ) with respect to V must be small because the
parameter may be relatively large, that is, p → 1.
(2) The norm of L−1(∂N/∂V ) must be smaller than one so that the series converges.
4 Mathematical Problems in Engineering
3. Application of Homotopy Perturbation Method on the NonlinearMagneto-Thermoelastic with Rotation Equations
In this section, we use the homotopy perturbation method to calculate the approximate
solutions of the following nonlinear magneto-thermoelastic with rotation equations:
(1 + σ1)utt + Ωut − uxx
(1 − σ2 + 2γux + 3δu2
x
)− β1θx − β2(θux)x = 0,(
θ − aux − 1
2bu2
x
)t
− [(1 + αux)θx]x = 0,
(3.1)
where γ, β1, β2, a, b, α are arbitrary constants, σ1, σ2 are the sensitive parts of the magnetic
field, and Ω is the rotation parameter, with the initial conditions
u(x, 0) = θ(x, 0) = A(1 − cos(x)), ut(x, 0) = θt(x, 0) = 0, (3.2)
where A is an arbitrary constant and the boundary conditions
u(0, t) = θ(0, t) = 0, ut(0, t) = θt(0, t) = 0. (3.3)
To investigate the traveling wave solution of (3.1), we first construct a homotopy perturbation
method as follows:
(1 − p
)[(1 + σ1)(Vtt − V0tt)] + p
[(1 + σ1)Vtt + ΩVt − Vxx
(1 − σ2 + 2γVx + 3δV 2
x
)−β1Θx − β2(ΘVx)x
]= 0,(
1 − p)(Θt −Θ0t) + p[Θt − aVxt − bVxVxt −Θxx − αVxxΘx − αVxΘxx] = 0,
(3.4)
where the initial approximations take the following form:
V0(x, t) = u0(x, t) = u(x, 0) = A(1 − cos(x)),
Θ0(x, t) = θ0(x, t) = θ(x, 0) = A(1 − cos(x)).(3.5)
According to the homotopy perturbation method, we can first use the embedding parameter
“p” as a small parameter and assume that the solution of (3.4) can be written as a power
series in “p” as the following:
V = V0(x, t) + pV1(x, t) + p2V2(x, t) + p3V3(x, t) + · · · ,
Θ(x, t) = Θ0(x, t) + pΘ1(x, t) + p2Θ2(x, t) + p3Θ3(x, t) + · · · ,(3.6)
where Vj and Θj , j = 1, 2, 3, . . . are functions to be determined.
Mathematical Problems in Engineering 5
Substituting from (3.6) into (3.4) and arranging the coefficients of “p” powers, we have
(1 + σ1)V0,tt +(σ1V1,tt − 3δV 2
0,xV0,xx − β2V0,xxΘ0 − 2γV0,xV0,xx + σ2V0,xx − β1Θ0,x
−V0,xx − β2V0,xΘ0,x + ΩV0,t + V1,tt
)p
+(σ1V2,tt − 2γV1,xV0,xx + V2,tt − 2γV0,xV1,xx − 6δV0,xV1,xV0,xx − V1,xx − β2V1,xxΘ0
−β1Θ1,x − β2V0,xΘ1,x − β2V1,xΘ0,x − β2V0,xxΘ1 + ΩV1,t − 3δV 20,xV1,xx + σ2V1,xx
)p2
+(V3,tt − β2V2,xΘ0,x − β2V2,xxΘ0 − 2γV2,xV0,xx − 2γV1,xV1,xx + σ1V3,tt − β1Θ2,x
− 6δV0,xV2,xV0,xx − 3δV 21,xV0,xx + σ2V2,xx + ΩV2,t − V2,xx − 3δV 2
0,xV2,xx − 2γV0,xV2,xx
−β2V1,xxΘ1 − β2V0,xxΘ2 − β2V0,xΘ2,x − 6δV0,xV1,xV1,xx − β2V1,xΘ1,x
)p3 + · · · = 0,
Θ0,t + (−aV0,xt − αV0,xΘ0,xx + Θ1,t − αV0,xxΘ0,x − 2bV0,xV0,xt −Θ0,xx)p
+ (Θ2,t −Θ1,xx − aV1,xt − αV1,xxΘ0,x − 2bV1,xV0,xt − αV0,xΘ1,xx − αV1,xΘ0,xx
−2bV0,xV1,xt − αV0,xxΘ1,x)p2
× (−αV1,xxΘ1,x − αV2,xxΘ0,x − 2bV2,xV0,xt − αV0,xxΘ2,x − 2bV1,xV1,xt − aV2,xt
+Θ3,t − 2bV0,xV2,xt − αV0,xΘ2,xx − αV1,xΘ1,xx − αV2,xΘ0,xx −Θ2,xx)p3 + · · · = 0.
(3.7)
In order to obtain the unknowns of Vj and Θj , (j = 1, 2, 3, . . .), we construct and solve the
following system considering the initial conditions (3.2):
σ1V1,tt − 3δV 20,xV0,xx − β2V0,xxΘ0 − 2γV0,xV0,xx + σ2V0,xx − β1Θ0,x − V0,xx
− β2V0,xΘ0,x + ΩV0,t + V1,tt = 0,
σ1V2,tt − 2γV1,xV0,xx + V2,tt − 2γV0,xV1,xx − 6δV0,xV1,xV0,xx − V1,xx − β2V1,xxΘ0
− β1Θ1,x − β2V0,xΘ1,x − β2V1,xΘ0,x − β2V0,xxΘ1 + ΩV1,t − 3δV 20,xV1,xx + σ2V1,xx = 0,
V3,tt − β2V2,xΘ0,x − β2V2,xxΘ0 − 2γV2,xV0,xx − 2γV1,xV1,xx + σ1V3,tt − β1Θ2,x
− 6δV0,xV2,xV0,xx − 3δV 21,xV0,xx + σ2V2,xx + ΩV2,t − V2,xx − 3δV 2
0,xV2,xx − 2γV0,xV2,xx
− β2V1,xxΘ1 − β2V0,xxΘ2 − β2V0,xΘ2,x − 6δV0,xV1,xV1,xx − β2V1,xΘ1,x = 0,
− aV0,xt − αV0,xΘ0,xx + Θ1,t − αV0,xxΘ0,x − 2bV0,xV0,xt −Θ0,xx = 0,
Θ2,t −Θ1,xx − aV1,xt − αV1,xxΘ0,x − 2bV1,xV0,xt − αV0,xΘ1,xx − αV1,xΘ0,xx
− 2bV0,xV1,xt − αV0,xxΘ1,x = 0,
− αV1,xxΘ1,x − αV2,xxΘ0,x − 2bV2,xV0,xt − αV0,xxΘ2,x − 2bV1,xV1,xt − aV2,xt + Θ3,t
− 2bV0,xV2,xt − αV0,xΘ2,xx − αV1,xΘ1,xx − αV2,xΘ0,xx −Θ2,xx = 0.
(3.8)
6 Mathematical Problems in Engineering
Consequently, we deduce after some calculations the following results:
u = limp→ 1
V = V0 + V1 + V2 + · · · ,
θ = limp→ 1
Θ = Θ0 + Θ1 + Θ2 + · · · ,(3.9)
where
V0 = A(1 − cosx),
V1 =t2
2(4 + 4σ1)
[4γA2 sin 2x + 3δA3 cosx − 3δA3 cos 3x + 4β2A
2 cosx
−4β2A2 cos 2x4σ2A cosx + 4β1A sinx + 4A cosx
],
V2 =t4
32(σ1 + 1)2
{(−8
3β2
2A3 − 8
3β2A
2 − 8
3γ2A3 − 4δA3 + 4A3σ2δ +
8
3σ2A − 9
2δ2A5 − 4
3Aσ2
2
−4
3A +
8
3A2σ2β2 − 4β2A
4δ
)cosx
−(
4
3β1A + δA3β1 +
4
3γA3β2 − 4
3Aσ2β1 +
4
3β2A
2β1
)sinx
+(−8γA2 + 8A2σ2γ − 8β2A
3γ +4
3β2A
2β1 − 16δA4γ
)sin 2x
+(
12γA3β2 + 3δA3β1
)sin 3x + 20δA4γ sin 4x
+(−20
3A2σ2β2 +
20
3β2
2A3 +
8
3γA2β1 +
20
3β2A
2 + 6δA4β2
)cos 2x
+(
63
4δ2A5 − 4β2
2A3 + 12δA3 + 12β2A
4δ + 8γ2A3 − 12A3σ2δ
)cos 3x
−14β2A4δ cos 4x − 45
4δ2A5 cos 5x
}
+t3
32(σ1 + 1)2
{(−16
3ΩAβ1 − 16
3β1Aσ1 − 8
3β2ασ1A
3 − 16
3β1A − 8
3β2A
3α
)sinx
− 16
3ΩA2γ sin 2x +
(8β2A
3ασ1 + 8β2A3α)
sin 3x
+(
16
3ΩAσ2 − 4ΩA3δ − 16
3ΩA2β2 − 16
3ΩA
)cos(x)
+(
16
3β2A
2 +32
3β1A
2ασ1 +16
3ΩA2β2 +
16
3β2A
2σ1 +32
3β1A
2α
)cos 2x
+ 4ΩA3δ cos 3x
},
Mathematical Problems in Engineering 7
Θ0 = A(1 − cos(x)),
Θ1 = A2tα sin(2x) +A cos(x)t,
Θ2 =A
2(1 + σ1)
(t3
3
(12αA2γcos3(x) + 2αAcos2(x)β1 + 36αA3δ sin(x)cos3(x)
+ 12αA2β2 sin(x)cos2(x) + 2αAσ2 sin(x) cos(x) − 24αA3δ sin(x) cos(x)
− 2αA2β2 sin(x) cos(x) − 2αA sin(x) cos(x) − αAβ1 − 10αγA2 cos(x)
−4αβ2A2 sin(x)
)
+t2
2
(− 2 cos(x) + 18aδA2 sin(x)cos2(x) + 16bA2γ sin(x)cos2(x)
+ 8aβ2A sin(x) cos(x) + 2aσ2 sin(x) + 2aβ1 cos(x) + 24α2A2σ1cos3(x)
− 36bA3δcos4(x) − 16bA2β2cos3(x) + 48bA3δcos2(x) + 4bA2β2cos2(x)
+ 8cos2(x)aγA − 4 cos (x)2bAσ2 − 20α2A2 cos(x) + 24 cos (x)3α2A2
+ 4bAcos2(x) − 4bA − 6aδA2 sin(x) − 20αA sin(x) cos(x)
− 2aβ2A sin(x) − 4bA2β2 − 12bA3δ + 4bAσ2 − 8bA2 sin(x)γ
+ 16bA2β2 cos(x) − 20α2A2 cos(x) σ1 − 4aγA − 2 cos(x)σ1
−2a sin(x) + 4bAβ1 sin(x) cos(x) − 20αA sin(x) cos(x)σ1
)).
(3.10)
Now we make calculations for the results obtained by the homotopy perturbation method
using the Maple software package with the following arbitrary constants:
a = 0.5, A = 0.001, b = 0.5, α = 1, β1 = β2 = 0.05, γ = 1, δ = 0.8. (3.11)
The results obtained in (3.9) are displayed graphically in Figures 1–4.
3.1. Special Cases
(1) If we take into our consideration the first iteration (i.e., u = V0+V1 and θ = Θ0+Θ1).See Figures 5, 6, 7, and 8.
(2) If the magnetic field and rotation are neglected, the components of the displacement
u and temperature θ take the following forms. See Figures 9 and 10.
8 Mathematical Problems in Engineering
−8000
−4000
0
4000
8000
u
xt
02
46
810
1214 0
1020
3040
50
(a)
xt
02
46
810
1214 0
1020
3040
50
−60
−20
20
60
(b)
Figure 1: Variations of the displacement u and temperature θ for various values of the axis x and time twhen Ω = 0.1, σ1 = 0.2, σ2 = 0.1.
u
x
02
46
810
1214 0
12
34
5
Ω
0
0.0005
0.001
0.0015
0.002
(a)
x
02
46
810
1214 0
12
34
5
Ω
0.0002
0.0006
0.0014
0.0018
0.001
(b)
Figure 2: Variations of the displacement u and temperature θ for various values of the axis x and rotationΩ when t = 0.1, σ1 = 0.2, σ2 = 0.1.
1
u
x
02
46
810
1214 0
24
68
100
0.0005
0.001
0.0015
0.002
(a)
x
02
46
810
1214 1
02
46
810
0.0002
0.0006
0.0014
0.0018
0.001
(b)
Figure 3: Variations of the displacement u and temperature θ for various values of the axis x and magneticfield σ1 when t = 0.1, Ω = 0.1, σ2 = 0.1.
Mathematical Problems in Engineering 9
u
x
02
46
810
1214 0
24
68
10
0
0.001
0.002
2
(a)
x
02
46
810
1214 0
24
68
10
2
0.0002
0.0006
0.0014
0.0018
0.001
(b)
Figure 4: Variations of the displacement u and temperature θ for various values of the axis x and magneticfield σ2 when t = 0.1, Ω = 0.1, σ1 = 0.1.
u
xt
02
46
810
1214 0
1020
3040
50
−0.8
−0.4
0
0.4
0.8
(a)
xt
02
46
810
1214 0
1020
3040
50
−0.4
0
0.4
0.2
−0.2
(b)
Figure 5: Variations of the displacement u and temperature θ for various values of the axis x and time twhen Ω = 0.1, σ1 = 0.2, σ2 = 0.1.
u
x
02
46
810
1214 0
12
34
5
Ω
0
0.0005
0.001
0.0015
0.002
(a)
01
23
45
Ω
0.0002
0.0006
0.0014
0.0018
0.001
x
02
46
810
1214
(b)
Figure 6: Variations of the displacement u and temperature θ for various values of the axis x and rotationΩ when t = 0.1, σ1 = 0.2, σ2 = 0.1.
10 Mathematical Problems in Engineering
1
02
46
810
u
x
02
46
810
1214
0
0.0005
0.001
0.0015
0.002
(a)
0.0002
0.0006
0.0014
0.0018
0.001
x
02
46
810
1214
1
02
46
810
(b)
Figure 7: Variations of the displacement u and temperature θ for various values of the axis x and magneticfield σ1 when t = 0.1, Ω = 0.1, σ2 = 0.1.
x
02
46
810
1214 0
24
68
10
2
u
0
0.0005
0.001
0.0015
0.002
(a)
x
02
46
810
1214 0
24
68
10
2
0.0002
0.0006
0.0014
0.0018
0.001
(b)
Figure 8: Variations of the displacement u and temperature θ for various values of the axis x and magneticfield σ2 when t = 0.1, Ω = 0.1, σ1 = 0.1.
4. Basic Idea of Variational Iteration Method
Consider the following nonhomogeneous nonlinear system of partial differential equations:
L1u(x, t) +N1(u(x, t), θ(x, t)) = f(x, t), (4.1)
L2θ(x, t) +N2(u(x, t), θ(x, t)) = g(x, t), (4.2)
Mathematical Problems in Engineering 11
u
x t
02
46
810
1214 0
1020
3040
50
−1
−0.5
0
0.5
1
(a)
xt
02
46
810
1214 0
1020
3040
50
−0.04
−0.02
0
0.02
0.04
(b)
Figure 9: Variations of the displacement u and temperature θ for various values of the axis x and time t(u = V0 + V1 + V2 and θ = Θ0 + Θ1 + Θ2) when Ω = σ1 = σ2 = 0.
x t
02
46
810
1214 0
1020
3040
50
u
−0.04
−0.02
0
0.02
0.04
(a)
x t
02
46
810
1214 0
0
10
1
2030
4050
−1
(b)
Figure 10: Variations of the displacement u and temperature Θ for various values of the axis x and time t(u = V0 + V1 and θ = Θ0 + Θ1) when Ω = σ1 = σ2 = 0.
where L1, L2 are linear differential operators with respect to time, N1, N2 are nonlinear
operators, and f(x, t), g(x, t) are given functions.
According to the variational iteration method, we can construct correct functionals as
follows:
un+1(x, t) = un(x, t) +∫ t
0
λ1(τ)[L1un(x, τ) +N1
(un(x, τ), θn(x, τ)
)− f(x, τ)
]dτ, (4.3)
θn+1(x, t) = θn(x, t) +∫ t
0
λ2(τ)[L2θn(x, τ) +N2
(un(x, τ), θn(x, τ)
)− g(x, τ)
]dτ, (4.4)
12 Mathematical Problems in Engineering
where λ1 and λ2 are general Lagrange multipliers, which can be identified optimally via
variational theory [8–20]. The second term on the right-hand side in (4.3) and (4.4) is called
the corrections, and the subscript n denotes the nth order approximation, un and θn are
restricted variations. We can assume that the above correctional functionals are stationary
(i.e., δun+1 = 0 and δθn+1 = 0), then the Lagrange multipliers can be identified. Now we
can start with the given initial approximation and by the previous iteration formulas we can
obtain the approximate solutions.
5. Application of the Variational Iteration Method on the NonlinearMagneto-Thermoelastic with Rotation Equations
According to the variational iteration method and after some manipulation of (4.3) and (4.4),the correct functionals are as follows:
un+1(x, t) = un(x, t) +∫ t
0
λ1(τ)[(1 + σ1)un,tt(x, τ) + Ωun,t(x, τ)
− un,xx
(1 − σ2 + 2γun,x(x, τ) + 3δu2
n,x(x, τ))
−β1θn,x(x, τ) − β2
(θn(x, τ)un,x(x, τ)
)x
]dτ,
θn+1(x, t) = θn(x, t) +∫ t
0
λ2(τ)[θn,t(x, τ) − aun,xt(x, τ) − bun,x(x, τ)un,xt(x, τ)
−θn,xx(x, τ) − αun,xx(x, τ)θn,x(x, τ) − αun,x(x, τ)θn,xx(x, τ)]dτ,
(5.1)
where un and θn are considered as a restricted variation, that is, δun+1 = 0 and δθn+1 = 0.
Consequently, the general Lagrange multipliers λ1 and λ2 take the following form:
λ1(τ) =τ − t
1 + σ1, λ2(τ) = −1. (5.2)
By the substitution of the identified Lagrange multipliers (5.2) into (5.1), we have the
following iteration relations:
un+1(x, t) = un(x, t) +∫ t
0
τ − t
1 + σ1
[(1 + σ1)un,tt(x, τ) + Ωun,t(x, τ)
− un,xx
(1 − σ2 + 2γun,x(x, τ) + 3δu2
n,x(x, τ))
−β1θn,x(x, τ) − β2(θn(x, τ)un,x(x, τ))x]dτ,
Mathematical Problems in Engineering 13
θn+1(x, t) = θn(x, t) −∫ t
0
[θn(x, τ) − aun,xt(x, τ) − bun,x(x, τ)un,xt(x, τ) − θn,xx(x, τ)
−αun,xx(x, τ)θn,x(x, τ) − αun,x(x, τ)θn,xx(x, τ)]dτ, n ≥ 0.
(5.3)
With help of Maple or Mathematica, we get the following results:
u0 = θ0 = A(1 − cosx),
u1 = − t2
2(1 + σ1)
[−β1 sinx − 2Aγ cosx sin(x) + 2β2Acos2x − cosx − 3δA2 cosx
−β2A + σ2 cosx − β2A cosx + 3δA2cos3x]+A(1 − cosx),
θ1 = A[cosx + 2αA cosx sinx]t +A(1 − cosx),
u2 =
{− Acos6(x)
6720(1 + σ1)4
(−19440A5δ2β1γ + 5760A5δβ3
2 − 247860A7δ3β2 − 19440A6δ2β22
−19440A5δ2β2 − 17280δβ2γ2A5 + 19440A5δ2β2σ2
)+
[− Acos2(x)
6720(1 + σ1)4
(−6480A5δβ2
2γ + 2025A4δβ1β22 + 270A3δβ1σ2β2 + 3240A4δ2β1σ2
− 38880A6δ2β2γ − 8505A6δ3β1 − 135A2δβ1σ22 − 6480A4γβ2δ
− 135A2δβ1 + 6480A4δβ2γσ2 − 3240A5δ2β1β2 + 270A2δβ1σ2
− 270A3δβ1β2 − 3240A4δ2β1 − 2160A4δβ1γ2 + 45A2δβ3
1
)− Acos4(x)
6720(1 + σ1)4
(7200A4γβ2δ + 118800A6δ2β2γ − 4050A4δ2β1σ2 + 4050A4δ2β1
+ 4050A5δ2β1β2 + 3600A4δβ1γ2 − 3600A4δβ1β
22
+ 7200A5δβ22γ + 30375A6δ3β1 − 7200A4δβ2γσ2
)− Acos3(x)
6720(1 + σ1)4
(17280A5δ2β1β2 − 1440A4γβ2δ − 93960A7δ3γ + 16560A5δβ2
2γ
+ 1440A4δβ2γσ2 − 23760A6δ2β2γ + 1440A4δβ1β22 − 720A3γδ
+ 1440A3δβ1β2 + 23760A5δ2γσ2 − 720A3δγσ22 − 5760A5δγ3
− 23760A5γδ2 − 1440A3δβ1σ2β2 + 720A3δβ21γ + 1440A3γδσ2
)− Acos5(x)
6720(1 + σ1)4
(−19440A5δ2β1β2 + 189540A7δ3γ + 19440A6δ2β2γ − 17280A5δβ2
2γ
+ 19440A5γδ2 + 5760A5δγ3 − 19440A5δ2γσ2
)
14 Mathematical Problems in Engineering
− A cos(x)
6720(1 + σ1)4
(− 1080A3γδσ2 + 540A3γδ + 540A3δγσ2
2 − 180A3δβ21γ
− 1080A4δβ2γσ2 − 720A4δβ1β22 − 2160A5δ2β1β2 + 1080A4γβ2δ
+ 720A3δβ1σ2β2 + 6480A6δ2β2γ − 2340A5δβ22γ − 6480A5δ2γσ2
− 720A3δβ1β2 + 14580A7δ3γ + 1440A5δγ3 + 6480A5γδ2)
e899
+243A8δ3γ
141(1 + σ1)4cos7(x) − Acos6(x)
6720(1 + σ1)4
(−90720A6δ2β2γ − 25515A6δ3β1
)
− A
6720(1 + σ1)4
(720A5δβ2
2γ + 45A2δβ1σ22 + 405A6δ3β1 + 720A4γβ2δ
− 720A4δβ2γσ2 + 90A3δβ1β2 + 45A4δβ1β22 + 2160A6δ2β2γ
+ 270A5δ2β1β2 + 45A2δβ1 − 90A3δβ1σ2β2 + 270A4δ2β1
− 270A4δ2β1σ2 − 90A2δβ1σ2 + 180A4δβ1γ2)]
sin(x)
− Acos5(x)
6720(1 + σ1)4
(204120A8δ4 − 7200A4δβ1γβ2 + 52245A6δ3 + 2025δ2A4
− 52245A6δ3σ2 + 3600A4γ2δ − 80055A6δ2β22 − 3600A5δβ3
2
+ 3600δβ2γ2A5 + 52245A7δ3β2 + 82080δ2γ2A6 + 2025A4δ2σ2
2
+ 3600A4δβ22σ2 − 4050A5δ2β2σ2 − 4050A4δ2σ2 − 2025A4δ2β2
1
+ 4050A5δ2β2 − 3600A4δβ22 − 3600A4δγ2σ2
)− 243A8δ3β2cos8(x)
141(1 + σ1)4
− Acos4(x)
6720(1 + σ1)4
(25920δβ2γ
2A5 + 1440A4δβ22 + 166860A7δ3β2 + 33480A5δ2β2
− 1440A4δβ22σ2 + 1440A3γβ1δ − 720A3δβ2
1β2 + 33480A6δ2β22
− 1440A3δβ1γσ2 − 7920A5δβ32 + 1440A4δβ1γβ2 + 27000A5δ2β1γ
− 33480A5δ2β2σ2 − 1440A3δβ2σ2 + 720A3δβ2σ22 + 720δβ2A
3)
− Acos3(x)
6720(1 + σ1)4
(2025A4δ2β2
1 + 6480A4δ2σ2 + 5040A4δγ2σ2 + 135A3δβ21β2
− 3240δ2A4 + 7920A4δβ1γβ2 − 70470A8δ4 − 32805A6δ3
− 4905A4δβ22σ2 + 135A2δβ2
1 + 270A3δβ2σ2 + 6480A5δ2β2σ2
− 135A3δβ2σ22 − 135A2δσ2
2 + 135A2σ2δ + 45A2δσ32 − 45A2δ
− 32805A7δ3β2 − 45900δ2γ2A6 + 32805A6δ3σ2 + 37800A6δ2β22
Mathematical Problems in Engineering 15
− 135δβ2A3+4995A5δβ3
2−135A2δβ21σ2−6480A5δ2β2−5040δβ2γ
2A5
+ 4905A4δβ22 − 5040A4γ2δ − 3240A4δ2σ2
2
)− 6561A9δ4cos9(x)
4481(1 + σ1)4
− Acos2(x)
6720(1 + σ1)4
(540A3δβ2
1β2 + 1980A5δβ32 − 9180A5δ2β1γ − 900δβ2A
3
+ 1800A4δβ22σ2 − 15120A5δ2β2 − 10080δβ2γ
2A5 − 15120A6δ2β22
+ 1440A3δβ1γσ2 − 1440A3γβ1δ − 37260A7δ3β2 − 1440A4δβ1γβ2
+ 15120A5δ2β2σ2 + 1800A3δβ2σ2 − 900A3δβ2σ22 − 1800A4δβ2
2
)− A
6720(1 + σ1)4
(180A4δβ1γβ2 + 360A4δβ2
2 + 180δβ2A3 − 360A4δβ2
2σ2
+ 1080A6δ2β22 + 1080A5δ2β2 + 1620A7δ3β2 − 180A3δβ1γσ2
+ 180A3γβ1δ + 180A5δβ32 + 540A5δ2β1γ − 1080A5δ2β2σ2
− 360A3δβ2σ2 + 180A3δβ2σ22 + 720δβ2γ
2A5)
− Acos7(x)
6720(1 + σ1)4
(45360A6δ2β2
2 + 25515A6δ3σ2 − 25515A6δ3 − 45360δ2γ2A6
− 240570A8δ4 − 25515A7δ3β2
)− A cos(x)
6720(1 + σ1)4
(− 270A4δ2β2
1−2430A4δ2σ2−1620A4δγ2σ2−90A3δβ21β2+1215δ2A4
− 1440A4δβ1γβ2 + 8505A8δ4 + 6075A6δ3 + 1305A4δβ22σ2
− 90A2δβ21 − 270A3δβ2σ2 − 2430A5δ2β2σ2 + 135A3δβ2σ
22
+ 135A2δσ22 − 135A2σ2δ − 45A2δσ3
2 + 45A2δ + 6075A7δ3β2
+ 8100δ2γ2A6 − 6075A6δ3σ2 − 3105A6δ2β22 + 135δβ2A
3
− 1395A5δβ32 + 90A2δβ2
1σ2 + 2430A5δ2β2 + 1620δβ2γ2A5
− 1305A4δβ22 + 1620A4γ2δ + 1215A4δ2σ2
2
)}t8
+
{− Acos6(x)
6720(1 + σ1)4
(72576A5δ2β2σ1 + 72576A5δ2β2
)
+
[− Acos5(x)
6720(1 + σ1)4
(−99792A5γδ2σ1 − 99792A5γδ2
)
− Acos2(x)
6720(1 + σ1)4
(1008A3δβ1β2 + 1344A2γ2β1 − 1344A2γβ2σ2σ1 − 1008A2δβ1σ2
16 Mathematical Problems in Engineering
+ 1344A3γβ22 + 1344A2γβ2σ1 + 1344A3γβ2
2σ1 + 1344A2γβ2
− 1008A2δβ1σ1σ2 + 1344A2γ2β1σ1 + 12096A4δ2β1
+ 1008A3δβ1σ1β2 + 12096A4δ2β1σ1 − 1344A2γβ2σ2
+ 40320A4γβ2δσ1 + 40320A4γβ2δ + 1008A2δβ1
+ 1008A2δβ1σ1
)− A
6720(1 + σ1)4
(− 448A2γβ2 + 448A2γβ2σ2σ1 − 448A2γβ2σ1 − 448A3γβ2
2σ1
+ 448A2γβ2σ2 + 336A2δβ1σ2 − 1008A4δ2β1 − 336A3δβ1β2
− 336A2δβ1 − 336A3δβ1σ1β2 + 336A2δβ1σ1σ2 − 336A2δβ1σ1
− 4032A4γβ2δ − 224A2γ2β1 − 448A3γβ22 − 4032A4γβ2δσ1
− 224A2γ2β1σ1 − 1008A4δ2β1σ1
)− A cos(x)
6720(1 + σ1)4
(2688A3δβ1β2 + 1680A3γβ2
2σ1 − 1792A3γ3 + 2688A3δβ1σ1β2
+ 6720A3γδσ2σ1 − 6720A3γδσ1 + 6720A3γδσ2 − 31248A5γδ2
− 224A2γβ2σ1 + 224A2γβ2σ2σ1 + 1680A3γβ22 + 224A2γβ2σ2
− 1792A3γ3σ1 − 6720A3γδ − 224A2γβ2 − 112Aγ
+ 112Aγβ21 + 112Aγβ2
1σ1 − 112Aγσ22σ1 + 224Aγσ2σ1
− 112Aγσ22 + 224Aγσ2 − 112Aγσ1 − 6720A4γβ2δσ1
− 31248A5γδ2σ1 − 6720A4γβ2δ)
− Acos3(x)
6720/(1 + σ1)4
(− 3584A3γβ2
2σ1 + 9408A4γβ2δσ1 + 118944A5γδ2 + 9408A4γβ2δ
+ 3584A3γ3σ1 + 9408A3γδ + 9408A3γδσ1 + 118944A5γδ2σ1
− 3584A3γβ22 − 5376A3δβ1β2 + 3584A3γ3 − 9408A3γδσ2
− 5376A3δβ1σ1β2 − 9408A3γδσ2σ1
)− Acos4(x)
6720(1 + σ1)4
(−15120A4δ2β1 − 47040A4γβ2δ − 15120A4δ2β1σ1
− 47040A4γβ2δσ1
)]sin(x)
− Acos7(x)
6720(1 + σ1)4
(95256A6δ3σ1 + 95256A6δ3
)
Mathematical Problems in Engineering 17
− Acos3(x)
6720(1 + σ1)4
(− 24192A4δ2σ2 − 24192A4δ2σ1σ2 + 24192δ2A4 − 1344A2γβ1β2σ1
+ 122472A6δ3 + 1008A3β2δσ1 − 1344A2γβ1β2 − 18312A4δβ22σ1
+ 1344A3γ2β2σ1 − 504A2δβ21 + 1344A2γ2σ1 − 1008A3δβ2σ2
+ 24192A5δ2β2σ1 + 504A2δσ1 + 24192A4δ2σ1 + 54024A4γ2δσ1
− 1344A2γ2σ2 + 504A2δσ22 − 1008A2σ2δ + 1008A2σ2δσ1
+ 504A2δ − 1008A3δβ2σ1σ2 − 1344A2γ2σ2σ1 − 122472A6δ3σ1
+ 1008δβ2A3 + 1344A2γ2 + 504A2δσ1σ
22 − 504A2δβ2
1σ1
+ 24192A5δ2β2 − 18312A4δβ22 + 45024A4γ2δ + 1344A3γ2β2
)− Acos5(x)
6720(1 + σ1)4
(− 195048A6δ3 + 15120A4δ2σ1σ2 + 15120A4δ2σ2 − 195048A6δ3σ1
− 15120A5δ2β2 + 13440A4δβ22 − 33600A4γ2δσ1
− 33600A4γ2δ − 15120δ2A4 − 15120A4δ2σ1 − 15120A5δ2β2σ1
+ 13440A4δβ22σ1
)− A
6720(1 + σ1)4
(− 112Aγβ1σ1 − 1344δβ2A
3 − 112A2γβ1β2 − 1008A3γβ1δσ1
+ 112Aγβ1σ2σ1 + 1344A3δβ2σ1σ2 − 896A3γ2β2 − 112A2γβ1β2σ1
− 1344A3β2δσ1 − 1344A4δβ22σ1 − 1008A3γβ1δ − 112Aγβ1
+ 112Aγβ1σ2 − 896A3γ2β2σ1 − 4032A5δ2β2 − 1344A4δβ22
+ 1344A3δβ2σ2 − 4032A5δ2β2σ1
)− Acos4(x)
6720(1 + σ1)4
(− 7168A3γ2β2 − 5376A4δβ2
2σ1 − 9408A3γβ1δ − 9408A3γβ1δσ1
− 5376A3β2δσ1 − 7168A3γ2β2σ1 − 5376δβ2A3 − 124992A5δ2β2
− 5376A4δβ22 + 5376A3δβ2σ2 + 5376A3δβ2σ1σ2−124992A5δ2β2σ1
)− A cos(x)
6720(1 + σ1)4
(9072A4δ2σ2 + 9072A4δ2σ1σ2 − 9072δ2A4 + 896A2γβ1β2σ1
− 22680A6δ3 − 1008A3β2δσ1 + 896A2γβ1β2 + 4872A4δβ22σ1
− 1120A3γ2β2σ1 + 336A2δβ21 − 1120A2γ2σ1 + 1008A3δβ2σ2
− 9072A5δ2β2σ1 − 504A2δσ1 − 9072A4δ2σ1 − 13440A4γ2δσ1
18 Mathematical Problems in Engineering
+ 1120A2γ2σ2 − 504A2δσ22 + 1008A2σ2δ + 1008A2σ2δσ1
− 504A2δ + 1008A3δβ2σ1σ2 + 1120A2γ2σ2σ1 − 22680A6δ3σ1
− 1008δβ2A3 − 1120A2γ2 − 504A2δσ1σ
22 + 336A2δβ2
1σ1
− 9072A5δ2β2 + 4872A4δβ22 − 13440A4γ2δ − 1120A3γ2β2
)− Acos2(x)
6720(1 + σ1)4
(− 224Aγβ1σ2σ1 + 224Aγβ1σ1 + 224A2γβ1β2 + 56448A5δ2β2
+ 6720A4δβ22 + 9072A3γβ1δσ1 + 224Aγβ1 + 6720δβ2A
3
+ 9072A3γβ1δ + 224A2γβ1β2σ1 + 7168A3γ2β2 + 56448A5δ2β2σ1
+ 6720A3β2δσ1 + 7168A3γ2β2σ1 − 6720A3δβ2σ1σ2
− 224Aγβ1σ2 + 6720A4δβ22σ1 − 6720A3δβ2σ2
)}t6
+
{− Acos3(x)
6720(1 + σ1)4
(− 1008A2β2β1ασ
21 − 4032A2β2
2σ1 − 1008A2β2β1α − 2016A2β2β1ασ1
− 2016A2β22 − 2016A2β2
2σ21
)− A cos(x)
6720(1 + σ1)4
(1344A2β2
2σ21 + 672A2αβ2β1 + 1344A2β2
2 + 2688A2β22σ1
+ 1344A2β2β1ασ1 + 672A2β2β1ασ21
)− A
6720(1 + σ1)4
(− 504δβ2A
3 − 336A2β22σ1 − 168Aβ2σ
21 + 336Aβ2σ2σ1
+ 168Aβ2σ2 − 1008A3β2δσ1 − 168β2A − 504A3β2δσ21
− 168A2β22 − 672A3β2γασ
21 − 168A2β2
2σ21 − 336Aβ2σ1
+ 168Aβ2σ2σ21 − 672A3β2γα − 1344A3β2γασ1
)− Acos2(x)
6720(1 + σ1)4
(5544A3β2δσ
21 + 5376A3β2γασ
21 − 672Aβ2σ2σ1 − 336Aβ2σ2
+ 336β2A + 672A2β22σ1 + 336A2β2
2 + 5376A3β2γα
+ 672Aβ2σ1 − 336Aβ2σ2σ21 + 336Aβ2σ
21 + 10752A3β2γασ1
+ 336A2β22σ
21 + 5544δβ2A
3 + 11088A3β2δσ1
)+
[− A cos(x)
6720(1 + σ1)4
(672β2Aβ1σ1 + 336β2Aβ1σ
21 + 2688A3β2
2ασ21
+ 2688A3β22α + 336β2Aβ1 + 5376A3β2
2ασ1
)
Mathematical Problems in Engineering 19
− Acos2(x)
6720(1 + σ1)4
(2016A2γβ2 + 24192A4β2δασ1 + 2016A2γβ2σ
21 + 2016A2β2σ1α
+ 1008A3β22α + 12096A4β2δασ
21 + 1008A3β2
2ασ21 + 4032A2γβ2σ1
− 1008A2β2σ2ασ21 + 2016A3β2
2ασ1 − 2016A2β2σ2ασ1
+ 12096A4β2δα + 1008A2β2σ21α + 1008A2β2α − 1008A2β2σ2α
)− A
6720(1 + σ1)4
(− 336A2γβ2 − 336A2β2α − 2016A4β2δασ1 − 1008A4β2δασ
21
− 1008A4β2δα − 336A2β2σ21α − 672A2γβ2σ1 + 336A2β2σ2α
− 336A3β22ασ
21 + 672A2β2σ2ασ1 + 336A2β2σ2ασ
21 − 336A3β2
2α
− 672A3β22ασ1 − 336A2γβ2σ
21 − 672A2β2σ1α
)− Acos4(x)
6720(1 + σ1)4
(−30240A4β2δασ1 − 15120A4β2δα − 15120A4β2δασ
21
)
− Acos3(x)
6720(1 + σ1)4
(−5376A3β2
2ασ21 − 5376A3β2
2α − 10752A3β22ασ1
)]sin(x)
− Acos4(x)
6720(1 + σ1)4
(− 5376A3β2γα − 12096A3β2δσ1 − 6048δβ2A
3 − 6048A3β2δσ21
− 10752A3β2γασ1 − 5376A3β2γασ21
)}t5
+
{− cos2(x)
6720(1 + σ1)4
(5600Aβ2σ2σ1 − 26040A3β2δσ
21 − 5600Aβ2σ1 − 2800A2β2
2
− 2800A2β22σ
21 − 2800β2A − 52080A3β2δσ1 + 2800Aβ2σ2σ
21
− 26040δβ2A3 − 2240Aγβ1σ1 − 1120Aγβ1 − 5600A2β2
2σ1
− 2800Aβ2σ21 − 1120Aγβ1σ
21 + 2800Aβ2σ2
)− A cos(x)
6720(1 + σ1)4
(280 + 560σ1 − 560σ2 + 22680δ2A4 − 560σ2σ
21 + 560β2A
+ 1120Aβ2σ1 − 560Aβ2σ2 + 560Aβ2σ21 − 1960A2β2
2σ21 +8400A3β2δσ
21
+ 16800A3β2δσ1 + 5600A2γ2σ21 + 560σ2
2σ1 + 280σ22σ
21 −560Aβ2σ2σ
21
+ 280σ22 +11200A2γ2σ1+16800A2δσ1+45360A4δ2σ1+22680A4δ2σ2
1
− 8400A2σ2δ − 8400A2σ2δσ21 − 16800A2σ2δσ1+8400A2δσ2
1 +280σ21
+ 8400A2δ − 1120σ2σ1 + 8400δβ2A3 + 5600A2γ2
− 1960A2β22 − 1120Aβ2σ2σ1 − 3920A2β2
2σ1
)
20 Mathematical Problems in Engineering
− Acos4(x)
6720(1 + σ1)4
(23520A3β2δσ
21 + 47040A3β2δσ1 + 23520δβ2A
3)
− Acos3(x)
6720(1 + σ1)4
(10080A2σ2δ − 6720A2γ2 − 13440A2γ2σ1 + 10080A2σ2δσ
21
+ 3360A2β22 − 10080A2δσ2
1 − 60480δ2A4 − 10080A2δ
− 6720A2γ2σ21 − 120960A4δ2σ1 − 20160A3β2δσ1 − 10080δβ2A
3
− 20160A2δσ1 + 20160A2σ2δσ1 + 6720A2β22σ1 + 3360A2β2
2σ21
− 60480A4δ2σ21 − 10080A3β2δσ
21
)+
[− A cos(x)
6720(1 + σ1)4
(− 560β2Aβ1 + 23520A3γδσ2
1 + 23520A3γδ − 1120β2Aβ1σ1
+ 3360Aγσ21 − 3360Aγσ2σ
21 + 6720Aγσ1 − 560β2Aβ1σ
21
+ 47040A3γδσ1 + 3360A2γβ2σ21 − 3360Aγσ2 + 3360A2γβ2
+ 6720A2γβ2σ1 − 6720Aγσ2σ1 + 3360Aγ)
− A
6720(1 + σ1)4
(560β1σ1 − 280σ2β1 − 280σ2β1σ
21 − 560σ2β1σ1 + 280β1
+ 1680A2δβ1σ1 + 280β2Aβ1σ21 + 560β2Aβ1σ1 + 2800A2γβ2
+ 280β1σ21 + 2800A2γβ2σ
21 + 5600A2γβ2σ1 + 840A2δβ1σ
21
+ 280β2Aβ1 + 840A2δβ1
)− Acos3(x)
6720(1 + σ1)4
(−33600A3γδ − 67200A3γδσ1 − 33600A3γδσ2
1
)
− Acos2(x)
6720(1 + σ1)4
(− 10080A2γβ2 − 20160A2γβ2σ1 − 5040A2δβ1σ1 − 2520A2δβ1σ
21
− 10080A2γβ2σ21 − 2520A2δβ1
)]sin(x)
− Acos5(x)
6720(1 + σ1)4
(37800δ2A4 + 37800A4δ2σ2
1 + 75600A4δ2σ1
)
− A
6720(1 + σ1)4
(4200A3β2δσ
21 + 2800A2β2
2σ1 − 1400Aβ2σ2σ21 + 1400Aβ2σ
21
− 2800Aβ2σ2σ1 + 1400β2A + 1400A2β22σ
21 + 560Aγβ1σ
21
+ 1400A2β22 + 8400A3β2δσ1 + 4200δβ2A
3 − 1400Aβ2σ2
+ 2800Aβ2σ1 + 1120Aγβ1σ1 + 560Aγβ1
)}t4
Mathematical Problems in Engineering 21
+
{[− A cos(x)
6720(1 + σ1)4
(2240ΩAγ + 4480ΩAγσ1 + 2240ΩAγσ2
1
)
− Acos2(x)
6720(1 + σ1)4
(−6720A2β2σ
31α − 20160A2β2σ
21α − 6720A2β2α − 20160A2β2σ1α
)
− A
6720(1 + σ1)4
(6720A2β2σ1α + 1120Ωβ1σ
21 + 1120β1σ
31 + 1120β1
+ 3360β1σ21 + 2240Ωβ1σ1 + 3360β1σ1 + 1120Ωβ1
+ 6720A2β2σ21α + 2240A2β2σ
31α + 2240A2β2α
)]sin(x)
− Acos2(x)
6720(1 + σ1)4
(− 4480ΩAβ2σ1 − 2240ΩAβ2σ
21 − 2240β2A − 6720Aβ2σ
21
− 13440Aβ1ασ21 − 2240Aβ2σ
31 − 4480Aβ1α − 2240ΩAβ2
− 13440Aβ1ασ1 − 6720Aβ2σ1 − 4480Aβ1ασ31
)− A
6720(1 + σ1)4
(1120β2A + 1120ΩAβ2σ
21 + 1120Aβ2σ
31 + 2240Aβ1α
+ 2240Aβ1ασ31 + 1120ΩAβ2 + 3360Aβ2σ
21 + 2240ΩAβ2σ1
+ 6720Aβ1ασ1 + 3360Aβ2σ1 + 6720Aβ1ασ21
)− A cos(x)
6720(1 + σ1)4
(3360ΩA2δ + 1120ΩAβ2 − 2240Ωσ2σ1 − 1120Ωσ2
− 1120Ωσ2σ21 + 1120Ω + 2240Ωσ1 + 2240ΩAβ2σ1
+ 1120Ωσ21 + 6720ΩA2δσ1 + 3360ΩA2δσ2
1 + 1120ΩAβ2σ21
)− Acos3(x)
6720(1 + σ1)4
(−6720ΩA2δσ1 − 3360ΩA2δσ2
1 − 3360ΩA2δ)}
t3
+
{− Acos3(x)
6720(1 + σ1)4
(30240A2δσ2
1 + 10080A2δσ31 + 10080A2δ + 30240A2δσ1
)
− A
6720(1 + σ1)4
(−10080Aβ2σ
21 − 3360β2A − 3360Aβ2σ
31 − 10080Aβ2σ1
)
+
[− A
6720(1 + σ1)4
(−10080β1σ
21 − 3360β1 − 3360β1σ
31 − 10080β1σ1
)
− A cos(x)
6720(1 + σ1)4
(−20160Aγσ1 − 6720Aγσ3
1 − 6720Aγ − 20160Aγσ21
)]sin(x)
− Acos2(x)
6720(1 + σ1)4
(20160Aβ2σ1 + 20160Aβ2σ
21 + 6720Aβ2σ
31 + 6720β2A
)
22 Mathematical Problems in Engineering
− A/ cos(x)
6720(1 + σ1)4
(− 30240A2δσ2
1 − 10080Aβ2σ1 + 3360σ2σ31 − 10080A2δ
+ 10080σ2σ1 − 3360Aβ2σ31 − 3360 − 30240A2δσ1 − 3360β2A
+ 10080σ2σ21 + 3360σ2 − 10080σ2
1 − 10080A2δσ31
− 10080Aβ2σ21 − 3360σ3
1 − 10080σ1
)}t2
− A cos(x)
6720(1 + σ1)4
(6720 + 6720σ4
1 + 40320σ21 + 26880σ3
1 + 26880σ1
)− A
6720(1 + σ1)4
(−6720 − 26880σ1 − 26880σ3
1 − 6720σ41 − 40320σ2
1
),
(5.4)
θ2 =
{− A cos (x)5
24(1 + σ1)2
(−540A4δα2 − 540A4δα2σ1 + 216A4bδβ2
)
− A cos (x)3
24(1 + σ1)2
(− 24A3bβ2
2 + 36A3β2α2 + 24A2bσ2β2 + 36A3β2α
2σ1 + 36A2α2σ1
+ 36A2αγσ1 + 702A4δα2σ1 − 36A2σ2α2σ1 − 36A2σ2α
2 + 36A2α2
− 24A2bβ1γ − 288A4bδβ2 − 24A2bβ2 + 36A2αγ + 702A4δα2)
+
[− Acos2(x)
24(1 + σ1)2
(− 24A2 bγσ2 + 36A2β1α
2σ1 − 24A2bβ1β2 + 36A2β1α2 + 36A2β2α
+ 24A3bγβ2 + 24A2bγ + 180A4bγδ + 36A2β2σ1α)+
9A5bγδcos4(x)
(1 + σ1)2
− A cos(x)
24(1 + σ1)2
(− 6αAσ1 + 6bAβ1 + 18A3bβ1δ − 72αA3δ − 96A3γα2σ1 + 48A3bγβ2
− 6A2β2σ1α − 6A2β2α − 96A3γα2 + 6αAσ2 − 6αA − 6Abβ1σ2
− 72αA3δσ1 + 6αAσ2σ1 + 6A2bβ1β2
)− Acos3(x)
24(1 + σ1)2
(192A3γα2σ1 + 192A3γα2 + 108αA3δ − 96A3bγβ2
+ 108αA3δσ1 − 54A3bβ1δ)
− A
24(1 + σ1)2
(− 12A2bγ − 12A2β2α − 12A3bγβ2 − 36A4bγδ − 6A2β1α
2 − 12A2β2σ1α
+12A2bγσ2 − 6A2β1α2σ1
)]sin(x)
Mathematical Problems in Engineering 23
− Acos2(x)
24(1 + σ1)2
(3Ab + 72A4bδβ2 + 48A3bγ2 + 3Abσ2
2 + 189A5bδ2 + 6Aβ1ασ1
− 6A2bσ2β2 + 6Aβ1α − 3Abβ21 + 192A3β2α
2 + 192A3β2α2σ1
+ 72A3bδ + 6A2bβ2 − 45A3bβ22 − 72A3bδσ2 − 6Abσ2
)− A cos(x)
24(1 + σ1)2
(72A4bδβ2 − 198A4δα2σ1 − 30A3β2α
2 − 30A2α2σ1 + 24A3bβ22
− 30A3β2α2σ1 + 30A2σ2α
2 − 24A2bσ2β2 + 12A2bβ1γ + 30A2σ2α2σ1
− 198A4δα2 − 30A2αγσ1 + 24A2bβ2 − 30A2αγ − 30A2α2)
− Acos4(x)
24(1 + σ1)2
(− 405A5bδ2 − 192A3β2α
2 + 54A3bδσ2 − 54A4bδβ2 − 48A3bγ2
+ 48A3bβ22 − 54A3bδ − 192A3β2α
2σ1
)− 81A6bδ2cos6(x)
8(1 + σ1)2
− A
24(1 + σ1)2
(− 12A3bγ2 − 18A4bδβ2 − 3Ab − 3Abσ2
2 − 3A3bβ22
+ 6A2bσ2β2 − 3Aβ1α − 27A5bδ2 − 3Aβ1ασ1 − 24A3β2α2σ1 − 18A3bδ
− 6A2bβ2 + 18A3bδσ2 − 24A3β2α2 + 6Abσ2
)}t4
+
{−A
(−48A2αγ − 48A2αγσ1
)cos3(x)
24(1 + σ1)2− A
(40A2αγ + 40A2αγσ1
)cos(x)
24(1 + σ1)2
− A(4Aβ1α + 4Aβ1ασ1
)24(1 + σ1)2
− A(−8Aβ1α − 8Aβ1ασ1
)cos2(x)
24(1 + σ1)2
+
[−A
(−48A2β2σ1α − 48A2β2α)
24(1 + σ1)2cos2(x) − A
(−144αA3δσ1 − 144αA3δ)cos3(x)
24/(1 + σ1)2
− A cos(x)
24(1 + σ1)2
(8αA + 96αA3δσ1 − 8αAσ2σ1 − 8αAσ2 + 8A2β2α + 96αA3δ
+ 8A2β2σ1α + 8αAσ1
)− A
24(1 + σ1)2
(16A2β2α + 16A2β2σ1α
)]sin(x)
}t3
+
{− A cos(x)
24(1 + σ1)2
(12 − 48A2bβ2 + 120A2α2σ2
1 − 48A2σ1bβ2 + 24σ1 + 12σ21
− 12σ1aβ1 + 240A2α2σ1 − 12aβ1 + 120A2α2)
− Acos3(x)
24(1 + σ1)2
(48A2bβ2 − 144A2α2σ2
1 − 288A2α2σ1 − 144A2α2 + 48A2σ1bβ2
)
24 Mathematical Problems in Engineering
− A
24(1 + σ1)2
(24Aaγ + 36A3bδ + 12A2bβ2 − 12Abσ2 + 12Ab + 36A3σ1bδ
+ 24Aσ1aγ + 12Aσ1b − 12Aσ1bσ2 + 12A2σ1bβ2
)+
[− Acos2(x)
24(1 + σ1)2
(−48A2bγ − 108aδA2 − 108σ1aδA
2 − 48A2σ1bγ)
− A
24(1 + σ1)2
(12σ1aβ2A + 36σ1aδA
2 − 12aσ2 + 12aβ2A − 12σ1aσ2
+ 12σ1a + 24A2bγ + 36aδA2 + 12a + 24A2σ1bγ)
− A cos(x)
24(1 + σ1)2
(240αAσ1 + 120αA − 48aβ2A − 12bAβ1 − 48σ1aβ2A
+ 120αAσ21 − 12σ1bAβ1
)]sin(x)
− Acos2(x)
24(1 + σ1)2
(− 12Ab + 12Abσ2 − 144A3σ1bδ − 144A3bδ − 12A2bβ2 − 48Aaγ
− 12A2σ1bβ2 − 48Aσ1aγ − 12Aσ1b + 12Aσ1bσ2
)− A
(108A3bδ + 108A3σ1bδ
)cos4(x)
24(1 + σ1)2
}t2
×{A(−48αA − 96αAσ1 − 48αAσ2
1
)cos(x) sin(x)
24(1 + σ1)2+ −A
(−24 − 48σ1 − 24σ21
)cos(x)
24(1 + σ1)2
}t
− A(24 + 48σ1 + 24σ2
1
)cos(x)
24(1 + σ1)2− A
(−24 − 48σ1 − 24σ21
)24(1 + σ1)2
.
(5.5)
Now we make calculations for the results obtained by the variational iteration method using
the Maple software package with the following arbitrary constants:
a = 0.5, A = 0.001, b = 0.5, α = 1, β1 = β2 = 0.05, γ = 1, δ = 0.8. (5.6)
5.1. Special Case
(1) If we take into our consideration the first iteration (i.e., u = u1 and θ = θ1). See
Figures 15, 16, 17, and 18.
(2) If the magnetic field and rotation are neglected, the components of the displacement
u2 and temperature θ2 take the following forms. See Figures 19 and 20.
Mathematical Problems in Engineering 25
xt
02
46
810
1214 0
1020
3040
50
−150−100−50
050
100150
u
(a)
xt
02
46
810
1214 0
1020
3040
50
−1
−0.5
0
0.5
1
1.5
(b)
Figure 11: Variations of the displacement u2 and temperature θ2 for various values of the axis x and time twhen Ω = 0.1, σ1 = 0.2, σ2 = 0.1.
x
02
46
810
1214 0
12
34
5
Ω
u
0
0.0005
0.001
0.0015
0.002
(a)
x
02
46
810
1214
01
23
45
Ω
0.0002
0.0006
0.0014
0.0018
0.001
(b)
Figure 12: Variations of the displacement u2 and temperature θ2 for various values of the axis x and rotationΩ when t = 0.1, σ1 = 0.2, σ2 = 0.1.
6. Discussion
With the view of illustrating the theoretical results obtained in the preceding sections, a
numerical result is calculated for the homotopy perturbation method and variational iteration
method.
Figures (1–10) illustrate the influences of time t, rotation Ω, and sensitive pats of the
magnetic field σ1 and σ2 for the iterations (u = V0 + V1 + V2 and θ = Θ0 + Θ1 + Θ2) and
(u = V0 + V1 and θ = Θ0 + Θ1), and if the rotation and magnetic field neglected, respectively,
respect to the coordinate x for the homotopy perturbation method. Figures (11–20) illustrate
the influences of time t, rotation Ω, and sensitive pats of the magnetic field σ1 and σ2 for the
iterations (u = V2, θ = θ2 and u = V1 and θ = θ1), and if the rotation and magnetic field have
been neglected, respectively, respect to the coordinate x for the variational iteration method.
From Figures 1 and 11, it is concluded that the displacement u and temperature θ
start from their maximum values, decrease and increase periodically with an increasing of
26 Mathematical Problems in Engineering
1
02
46
810
u
x
02
46
810
1214
0
0.0005
0.001
0.0015
0.002
(a)
0.0002
0.0006
0.0014
0.0018
0.001
x
02
46
810
1214 1
02
46
810
(b)
Figure 13: Variations of the displacement u2 and temperature θ2 for various values of the axis x andmagnetic field σ1 when t = 0.1, Ω = 0.1, σ2 = 0.1.
x
02
46
810
1214 0
24
68
10
2
u
0
0.001
0.002
(a)
x
02
46
810
1214 0
24
68
10
2
0.0002
0.0006
0.0014
0.0018
0.001
(b)
Figure 14: Variations of the displacement u2 and temperature θ2 for various values of the axis x andmagnetic field σ2 when t = 0.1, Ω = 0.1, σ1 = 0.1.
x t
02
46
810
1214 0
1020
3040
50
u
−0.8−0.4
00.4
0.8
(a)
x t
0 2 4 6 8 10 12 14 010
2030
4050
−0.04
−0.02
0
0.02
0.04
(b)
Figure 15: Variations of the displacement u1 and temperature θ1 for various values of the axis x and time twhen Ω = 0.1, σ1 = 0.2, σ2 = 0.1.
Mathematical Problems in Engineering 27
x
02
46
810
1214 0
12
34
5
Ω
u
0
0.0005
0.001
0.0015
0.002
(a)
x
02
46
810
1214
0.0002
0.0006
0.0014
0.0018
0.001
01
23
45
Ω
(b)
Figure 16: Variations of the displacement u1 and temperature θ1 for various values of the axis x and rotationΩ when t = 0.1, σ1 = 0.2, σ2 = 0.1.
10
24
68
10
u
x
02
46
810
1214
0
0.0005
0.001
0.0015
0.002
(a)
0.0002
0.0006
0.0014
0.0018
0.001
x0
24
68
1012
14
1
02
46
810
(b)
Figure 17: Variations of the displacement u1 and temperature θ1 for various values of the axis x andmagnetic field σ1 when t = 0.1, Ω = 0.1, σ2 = 0.1.
the coordinate x, also, it is obvious that their values take the minimum values and increases
with the increasing values of the time t. From Figures 2, 3, 4, 12, 13, 14, it is seen that the
components of the displacement u and temperature θ begin from the minimum values near
zero increase and then decrease periodically with the coordinate x, it is clear also that there
are a sligh increasing with an increasing of the sensitive parts of the magnetic field, also, one
can see that u and θ decrease with an increasing of the rotation Ω.
Figures 5–8 and 15–18 display the first iteration with respect to the homotopy
perturbation method and variational iteration method on the influences of the parameters
time t, rotation Ω, and sensitive pats of the magnetic field σ1 and σ2 to obtain the displacement
and the temperature components on the medium due to the harmonic wave propagation. It
is shown that the increasing of the coordinate x sensitive an increasing and dereasing on
them periodically due to appearance of the pairs (cos, sin) in the initial condition and the
approximate solutions; it is also clear that the components begin from their minimum values
and increase absolutely with the variation of the time t. With the variations of the rotation
and magnetic field tends to slightly affect on the displacment and the temperature.
28 Mathematical Problems in Engineering
x
02
46
810
1214 0
24
68
10
2
u
0
0.0005
0.001
0.0015
0.002
(a)
x
02
46
810
1214 0
24
68
10
2
0.0002
0.0006
0.0014
0.0018
0.001
(b)
Figure 18: Variations of the displacement u1 and temperature θ1 for various values of the axis x andmagnetic field σ2 when t = 0.1, Ω = 0.1, σ1 = 0.1.
−400
−200
0
200
400
u
x t
02
46
810
1214 0
1020
3040
50
(a)
−1−0.5
00.5
1.52
1
xt
02
46
810
1214 0
1020
3040
50
(b)
Figure 19: Variations of the displacement u2 and temperature θ2 for various values of the axis x and time t(u = u2 and θ = θ2) when Ω = σ1 = σ2 = 0.
u
x t
02
46
810
1214 0
1020
3040
50
−1
−0.5
0
0.5
1
(a)
xt
02
46
810
1214 0
1020
3040
50
−0.04
−0.02
0
0.02
0.04
(b)
Figure 20: Variations of the displacement u1 and temperature θ1 for various values of the axis x and time t(u = u1 and θ = θ1) when Ω = σ1 = σ2 = 0.
Mathematical Problems in Engineering 29
It seems too that there are a clear differs between the results obtained by the HPM
and the corresponding results obtained by VIM resultant to the appearance of the high order
of time in VIM tends to the high values of the approximate solution comparing with the
results obtained by HPM. Because of the results obtained, we concluded that the homotopy
perturbation method is more effective and powerful than the variational iteration method.
On the other hand, from Figures 9, 10, 19, and 20, it is obvious that if the rotaion and
magnetic field are neglected, the approximate solutions by HPM and VIM in first iteration
are the same in both methods and agree with the results obtained by Sweilam and Khader
[1].
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 958101, 21 pagesdoi:10.1155/2012/958101
Research ArticleSimplicial Approach to Fractal Structures
Carlo Cattani, Ettore Laserra, and Ivana Bochicchio
Department of Mathematics, University of Salerno, Via Ponte don Melillo, 84084 Fisciano, Italy
Correspondence should be addressed to Ivana Bochicchio, [email protected]
Received 28 July 2011; Accepted 16 November 2011
Academic Editor: Cristian Toma
Copyright q 2012 Carlo Cattani et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
A fractal lattice is defined by iterative maps on a simplex. In particular, Sierpinski gasket and vonKoch flake are explicitly obtained by simplex transformations.
1. Introduction
Simplicial calculus [1–3] has been since the beginning a suitable tool for investigating discrete
models in many physical problems such as discrete models in space-time [4–9] complex
networks [10–13], molecular crystals, aggregates and diamond lattices [14–17], computer
graphics [18, 19], and more recently signal processing and computer vision, such as stereo
matching and image segmentation [20, 21].In some recent papers [22–25], fractals [26–29] generated by simplexes, also called
fractal lattices, were proposed for the analysis of nonconventional materials as some kind of
polymers [24, 25] or nanocomposites [22, 23, 30, 31] having extreme physical and chemical
properties. Moreover, the analysis of complex traffic on networks [32, 33] and image analysis
[20, 21] based on fractal geometry and simplicial lattices has focussed on the importance of
these methods in handling modern challenging problems.
However, only a few attempts were made in order to define the fractal lattice
(structure) by an iterated system of functions on simplexes [34, 35]. The main scheme for
affine contraction has been given in [35], whereas some generation of fractals by simplicial
maps can be found in [34].In this paper, we define a method based on simple algorithms for the generation
of fractal-like structures by continuously deforming a simplex. This algorithm is based on
a well-defined analytical map, which can be used to finitely describe fractals. Instead of
recursive law, or nested maps (see, e.g., [1, 2, 15]), we propose a method which can be more
easily implemented.
2 Mathematical Problems in Engineering
In the following, we will study an m-dimensional fractal structure defined by the
transformation group of a simplicial complex. Starting from a simplex, it will define the group
of transformation on it, so that the intrinsic (affine) metric remains scale invariant. The group
of transformations (isometries and homotheties) will be characterized by matrices acting on
the skeleton of the simplex. We will derive the basic properties of the fractal lattice and give
a suitable definition of self-similarity on lattices. The concept of self-similarity is shown to
be fulfilled by some classical transformation on simplices (homothety) and, simplicial based,
fractals as the Sierpinski tessellations and the von Koch flake.
2. Euclidean Simplexes
In the ordinary Euclidean space Rn, we assume that there exists a triangulation of Rn, in the
sense that there is at least a finite set of n+1 points geometrically independent (simplexes). A
simplex will be considered both as a set of points and as the convex subspace of Rn, defined
by the geometrical support of the simplex. Union of n-adjacent simplexes is an n-polyhedron
P [4, 18, 19].The euclidean m-simplex σm, of independent vertices V0, V1, . . . , Vm, is defined [1–3]
as the subset of Rn,
σm def=
{P ∈ Rn | P
m∑i=0
λiVi withm∑i=0
λi = 1, 0 ≤ λi ≤ 1
}. (2.1)
Let us denote with [σm] = [V0, V1, . . . , Vm] the set of points which form the skeleton of σm, and
let #σm = m + 1 be the cardinality of the set of points. The p-face of σm, with p ≤ m, is any
simplex σp such that [σp] ∩ [σm]/= ∅, and we write σp � σm.
The number of p-faces of σm is (m+1p+1 ).
The m-dimensional simplicial complex Σm is defined as the finite set of p simplexes
(p ≤ m) such that
(1) for all σk ∈ Σm if σh � σk, then σh ∈ Σm,
(2) for all σk, σh ∈ Σm, then either [σh] ∩ [σk] = ∅ or [σh] ∩ [σk] = [σj] with σj ∈ Σm.
The set of points P such that P ∈ σp, p ≤ m, and σp ∈ Σm is the geometric support
of Σm also called m-polyhedron Mm. The p-skeleton of Σm is [Σm]p def= [σp] for all σp ∈ Σm.
The boundary ∂Σm of Σm is the complex Σm−1 such that each σm−1 ∈ Σm−1 is face of only one
m-simplex of Σm. A finite set of simplexes is also called lattice (or tessellation).
2.1. Barycentric Coordinates and Barycentric Bases
In each simplex, it is possible to define the barycentric basis as follows: given the m-simplex
σm with vertices V0, . . . , Vm, the barycentric basis is the set of (m + 1) vectors
eidef= Vi − Gm, (2.2)
Mathematical Problems in Engineering 3
based on the barycenter
Gm def= G(σm) =m∑i=0
1
m + 1Vi. (2.3)
These vectors ei belong to the n-dimensional vector space E isomorphic to Rn. Moreover, they
are linearly dependent, since according to their definition, it is
m∑i=0
ei = 0. (2.4)
Each point P ∈ σm can be characterized by a set of barycentric coordinates (λ0, . . . , λm)such that
0 ≤ λi ≤ 1,m∑i=0
λi = 1, i = 0, . . . , m, (2.5)
and P − Gm =∑m
i=0 λiei
(2.2),(2.4)=⇒ P =
∑mi=0 λ
iVi. Therefore, each point of σm can be formally
expressed as a linear combination of the skeleton [σm].The dual space is defined as the linear map of the vector space E into R as
⟨ei, ek
⟩=δi
k, (2.6)
with [14]
δik
def= δik −
1
m + 1=
⎧⎪⎪⎨⎪⎪⎩− 1
m + 1, i /= k,
+m
m + 1, i = k,
(2.7)
δik
being the Kroneker symbol. According to the definition (2.7), it is
m∑i=0
δik =
m∑k=0
δik = 0. (2.8)
In addition, the metric tensor in σm is defined as [5]
gijdef= −1
2δhi δ
kj
2hk,
(i, j, h, k = 0, 1, . . . , m
)(2.9)
being 2hk
def= (Vk − Vh)2 = (ek − eh)
2.
4 Mathematical Problems in Engineering
2.2. Measures of the m-Simplex
Let
Lijdef= Vj − Vi
(= ej − ei
), lij
def=⟨Lij ,Lij
⟩, (2.10)
by using the ordinary wedge product of the vectors ej1 , . . . , ejp , we can define the p-form ω,
ω =1
p!
∑j1,...,jm
ωj1...jpej1 ∧ · · · ∧ ejp , (2.11)
whose affine components are ωj1...jp def= 〈ω, ej1 ∧ ej2 ∧ · · · ∧ ejp〉 [14].The euclidean measure of the m-simplex σ (volume) is [14]
εΩ2 def=1
m!|L01 ∧ · · · ∧ L0m|, (2.12)
from where, it follows that
Ω2 =(
1
m!
)3 ∑j1 ,...,jm
k1,...,km
εj1...jmεk1...kmm∏a=1
l2jaka , (2.13)
being
εj1...jmdef= ±1, (2.14)
according to the even/odd permutation j0, j1, . . . , jm of the indices 0, 1, . . . , m.
In particular, the volume of each p-face σi1...im−p (see also [9]) is
Ω2i1...im−p =
(−1
2
)p( 1
p!
)3 ∑j1 ,...,jp
k1,...,kp
εj1...jp εk1...kp
p∏a=1
l2jaka(0 < p ≤ m
), (2.15)
where j1, . . . , jp, k1, . . . , kp /= i1, . . . , im−p.
3. m-Dimensional Homothety
Let I(σi) be the subspace of Rm to which σi belongs; it can be easily proved that [14]
∀ v ∈ I(σi)⟨ni,v
⟩= 0 ( i fixed ), (3.1)
Mathematical Problems in Engineering 5
where the normal vector ni is defined as
ni def= −mΩΩi
ei, hi def=mΩΩi
. (3.2)
The above definition of vector orthogonal to a (m − 1)-face allows us to characterize
the m-parallelism of simplexes as follows. Let σ,�σ be two simplexes in Rm; let σi,
�σi be the ith
(m − 1)-faces of σ and�σ, respectively, and let ni,
�ni
be their normal vectors, then we say that
σ is m-parallel to�σ ( σ‖m�
σ) if and if only σi‖�σi, that is, ni =�ni(i = 0, . . . , m).
Let ϕ be a map
ϕ : Rm −→ Rm, σϕ −→ �
σ (3.3)
such that
(1) ϕ is a bijective simplicial map on σ,
(2) the s-adjacent faces of σ correspond (under the map ϕ) to s-adjacent faces of�σ,
(3) σ and�σ are m-parallel.
We also assume that this transformation depends on the edge vectors and in particular
on the edge lengths, so that any quantity, defined on the simplex, transforming under the action ofϕ, is a function of the edge lengths. Furthermore, we assume the following conditions:
(4) there exists a fixed point under the action of ϕ:
∃O ∈ Rm | ϕ(O) ≡ O, (3.4)
(5) each (m − 1)-face σi translates of an amount t ∈ [0,∞).Let us choose as a fixed point one of the vertices, for example, V0. We define this
bijective simplicial map applying any P ∈ σ into�
P ∈ �σ (t ∈ [0,∞)) as
�
Pdef= P + t
Ω0
mΩ
m∑i=0
λiL0i; (3.5)
in particular, this function acts on any vertex Vi as
�
V i = Vi + tΩ0
mΩL0i,
(�
V 0 = V0
), (3.6)
so that we can easily prove that all the previous conditions are easily satisfied [14]. According
to the above equations, each edge transforms as
�
Lij =(
1 + tΩ0
mΩ
)Lij , (3.7)
where�
Lij =�
V j −�
V i.
6 Mathematical Problems in Engineering
3.1. Variation Law of the p-Faces of σ
The variation law of the edge lengths, resulting from (3.7), is given by the formula
�
l ij =(
1 + tΩ0
mΩ
)lij , (3.8)
where lij is the length of the edge Lij , and�
l ij is the length of the edge�
Lij .
According to (2.13), the volume Ω is a homogeneous function of degree m of the m(m+1)/2 variable {l2ij}i<j , so that its variation law is
�
Ω =(
1 + tΩ0
mΩ
)m
Ω, (3.9)
and for any p-face,
�
Ωi1...im−p =(
1 + tΩ0
mΩ
)p
Ωi1...im−p(0 < p < m
); (3.10)
analogously, taking into account the definition (5.5)2, we have the transformation law of hi:
�
hi = m
(1 + t
Ω0
mΩ
)hi. (3.11)
There follows, for the fundamental vectors of�σ, that
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
�e i
def=�
V i −�
G =(
1 + tΩ0
mΩ
)ei,
�ni
= ni,
�ei
=
�
Ωi/Ωi�
Ω/Ωei =
(1 + t
Ω0
mΩ
)−1
ei.
(3.12)
4. Self-Similar Structure
Let (Rn, d) be the complete metric space with the standard Euclidean metric d, and let K(Rn)be the set
K(Rn) ={K ⊆ Rn : K is a nonempty compact set
}. (4.1)
The iterated function system (IFS)
{wi} = (Rn, d,w1, w2, . . . , wn) (4.2)
Mathematical Problems in Engineering 7
is the finite set of contractions wi on the complete metric space (Rn, d), being the contraction
w defined as
d(w(x), w
(y)) ≤ cd
(x,y
), ∀x,y ∈ Rn, (4.3)
with c contraction coefficient.
For each A ∈ K(Rn), the (IFS) contracting mapping is
w : A ∈ K(Rn) −→ w1(A)⋃
· · ·⋃
wn(A) ∈ K(Rn), (4.4)
with contraction coefficient c = max{c1, . . . , cn}. Each function wi usually is linear, or more
generally an affine transformation, but sometimes it can be nonlinear, including projective
and Mobius transformations [27].According to the Banach fixed-point theorem (see, e.g., [36]), every contraction
mapping on a nonempty complete metric space has a unique fixed point, so that there exists
a unique compact (i.e., closed and bounded) fixed set A such that A = w(A). The set A is
also known as the fixed set of the Hutchinson operator [28].One way of constructing such fixed set is to start with an initial set A and by iterating
the actions of w. Hence,
A =⋃
i1,...,ih=1,...,n
wi1 ◦ · · · ◦wih(A), (4.5)
so that A is a self-similar set, expressed as the finite union of its conformal copies, each one
reduced by a factor ch.
The attractor A of IFS is characterized by a similarity dimension as follows.
Definition 4.1. Given an IFS of n contraction mappings with the same contraction coefficient
c, the similarity dimension is defined as
s =log n
log 1/c
(= − log n
log c
). (4.6)
Sets having noninteger similarity dimensions are called fractal sets, or simply fractals.
There follows that the iterated function systems are a method of constructing fractals; the
resulting constructions are always self-similar such that w(μx) = μHw(x). Hence, each map
w is also called a self-similar map [27].
5. Fractal Structures from Simplicial Maps
In this section, some examples of self-similar (scale invariant) structures obtained by IFS on
simplexes are given in R2. In particular, the IFS will be defined by affine transformations, as
conformal maps of the affine metrics.
8 Mathematical Problems in Engineering
In the following, we will introduce some self-similar maps defined both on 2-simplexes
and 1-simplexes, so that, from (4.1),
K(R2)={σ2; σ1;σ0
},
w : K(R2) −→ K
(R2).
(5.1)
In particular, let σ2 be the simplex [V1, V2, V3], then it is
K(R2)= {[V1, V2, V3]; [V1, V2], [V1, V3], [V2, V3]; [V1], [V2], [V3]}, (5.2)
so that a map w on K(R2) could be the more general function defined on any face of σ2.
Examples. If the skeleton of σ2 is the set of vertices {V1, V2, V3} with V1 = (x1,y1),V2 = (x2,y2),and V3 = (x3,y3), the affine map w is defined by the matrix
W =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
a11 a12 a13 a14 a15 a16
a21 a22 a23 a24 a25 a26
a31 a32 a33 a34 a35 a36
a41 a42 a43 a44 a45 a46
a51 a52 a53 a54 a55 a56
a61 a62 a63 a64 a65 a66
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(5.3)
and the constant vector
U = (u1, u2, u3, u4, u5, u6). (5.4)
The function w maps a 2-simplex into a 2-simplex whereas, by a matrix product, the vector
X =(x1,y1,x2,y2,x3,y3
)(5.5)
is mapped into the vector
WX +U, (5.6)
Mathematical Problems in Engineering 9
so that the skeleton of w(σ2) is given by the vector WX + U. For instance, a rotation with
fixed point V1 is given by the matrix
W =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
1 0 0 0 0 0
0 1 0 0 0 0
0 0 a33 a34 0 0
0 0 a43 a44 0 0
0 0 0 0 a55 a56
0 0 0 0 a65 a66
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (5.7)
with
a33a44 − a34a43 = ±1, a55a66 − a56a65 = ±1, (5.8)
and the vector U = {0, 0, 0, 0, 0, 0}.
Some more special maps will be given in the following where, in particular, we
consider, without restriction, some special maps on the 1-faces of σ2 such that
w(σ2)= w1
(σ2
1
)∪w2
(σ2
2
)∪w3
(σ2
3
), #w
(σ2)= 3. (5.9)
In this case, the matrix W , acting on σ2, follows from the direct sum of lower-order matrices
acting on σ1 simplexes, as follows:
(a) the first vertex V1 remains fixed, and the map w on σ2 is a consequence of the
transformation of the simplex σ1 = [V2, V3], that is, by defining
I =
(1 0
0 1
), W1 =
⎛⎜⎜⎜⎜⎜⎝a33 a34 a35 a36
a43 a44 a45 a46
a53 a54 a55 a56
a63 a64 a65 a66
⎞⎟⎟⎟⎟⎟⎠, (5.10)
it is
W = I ⊕W1 =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
1 0 0 0 0 0
0 1 0 0 0 0
0 0 a33 a34 a35 a36
0 0 a43 a44 a45 a46
0 0 a53 a54 a55 a56
0 0 a63 a64 a65 a66
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (5.11)
10 Mathematical Problems in Engineering
(b) the second vertex V2 remains fixed, and the map w on σ2 is a consequence of the
transformation of the simplex σ1 = [V1, V3], so that
W2 =
⎛⎜⎜⎜⎜⎜⎝a11 a12 a15 a16
a21 a22 a25 a26
a51 a52 a55 a56
a61 a62 a65 a66
⎞⎟⎟⎟⎟⎟⎠ ,
W =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
a11 a12 0 0 a15 a16
a21 a22 0 0 a25 a26
0 0 1 0 0 0
0 0 0 1 0 0
a51 a52 0 0 a55 a56
a61 a62 0 0 a65 a66
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,
(5.12)
(c) the third vertex V3 remains fixed, and the map w on σ2 is a consequence of the
transformation of the simplex σ1 = [V1, V2], that is,
W = W3 ⊕ I =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
a11 a12 a13 a14 0 0
a21 a22 a23 a24 0 0
a31 a32 a33 a34 0 0
a41 a42 a43 a44 0 0
0 0 0 0 1 0
0 0 0 0 0 1
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (5.13)
being
W3 =
⎛⎜⎜⎜⎜⎜⎝a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
⎞⎟⎟⎟⎟⎟⎠. (5.14)
In the following, we will characterize the transformation on a 2-simplex as a result
of iterative maps on its boundary 1-simplexes. These maps on 1-simplexes are defined by
the matrices W1,W2, and W3, applied to the vectors of coordinates of [V2, V3], [V1, V3], and
[V1, V2], respectively.
Mathematical Problems in Engineering 11
C C
B
A
B
C
B
A C
B
Figure 1: Homothety map.
5.1. Homothety
Let us consider the 2-simplex σ2 = {A,B,C} and the map (Figure 1)
σ2 = {A,B,C} =⇒ w(σ2)={A,B′, C′}, (5.15)
such that nC = ±nC′ . This map, according to (5.9), is obtained as a combination of 3 maps
acting on the faces of σ2, since
w(σ2)= w1([A,B]) ∪w2([B,C]) ∪w3([A,C]). (5.16)
This map is a scale invariant, since there results
2AB = λ 2
A′B′ , (0 ≤ λ), (5.17)
2BC = λ 2
B′C′ , and 2AC = λ 2
A′C′ , as well.
So that when λ < 1, we have a contraction and a dilation when λ > 1.
Moreover, according to (2.9), the metric g ′ij of the transformed simplex is given by a
conformal transformation g ′ij = λgij .
5.2. Sierpinski Gasket
As a first example of fractal defined by IFS on simplexes, we will consider the Sierpinski
gasket. To this end, let us introduce an orthogonal coordinate system 0xy in R2 and three
homothety maps w1, w2, and w3. Each wi is uniquely and completely determined once we
know as it acts on the paired points A = (xA,yA), B = (xB, yB), and C = (xC, yC), vertices of
the 2-simplex [σ2] = [A,B,C].
12 Mathematical Problems in Engineering
In order to define the Sierpinski gasket by IFS of maps, we consider a sequence of
maps that, at each step, shrink the area of σ2 by a factor 0.25 and move the edges by a suitable
homothety (Figure 2). In particular, the 3 maps are explicitly defined as follows:
w1 :
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
xA
yA
xB
yB
xC
yC
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⇒ M ·
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
xA
yA
xB
yB
xC
yC
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,
w2 :
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
xA
yA
xB
yB
xC
yC
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⇒ M ·
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
xA
yA
xB
yB
xC
yC
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
0
1/2
0
1/2
0
1/2
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,
w3 :
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
xA
yA
xB
yB
xC
yC
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⇒ M ·
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
xA
yA
xB
yB
xC
yC
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
1/2
0
1/2
0
1/2
0
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,
(5.18)
where M is the matrix
M =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
1/2 0 0 0 0 0
0 1/2 0 0 0 0
0 0 1/2 0 0 0
0 0 0 1/2 0 0
0 0 0 0 1/2 0
0 0 0 0 0 1/2
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (5.19)
Once we get the vertices of w(σ2), we can easily define the map for each point P of the
σ2 convex domain.
Mathematical Problems in Engineering 13
Figure 2: Fundamental maps.
Comment. In fact, let λ1, λ2, and λ3 be the barycentric coordinates of a given point P inside
σ2, as given by (2.1), then we can write the barycentric expansion of P ≡ (x,y) in terms of the
coordinates of vertices A,B, and C as
x = λ1xA + λ2xB + λ3xC,
y = λ1yA + λ2yB + λ3yC.(5.20)
Substituting λ3 = 1 − λ1 − λ2 into the above and rearranging, this linear transformation can be
written as
H ·Λ = P − C, (5.21)
14 Mathematical Problems in Engineering
where Λ is the vector of barycentric coordinates, and H is the matrix
H =
(xA − xC xB − xC
yA − yC yB − yC
). (5.22)
Since H is invertible, we can easily obtain the barycentric coordinates of P = (x,y):
λ1 =
(yB − yC
)(x − xC) + (xC − xB)
(y − yC
)(yB − yC
)(xA − xC) + (xC − xB)
(yA − yC
) ,λ2 =
(yC − yA
)(x − xC) + (xA − xC)
(y − yC
)(yC − yA
)(xB − xC) + (xA − xC)
(yB − yC
) ,λ3 = 1 − λ1 − λ2.
(5.23)
According to (5.9) each map wi, i = {1, 2, 3} is a contraction (dilation) of the σ2 faces,
such that the union gives rise to a 2-simplex (Figure 2). Any P ∈ [σ2] is mapped into�
P ∈ �σ =
wi([σ2]) as
�
P = P − 1
2
m∑i=0
λiL0i. (5.24)
Moreover, each vertex in the wi([σ2]), i = 1, 2, 3 can be expressed as in (3.6)
�
V i = Vi − 1
2L0i, (5.25)
so that
�
Lij =1
2Lij ,
�
l ij =1
2lij ,
�
Ω =1
4Ω.
(5.26)
Reiterating this process for each remaining triangle, at the step k, we will obtain the
compact set Tk given by 3k triangles whose edges are contracted by (1/2k). In other words,
�
Lij =1
2kLij ;
�
l ij =1
2klij ,
�
Ω =1
22kΩ.
(5.27)
Finally, we note that through the three simplicial maps in R2, provided with the natural
metric d, we are able to construct the IFS (R2, d,w1, w2w3) that has the well-known Sierpinski
Mathematical Problems in Engineering 15
Figure 3: Sierpinski gasket.
gasket T =⋂
k Tk as fractal attractor. So, we have obtained the Sierpinski gasket, as the
combination of homothety maps (Figure 2). The iterating function will generate the known
fractal-shaped curve (Figure 3).The Sierpinski gasket supplies one of the most simple cases of construction of fractals
through simplicial maps. In fact, the fractal structure is obtained acting on the 2-simplex only
with homothetic transformations. Sometimes a fractal object can be constructed not only
acting on simplexes with one map, but considering the compositions of different suitable
transformations. Hereafter, in order to obtain another fractal object, we will consider, in
details, some more elementary maps: the translation and the rotation (which are special cases
of the matrix W).
6. Von Koch Curve
The von Koch curve [27, 28] can be obtained as a combination of homothety, translation, and
rotation maps, so that the von Koch snowflake is obtained by their iteration.
6.1. Translation
Let the translation operator be defined as the operator
T : Rm −→ Rm, σT −→ �
σ (6.1)
such that
T(P) = P + v, (6.2)
16 Mathematical Problems in Engineering
where v = (v1,v2, . . . ,vm) is a given vector of Rm, then the image of a simplex σ under the
function T is the translation of σ by T so that any vertex Vi is transformed into
�
V i = Vi + v. (6.3)
Since in a Euclidean space, any translation is an isometry, we have no variation of the edge
lengths of σ.
According to the definitions (5.3), (5.4) in R2, it is
U = (v1, v2,v1, v2,v1, v2), v1 = Cnst., v2 = Cnst., (6.4)
being W the zero matrix.
6.2. Rotation
Rotation is characterized by having a fixed point, however, like the translation which is an
isometry. This is like the previous maps on simplexes that can be defined by a suitable matrix
(5.3). R2 rotation is defined by (5.7), which however can be expressed by a single parameter
(rotation angle). Hence, in two dimensions, a rotation with fixed point V0 is the operator
R : R2 −→ R2, σR −→ �
σ (6.5)
such that
R(P) = V0 + Rθ(P − V0), (6.6)
where Rθ is the matrix
Rθ =
(cos θ − sin θ
sin θ cos θ
), (6.7)
so that (5.7), when applied to the simplex σ2 = [V0, V1, V2] with one fixed vertex, becomes
W =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
1 0 0 0 0 0
0 1 0 0 0 0
0 0 cos θ − sin θ 0 0
0 0 sin θ cos θ 0 0
0 0 0 0 cos θ − sin θ
0 0 0 0 sin θ cos θ
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (6.8)
Mathematical Problems in Engineering 17
With respect to an orthogonal coordinate system with origin O, for any P ∈ σ, we
define the rotation as the bijective simplicial map which applies P ∈ σ into�
P ∈ �σ,
�
Pdef= [Rθ(P − v)] + v, (6.9)
where v = V0 −O; in particular, the vertices V0, V1, and V2 are transformed into
�
V 0 = V0,�
V i = [Rθ(L0i +O)] + (V0 −O), (i = 1, 2). (6.10)
6.3. Von Koch Snowflake
Let 0xy be an orthogonal coordinate system for R2, and let σ2 = [A,B,C] be a two simplex
under the homothety map. According to (5.16), this map can be realized by a composition of
maps on the 1-simplexes σ1 = [A,B], σ1 = [B,C], and σ1 = [A,C]. The coordinates of vertices
are A = (xA,yA), B = (xB,yB), and C = (xC,yC), respectively. In the following we will
give both the construction of the Koch curve as IFS on σ1 and the construction of the Koch
snowflake as IFS on σ2.
6.3.1. Von Koch Curve
Koch curve can be classically constructed by starting with a line segment, then recursively
altering the shape as follows: divide the line segment into three segments of equal length;
draw an equilateral triangle that has the middle segment from step 1 as its base and points
outward; remove the line segment that is the base of the triangle from step 2 (see Figure 5).Following the classical construction, we consider the following maps on 1-simplexes:
w1 :
⎛⎜⎜⎜⎜⎜⎝xA
yA
xB
yB
⎞⎟⎟⎟⎟⎟⎠ =⇒ M ·
⎛⎜⎜⎜⎜⎜⎝xA
yA
xB
yB
⎞⎟⎟⎟⎟⎟⎠,
w2 :
⎛⎜⎜⎜⎜⎜⎝xA
yA
xB
yB
⎞⎟⎟⎟⎟⎟⎠ =⇒ M ·
⎛⎜⎜⎜⎜⎜⎝xA
yA
xB
yB
⎞⎟⎟⎟⎟⎟⎠ +
⎛⎜⎜⎜⎜⎜⎝2/3
0
2/3
0
⎞⎟⎟⎟⎟⎟⎠,
(6.11)
where M is the matrix
M =
⎛⎜⎜⎜⎜⎜⎝1/3 0 0 0
0 1/3 0 0
0 0 1/3 0
0 0 0 1/3
⎞⎟⎟⎟⎟⎟⎠. (6.12)
18 Mathematical Problems in Engineering
Hence, w1 is a factor of the homothety w having A as a fixed vertex, while w2 leaves B
unchanged:
w1(P) = P − 2
3λiL0i, L0i = Vi −A,
w2(P) = P − 2
3λiL0i, L0i = Vi − B.
(6.13)
Moreover, each vertex in the wi([σ1]), i = 1, 2, can be expressed as in(3.6)
�
V i = Vi − 2
3L0i. (6.14)
Let us now consider the transformation, on two steps, which first rotates w1([A,B])of an angle θ = 60◦ around the fixed point A, and then it translates the rotated simplex by a
vector v = (1/3, 0).So that we obtain
w3 :
⎛⎜⎜⎜⎜⎜⎝xA
yA
xB
yB
⎞⎟⎟⎟⎟⎟⎠ =⇒ M′ ·
⎛⎜⎜⎜⎜⎜⎝xA
yA
xB
yB
⎞⎟⎟⎟⎟⎟⎠ +
⎛⎜⎜⎜⎜⎜⎝1/3
0
1/3
0
⎞⎟⎟⎟⎟⎟⎠, (6.15)
where M′ is the matrix
M′ =
⎛⎜⎜⎜⎜⎜⎝1/6 −√3/6 0 0√
3/6 1/6 0 0
0 0 1/6 −√3/6
0 0√
3/6 1/6
⎞⎟⎟⎟⎟⎟⎠. (6.16)
Finally, let us apply the transformation which first rotates w1([A,B]) of an angle θ =120◦ around the fixed point A, and then it translates the rotated simplex by the vector v =(2/3, 0). Accordingly, it is
w4 :
⎛⎜⎜⎜⎜⎜⎝xA
yA
xB
yB
⎞⎟⎟⎟⎟⎟⎠ =⇒ M′′ ·
⎛⎜⎜⎜⎜⎜⎝xA
yA
xB
yB
⎞⎟⎟⎟⎟⎟⎠ +
⎛⎜⎜⎜⎜⎜⎝2/3
0
2/3
0
⎞⎟⎟⎟⎟⎟⎠, (6.17)
Mathematical Problems in Engineering 19
Figure 4: Image of w([σ1]) =⋃
i = 1,...,4 wi([σ1]), where the 1-simplex σ1 is the unitary interval.
where M′′ is the matrix
M′′ =
⎛⎜⎜⎜⎜⎜⎝−1/6 −√3/6 0 0√
3/6 −1/6 0 0
0 0 −1/6 −√3/6
0 0√
3/6 −1/6
⎞⎟⎟⎟⎟⎟⎠. (6.18)
Since, as previously shown, rotation and translation are isometries, for each wi([σ1]),i = 1, 2, 3, 4, we obtain
�
Lij =1
3Lij ,
�
l ij =1
3lij ,
�
Ω =1
3Ω.
(6.19)
In order to visualize the von Koch pattern, let us consider the 1-simplex {A,B} ={{0, 0}, {1, 0}}; since the point A has been chosen as the origin of the reference system, and
w3 and w4 are obtained as rotation leaving fixed the origin, the transformed instances can be
easily computed so that, at the first step, the IFS maps on 1-simplexes can be drawn (Figure 4).Reiterating this process for each remaining segment, at the step k we will obtain the
compact set Tk made of 22k segments whose sides are contracted by a factor (1/3)k. The IFS
map (R1, d,w1, w2, w3, w4) gives us the Koch curve L =⋂
k Lk with similarity dimension
equal to
s =log 4
log 1/(1/3)=
log 4
log 3. (6.20)
This is also the similarity dimension of the Koch snowflake [27, 28]. So, the Koch curve
(Figure 5) is obtained as a combination of IFS simplicial maps generating the known fractal-
shaped curve.
6.3.2. Koch Flake
According to (5.16) and to the examples previously given, Koch flake (snowflake) can be
constructed in a non-classical approach as IFS of maps on a 2-simplex. Koch snowflake can
be seen as the image of a suitable system of iterated homotheties acting on a 2-simplex, given
by suitable translations of the boundary 1-simplexes.
20 Mathematical Problems in Engineering
Figure 5: Koch curve.
In this process, the total length of each side of a triangle increases by one-third, and
thus, the total length at the kth step will be (4/3)k of the original triangle perimeter.
7. Conclusion
In this paper, a nonclassical approach to fractal generation based on IFS of maps on simplexes
has been given. Some of the most popular fractals, as the Sierpinski gasket and the von Koch
flake, were obtained by iterative maps on simplexes. All maps were also intrinsically defined
by using the affine (barycentric) coordinates and some basic measures on simplexes. The
method proposed in this paper could be used to generate some new classes of fractals in any
dimension, by simply defining suitable IFS on simplexes, thus opening new perspectives in
fractal lattice geometry.
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 371890, 10 pagesdoi:10.1155/2012/371890
Research ArticleGaussian Curvature in Propagation Problems inPhysics and Engineering
Ezzat G. Bakhoum
Department of Electrical and Computer Engineering, University of West Florida,11000 University Parkway, Pensacola, FL 32514, USA
Correspondence should be addressed to Ezzat G. Bakhoum, [email protected]
Received 1 September 2011; Accepted 9 October 2011
Academic Editor: Cristian Toma
Copyright q 2012 Ezzat G. Bakhoum. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
The computation of the Gaussian curvature of a surface is a requirement in many propagationproblems in physics and engineering. A formula is developed for the calculation of the Gaussiancurvature by knowledge of two close geodesics on the surface, or alternatively from the projection(i.e., image) of such geodesics. The formula will be very useful for problems in general relativity,civil engineering, and robotic navigation.
1. Introduction
In many propagation problems in physics and engineering, it becomes necessary to compute
the Gaussian curvature of a two-dimensional surface. In physics, this becomes necessary in
the applications of general relativity, where it is sometimes desired to calculate the Gaussian
curvature at a point in space from the observed geodesic paths of planets or light rays [1, 2]. In
engineering, engineers who are involved in the design of structures such as geodesic domes
frequently require a practical formula for computing the Gaussian curvature, where relations
exist between the Gaussian curvature at any point on the surface of the structure and the
stability of such a structure [3]. In certain other engineering applications, such as computer
vision and robotic navigation, engineers sometimes find themselves facing the complicated
problem of having to compute the Gaussian curvature of a surface in order to calculate 3-
dimensional depth data (or range) [4–6].From the basic principles of differential geometry, the Gaussian curvature G at any
point of a two-dimensional surface S is given by
G = k1k2, (1.1)
2 Mathematical Problems in Engineering
S
S′
O
p
Figure 1: A general curve S that is embedded in a surface of revolution and a copy S′ that is separatedfrom S by a small rotation.
where k1 and k2 are the maximum and the minimum normal curvatures [6]. Unfortunately,
in many practical situations, k1 and k2 are simply unknown. In the following section, we
will derive a formula for computing the Gaussian curvature at any point on a surface by
knowledge of two close geodesics on the surface, or alternatively from the projection (i.e.,
image) of such geodesics (this is very important in applications such as general relativity and
robotic navigation, where no direct knowledge of the geodesics exists, but only an image of
the geodesics is available). A simple test of the formula is given in Section 3 (the test shows
that G, as computed from the formula, must vanish in an Euclidean 2-space). In Section 4, it
is proven that the Gaussian curvature is a projective invariant and hence can be calculated
from any projected image of two geodesics.
2. Calculation of the Gaussian Curvature from Geodesic Deviation
It is well known that any general 2-dimensional surface is topologically equivalent at any
given point to a surface of revolution [6]. Hence, two close geodesics on the surface, when
considered only within a small surface patch, can be treated as embedded in a surface
of revolution. Such curves, however, will not necessarily be geodesics in the surface of
revolution. Consider now a surface of revolution, where the smooth curve S is a general
curve that is embedded in the surface (Figure 1). S′ is a copy of S that is obtained by rotating
S through a small angle θ.
pr is a position vector, defined over a circular ring passing by S-S′. Let us select two
parameters u and v, such that u varies as we travel along the curve S, but v is constant, and
v varies as we pass from one curve to another, but u is constant. Obviously,
u = s, v = θ, (2.1)
Mathematical Problems in Engineering 3
where θ is the rotation angle of the axis from S to S′. Given such parameters on any surface
in space, it can be shown that [7]
(∇2
sθδρ)dθ −
(∇2
θsδρ)dθ =
∑ζ,μ,ν
Rρ
ζμνδζδνpμ, (2.2)
where δr is the unit tangent vector to the curve, Rabcd
is the mixed curvature tensor [7], and
the symbol ∇ is the covariant derivative operator [6, 7]. If the curve S was a geodesic in the
surface, we must have had [6, 7]
∇sδr = 0, (2.3)
since the covariant derivative of the unit tangent vector to a geodesic vanishes along the curve
[6, 7]. Since S is a general curve, however, then ∇sδr will be the components of a vector of
finite length, normal to the vector δr [6]. On the other hand, due to circular symmetry in a
surface of revolution, the vector ∇sδr , clearly, is parallel transported [6, 7] along a circular
ring in the surface. Hence, we must conclude that
∇θ(∇sδr) = ∇2
θsδr = 0, (2.4)
at any point on S. Further, given the parameters s and θ, it can be shown that [7]
(∇2
sθδr)dθ = ∇2
spr, (2.5)
for any 2-dimensional surface. From (2.4) and (2.5), (2.2) is rewritten as
∇2sp
ρ =∑ζ,μ,ν
Rρ
ζμνδζδμpν, (2.6)
where ζ, μ, ν = 1, 2. Moreover, it has be shown that [7]
Rabcd
= G(δac gbd − δa
dgbc
), (2.7)
for a smooth 2-dimensional manifold, where G is the Gaussian curvature, δab
is the Kronecker
delta, and gab are the components of the metric tensor at any point on the surface. Substituting
from (2.7) into (2.6) and carrying out the summation, we obtain
∇2sp
ρ +Gpρ = Gδρ∑μ,ν
gμνδμpν, (2.8)
where we have used the identity
∑μ,ν
gμνδμδν = 1. (2.9)
4 Mathematical Problems in Engineering
Axis of rotation
O
S′
S
r′
r
Figure 2: Unit tangent vectors to S and S′, respectively.
Equation (2.8) is analogous to the equation of geodesic deviation [6, 7]. Once again, if S was
a geodesic in the surface, we must have had an orthogonality condition
∑μ,ν
gμνδμpν = 0, (2.10)
and (2.8) would have reduced to the well-known equation of deviation of two geodesics in
a Riemannian 2-manifold. Equation (2.8), in its given form, will not allow the computation
of the Gaussian curvature G, since the metric tensor components, as well as all the covariant
derivatives on the surface, are unknown. However, (2.8) can be further reduced as follows:
for an infinitesimal rotation dθ,
pr =∂xr
∂θdθ. (2.11)
Thus,
∂pr
∂s=
∂δr
∂θdθ. (2.12)
Now,
δrS′ = δr
S +∂δr
S
∂θdθ
= δrS +
∂pr
∂s,
(2.13)
where δrS, δr
S′ are unit tangent vectors at S and S′, respectively, separated by a rotation dθ
(Figure 2).
Mathematical Problems in Engineering 5
Given that, for any vector pr on the surface [7],
∇spρ =
∂pρ
∂s+∑μ,ν
Γρμνpμ∂xν
∂s, (2.14)
where Γabc
is a Christoffel symbol of the second kind, we can always select coordinates such
that Christoffel symbols vanish at the origin [6, 7] (e.g., we can select coordinates on the
surface, at the location of the vector pr). Then, the vector∑
μ,νΓρμνp
μδν is generally very small
in the vicinity of the origin, and can be neglected (i.e., a linear approximation of ∇spρ is
assumed here. This approximation will be further justified in the following discussion and in
Section 3). Therefore, let
ηr =(δrS′ − δr
S
)=
∂pr
∂s≈ ∇sp
r. (2.15)
Further, let us define the deviation angle, ψ, as the angle between the two unit tangent vectors
δrS, δr
S′ , at any point along the curve S. Generally, the angle between two curves is given by
[7]
cosψ =∑μ,ν
gμνdxμ
ds· dx
ν
ds′, (2.16)
but since s = s′ is the length of the curve, and having δr = dxr/ds, we can write
cosψ =∑μ,ν
gμν(δμ)S(δν)S′ . (2.17)
Hence, from (2.15) and (2.17),
∑μ,ν
gμνημην =
∑μ,ν
gμν[δμ
S′δνS′ + δ
μ
SδνS − 2δ
μ
S′δνS
]= 2
(1 − cosψ
)≈∑μ,ν
gμν(∇sp
μ)(∇sp
ν).
(2.18)
We also see that
cosψ ≈∑μ,ν
gμνδμ[δν +∇sp
ν]
≈ 1 +∑μ,ν
gμνδμ(∇sp
ν).
(2.19)
6 Mathematical Problems in Engineering
Now, consider (2.8) and the summation
∑μ,ν
gμνpμ(∇2
spν)+G
∑μ,ν
gμνpμpν = G
∑α,β
gαβpαδβ
[∑μ,ν
gμνδμpν
]
= G
[∑μ,ν
gμνδμpν
]2
,
(2.20)
and let
P =√∑
μ,ν
gμνpμpν (2.21)
denote the Euclidean norm of the vector pr ; thus,
d
dsP 2 = 2
∑μ,ν
gμνpμ(∇sp
ν), (2.22)
d2
ds2P 2 = 2
∑μ,ν
gμν[pμ(∇2
spν)+(∇sp
μ)(∇sp
ν)]. (2.23)
Substitution from (2.18), (2.21), and (2.23) into (2.20) gives
1
2
d2P 2
ds2− 2
(1 − cosψ
)+GP 2 = G
[∑μ,ν
gμνδμpν
]2
. (2.24)
To evaluate the last term in (2.24), we rewrite (2.8) as
pρ = δρ∑μ,ν
gμνδμpν − 1
G∇2
spρ. (2.25)
Now, from (2.22) and (2.25), we have
dP 2
ds= 2
∑μ,ν
gμν
⎡⎣δμ∑α,β
gαβpαδβ − 1
G∇2
spμ
⎤⎦(∇spν)
= 2
(∑μ,ν
gμνpμδν
)(∑μ,ν
gμνδμ(∇sp
ν))
− 2
G
∑μ,ν
gμν(∇2
spμ)(
δνs′ − δν
s
).
(2.26)
Mathematical Problems in Engineering 7
Each of the components in the last term of (2.26) vanishes identically. To prove this, we
evaluate each of the components for each of the curves, S and S′, by substitution from (2.6).We have
∑μ,ν
gμνδμ(∇2
spν)=∑α
δα
⎡⎣ ∑ρ,ζ,μ,ν
gαρRρ
ζμνδζδμpν
⎤⎦=
∑α,ζ,μ,ν
Rαζμνδαδζδμpν
=∑
α,ζ,μ,ν
G(gαμgζν − gανgζμ
)δαδζδμpν.
(2.27)
A straightforward summation shows that the right-hand side of (2.27) vanishes. We therefore
conclude that ∇2sp
r is in the direction normal to the curve. In plus ∇spr is in the direction of
the tangent to the curve.
Finally, substitution from (2.19) into the first term of (2.26) gives
dP 2
ds= 2
(∑μ,ν
gμνpμδν
)(cosψ − 1
), (2.28)
or
∑μ,ν
gμνpμδν =
−dP 2/ds
2(1 − cosψ
) , (2.29)
and hence (2.24) is further reduced to
G =(1/2)d2P 2/ds2 − 2
(1 − cosψ
)[(dP 2/ds)/2(1 − cosψ)
]2 − P 2, (2.30)
where G is the Gaussian curvature of the surface at the location of the vector P . In the
following section, we prove that the Gaussian curvature given by (2.30) must vanish in an
Euclidean 2-space. In Section 4, it is further proven that G is a projective invariant and hence
can be calculated from any projected image of the curves S and S′.
3. Investigation of the Behavior of G in an Euclidean Space
Here, we illustrate by a simple example that the Gaussian curvature G, given by (2.30), must
vanish in an Euclidean 2-space.
Consider a right circular cone, shown in Figure 3 .
P is the Euclidean norm of the position vector, and θ is the rotation angle (as discussed
in the above text).
8 Mathematical Problems in Engineering
r
r
P
s
ss
r
r
Figure 3: A right circular cone and the corresponding geometry.
From the Figure 3 , we see that
P 2 = 2r2(1 − cos θ),
r = s sinα,(3.1)
where s is the length of the generator. Hence,
P 2 = 2(1 − cos θ)(s2sin2α
),
dP 2
ds= 4ssin2α(1 − cos θ).
(3.2)
Thus
d2P 2
ds2= 4sin2α(1 − cos θ). (3.3)
For an infinitesimal rotation, cos θ is expressed by the first two terms of its power series, that
is,
cos θ ≈ 1 − θ2
2!, (3.4)
and thus, (3.3) is written as
d2P 2
ds2= 2sin2αθ2. (3.5)
Mathematical Problems in Engineering 9
Furthermore, we can see that
r2θ2 = 2s2(1 − cosψ
), (3.6)
where ψ is the deviation angle, or
(1 − cosψ
)=
1
2sin2αθ2. (3.7)
From (3.5) and (3.7), we have
d2P 2
ds2= 4
(1 − cosψ
). (3.8)
By comparison of (2.30) and (3.8), we immediately see that G must vanish in an Euclidean
2-space. This proves the correctness of (2.30).
4. Proof That the Gaussian Curvature G Is a Projective Invariant
Now, we will reach our final goal by demonstrating that G, formulated by (2.30), can be
measured directly in the image plane.
We rewrite (2.8) as
G = − ∇2sp
ρ
pρ − δρ∑μ,νgμνδμpν
. (4.1)
From (2.27), we saw that ∇2sp
r is a vector in the direction normal to the curve. Now, by using
(2.9) and taking the summation
∑μ,ν
gμνδμpν −
∑μ,ν
gμνδμδν
[∑μ,ν
gμνδμpν
]= 0, (4.2)
it is easy to see that the denominator in (4.1) is also a vector in the direction normal to the
curve.
If we now let
∇2sp
ρ = αvρ,
pρ − δρ∑μ,ν
gμνδμpν = βvρ,
(4.3)
where α, β are scalars, and vρ is a vector in the direction normal to the curve, then
G = −αβ. (4.4)
10 Mathematical Problems in Engineering
Now, consider the orthographic projection of the two vectors in (4.3), written as
∇2sp
ρ =∑σ
Jρσ
(∇2
spσ)= α
∑σ
Jρσv
σ,
pρ − δρ∑μ,ν
gμνδμpν =∑σ
Jρσ
(pσ − δσ
∑μ,ν
gμνδμpν
)
= β∑σ
Jρσv
σ,
(4.5)
where Jab
is a transformation Jacobian between a coordinate system on the surface and a
coordinate system in the image plane.
If measured in the image plane, the Gaussian curvature G is now given by
G = − ∇2sp
ρ
pρ − δρ∑μ,νgμνδμpν
= −αβ, (4.6)
as we can easily see from (4.5).Hence,
G = G. (4.7)
The Gaussian curvature is therefore a projective invariant. It should be noted that, while
orthographic projection is assumed, the image plane may be placed in any arbitrary position
with respect to the curve S, and for all such positions, the Gaussian curvature K holds the
same numerical value. Equation (2.30) should be used in the image plane to obtain a correct
measurement of K.
References
[1] S. K. Blau, “Gravity probe B concludes its 50-year quest,” Physics Today, vol. 64, no. 7, pp. 14–16, 2011.[2] E. G. Bakhoum and C. Toma, “Relativistic short range phenomena and space-time aspects of pulse
measurements,” Mathematical Problems in Engineering, vol. 2008, Article ID 410156, 20 pages, 2008.[3] T. Rothman, “Geodesics, domes, and spacetime,” in Science a la Mode, Princeton University Press, 1989.[4] R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis, Wiley, New York, NY, USA, 1973.[5] B. K. Horn, Robot Vision, The MIT Press, 1987.[6] M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ, USA,
1976.[7] J. L. Synge and A. Schild, Tensor Calculus, Dover, New York, NY, USA, 1978.
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 653939, 28 pagesdoi:10.1155/2012/653939
Research ArticleSolving Linear Coupled FractionalDifferential Equations by Direct OperationalMethod and Some Applications
S. C. Lim,1 Chai Hok Eab,2 K. H. Mak,1 Ming Li,3 and S. Y. Chen4
1 Faculty of Engineering, Multimedia University, Selangor Darul Ehsan, 63100 Cyberjaya, Malaysia2 Department of Chemistry, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand3 School of Information Science & Technology, East China Normal University, No. 500, Dong-Chuan Road,Shanghai 200241, China
4 College of Computer Science, Zhejiang University of Technology, Hangzhou 310023, China
Correspondence should be addressed to S. C. Lim, [email protected]
Received 20 July 2011; Accepted 7 September 2011
Academic Editor: Carlo Cattani
Copyright q 2012 S. C. Lim et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
A new direct operational inversion method is introduced for solving coupled linear systems ofordinary fractional differential equations. The solutions so-obtained can be expressed explicitly interms of multivariate Mittag-Leffler functions. In the case where the multiorders are multiples ofa common real positive number, the solutions can be reduced to linear combinations of Mittag-Leffler functions of a single variable. The solutions can be shown to be asymptotically oscillatoryunder certain conditions. This technique is illustrated in detail by two concrete examples, namely,the coupled harmonic oscillator and the fractional Wien bridge circuit. Stability conditions andsimulations of the corresponding solutions are given.
1. Introduction
Fractional differential equations are well suited to model physical systems with memory or
fractal attributes. This is particularly true in the fields of condensed matter physics, where
fractional differential equations have been used to model various anomalous transport and
relaxation phenomena [1–9]. Coupled fractional differential equations (CFDEs) of nonlinear
type are widely used in studying various chaotic systems [10] such as the Lorentz system
[11], fractional Chuah’s circuit [12], fractional Rossler system [13], and fractional Duffing
system [14]. Since in most cases no analytic solutions for such nonlinear CFDEs exist, it is
necessary to resort to numerical approximations and simulations [15–21]. Even for linear
2 Mathematical Problems in Engineering
CFDEs with unequal multiorders, analytic solutions, if they exist, are difficult to obtain and
very often numerical methods have to be used.
In this paper we introduce a direct operational method to solve a system of linear
inhomogeneous CFDEs. We will restrict our discussion to a system of linear nonhomoge-
neous ordinary differential equations of arbitrary fractional-orders. These equations based
on two types of fractional derivatives will be considered, namely, the Caputo and Riemann-
Liouville fractional derivatives. The main idea is to reexpress the coupled fractional equations
by incorporating the initial conditions based on the definitions of these derivatives. The
solutions obtained can be expressed in terms of multivariate Mittag-Leffler functions. When
each order of the CFDEs is an integer multiple of a certain common real positive number, it
is possible to further reduce the solutions to the single-variate Mittag-Leffler functions. For
such cases, we study the conditions for the existence of asymptotic periodic solutions.
In the next section we consider two types of coupled fractional differential equations of
Caputo and Riemann-Liouville type, and a direct operational method is introduced to solve
these equations. Subsequent sections deal with the applications of the coupled fractional
differential equations to two physical systems, namely, the coupled fractional oscillator and
the fractional Wien bridge circuit, as examples to illustrate the proposed method.
2. Linear-Coupled Fractional Differential Equations
We consider two types of fractional derivatives [22–26]:
Caputo Dα∗f(t) = Im−αDmf(t), (2.1a)
Riemann-Liouville Dα#f(t) = DmIm−αf(t), (2.1b)
where the fractional integral is defined for γ > 0 as
Iγf(t) =1
Γ(γ) ∫∞
0
(t − τ)γ−1f(τ)dτ. (2.2)
When referring to either definition, we simply use the notation Dα.
Let us consider a linear-coupled system of inhomogeneous fractional differential equa-
tions of the form
DαX(t) = B(t) +AX(t), (2.3)
where X = (x1, . . . ,xn) and B = (b1, . . . , bn) are vectors of dimension n, A = (aij), i, j = 1, . . . , n
is an (n×n)-matrix, and Dα is the fractional differential operator given by the diagonal matrix
operator:
Dα =
⎛⎜⎜⎜⎜⎜⎜⎝Dα1 0 · · · 0
0 Dα2 · · · 0
......
. . ....
0 0 · · · Dαn
⎞⎟⎟⎟⎟⎟⎟⎠. (2.4)
Mathematical Problems in Engineering 3
The orders αi are real positive numbers with mi − 1 < αi < mi, where mi is a positive integer
for each i = 1, 2, . . . , n. The boundary conditions for (2.3) are given by
[Dki∗ xi
](0) = cki∗i for ki = 0, 1, 2, . . . , (mi − 1), (2.5a)[
Dαi−ki# xi
](0) = cki#i for ki = 1, 2, . . . , mi, (2.5b)
and i = 1, 2, . . . , n. We remark that the mi’s generally differ in value. Let its maximum and
minimum values be denoted by m and mo, respectively.
For a single fractional differential equation, its solution can be obtained by integral
transform methods such as the Fourier, Laplace, and Mellin transforms (see, e.g., [24, Chapter
4]). However, in the case of a system of CFDEs, it is necessary to employ specific techniques
appropriate to the given problem, that is, the form of matrix A and the type of fractional
derivative involved. There exist several methods (see [24, Chapters 5 and 6]) for solving
such problems. Here we want to develop a technique which is more direct, similar to Green’s
function method.
Let the operator be
L = Dα −A. (2.6)
The solution of (2.3) can then be expressed as
X(t) = L−1B. (2.7)
Unfortunately, the inverse of L may not exist. However, the right-inverse G does exist for
both the Riemann-Liouville and Caputo cases with
LG = 1/=GL. (2.8)
The main task now is to determine the right-inverse of L associated with (2.3) for both
Riemann-Liouville and Caputo fractional derivatives. We remark that our treatment is rather
formal, aiming mainly to provide an alternative direct operational method to the usual
Laplace transform technique in solving CFDEs. In particular, the existence of solutions will be
assumed, and all operators considered are assumed to be well defined in a certain appropriate
function space.
2.1. The Right-Inverse Operator
For the Caputo derivative Dα∗ with arbitrary α and m,
IαDα∗ f(t) = IαIm−αDmf(t) = ImDmf(t)
= f(t) −m−1∑k=0
tk
Γ(k + 1)
[Dkf
](0),
(2.9a)
4 Mathematical Problems in Engineering
and similarly for the Riemann-Liouville derivative,
IαDα#f(t) = IαDmIm−αf(t)
= f(t) −m∑k=1
tα−k
Γ(α − k + 1)
[Dα−k
# f](0).
(2.9b)
Applying the fractional integral operator Iα to (2.3) and using the initial conditions (2.5a) and
(2.5b) gives
X = IaAX + IaB +W, (2.10)
where W is given by
w∗i =mi−1∑k=0
tk
Γ(k + 1)ck∗i,
w#i =mi∑k=1
tα−k
Γ(α − k + 1)ck#i.
(2.11)
Let
Q = [1 − IαA]. (2.12)
Now by rearranging (2.10) one gets
QX = IαB +W. (2.13)
The operator Q has an inverse K. One possible representation of K is given by
K =∞∑0
(IαA)p. (2.14)
This form may not be a simple one, since in most cases Iα and A do not commute. However,
the verification is straightforward:
QK = KQ =∞∑p=0
(IαA)p −∞∑p=0
(IαA)p+1 = 1. (2.15)
The other possible representation is
K = ΨQ∗, (2.16)
Mathematical Problems in Engineering 5
where Q∗ is the adjoint of Q, that is, Q∗Q = detQ, and the Ψ is the inverse operator of detQsuch that ΨdetQf(t) = f(t). It is quite simple to verify that this representation is the inverse
of Q, and this will be shown in Section 4.
Now we define
G = KIα. (2.17)
One can easily verify that G is the right inverse of L. Thus, the solution is given by
X = GB +KW. (2.18)
The detailed evaluation can be carried out by using different techniques, a few of which will
be considered here.
2.2. System with Constant Inhomogeneous Terms
When the inhomogeneous term is a constant, we can absorb this term in the following way,
though it may not be immediately obvious. For the Caputo case, let
X∗ = X∗ −A−1B, (2.19)
with
x∗i(0) = x∗i(0) +(a−1
)ijbj . (2.20)
The initial conditions have to be transformed accordingly and they become
c0∗i = c0
∗i +(a−1
)ijbj . (2.21)
In the Riemann-Liouville case, we cannot absorb the term B into X as in the Caputo
case. However, if one compares the initial condition terms w#i and the term IαB, one can see
that if we modify
w#i =mi∑k=0
tα−k
Γ(α − k + 1)ck#i, (2.22)
with c0#i = bi, then the solution can be written as if it is a solution of a homogeneous linear
equation.
Thus the inhomogeneous linear fractional differential equation with constant source
term bi can be transformed into a homogeneous linear fractional differential equation. The
solution of the transformed equation can then be written as
X = KW. (2.23)
6 Mathematical Problems in Engineering
When bi is time dependent with power up to mi, the above modification can still apply to
the Caputo case. In the Riemann-Liouville case, if bi(·) are analytic functions, then the same
modification as above holds if (2.22) is altered with the summation from −∞ to mi − 1.
In subsequent sections, we will consider the solution of (2.3) according to the above
modifications.
3. System with Equal Fractional Orders
In the case where all αi = α, then Iα = Iα1, and
K =∞∑p=0
AnInα. (3.1)
It would be convenient if we introduce the Mittag-Leffler function with matrix argument
[27–29]:
Eα,β(Z) =∞∑n=0
Zn
Γ(nα + β
) . (3.2)
Then
Kδ(t) = t−1Eα,0(Atα). (3.3)
Here we have used the following definition of the Dirac delta function:
δ(t) = limε→ 0
tε−1+
Γ(ε). (3.4)
The matrix A can always be decomposed into Jordan normal form. However, we consider
only the case where it can be diagonalized:
Λ = P−1AP,
K = P−1KP,(3.5)
with eigenvalues λj as the diagonal elements of Λ, and
Kijδ(t) = δijt−1Eα,0
(λjt
α), (3.6)
X∗ =m−1∑k=0
PtkEα,k+1(Λtα)P−1Ck∗ ,
X# =m−1∑k=−1
Ptα−k−1Eα,α−k(Λtα)P−1Ck# .
(3.7)
Mathematical Problems in Engineering 7
Thus, the solutions of the fractional differential equation under consideration based on both
types of fractional derivatives are simply linear combinations of Mittag-Leffler functions.
3.1. Coupled Oscillator with Equal Fractional Orders
Here we demonstrate our method by considering a linear-coupled oscillator system given by
Dα1x1(t) = −ω2x1(t) +ω2(x2(t) − x1(t)),
Dα2x2(t) = −ω2x2(t) +ω2(x1(t) − x2(t)),(3.8)
where ω2 and ω2 are nonnegative real numbers. The initial conditions are
x∗j(0) = c0∗j , Dx∗j(0) = c1
∗j ,
Dαj−1x#j(0) = c0#j , Dαj−2x#j(0) = c1
#j .(3.9)
Thus
A = −(ω2 +ω2 −ω2
−ω2 ω2 +ω2
). (3.10)
For simplicity, we use the following notation:
a11 = a22 = −(ω2 +ω2
)= −η,
a12 = a21 = ω2 = ε.
(3.11)
We first consider the simpler case with α1 = α2 = α in this section. Since A is symmetric, it can
be diagonalized to give two eigenvalues:
λ± = −η ± ε =
⎧⎨⎩−ω2
−ω2 −ω2.(3.12)
Both eigenvalues are nonpositive and
P =1√2
(1 1
−1 1
). (3.13)
8 Mathematical Problems in Engineering
0 5 10 15 20 25 30
−1
−0.5
0
0.5
1
t
= 1.3x(t)
(a)
0 5 10 15 20 25 30
−1
−0.5
0
0.5
1
t
= 1.5
x(t)
(b)
= 1.7
0 5 10 15 20 25 30
−1
−0.5
0
0.5
1
t
x(t)
(c)
= 1.9
0 5 10 15 20 25 30
−1
−0.5
0
0.5
1
t
x(t)
(d)
Figure 1: Coupled oscillator with equal fractional orders. Parameters: ω = 1, ω = 0.5, x∗1(0) = 1.0, x∗2(0) =0, Dx∗1(0) = 0, Dx∗2(0) = 0.1. Legend: x∗1 (solid line), x∗2 (dashed line).
Just as we have shown in (3.7), the solutions are again linear combinations of Mittag-Leffler
functions:
x∗1 =(c0∗1 + c0
∗2
)Eα,1(λ−tα) +
(c0∗1 − c0
∗2
)Eα,1(λ+t
α) +(c1∗1 + c1
∗2
)tEα,2(λ−tα)
+(c1∗1 − c1
∗2
)tEα,2(λ+t
α),(3.14a)
x#1 =(c1
#1 + c1#2
)tα−1Eα,α(λ−tα) +
(c1
#1 − c1#2
)tα−1Eα,α(λ+t
α) +(c2
#1 + c2#2
)tα−2Eα,α−1(λ−tα)
+(c2
#1 − c2#2
)tα−2Eα,α−1(λ+t
α).
(3.14b)
Figure 1 shows simulations of the Caputo solution (3.14a) for orders 1 < α ≤ 2. An
interesting feature of fractional oscillators in general is the presence of damping internal to the
system, that is, an inherent decay in the amplitude which is not associated with any external
friction. The variation in the amount of internal damping can be clearly seen as the order
increases. In the limiting cases we obtain exponential decay (α = 1) and undamped oscillation
(α = 2).
Mathematical Problems in Engineering 9
4. The Adjoint Method
As we mentioned in Section 2.1, (2.16), the main task now is to calculate the inverse of detQ.
Note that (2.16) can be reexpressed as
K = ψ[1 − IαA]∗, (4.1)
so that
G = ψ[1 − IαA]∗Iα. (4.2)
Also recall that
detQ = det(1 − IαA). (4.3)
We know that any finite dimension determinant can be evaluated; however, it can be easily
obtained only for a few lower-dimensional cases. We will compute it explicitly for a two-
dimensional system.
4.1. Two-Dimensional System
In a two-dimensional system the determinant is easy to calculate, and we have
detQ = 1 − a11Iα1 − a22I
α2 + Iα1+α2 detA. (4.4)
The inverse is given by
ψ =∞∑r=0
[a11Iα1 + a22I
α2 − Iα1+α2 detA]r (4.5)
=∞∑r=0
∑k1+k2+k3=r
r!
k1!k2!k3![a11I
α1]k1[a22Iα2]k2[−Iα1+α2 detA]k3 . (4.6)
Its kernel is the multivariate Mittag-Leffler function of the kind given by (B.3), Appendix B:
ψδ(t) = t−1Eα1,α2,α1+α2,0(a11tα1 , a22t
α2 ,−detAtα1+α2)
= εα1,α2,α1+α2,0(a11, a22,−detA : t).(4.7)
The adjoint of Q is
Q∗ =
(1 − a22I
α2 a12Iα1
a21Iα2 1 − a11I
α1
). (4.8)
10 Mathematical Problems in Engineering
In the following we evaluate explicitly only for the 11- and 12-elements, while the 22- and
21-elements can be obtained by just interchanging subscripts. The kernels of K are given by
K11δ(t) = εα1,α2,α1+α2,0(a11, a22,−detA : t) − a22εα1,α2,α1+α2,α2(a11, a22,−detA : t),
K12δ(t) = a12εα1,α2,α1+α2,α1(a11, a22,−detA : t).(4.9)
4.2. The Solutions
The solutions based on the adjoint method are given by
x∗1 =m1−1∑k=0
ck∗1[εα1,α2,α1+α2,k+1(a11, a22,−detA : t)
−a22εα1,α2,α1+α2,α2+k+1(a11, a22,−detA : t)]
+ a12
m2−1∑k=0
ck∗2εα1,α2,α1+α2,α1+k+1(a11, a22,−detA : t),
(4.10a)
x#1 =m1−1∑k=0
ck#1[εα1,α2,α1+α2,α1−k+1(a11, a22,−detA : t)
−a22εα1,α2,α1+α2,α2+α1−k+1(a11, a22,−detA : t)]
+ a12
m2−1∑k=0
ck#2εα1,α2,α1+α2,α1+α2−k+1(a11, a22,−detA : t).
(4.10b)
5. Laplace Transform Method
In this section we briefly discuss how the widely used Laplace transform technique can be
employed to determine Green’s function G and the operator K. There is no intention here to
provide a detailed discussion of this method. Instead, it will be discussed as a complement to
the adjoint method introduced above, so as to allow one to see the relation between the direct
operational method presented here and usual Laplace transform method.
Without loss of generality (see Section 2.2) the system of CFDEs is assumed to be
homogeneous. We begin by calculating the Laplace transform of detQδ(t) using (4.5):
∫∞
0
detQδ(t)e−stdt = 1 − a11s−α1 − a22s
−α2 + detAs−(α1+α2) (5.1)
and the Laplace transform of the adjoint kernel Q∗δ(t):
∫∞
0
Q∗11δ(t)e
−stdt = 1 − a22s−α2 ,
∫∞
0
Q∗12δ(t)e
−stdt = a12s−α1 ,
∫∞
0
Q∗21δ(t)e
−stdt = a21s−α2 ,
∫∞
0
Q∗22δ(t)e
−stdt = 1 − a11s−α1 .
(5.2)
Mathematical Problems in Engineering 11
Next, the Laplace transforms of the part related to the initial conditions from (2.11) are
∫∞
0
w∗i(t)e−stdt =mi−1∑k=0
s−k−1ck∗i,
∫∞
0
w#i(t)e−stdt =mi∑k=1
s−αi+k−1ck#i.
(5.3)
Thus the Laplace transforms of the solutions become
x∗1(s) =[1 − a22s
−α2]∑m1−1
k=0s−k−1ck∗1 + a12s
−α1∑m2−1
k=0s−k−1ck∗2
1 − a11s−α1 − a22s−α2 + det As−(α1+α2), (5.4a)
x#1(s) =[1 − a22s
−α2]∑m1
k=1s−α1+k−1ck#1 + a12s
−α1∑m2
k=1s−α2+k−1ck#2
1 − a11s−α1 − a22s−α2 + det As−(α1+α2). (5.4b)
x∗2(s) and x#2(s) can be obtained just by interchanging 1 ↔ 2. From the complexity of the
Laplace transforms, one sees that it is virtually impossible to obtain the analytic solutions
by direct application of the inverse Laplace transform. To obtain the solution of this type
one has to use the Laplace transform of the multivariate Mittag-Leffler function [27–29],which then gives the identity for getting the Laplace inversion of (5.4a) and (5.4b). This is
one main advantage of the direct operational inversion method proposed here as it will give
the solution directly.
Clearly, it is important that both methods produce equivalent solutions. This is verified
explicitly in Appendix C for a 2-dimensional system.
6. Multiple Fractional-Order System
In physics and engineering problems the fractional-orders can often be approximated by
rational numbers, that is, αi = pi/qi, for some pi, qi ∈ . Thus one gets αi = μi/q, where
q is the least common multiple of q1, q2, . . . , qn with some μi ∈ . However, we can also
consider the more general case with αj = μjα0, for some μj ∈ . Here, α0 ∈ + can be either
rational or irrational. In this section we show how such a system of CFDEs with these multiple
fractional-orders can be solved.
Referring to (4.4), if we assign the symbol ξ = Iα0 , the expansion of the determinant
will be a polynomial of order μ = μ1 + μ2 + μ3 + · · · + μn in ξ. By the fundamental theorem of
algebra, it must have in general μ complex roots, that is, ζj for j = 1, 2, 3, . . . , μ. Note that since
all coefficients of the polynomial are real, if any root ζp is complex, its complex conjugate ζpis also a root. That means that the complex roots occur in pairs. For convenience we write
λj = 1/ζj , for j = 1, 2, 3, . . . , μ. We can then factorize the polynomial
detQ =μ∏j=1
(1 − λjξ
). (6.1)
12 Mathematical Problems in Engineering
If all roots are distinct, the inverse can be written as a partial fraction:
ψ =μ∑j=1
hj
1 − λjξ
=μ∑j=1
hj
∞∑k=0
λkj I
kα0 .
(6.2)
6.1. Two-Dimensional System
We will explore this method further for two dimensions. Using (4.5) and the adjoint in (4.8),the solution is given by
x∗1 =μ1+μ2∑j=1
hj
{m1−1∑k=0
ck∗1
[tkEα0,k+1
(λjt
α0) − a22t
μ2α0+kEα0 ,μ2α0+k+1
(λjt
α0)]
+a12
m2−1∑k=0
ck∗2tμ1α0+kEα0,μ1α0+k+1
(λjt
α0)}
,
x#1 =μ1+μ2∑j=1
hj
{m1∑k=1
ck#1
[tμ1α0−kEα0 ,μ1α0−k+1
(λjt
α0) − a22t
(μ1+μ2)α0−kEα0,(μ1+μ2)α0−k+1
(λjt
α0)]
+a12
m2∑k=1
ck#2t(μ1+μ2)α0−kEα0 ,(μ1+μ2)α0−k+1
(λjt
α0)}
.
(6.3)
To simplify the problem we consider the case where 0 < max(μ1α0, μ2α0) ≤ 1, that is,
m1 = m2 = 1. The extension to the general case as above is straightforward:
x∗1 =μ1+μ2∑j=1
hj
{c0∗1
[Eα0,1
(λjt
α0) − a22t
μ2α0Eα0,μ2α0+1
(λjt
α0)]
+ a12c0∗2t
μ1α0Eα0,μ1α0+1
(λjt
α0)}
,
x#1 =μ1+μ2∑j=1
hj
{c1
#1tμ1α0−1Eα0,μ1α0
(λjt
α0) − (a22c
1#1 − a12c
1#2
)t(μ1+μ2)α0−1Eα0,(μ1+μ2)α0
(λjt
α0)}
.
(6.4)
Using the following formula:
zqEα,qα+γ (z) = Eα,γ(z) −q−1∑p=0
zp
Γ(pα + γ
) , (6.5)
Mathematical Problems in Engineering 13
(6.4) can be written as
x∗1 = f∗1 + g∗1,
x#1 = f#1 + g#1,(6.6)
where
f∗1 =μ1+μ2∑
j
hjfj
∗1, g∗1 =μ1+μ2∑
j
hjgj
∗1, (6.7a)
f#1 =μ1+μ2∑
j
hjfj
#1, g#1 =μ1+μ2∑
j
hjgj
#1. (6.7b)
fj
∗1 =[c0∗1
(1 − a22λ
−μ2
j
)+ a12c
0∗2λ
−μ1
j
]Eα0,1
(λjt
α0), (6.8a)
fj
#1 =[c1
#1λ1−μ1
j −(a22c
01 − a12c
1#2
)λ
1−μ1−μ2
j
]tα0−1Eα0,α0
(λjt
α0). (6.8b)
gj
∗1 = a22c0∗1λ
−μ2
j
μ2−1∑p=0
λp
j tpα0
Γ(pα0 + 1
) − a12c0∗2λ
−μ1
j
μ1−1∑p=0
λp
j tpα0
Γ(pα0 + 1
) , (6.9a)
gj
#1 = −c1#1λ
1−μ1
j
μ1−2∑p=0
λp
j tpα0+α0−1
Γ(pα0 + α0
) +(a22c
1#1 − a12c
1#2
)λ
1−μ1−μ2
j
μ1+μ2−2∑p=0
λp
j tpα0+α0−1
Γ(pα0 + α0
) . (6.9b)
6.2. Solutions with Asymptotic Oscillations
It is clear from the previous expansion of the Caputo terms that for the solution x∗1, as t → ∞,
any possible oscillation arises from the pair of complex roots (λj, λj), while all (negative) real
roots must result in an asymptotic decay. Note that each term in (6.9a) with p > 0 will grow
asymptotically as the power law tpα0 . However, when we combine the terms and reexpress
the equation explicitly as
g∗1 =μ1+μ2∑j=1
hjgj
∗1
= a22c0∗1
μ2−1∑p=0
⎡⎣μ1+μ2∑j=1
hjλp−μ2
j
⎤⎦ tpα0
Γ(pα0 + 1
) − a12c0∗2
μ1−1∑p=0
⎡⎣μ1+μ2∑j=1
hjλp−μ1
j
⎤⎦ tpα0
Γ(pα0 + 1
) ,(6.10)
14 Mathematical Problems in Engineering
for the particular case with μ1 = 1, μ2 = 2, one has
g∗1 = a22c0∗1
1∑p=0
⎡⎣ 3∑j=1
hjλp−2
j
⎤⎦ tpα0
Γ(pα0 + 1
) − a12c0∗2
⎡⎣ 3∑j=1
hjλ−1j
⎤⎦ 1
Γ(1)(6.11)
= a22c0∗1
⎧⎨⎩⎡⎣ 3∑
j=1
hjλ−2j
⎤⎦ 1
Γ(1)+
⎡⎣ 3∑j=1
hjλ−1j
⎤⎦ tα0
Γ(α0 + 1)
⎫⎬⎭ − a12c0∗2
⎡⎣ 3∑j=1
hjλ−1j
⎤⎦ 1
Γ(1).
(6.12)
The second term is the only term that grows asymptotically as ∼ tα0 , which does not
contribute since [∑3
j=1 hjλ−1j ] is zero. The verification of this result is given here for the general
case of n-roots. Let us consider general partial fractions:
1
(1 − λ1x)(1 − λ2x) · · · (1 − λnx)=
h1
(1 − λ1x)+
h2
(1 − λ2x)+ · · · + hn
(1 − λnx). (6.13)
We have
h1 + h2 + · · · + hn = 1, (6.14.0)
h1(λ2 + λ3 + · · ·λn) + h2(λ3 + λ4 + · · ·λn + λ1) + · · · + hn(λ1 + λ2 + · · ·λn−1) = 0,
(6.14.1)
... (...)
h1λ2λ3 · · ·λn + h2λ3λ4 · · ·λnλ1 + · · · + hnλ1λ2 · · ·λn−1 = 0. (6.14.n−1)
We can rewrite (6.14.n−1) as
h1
λ1+h2
λ2+h3
λ3+ · · · + hn
λn= 0. (6.15)
Using (6.15) with n = 3 in (6.12), we have
g∗1 = a22c0∗1
⎡⎣ 3∑j=1
hjλ−2j
⎤⎦, (6.16)
which is a constant. Thus for this particular case with the Caputo derivative one can have
asymptotic oscillations. In general, CFDEs based on the Caputo derivatives will not oscillate
asymptotically, since one cannot find any rule for the power law terms to cancel out.
However, this is possible for some special cases under suitable conditions on the elements
of A.
Similar consideration can be given to Riemann-Liouville system. However, now gj
#1
approaches a constant or possible zero as t → ∞ if 0 < max(μ1α0, μ2α0) ≤ 1, and
Mathematical Problems in Engineering 15
min(μ1, μ2) ≤ 2. Thus the possible asymptotically stable oscillations are due to the term fj
#1
with its corresponding complex conjugate. If one looks at (6.8b), one can write explicitly the
asymptotic expansion:
tα0−1Eα0,α0
(λjt
α0)=λ(1−α0)/α0
j
α0eλ
1/α0j t + tα0−1O
(1
tα0
)−→
λ(1−α0)/α0
j
α0eλ
1/α0j t +O
(1
t
)
−→λ(1−α0)/α0
j
α0eλ
1/α0j t
.
(6.17)
One has to impose the condition that λ1/α0
j be a purely imaginary number. Let us denote the
inversion of the root by λj = |λj |eiθ; we must have
θ = ±α0π
2. (6.18)
Furthermore, all the other roots that will not give rise to oscillation must have a negative real
part so that their contributions will be asymptotically zero.
Assume that there is only one pair of complex roots that satisfies (6.18). The coefficient
of the exponential in (6.17) after substitution into (6.8b) gives the jth term of (6.7b) as
hj
[c1
#1λ1−μ1
j −(a22c
1#1 − a12c
1#2
)λ
1−μ1−μ2
j
]λ(1−α0)/α0
j
α0= rje
iϕ, (6.19)
and we then have the asymptotic solution:
x1 −→ rjei|λj |t+iϕ + rje
−i|λj |t−iϕ = 2rj cos(∣∣λj
∣∣t + ϕ). (6.20a)
x2 can be evaluated in a similar way; it has the same period but different modulus and phase:
x2 −→ 2r ′j cos(∣∣λj
∣∣t + ϕ′). (6.20b)
We omit the determination of the roots for each system, which can be computed without any
difficulty.
The oscillation condition (6.18) was first derived by Matignon [30] who also showed
that an identical condition existed for the eigenfunction Eα,1(z) of the Caputo derivative. This
will be elaborated in the context of a physical system in Section 7.
7. Wien Bridge System
In this section we apply the solution methods discussed earlier to model a fractional-order
Wien bridge oscillator (Figure 2). The Wien bridge is a common electronic circuit that can
generate a sinusoidal output signal without requiring an oscillatory input. The resistor-
capacitor pairs form a frequency-selective network, hence allowing the selection of output
16 Mathematical Problems in Engineering
−
+
R
RC
vo
C
(a)
R
R
C
C
vo
v1(t)
+
−
v2(t)
+
−
(b)
Figure 2: Wien bridge oscillator: (a) circuit schematic with operational amplifier and (b) simplified circuitdiagram for voltage analysis.
sine wave frequency by varying the circuit parameters. Ahmad et al. [31] first proposed
a generalization of this circuit using fractional-order capacitors. Since the authors did not
obtain the solutions explicitly, we briefly show how analytic solutions for such a system
can be obtained within our present framework. Also, we show solutions based on both the
Caputo and the Riemann-Liouville derivatives. (Note that reference [31] does not mention
the type of derivative used; we were informed by one of the authors, Professor Ahmad, that
they used the Riemann-Liouville fractional derivative.)It is well known that a fractional differential equation of order 0 < α < 1 is usually used
to describe relaxation phenomena [2]. In the case of Wien bridge system, however, oscillation
is achieved via the active elements and feedback provided in the circuit (see Section 7.3).In the following we use normalized voltages xi = vi/Vsat where Vsat is the amplifier
saturation voltage and time axes (normalized with respect to time constant τ = RC).Using basic circuit analysis, it can be shown that the capacitor voltages are related via a 2-
dimensional CFDE:
DαX = AX + B, (7.1)
where
A =
(a − 2 −1
a − 1 −1
), B =
(b
b
), (7.2a)
(a, b) =
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩(0, 1), Kx1 ≥ 1,
(K, 0), −1 < Kx1 < 1,
(0, 1), Kx1 ≤ 1.
(7.2b)
Mathematical Problems in Engineering 17
Here K is the amplifier gain (i.e., vo = Kv1). In the linear region of the amplifier, −1 < Kx1 < 1
and (7.2a) simplifies to
A =
(K − 2 −1
K − 1 −1
), B =
(0
0
), (7.3)
Thus we have to solve the homogeneous linear fractional-order differential system with
a11 = K − 2, a21 = K − 1, a12 = −1, a22 = −1. (7.4)
For fractional capacitors, the real orders are restricted to
0 < α1 ≤ 1, 0 < α2 ≤ 1. (7.5)
We remark that the boundary conditions associated with the Caputo derivative seem more
“physical” as they can be verified by experiments, whereas for the Riemann-Liouville case,
the fractional derivative boundary conditions cannot be measured. However, the Riemann-
Liouville operators are popular with mathematicians and theoretical physicists. We can write
the initial conditions explicitly as
x1(0) = c0∗1, x2(0) = c0
∗2,
Dα1−1# x1(0) = c1
#1, Dα2−2# x2(0) = c1
#2.(7.6)
7.1. Solution Using the Adjoint Method
Substituting (7.4) into the solution (4.10a) we get for the Caputo case
x∗1 = c0∗1[εα1,α2,α1+α2,1(K − 2,−1,−1 : t) + εα1,α2,α1+α2,α2(K − 2,−1,−1 : t)]
− c0∗2εα1,α2,α1+α2,α1(K − 2,−1,−1 : t),
x∗2 = c0∗2[εα1,α2,α1+α2,1(K − 2,−1,−1 : t) − (K − 2)εα1,α2,α1+α2,α1(K − 2,−1,−1 : t)]
+ (K − 1)c0∗1εα1,α2,α1+α2,α2(K − 2,−1,−1 : t).
(7.7)
Similarly, for the Riemann-Liouville case, substituting (7.4) into the solution (4.10b), we get
x#1 = c1#1[εα1,α2,α1+α2,α1(K − 2,−1,−1 : t) + εα1,α2,α1+α2,α2+α1−1(K − 2,−1,−1 : t)]
− c1#2εα1,α2,α1+α2,α1+α2−1(K − 2,−1,−1 : t),
x#2 = c1#2[εα1,α2,α1+α2,α2(K − 2,−1,−1 : t) − (K − 2)εα1,α2,α1+α2,α2+α1−1(K − 2,−1,−1 : t)]
− c1#1(K − 1)εα1,α2,α1+α2,α1+α2−1(K − 2,−1,−1 : t).
(7.8)
18 Mathematical Problems in Engineering
7.2. Solution Using the Laplace Transform
Substituting (7.4) into the Laplace transform solution (5.4a), we obtain
x∗1(s) =[1 + s−α2]s−1c0
∗1 − s−α1−1c0∗2
1 − (K − 2)s−α1 + s−α2 + s−(α1+α2), (7.9a)
x∗2(s) =[1 − (K − 2)s−α1]s−1c0
∗2 + (K − 1)s−α2−1c0∗1
1 − (K − 2)s−α1 + s−α2 + s−(α1+α2). (7.9b)
Similarly, for the Laplace transform of the solution (5.4b),
x#1(s) =[1 + s−α2]s−α1c1
#1 − s−(α1+α2)c1#2
1 − (K − 2)s−α1 + s−α2 + s−(α1+α2),
x#2(s) =[1 − (K − 2)s−α1]s−α2c1
#2 + (K − 1)s−(α1+α2)c1#1
1 − (K − 2)s−α1 + s−α2 + s−(α1+α2).
(7.10)
In the following subsections, we study the conditions under which asymptotically stable
oscillations are possible for a fractional Wien bridge oscillator and also present numerical
simulations of the capacitor voltages.
7.3. Equal-Order Fractional Wien Bridge
The classical Wien bridge oscillator produces a stable sinusoidal output when its amplifier
gain K = 3. The amplitude and frequency of the sinusoid are a function of the initial
capacitor voltages and the circuit time constant. For a fractional capacitor, the current-voltage
relationship is dependent on both capacitor value and order; hence an additional degree of
freedom is introduced into the Wien bridge circuit. We consider first the simple case with
equal real fractional orders α1 = α2 = α. Equation (7.7) then simplifies to
x∗1 = c0∗1εα,α,2α,1(K − 2,−1,−1 : t) +
(c0∗1 − c0
∗2
)εα,α,2α,α(K − 2,−1,−1 : t),
x∗2 = c0∗2εα,α,2α,1(K − 2,−1,−1 : t) +
[c0∗1(K − 1) − c0
∗2(K − 2)]εα,α,2α,α(K − 2,−1,−1 : t).
(7.11)
To gain further insight into the system’s behaviour, it is advantageous to express the solution
in a simpler form using only 1-parameter Mittag-Leffler functions Eα,1(λitα). From (7.9a),
x∗1(s) = sα−1c0∗1s
α +(c0∗1 − c0
∗2
)s2α − (K − 3)sα + 1
=σ1s
α−1
sα − λ1+
σ2sα−1
sα − λ2, (7.12)
where the σi ∈ are constants to be determined from partial fraction decomposition (see
equivalent method in Section 6). The inverse transform yields
x∗1(t) = σ1Eα,1(λ1tα) + σ2Eα,1(λ2t
α). (7.13)
Mathematical Problems in Engineering 19
0 5 10 15 20
−0.4
−0.2
0
0.2
0.4
t
= 0.3
x(t)
(a)
= 0.5
0 5 10 15 20
−0.4
−0.2
0
0.2
0.4
t
x(t)
(b)
= 0.7
0 5 10 15 20
−0.4
−0.2
0
0.2
0.4
t
x(t)
(c)
0 5 10 15 20
−0.4
−0.2
0
0.2
0.4
t
= 1
x(t)
(d)
Figure 3: Caputo model of fractional Wien bridge with equal orders—comparison of phase and amplitudefor capacitor voltages. Parameters: x∗1(0) = x∗2(0) = 0.03. Legend: x∗1 (solid line); x∗2 (dashed line).
The solution for x∗2(t) can be found in a similar manner. For brevity, we present only
numerical simulations of the Caputo solutions (one plot of the Riemann-Liouville solution
is presented for comparison). In order for the Wien bridge to produce sustained oscillations,
we need to impose condition (6.18) on the complex roots λi. For the current system, this
translates to the following expression for K:
K = 3 + 2 cos(απ
2
). (7.14)
Hence, the amplifier gain K is no longer a constant as in the case of the classical Wien bridge
but a function of capacitor order α. Simulations of (7.13) were plotted using Mathematica.
Figure 3 shows plots of x∗1 and x∗2 for α = 0.3, 0.5, 0.7, and 1.0. There is a clear
dependence of waveform amplitude on the fractional-order. The plot for α = 1 corresponds
to the classical Wien bridge (with ordinary capacitors) and is included for comparison. It is
important to keep in mind that the time axes are normalized and oscillation frequency ωα
actually varies with order as
ωα = (RC)−1/α. (7.15)
20 Mathematical Problems in Engineering
0 5 10 15 20
−0.4
−0.2
0
0.2
0.4
t
x1(t)
RC = 0.9
(a)
0 5 10 15 20
−0.4
−0.2
0
0.2
0.4
t
x1(t)
= 0.3
(b)
Figure 4: Variation of waveform characteristics for x1(t). Parameters: x∗1(0) = x∗2(0) = 0.05, (a) Capacitororder affects both frequency and amplitude. Legend: α = 0.3 (solid), 0.5 (dashed), 0.7 (dash-dotted), 1.0(dotted) and (b) Time constant affects frequency while amplitude remains constant. Legend: RC = 0.8(solid), 0.9 (dashed), and 1.0 (dash-dotted), and 1.1 (dotted).
0 5 10 15 20
t
= 0.3
−0.4
−0.2
0
0.2
0.4
x1(t)
Figure 5: Comparison of x1(t) waveform for both derivative models. Solution for Riemann-Liouville modeldiverges as t → 0. Parameters: x∗1(0) = x∗2(0) = 0.03. Legend: Caputo (solid line), and Riemann-Liouville(dashed line).
This can be seen in Figure 4(a). It is interesting to note that the values of resistance and
capacitance have no effect on the output waveform amplitude (Figure 4(b)). Hence, the
frequency of oscillation can be controlled by both the value C and order α of the capacitors.
As noted in [31], a clear advantage of this is that high frequencies can be obtained by reducing
the order of the capacitors rather than their value, which can remain sufficiently large.
In Figure 5 we see that the Riemann-Liouville solutions (7.8) are very similar to the
Caputo solutions in terms of frequency and amplitude but differ in phase due to the second
parameter β of the Mittag-Leffler function. Of particular concern is the fact that the former
tends to diverge at the origin since the fractional initial conditions (2.5b) do not correspond to
measurable physical quantities. This is an important distinction between the two definitions
and has to be taken into consideration when modeling physical systems.
Mathematical Problems in Engineering 21
7.4. Multiorder Fractional Wien Bridge
The Wien bridge circuit can be further generalized by allowing the orders to assume values
αj = μjα as detailed in Section 6. We use the case α2 = 2α1 as a starting point. Substituting
α1 = α and α2 = 2α into (7.7), (7.9a) and (7.9b), we have
x∗1 = c0∗1[εα,2α,3α,1(K − 2,−1,−1 : t) + εα,2α,3α,2α(K − 2,−1,−1 : t)]
− c0∗2εα,2α,3α,α(K − 2,−1,−1 : t),
x∗2 = c0∗2[εα,2α,3α,1(K − 2,−1,−1 : t) − (K − 2)εα,2α,3α,α(K − 2,−1,−1 : t)]
+ (K − 1)c0∗1εα,2α,3α,2α(K − 2,−1,−1 : t).
(7.16)
As with the equal-order bridge, we express the solution in Laplace domain and use partial
fractions to obtain a more tractable form:
x∗1(s) = sα−1
(s2α + 1
)c0∗1 − sαc0
∗2
s3α − (K − 2)s2α + sα + 1=
3∑k=1
σksα−1
sα − λk, (7.17)
x∗1(t) =3∑
k=1
σkEα,1(λktα). (7.18)
Only solutions with one negative real and two complex-conjugate roots will be of
concern to our present discussion. To justify this, we note that the alternative case of three real
roots is of no physical interest as it does not produce oscillatory solutions. With the exception
of the exponentially decaying term (due to the negative real root, see asymptotic analysis
in Section 6.2), the solution is hence similar to the case with equal capacitor orders; that is,
the output of the fractional Wien bridge can be expressed as a linear combination of Mittag-
Leffler functions. Unfortunately, the relationship between K and α is not as simple as in the
equal-order case (7.14). Although K is still a function of α, its form is sufficiently complex
that a more convenient alternative is to define K implicitly, that is, find α = φ(K), and use
polynomial curve-fitting as shown in Figure 6.
Using the method of least-squares, we obtain a third-order approximation for K:
K ≈ 4.611 − 4.821α2 + 0.008α3. (7.19)
Two restrictions apply to the usable range of K and α. The first is the requirement that
the real root be negative. Plotting the denominator of (7.17) as a function of sα for various
amplifier gains, we obtain the relationship in Figure 7. For 0 < K < K0 ≈ 4.611, we have
λ1 ∈ − and λ2 = λ3 ∈ as required. Therefore, this serves as an upper limit to the amplifier
gain. The lower limit can be determined by recalling that 0 < α2 = 2α1 < 1 so that 0 < α < 0.5.
The result of these restrictions is also shown in Figure 6.
Within the stipulated range, the values of gain calculated from the polynomial curve
(7.19) are sufficiently accurate to create oscillatory solutions, as demonstrated in the following
simulations (Figure 8).
22 Mathematical Problems in Engineering
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
KPossible values for , K
(a)
0 0.1 0.2 0.3 0.4 0.53.4
3.6
3.8
4
4.2
4.4
4.6
K()
(b)
Figure 6: Determination of amplifier gain K: (a) restrictions on possible values of K and α; (b) third-orderleast-squares approximation of K(α) for x ∈ (0, 0.5).
K = 0
K 0
2
4
−2 −1 0 1 2 3
s
Ch
ara
cter
isti
ceq
uati
on
−20
−10
0
10
20
Figure 7: Properties of roots of (7.19) (shown as horizontal intercepts) for parameter K.
As mentioned earlier, it is possible to extend this procedure to any case where one
order is an integer multiple of the other. Consider the solution of x∗1 for α1 = α, α2 = υα:
x∗1(s) = sα−1(sυα + 1)c0
∗1 − s(υ−1)αc0∗2
s(υ+1)α − (K − 2)sυα + sα + 1=
υ+1∑k=1
σksα−1
sα − λk, (7.20)
x∗1(t) =μ+1∑k=1
σkEα,1(λktα). (7.21)
Indeed, unless we are concerned about obtaining the actual Laplace solution in partial
fraction form, the value of K that produces asymptotic oscillations can be found by simply
studying the roots of the denominator in (7.20) (a polynomial in sα of order υ + 1) and
imposing suitable conditions as previously shown so that at least one pair of Mittag-Leffler
terms has an eigenvalue that satisfies (6.18). For example, when υ = 3, a possible solution
contains 4 roots in 2 complex-conjugate pairs. One can adjust the amplifier gain such that
Mathematical Problems in Engineering 23
−0.2
−0.1
0
0.1
0.2
t
1 = 0.3, 2 = 0.6, K = 4.18952
0 2 4 6 8 10 12 14
x(t)
−0.2
−0.1
0
0.1
0.2
t
0 2 4 6 8 10 12 14
x(t)
= 0.35, 2 = 0.7, K = 4.04346
−0.2
−0.1
0
0.1
0.2
t
0 2 4 6 8 10 12 14
x(t)
1 = 0.4, 2 = 0.8, K = 3.87881
−0.2
−0.1
0
0.1
0.2
t
0 2 4 6 8 10 12 14
x(t)
1 = 0.45, 2 = 0.9, K = 3.6971
(a)
−0.2
−0.1
0
0.1
0.2
t
0 2 4 6 8 10 12 14
x(t)
1 = 0.5, 2 = 1, K = 3.5
(b)
Figure 8: Caputo model of fractional Wien bridge with α2 = 2α1. The effect of the nonoscillatory termin (7.20) can be observed as an initial offset that decays asymptotically as t → ∞. Parameters: x∗1(0) =x∗2(0) = 0.03, (a) Comparison of phase and amplitude for capacitor voltages. (b) The limiting case of oneordinary capacitor and one semicapacitor (order 1/2). Legend: and x∗1 (solid line), x∗2 (dashed line).
the roots satisfy |θ| = α0π/2 for the first pair and |θ| > α0π/2 for the second pair, hence
resulting in sustained oscillation and asymptotically decaying oscillation, respectively.
8. Concluding Remarks
We have proposed a new direct operational method for solving coupled linear fractional
differential equations of multiorders. This technique provides an alternative way for solving
24 Mathematical Problems in Engineering
some linear CFDEs, and the solutions so obtained can be expressed in terms of multivariate
Mittag-Leffler functions. For the special cases where each of the multiorders is an integer
multiple of a real positive number, the solutions can be further reduced to linear combinations
of Mittag-Leffler functions of a single variable. Conditions for asymptotically oscillatory
solutions are considered. Two examples, namely, the coupled fractional harmonic oscillator
and the fractional Wien bridge circuit, are given to illustrate our method. Simulations of
solutions and stability conditions are given. Note that to obtain the solution based on our
method requires the use of the Laplace transform of the multivariate Mittag-Leffler function,
which then gives the identity for getting the Laplace inversion for the solution. This is one
main advantage of the direct operational inversion method proposed here as it will give the
solution directly. We remark that our method does not actually simplify the computational
aspect of obtaining solutions, though intuitively it allows one to obtain the solution in explicit
form.
Here we would like to remark that there were attempts recently to transform CFDEs
with different multiorders into an equivalent system of CFDEs of a single order [32, 33].Such a method again does not reduce the amount of computation necessary to obtain the
solutions; instead, due to the increase in the number of the auxiliary equations in the latter
system, it is actually more tedious to obtain the full solutions. Our view on CFDEs is that, in
general, one still has to use numerical methods to obtain approximate solutions. The point
is to find a method that provides a more efficient way of doing so. We hope to look into
this aspect in a future work. Finally, it will be interesting to consider whether the above
method can be extended to nonlinear CFDEs. One expects that such a generalization will
not be straightforward.
Appendices
A. Mittag-Leffler Function and Related Functions
The Mittag-Leffler function [26, 28] and its generalizations are defined as follows:
Eα(z) =∞∑n=0
zn
Γ(nα + 1),
Eα,β(z) =∞∑n=0
zn
Γ(nα + β
) ,Eγ
α,β(z) =∞∑n=0
(γ)nzn
Γ(nα + β
)n!,
(A.1)
where
(γ)n= γ
(γ + 1
)(γ + 2
) · · · (γ + n − 1)=Γ(γ + n
)Γ(γ) . (A.2)
Mathematical Problems in Engineering 25
Note that
(γ)
0= 1, (0)n = 0 for n = 0, (0)0 = 1. (A.3)
Thus we have
E0α,β(z) =
1
Γ(β) . (A.4)
For convenience we define the following functions:
εα,β(λ : t) = tβ−1Eα,β(λtα), (A.5a)
εγ
α,β(λ : t) = tβ−1E
γ
α,β(λtα). (A.5b)
A.1. Asymptotic Expansion of Mittag-Leffler Function [8, 24, 26]
For 0 < α < 2,
Eα(z) = −N∑n=1
z−n
Γ(1 − nα)+ 0
(|z|−N+1
), z −→ ∞,
απ
2< arg(z) < 2π − απ
2,
Eα(z) =e1/α
α−
N∑n=1
z−n
Γ(1 − nα)+ 0
(|z|−N+1
), z −→ ∞,
∣∣argz∣∣ < απ
2.
(A.6)
Similarly one has
Eα,β(z) = −N∑n=1
z−n
Γ(β − nα
) + 0(|z|−N+1
), z −→ ∞,
απ
2< arg(z) < 2π − απ
2,
Eα,β(z) =exp
(z1/α
)α
−N∑n=1
z−n
Γ(1 − nα)+ 0
(|z|−N+1
), z −→ ∞,
∣∣argz∣∣ < απ
2.
(A.7)
B. Multivariate Mittag-Leffler Functions [27–29]
Let us adopt the following notations:
αi ∈ , β ∈ , zi ∈ , pi ∈ 0 = ∪ {0},α = (α1, α2, α3, . . . , αn), z = (z1, z2, z3, . . . , zn), p =
(p1, p2, p3, . . . , pn
),
zp =n∏i=1
zpii , p · α =
n∑i=1
piαi,[p]=
n∑i=1
pi,
(B.1)
26 Mathematical Problems in Engineering
and the binomial coefficient is generalized to
αi ∈ ,(k
p
)=
k!∏ni=1pi!
.(B.2)
The multivariate Mittag-Leffler functions is defined as:
Eα,β(z) =∞∑k=0
∑[p]=k
(k
p
)zp
Γ(p · α + β
) . (B.3)
C. Equivalence of Adjoint Method and Laplace Transform Solutions
We demonstrate the equivalence of the time-domain and frequency-domain (Laplace) solu-
tions for the 2-dimensional system as given by (4.10a) and (4.10b) and (5.4a) and (5.4b),respectively. The generalization to systems of higher dimension is straightforward and will
be omitted for brevity. The Laplace transform of the multivariate Mittag-Leffler function in
(A.5a) is
L[εα1,...,αn,β(a1, . . . , an : t)
]=
s−β
1 −∑ni=1 ais−αi
, (C.1)
with β > 0. The transform of (4.10a) and (4.10b) is then
L[x∗1] =
∑m1−1k=0
ck∗1s−(k+1) −∑m1−1
k=0ck∗1a22s
−(α2+k+1) +∑m2−1
k=0ck∗2a12s
−(α1+k+1)
1 − a11s−α1 − a22s−α2 + detAs−(α1+α2),
L[x#1] =
∑m1
k=1ck#1s
−(α1−k+1) −∑m1
k=1ck#1a22s
−(α1+α2−k+1) +∑m2
k=1ck#2a12s
−(α1+α2−k+1)
1 − a11s−α1 − a22s−α2 + detAs−(α1+α2),
(C.2)
which agrees precisely with (5.4a) and (5.4b).
Acknowledgments
The authors would like to thank Professor W. Ahmad for correspondence. S. C. Lim would
like to thank the Malaysian Ministry of Science, Technology and Innovation for the support
under its Brain Gain Malaysia (Back to Lab) Program. Li acknowledges the 973 plan under
the project no. 2011CB302802, and the NSFC under the project Grant nos. 60873264, 61070214.
S. Y. Chen acknowledges the NSFC under the project Grant no. 60870002 and Zhejiang
Provincial Natural Science Foundation (R1110679).
Mathematical Problems in Engineering 27
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 706326, 10 pagesdoi:10.1155/2012/706326
Research ArticleStudy of the Fractal and Multifractal ScalingIntervening in the Description of FractureExperimental Data Reported by the Classical Work:Nature 308, 721–722(1984)
Liliana Violeta Constantin1 and Dan Alexandru Iordache2
1 Physics Faculty, University of Bucharest, P.O. Box MG-11, 077125 Bucharest, Romania2 Physics Department, University “Politehnica” of Bucharest, Splaiul Independentei,060042 Bucharest, Romania
Correspondence should be addressed to Liliana Violeta Constantin,
Received 9 September 2011; Accepted 4 October 2011
Academic Editor: Cristian Toma
Copyright q 2012 L. V. Constantin and D. A. Iordache. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.
Starting from the experimental data referring to the main parameters of the fracture surfacesof some 300-grade maraging steel reported by the classical work published in Nature 308, 721–722(1984), this work studied (a) the multifractal scaling by the main parameters of the slit islandsof fracture surfaces produced by a uniaxial tensile loading and (b) the dependence of the impactenergy to fracture and of the fractal dimensional increment on the temperature of the studiedsteels heat treatment, for the fracture surfaces produced by Charpy impact. The obtained resultswere analyzed, pointing out the spectral (size) distribution of the found slit islands in the frame ofsome specific clusters (fractal components of the multifractal scaling) of representative points ofthe logarithms of the slit islands areas and perimeters, respectively.
1. Introduction: Complexity, Universality, Power Laws, andFractal Scaling
As it is well-known, one of the most important present topics refers to the obtainment of
scientific information about the complex materials and systems [1–3].The main founders of the complexity theory in physics have pointed out [4–7] (see
also the synthesis review [8, 9]) that several completely different complex systems (computer
arrays, complex random (Internet, particularly), robots, networks, social sciences, biology
(with some specific topics: colonies, swarms, immunology, brain, genetics, and proteomics),
2 Mathematical Problems in Engineering
economics, mathematics, glasses, agents, and cognition [10–13], etc.) have some common
features centered on their statistical behavior and the corresponding phase transforms [4, 5, 8, 9]and chemical reactions, particularly, as well as of some dynamic aspects [14–16], nonlinear
effects [17], and so forth. It results that these complex systems have certain universality
properties, which—due to their generality (see, e.g., [8])—can be described only by some
specific numbers (the so-called similitude numbers, or criteria [18–20]).How could it be possible to describe dimensional (physical, particularly) quantities
only by numbers? The answer is obtained from the examination of: (a) predictions of
Anderson [4, 5, 8, 9] relative to the “explosive” autocatalytic (exponential) growth following
the spontaneous symmetry breaking inside the specific complex systems (one finds that a
certain dimensional parameter p has to be described by its logarithm: ln p), (b) Dalton’s law
of “defined proportions”, intervening in the theory of chemical reactions (somewhat similar
to the phase transforms) [3]: dξ = −dν1/ν1 = −dν2/ν2 = · · · = +dνN/νN , where the sign
“−” corresponds to substances that disappear during the considered chemical reaction, while
the sign “+” corresponds to the appearing substances, finding that the degree of advance ξ
of the considered reaction can be expressed by means of ln νj , where νj is the amount (e.g.,
number of moles) of one of the substances participating in the chemical reaction, (c) statistical
expression of the thermodynamic entropy (describing the dissipative processes), given by
the Planck-Boltzmann’s expression: S = −k · ln℘, where k is the the Boltzmann’s constant,
where ℘ is the probability density, (d) Claude Shannon’s expression [21–23] of the individual
information quantity: = −a · ln℘ (a = constant).The simplest expression (the zero-order approximation) of the relation between a test
physical parameter t(u) and the uniqueness one u is, of course, the linear expression:
ln t = ln t1 + s · lnu, equivalent to the power law: t(u) = t1 · us. (1.1)
If the uniqueness parameter u corresponds to the size L of the considered complex system,
then the power law (1.1) particularizes into the fractal scaling
t(L) = t1 · Ls. (1.2)
When the relation ln t = f(lnL) is more intricate than the linear one, presenting, for example,
a certain curvature, then the existing experimental data can be divided in some groups of
pairs {tk1, Lk1; . . . tkn, Lkn} so that for each group, a specific linear relation is valid: ln tki =ln t1k+sk ln uki, equivalent to the fractal scaling: tki = t1k ·usk
ki. Because the prepower coefficient
t1k and the power exponent sk depend on the group k of chosen data, it results that the set of
relations {tki = t1k · uskki| k = 1,N} corresponds to a multifractal scaling [24, 25].
Some additional detailed studies of the different types of fractal and multifractal
scaling were accomplished in the frame of works [26–28].
2. Critical Findings Referring to the Work Nature, 308, 721-722(1984)
In 1984, Mandelbrot et al. [29] claimed that the fracture surfaces of metals are fractal (self-
similar) over a wide range of sizes, and introduced the experimental methods named “slit
island analysis” (SIA) and “fracture profile analysis” (FPA). As the large majority of papers
published by Nature (average impact factor 12.86 in 1985 and 24.82 in 1996), the above-
indicated work had a high scientific impact: we identified [30, 31] at least 26 papers and
Mathematical Problems in Engineering 3
books published only in the following 10 years (up to 1993, inclusively [30]), studying
the fractal character of the fracture surfaces. Despite of its large impact, the hypothesis of
Mandelbrot et al. [29] was somewhat restricted by the following studies: (a) even the papers
of Underwood [32], Pande et al. [33], Lung and Mu [34], and Huang et al. [35] affirmed
that the fracture surfaces of metals can be approximately considered to possess a certain fractal
character, (b) Underwood and Banerji [32] concluded that the slit island analysis itself was
imperfect in nature as a method for measuring the fractal dimension of fractured surfaces,
(c) Lung and Mu [34] found that the fractal dimension was largely affected by the measuring
ruler employed and postulated the concept of inherent measuring ruler, (d) Huang et al.
[35] pointed out that how to determine the fractal dimension of a fractured surface has
always been a problem of “argument”, and (e) Williford [36] tried to explain the obtained
results in terms of multifractals, but this explanation seemed not to be satisfactory for some
experimental results [37, 38], and so forth.
The detailed numerical analysis accomplished in the frame of this work pointed out
that the main missing elements of work [29] are the following:
(a) no justification of the indicated values of fractal dimensional increment from the
capture of Figure 1 [29],
(b) no analysis of the multifractal scaling of the logA = f(log P) dependence corre-
sponding to the slit islands areas and perimeters, respectively,
(c) the regression line: impact energy = f (fractal dimensional increment) from
Figure 3 [29] is obviously inexact, and it does not consider the corresponding
possible nonlinear dependence,
(d) the dependence of the fractal dimensional increment on the temperature of the heat-
treatment of the 300-grade maraging steel Charpy impact specimens studied by
Figure 3 [29] was not studied.
3. Procedure Intended to the Evaluation of the Fractal Dimension ofthe Slit Islands
In order to evaluate the slope of the regression line logA = f(log P), the numerical values
of the decimal logarithms logA, logP of the slit islands areas and perimeters, respectively,
(indicated by Figure 1 [29]) were firstly evaluated by means of the scanning procedure [39].We obtained s ≡ D′ ∼= 1.6225 = D − 1 = iF , in considerable disagreement with the values 1.28
and 1.26 indicated by the capture of Figure 1 [29].Starting from the interpretation provided by the monograph [40, pages 64–65] of
the experimental data obtained by means of the slit island method, according to whom
(a) the cross-section of area A of the fractured material is not fractal; therefore, this area is
proportional to the square of the slit island average radius R: A ∝ R2, while (b) the perimeter
P of the slit island is really fractal (of dimension D−1, where D is the dimension of the fracture
surface); therefore, P ∝ RD−1, and we have found that A ∝ P 2/(D−1) and the slope of the
logA = f(log P) plot is: s = 2/(D − 1). From this relation, we obtained, in good quantitative
agreement with the indicated fractal dimensional increment, iF = D − 1 values indicated in
the caption of Figure 1 [29] as well as with the results obtained by other similar works (e.g.,
[41]).
4 Mathematical Problems in Engineering
Table 1: Main features of the fractal: logA = c0+c1 logP and multifractal (parabolic): logA = co+c1 logP +c2(logP)2 scalings of the parameters A, P of the slit islands of fracture surfaces reported by Figure 1 [29].
The type of the logA = f(logP) correlation(scaling)
Regression line(fractal scaling)
Parabolic correlation(multifractal scaling)
c0 −1.776 −1.5686
c1 1.6265 (average slope) +2.39074
c2 0 −0.16965
Correlation coefficient 0.9655 0.9792
Average relative error (%) for all 41 studied slitislands
7.540% 7.346%
Average relative error for the 6 extreme (first 3and last 3) representative points of Figure 1[24, 25]
10.134% 8.733%
Apparent fracture surface fractal dimension:DM = 1 + slope
2.6265 2.4487 · · · 3.0514
Fracture surface fractal dimension according toour considerations (this work): Ds = 1 + 2/slope
2.23 1.975 · · · 2.380
4. Study of the Multifractal Scaling of the logA = f(logP) Dependence
Taking into account that all 6 extreme (first 3 and last 3) representative points of Figure 1
[29] are located under the regression line, we assumed that a nonlinear (even a parabolic)logA = f(log P) expression could agree better with the experimental data reported by this
figure. To check this assumption, we used the procedures of the well-known classical gradient
method [42–44] in order to find the parameters of the parabolic correlation
logA = c2
(log P
)2 + c1 logP + c0, (4.1)
which ensure the best fit of the experimental data of Figure 1 [29].The obtained results are synthesized by Table 1.
One finds that the explanation given by Williford [36], in terms of multifractals, of the
experimental data referring to the fracture surfaces is more realistic than the initial Mandel-
brot’s hypothesis. We have to underline that this explanation (multifractals) is supported also
by the results obtained by Carpinteri and Chiaia [24, 25] especially for concrete samples.
The new versions of Figures 1 and 3 [29], after our numerical conversion (using the
method of work [39]) of the experimental data indicated by these figures and the following
parabolic fit (for the logA = f(log P) pairs), and the least-squares fit (for the fractal dimen-
sional increment = f (impact energy)) are presented below in the frame of our Figures 1 and
2.
5. Towards the Fractal Components of the Multifractal Set of FractureSurfaces Slit Islands of the Maraging Steels Studied by [29]
Taking into account the practical continuous change of the slope of the logA = f(log P)plot, the definition of the fractal components of the multifractal set of fracture surfaces
slit islands is strongly related to the experimental accuracy of the logA, log P parameters. As
the accuracy of these parameters is not known, a certain image on these fractal components
Mathematical Problems in Engineering 5
100
101
102
103
104
105
100 101 102 103 104 105
Are
a(μ
m2)
Perimeter (μm)
Figure 1: The new (improved) version of Figure 1 [29] after [29], the numerical conversion (using method[39]) of the corresponding experimental data and the parabolic fit of the logA = f(logP) pairs.
0 50 100 150 200
0
0.1
0.2
0.3
Fra
ctal
dim
ensi
on
al
incr
emen
t
Impact energy (J)
430 C
370 C
340 C
360 C
315 C
300 C
Figure 2: The new (corrected) version of Figure 3 [29] after the numerical conversion (using the method[39]) of the corresponding experimental data, and the least-squares fit of the fractal dimensional incre-ment = f (impact energy) dependence data.
can be obtained starting from the identification of clusters of representative points logA,
log P .
We defined the logA, log P clusters starting from the distances between the nearest
representative points in the space logA, log P . If the distance between the nearest represen-
tative points belonging to 2 neighbor sets is considerably larger than that for the nearest such
points belonging to each set, these neighbor sets correspond to the desired clusters.
Using this procedure, we have identified 6 clusters in the logA, log P space of
Figure 1 [29], defined by the pairs of logA, log P coordinates corresponding to the marginal
representative points of each cluster.
These clusters of representative points in the logA, log P space are gathered around
some average (logP)i values (i = 1,N). For each cluster of representative points, the local
6 Mathematical Problems in Engineering
Table 2: The “spectral” (size) distribution of the clusters of representative points in the logA, logP plotinvolved by Figure 1 [26–29] as representing the fractal components of the multifractal scaling of thelogA = f(logP) dependence.
Interval of fractalincrement values
(0.975; 1.014) (1.034; 1.140) (1.123; 1.213) (1.205; 1.297) (1.283; 1.392) (1.392; 1.747)
Pairs of values ofthe slit islands PPerimeter (μm)and area (μm2)
(10.00; 5.62)· · ·
(17.15; 11.55)
(22.07; 27.38)· · ·
(74.99; 103.7)
(62.64; 237.1)· · ·
(154.0; 421.7)
(143.3; 930.6)· · ·
(316.2; 2458)
(283.9; 4068)· · ·
(649.4; 4371)
(649.; 23714)· · ·
(4698;83536)
Number ofrepresentativepoints in Figure 1[1–3]
3 8 10 8 6 6
Percentage ofrepresentativepoints
7.318% 19.512% 24.390% 19.512% 14.634% 14.634%
slope si = 2c2(logP)i + c1 of the multifractal scaling logA = c2(logP)2 + c1 logP + c0 and
the local fractal dimensional increment iFi = 2/si were evaluated, the obtained results being
synthesized by Table 2. The synthesis of these clusters features as well as the corresponding
fractal dimensions (or increments) corresponding to each cluster (as a specific representative
of the fractal components of the multifractal set of fracture surfaces slit islands) is presented
by Table 2.
One finds that the small values of the fractal dimension correspond to slit islands of
relatively small dimensions (perimeters of the magnitude order of μm), corresponding to
fracture surfaces not too curly, and even involving some surface breaks (which could explain
eventually the seldom values little less than 2 of the fractal dimension corresponding to some
small parts of the fracture surface).
6. Study of the Fractal Dimensional Increment ofthe Fracture Surfaces Produced by Impact onthe Temperature of the Steels Heat Treatment
Unlike the fracture surfaces produced by uniaxial tensile loading, whose characteristic
parameters were reported for the 300-grade maraging steel by Figure 1 [29], the last part
of this work (Figure 3 [29]) reports the main features of the fracture surfaces produced by
impact.
The evaluation of the slope s and intercept i of the regression line Eimp(J) = s · theat + i
describing the impact energy to fracture in terms of the temperature of the studied steels
heat treatment led us to the results: s ∼= −1.069 J/◦C, i ∼= 494.21 J with a correlation coefficient
r ∼= −0.9563 and a square mean relative error of 10.05%.
Similarly, the evaluation of the slope s′ and of the intercept i′ of the regression line
iF = s′ · theat + i′ describing the fractal dimensional increment of the fracture surface produced
by impact in terms of the temperature of the studied steel heat treatment leads to the results
s′ ∼= 1.25 · 10−3 ( ◦C)−1, i′ ∼= −0.260, with a correlation coefficient r ′ ∼= 0.9243 and a square mean
relative error of 10.971%.
One finds that, as it was expected, (a) the impact energy to fracture decreases (approx-
imately linearly, up to 450◦C) with the temperature of the studied steels heat treatment and
Mathematical Problems in Engineering 7
(b) the fracture surface deformation (from its ideal planar shape), measured by its fractal
dimensional increment, increases with the temperature of the heat treatment.
It was possible to obtain also the parameters of a more exact (than that performed in
the frame of Figure 3 [26–29]) regression line Eimp(J) = s′′ · iF + i′′ describing the dependence
of the impact energy to fracture on the corresponding fracture surface deformation (fractal
dimensional increment) s′′ ∼= −781.47 J, i′′ ∼= 258.40 J, correlation coefficient r ′′ ∼= −0.9442, and
square mean relative error 13,31%, but we consider these last results as less important than
the above-indicated ones, referring to the Eimp and iF = f(theat) dependencies.
7. Investigations on the Compatibility with the Experimental Data ofthe Fractal/Multifractal Descriptions of the Fracture Parameters
Taking into account the errors affecting practically all experimental data, the decision about
the compatibility (or incompatibility) of a certain hypothesis (e.g., the fractal character of the
fracture surfaces) has to be established using some statistical tests [45–47]. Unfortunately,
neither [29] nor [30, 32–38] studied statistically the compatibility of the investigated
hypothesis relative to the experimental data, and even these works did not indicate the errors
corresponding to the used experimental data.
In order to evaluate the error risk at the rejection of the compatibility of a certain
representative point relative to the studied correlation Yi = f(X), it is possible to use both
global (for the entire correlation) or local test, respectively. For example, the error risk can be
evaluated by means of the expression (see [44–48])
qk = exp
{− 1
2(1 − r2
k
)[(Yik − Yi,tk
s(Yik)
)2
+(Xk −Xtk
s(Xk)
)2
− 2rk
(Yik − Yi,tk
s(Yik)
)(Xk −Xtk
s(Xk)
)]},
(7.1)
where Yik and Xk are the impact energy and the fractal dimension corresponding to the
representative point (state) k(= 1, 2, . . .N), Yi,tk and Xtk are the impact energy and the fractal
dimension corresponding to the tangency point of the confidence ellipse centered in (Yik, Xk)with the studied correlation plot: Yi = f(X), while rk, s(Yik), and s(Xk) are the correlation
coefficient and the square mean errors corresponding to the individual values Yik and Xk.
Because these errors are not indicated by the studied work [29], we will try evaluate them
from other studies about the fracture energy.
The studies [31, 48] of the published works concerning the (multi)fractal correlations
of some mechanical (fracture) parameters with the specimen size points out the magnitude
orders of the errors corresponding to the fracture energy. The corresponding relative errors
are indicated in Table 3. One finds that for concrete specimens, the average relative errors
affecting the fracture energy is of (approximately) 7%.
Assuming that the relative errors affecting the values of the fractal dimension
are considerably less than those corresponding to the impact energy (approx. 10%), the
expression (7.1) leads to error risks somewhat larger than 2% associated to the rejection of
the compatibility hypothesis of the fractal/multifractal descriptions with the experimental
data. It results that the compatibility hypothesis cannot be rejected, but a more sure decision
needs imperatively the knowledge of the corresponding measurement errors.
8 Mathematical Problems in Engineering
Table 3: Relative errors corresponding to the experimental data concerning the fracture energy GF for dif-ferent concrete and rocks specimens.
Material [reference] Concrete [49] Dry concrete [50] Wet concrete [50] Red felsersandstone [50]
Limits of relative errors 4.68 · · · 9.76% 4.028 · · · 16.217% 2.502 · · · 11.585% 3.125 · · · 35.424%
Average relative error 6.062% 8.131% 6.473% 16.236%
8. Conclusions
The accomplished study of the numerical data involved by [29] points out the following main
original findings.
(1) The decision concerning the fractal (or multifractal) character of the fracture sur-
faces of metals needs a previous rigorous study by means of the numerical analysis
procedures.
(2) In this aim, both the errors corresponding to the geometrical parameters (perime-
ters and areas of the slit islands) and to the specific mechanical parameters (impact
energies), respectively, are necessary.
(3) Taking into account the considerable differences between the values of the fractal
dimension resulting from Figure 1 [29], or indicated in the caption of Figure 1 [29],or in Figure 3 [29], we consider that the correct calculation of the fractal dimension
corresponds to the interpretation from work [40], which considers that only the
perimeters of the slit islands present a fractal character: P ∝ RD−1, while the areas of
these slit islands present the usual second degree dependence on their radii A ∝ R2;
we have found that this interpretation [40] leads also to an agreement between the
data from Figure 1 [29] and the values of the fractal dimension indicated by this
work [29].
(4) The accomplished study indicates a multifractal nature of the fracture surfaces of
metals, the size distribution of the fractals (involved by this multifractal structure)being also evaluated by this work.
(5) The influence of the temperature of the studied maraging steels heat-treatment
on the (a) impact energy to fracture and (b) the fracture surface deformation,
measured by its fractal dimensional increment, were also studied, finding the
increase of the fracture surface deformation with the heat-treatment temperature,
particularly.
(6) Using the evaluated errors affecting the fracture energies of some concrete speci-
mens, we have found that the compatibility hypothesis of the fractal/multifractal
descriptions with the experimental data cannot be rejected, but a more sure decision
always needs an accurate knowledge of the corresponding measurement errors.
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 478295, 27 pagesdoi:10.1155/2012/478295
Research ArticleMultidimensional Wave Field Signal Theory:Transfer Function Relationships
Natalie Baddour
Department of Mechanical Engineering, University of Ottawa, Ottawa, ON, Canada K1N 6N5
Correspondence should be addressed to Natalie Baddour, [email protected]
Received 29 August 2011; Accepted 20 September 2011
Academic Editor: Carlo Cattani
Copyright q 2012 Natalie Baddour. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.
The transmission of information by propagating or diffusive waves is common to many fieldsof engineering and physics. Such physical phenomena are governed by a Helmholtz (real wave-number) or pseudo-Helmholtz (complex wavenumber) equation. Since these equations are linear,it would be useful to be able to use tools from signal theory in solving related problems. The aim ofthis paper is to derive multidimensional input/output transfer function relationships in the spatialdomain for these equations in order to permit such a signal theoretic approach to problem solving.This paper presents such transfer function relationships for the spatial (not Fourier) domain withinappropriate coordinate systems. It is shown that the relationships assume particularly simpleand computationally useful forms once the appropriate curvilinear version of a multidimensionalspatial Fourier transform is used. These results are shown for both real and complex wavenumbers.Fourier inversion of these formulas would have applications for tomographic problems in variousmodalities. In the case of real wavenumbers, these inversion formulas are presented in closed form,whereby an input can be calculated from a given or measured wavefield.
1. Introduction
The transmission of information over space and time is often governed by the theory
of waves. Many important physical phenomena are described by a Helmholtz equation,
for example, in the fields of electromagnetism, acoustics, and optics [1–4]. Other physical
phenomena are well described by the use of damped or diffusion waves, such as the pro-
pagation of heat or photonic waves. For example, ultrasound has long been used for medical
imaging [3] while the use of optical radiation via diffuse photon density waves for imaging
inhomogeneities in turbid media is a newer development [5–8]. Similarly, photothermal
tomographic imaging methods have also been used for nondestructive evaluation [9–12],and more recently for biomedical imaging [13, 14]. Furthermore, a combination of ultrasonic
2 Mathematical Problems in Engineering
and optical techniques known as photoacoustics has also gained prominence in recent years
[4, 15].Modelled as a linear process, the propagation and scattering of waves obey the prin-
ciples of superposition and homogeneity. One common method of studying linear processes
is to view them as linear systems with an input and output. The input/output relationships
of the system can then be characterized via the system transfer function, which implies
interpreting the problem as a signal theory problem and thus enabling the application of
powerful system theoretic results and concepts. This type of approach applied to propagating
or diffusion wave problems has already yielded promising results [16–18] and the goal of this
paper is to expand the scope of this approach.
The seeds of such an approach to the solution of tomographic-type problems can be
seen in [1, 8, 10, 19, 20]; however, in those papers there is no clear reference to any such
system theoretic concepts. As such, a unifying framework for a signal theoretic approach to
wavefields does not exist. The goal of this work is to propose a clear and cohesive signal
theoretic framework for working with acoustic, thermal, photonic, or other wavefield-related
problems. This would be particularly powerful, enabling the application of a vast array of
results from a signal processing point of view and in particular, recent novel results from
algebraic signal processing [21–27] should find ground-breaking applications.
With these ideas in mind, the goal of this exposition is to derive multidimensional
spatial transfer function relationships for the Helmholtz and pseudo-Helmholtz equation so
that the tools of linear system theory can be used to solve related problems.
We consider the input quantity to be the time- and space-dependent inhomogeneity
term (the “forcing” term in the relevant partial differential equation). The output quantity
is then taken as the resulting field present at any point in space. The goal is to develop
transfer function relationships in spatial multidimensions so that general results of the theory
of signals and systems may be used. It is known that the relationship between the input
(inhomogeneity term) and the output (resulting wavefield) is one of convolution with the
impulse response (Green’s function in classical terms). In the Fourier domain, the equivalent
relationship is a transfer function relationship, that is to say one of multiplication. In this
paper, key transfer function relationships are shown to hold for the spatial (not Fourier)domain with the use of appropriate coordinate systems. That is to say, the relationship
between input and output is shown to be one of multiplication (not convolution) even in the
spatial domain, hence yielding the signal theoretic construction that is the aim of the paper.
Rather than focusing on a specific application area (acoustics, medical imaging,
seismic, etc.), the end goal of this paper is a unifying framework. The backbone of
this framework is Fourier transforms in multidimensions which are also transcribed into
curvilinear coordinate systems. The representations in this paper are embedded in a time-
dependent, spatially one- to three-dimensional description.
The outline of this paper is as follows. Sections 2 to 7 establish the background
preliminaries, definitions, and sign conventions that are important to the developments in
the rest of the paper. The definition of multidimensional Fourier transforms is given in
Section 2. Section 3 presents the mathematical forms of the types of signal to which this
development applies, namely, any signal governed by a Helmholtz or pseudo-Helmholtz
equation. Section 4 introduces the Green’s and transfer functions for the (generalized)Helmholtz equation and points out that the spatial transfer function assumes a particularly
simple form in curvilinear coordinates. Section 5 explains the sign conventions that will be
employed in this paper to ensure consistency and simplicity of results. Sections 6 and 7
develop multidimensional Fourier transforms in curvilinear coordinates. Sections 8, 9, and 10
Mathematical Problems in Engineering 3
present the relevant transfer function relationships in curvilinear coordinates in 3D, 2D, and
1D, respectively, and form the core results of this paper. Section 11 summarizes the results
of the previous three sections in a tabular format that is easy to refer to. Sections 12 and 13
present some applications while Section 14 concludes the paper.
2. Fourier Transforms in n-Dimensions
The theory of Fourier transforms can be extended to Rn in a way that is completely analogous
to the treatment of one-dimensional Fourier transforms. Let elements of Rn be denoted by
�x = (x1,x2, . . . ,xn) or more generally �r, and elements of the corresponding Fourier dual space
Ωn be denoted by �ω = (ω1, ω2, . . . , ωn). Under the suitability of integration of f , then the
Fourier transform in Rn is defined for �ω in Ωn as
F( �ω) =∫
Rn
f(�x)e−j �ω·�xd �x. (2.1)
Under suitable conditions, the function f can be recovered from the inverse transform
through
f(�x) =1
(2π)n
∫Ωn
F( �ω)ej �ω·�xd �ω. (2.2)
Various other conventions are possible regarding the location of the positive and negative
signs and also the factors of 2π in (2.1) and (2.2).
3. The Helmholtz and Pseudo-Helmholtz Equation
All wave fields governed by the wave equation (such as acoustic or electromagnetic waves)lead to the Helmholtz equation once a Fourier transform is used to transform the time domain
to the frequency domain:
∇2u(�r, ω) + k2su(�r, ω) = −s(�r, ω), (3.1)
where k2s = ω2/c2
s . Here, s(�r, ω) is the temporal Fourier transform of the inhomogeneous
time- and space-dependent source term for the wave equation. From a signal theory
perspective, this is considered to be the input to the system. The variable u(�r, ω) represents a
physical variable that is governed by the wave equation, for example, acoustic pressure, and
is considered to be the output from a signal theory point of view. Both s(�r, ω) and u(�r, ω) are
functions of position, �r, and (temporal) frequency, ω. The variable cs represents the speed of
the wave, which for an acoustic wave would be the speed of sound and for an electromagnetic
wave would be the speed of light. For wavefields governed by the wave equation, k2s is a real
(and positive) quantity.
Other physical phenomena lead to a “pseudo-” Helmholtz equation upon Fourier
transformation of the time variable to a (temporal) frequency variable. For example, the
equation for a diffuse photon density wave (DPDW) which describes the photon density
4 Mathematical Problems in Engineering
u(�r, t) in a solid due to incident energy intensity s(�r, t) (optical source function) is given in
the time domain by [2]
D∇2u(�r, t) − 1
c
∂
∂tu(�r, t) − μau(�r, t) = −s(�r, t). (3.2)
In the above equation, μa is the optical absorption coefficient (m−1), c is the speed of
light in the turbid medium (m/s), and D is the optical diffusion coefficient (m). Taking the
temporal Fourier transform (FT) of (3.2) to transform time to frequency gives the following:
∇2u(�r, ω) + k2pu(�r, ω) = −s(�r, ω)
D, (3.3)
where k2p = −(μa/D) − (iω/cD) is the complex wavenumber. Equation (3.3) is a pseudo-
Helmholtz equation. The term “pseudo” is used because although (3.3) has the form of
a Helmholtz equation similar to (3.1), the wavenumber k2p is a complex variable, with
the imaginary part indicating a decaying or damped wave. In this development the
terms “Helmholtz” and “pseudo-Helmholtz” will generally be used interchangeably unless
otherwise noted. Similarly, the standard heat equation is given by
∇2u(�r, t) − 1
α
∂
∂tu(�r, t) = −s(�r, t), (3.4)
where s(�r, t) is the time- and space-dependent heat source, u(�r, t) describes the temperature
in the material as a function of time and space, and α is the thermal diffusivity of a material.
As done previously, we take the temporal FT of (3.4) to obtain
∇2u(�r, ω) + k2t u(�r, ω) = −s(�r, ω), (3.5)
where k2t = −iω/α is the complex wavenumber. Once again (3.5) has the form of a Helmholtz
equation, although it is a pseudo-Helmholtz equation to be exact since the wavenumber is
complex and indicates a damped wave.
Thus, we see that the general Helmholtz form of (3.1) can be used to describe several
different physical phenomena, from the propagation of light or acoustic waves to the heavily
damped nature of photonic or thermal waves. The exact form of the wavenumber in each case
is the best indicator as to the propagation characteristics of a wave with a real k2 indicating a
propagating wave and a complex k2 indicating a damped wave.
4. Green’s Function and Transfer Function for the Helmholtz Equation
Taking the full spatial Fourier transform of the Helmholtz (or pseudo-Helmholtz) equation
and rearranging yields
U( �ω,ω) = U(ωx, ωy, ωz, ω
)=
1(ω2
x +ω2y +ω2
z − k2)S( �ω,ω), (4.1)
Mathematical Problems in Engineering 5
where we have used the shorthand notation ( �ω) = (ωx, ωy, ωz) to denote a point in 3D spatial
Fourier frequency space. The wavenumber may be real or complex. A capital letter is used to
denote the full 3D spatial and temporal Fourier transform of a function, although it should
be clear from the arguments which is being indicated. In cases where it may not be clear, a
tilde (∼) will be used to denote the function in spatial Fourier space. For shorthand notation,
let ω2k= ω2
x +ω2y +ω2
z, so that ωk is the length of the spatial Fourier vector. The Fourier space
has the same spatial frequency dimension as the spatial dimension of the problem. So in 2D
space, a ωz spatial frequency variable would not be required and so forth.
By inverse spatial Fourier transformation of (4.1), the wavefield in n dimensions is
given by
u(�r, ω) =1
(2π)n
∫∞
−∞
S( �ω,ω)(ω2
k− k2
)ei �ω·�rd �ω. (4.2)
Using the definition of the Fourier transform of s(�r, ω), the above equation can be rewritten
as
u(�r, ω) =1
(2π)n
∫∞
−∞
ei �ω·�r(ω2
k− k2
) ∫∞
−∞s(�x, ω)e−i �ω·�xd�x d �ω. (4.3)
Let us define the spatial Green’s function in n-dimensional space as
g(�r | �x, ω) = g(�r − �x, ω) =1
(2π)n
∫∞
−∞
ei �ω·(�r−�x)
ω2k− k2
d �ω. (4.4)
This is the Green’s function for the (pseudo-) Helmholtz equation. By switching the order
of integration, (4.3) can be rewritten such that it can be clearly interpreted as a spatial
convolution of Green’s function with the input source:
u(�r, ω) =∫∞
−∞s(�x, ω)g(�r − �x, ω)d�x = s(�r, ω)∗�rg(�r, ω). (4.5)
The notation ∗�r has been used to denote a (multidimensional) space-only convolution. The
frequency domain equivalent to (4.5) is (4.1), which can be interpreted as a multiplication in
terms of the spatial transfer function as
U( �ω,ω) = G( �ω,ω)S( �ω,ω), (4.6)
where the Fourier transform of the spatial Green’s function is the spatial transfer function:
G( �ω,ω) = F[g(�r, ω)
]=
1(ω2
k− k2
) . (4.7)
The dependence of the spatial Green’s function on temporal frequency is via the wavenumber
k of the wavefield (which may be real or complex) and this is emphasized by writing G( �ω,ω)as a function of spatial and temporal frequencies.
6 Mathematical Problems in Engineering
4.1. Spatial Transfer Function in Curvilinear Coordinates
Equations (4.4) and (4.7) hold in any n-dimensional space with Green’s function of (4.4)being the n-dimensional spatial inverse Fourier transform of the spatial transfer function
of (4.7). More specifically, in 1D, ω2k
= ω2x, and in 2D ω2
k= ω2
x + ω2y. Transforming to
polar coordinates in the 2D spatial frequency domain via the transformation ωx = ρ cosψ,
ωy = ρ sinψ gives ω2k= ω2
x + ω2y = ρ2 where ρ is the radial frequency variable in polar
coordinates. Similarly, in 3D, ω2k= ω2
x+ω2y+ω
2z becomes ω2
k= ω2
x+ω2y+ω
2z = ρ2 where ρ is the
spherical radial frequency variable in spherical polar coordinates in spatial frequency space
via the coordinate transformation given by ωx = ρ sinψω cos θω, ωy = ρ sinψω sin θω and ωz =ρ cosψω. Thus, the spatial transfer functions assume particularly simple, radially symmetric
forms in polar coordinates for 2D space and spherical polar coordinates for 3D space. These
particularly simple forms for the 2D and 3D transfer functions in polar coordinates motivates
the development of the general 2D and 3D Fourier transforms in curvilinear coordinates
[28, 29] so that this simple form of the transfer function may be exploited.
5. Notation and Sign Conventions
Wavenumbers are defined by the squares of their quantities and arise as a result of taking
the Fourier transform of the corresponding propagation equation (be it acoustic, thermal,
or otherwise), which in turn leads to a Helmholtz or pseudo-Helmholtz equation. These
wavenumbers are defined so that the wave propagation equation transformed to the
temporal frequency domain all have the form of a pseudo-Helmholz equation as given
by (3.1) or (3.5) with a complex or real wavenumber. For the rest of the paper, we will
consider that kt represents a generic complex wavenumber while ks represents a generic real
wavenumber.
Rather than the squared wavenumber, the quantity of interest will prove to be the
wavenumber itself, namely, ks and kt, which are the square roots of the given squared
wavenumber in the Helmholtz equation. Each k can be considered as the sum of a real and an
imaginary part, so that kt = ktr + ikti with ktr denoting the real part of kt and kti denoting the
imaginary part. Since the square root of any k2 can be ±k, we will use the convention that for
a given complex k, the required square root of the corresponding k2 is defined such that the
imaginary part of k is negative. Hence, kt is chosen as the square root of k2t such that kti < 0 and
so forth. If this sign convention is adopted, then it was shown in [36] that the many results
for complex wavenumber use the same mathematical form of travelling wave solution as for
the real wavenumbers.This makes the notation and book-keeping considerably simpler. It is
noted that this sign convention is the opposite of what this author adopts in [30].
5.1. Sommerfeld Radiation Condition
To aid in the selection of a causal solution, the Sommerfeld radiation condition is required.
The Sommerfeld radiation condition states that the sources in the field must be sources not
sinks of energy. Therefore, energy radiated from sources must scatter to infinity and cannot
radiate from infinity into the field. Mathematically, a solution u(x), where x is the spatial
variable, to the Helmholtz equation is considered to be radiating if it satisfies
lim|x|→∞
|x|(n−1)/2
(∂
∂|x| + ik
)u(x) = 0, (5.1)
Mathematical Problems in Engineering 7
where n is the dimension of the space and k is the wavenumber in the Helmholtz equation.
This is the radiation equation based on an implied time variation of eiωt which is implicit in our
chosen definition of the Fourier transform, (2.1). For example, with the standard definition of
the Fourier transform, the transform of f ′(t) is iωF(ω), clearly implying the eiωt dependence.
Had a different definition of the Fourier transform been used, the implied time variation
would have been e−iωt, in which case the sign of i in (5.1) would be reversed.
6. 2D Fourier Transforms in Terms of Polar Coordinates
Given the desire to exploit the simple form of the spatial transfer function in curvilinear
coordinates, we consider 2D and 3D Fourier transform in terms of curvilinear coordinates.
Let us first consider the Fourier transforms in 2D in polar coordinates. In order to define this,
some preliminary definitions of Hankel transforms are required first.
6.1. Hankel Transforms
The Hankel transform of order n is defined by the integral [31]
�
Fn
(ρ)= Hn
(f(r)
)=∫∞
0
f(r)Jn(ρr)r dr, (6.1)
where Jn(z) is the nth order Bessel function. If n > −1/2, the transform is self-reciprocating
and the inversion formula is given by
f(r) = H−1n
{�
Fn
(ρ)}
=∫∞
0
�
Fn
(ρ)Jn(ρr)ρ dρ. (6.2)
6.2. Connection between the 2D Fourier Transform and Hankel Transform
To develop Fourier transforms in 2D in polar coordinates, both the function f(�r) and its
Fourier transform F( �ω) are expressed in polar coordinates. In general f(�r) = f(r, θ) is not
radially symmetric and is a function of both r and θ so that the θ dependence can be expanded
into a Fourier series due to the 2π periodicity of the function in θ:
f(r, θ) =∞∑
n=−∞fn(r)ejnθ, (6.3)
where the Fourier coefficients fn(r) can be found from
fn(r) =1
2π
∫2π
0
f(r, θ)e−jnθdθ. (6.4)
8 Mathematical Problems in Engineering
Similarly, the 2D Fourier transform F( �ω) = F(ρ, ψ) can also be expanded into its own Fourier
series so that
F(ρ, ψ
)=
∞∑n=−∞
Fn
(ρ)ejnψ, (6.5)
and where those Fourier coefficients are similarly found from
Fn
(ρ)=
1
2π
∫2π
0
F(ρ, ψ
)e−jnψdψ, (6.6)
The relationship between the Fourier coordinates in normal space fn(r) in (6.3) and
the spatial frequency space Fourier coordinates Fn(ρ) in (6.5) is desired. The details of this
are given in [28], and the results are summarized here. It is emphasized that the relationship
between fn(r) and Fn(ρ) is not a Fourier transform. The relationship between them is given
by
Fn
(ρ)= 2πi−n
∫∞
0
fn(r)Jn(ρr)r dr = 2πi−nHn
{fn(r)
}, (6.7)
where Hn is the nth order Hankel transform. The reverse relationship is given by
fn(r) =in
2π
∫∞
0
Fn
(ρ)Jn(ρr)ρ dρ =
in
2πHn
{Fn
(ρ)}
. (6.8)
The nth term in the Fourier series for the original function will Hankel transform into the
nth term of the Fourier series of the Fourier transform. However, it is an nth order Hankel
transform for the nth term, namely, all the terms are not equivalently transformed. The
mapping from fn(r) to Fn(ρ) is one of nth-order Hankel transform, which in general is not a
2D Fourier transform.
The operation of taking the 2D Fourier transform of a function is thus equivalent to
(1) first finding its Fourier series expansion in the angular variable and (2) then finding the
nth-order Hankel transform (of the radial variable to the spatial radial variable) of the nth
coefficient in the Fourier series. Since each of these operations involves integration over one
variable only with the others being considered parameters vis-a-vis the integration, the order
in which these operations are performed is interchangeable.
7. Spherical Hankel Transform
We introduce the spherical Hankel transform, which will form part of the 3D Fourier trans-
form in spherical coordinates.
Mathematical Problems in Engineering 9
7.1. Definition of the Spherical Hankel Transform
The spherical Hankel transform can then be defined as [32, 33]
�
Fn
(ρ)= Sn
{ff(r)
}=∫∞
0
f(r)jn(ρr)r2dr. (7.1)
Sn is used to specifically denote the spherical Hankel transform of order n. The inverse
transform is given by [29]
f(r) =2
π
∫∞
0
�
Fn
(ρ)jn(ρr)ρ2dρ. (7.2)
The spherical Hankel transform is particularly useful for problems involving spherical
symmetry.
7.2. Spherical Harmonics
The spherical harmonics are the solution to the angular portion of Laplace’s equation in
spherical polar coordinates and can be shown to be orthogonal. These spherical harmonics
are given by [32]
Yml
(ψ, θ
)=
√(2l + 1)(l −m)!
4π(l +m)!Pml
(cosψ
)eimθ, (7.3)
where Yml
is called a spherical harmonic function of degree l and order m, Pml
is an associated
Legendre function, 0 ≤ ψ ≤ π represents the colatitude and 0 ≤ θ ≤ 2π represents the
longitude. With the normalization of the spherical harmonics as given in (7.3), the spherical
harmonics are orthonormal so that∫2π
0
∫π
0
Yml Ym′
l′ sinψ dψ dθ = δll′δmm′ . (7.4)
Here δij is the kronecker delta and the overbar indicates the complex conjugate. It is
important to note that there are several different normalizations of the spherical harmonics
that are possible, that will differ from (7.3) and thus lead to a different version of (7.4).Several of these are nicely catalogued in [34], including which disciplines tend to use which
normalization. This is important to note since any result that uses orthogonality will differ
slightly depending on the choice of normalization.
The spherical harmonics form a complete set of orthonormal functions and thus form
a vector space. When restricted to the surface of a sphere, functions may be expanded on
the sphere into a series approximation much like a Fourier series. This is in fact a spherical
harmonic series. Any square-integrable function may be expanded as
f(ψ, θ
)=
∞∑l=0
l∑m=−l
fml Ym
l
(ψ, θ
), (7.5)
10 Mathematical Problems in Engineering
where
fml =
∫2π
0
∫π
0
f(ψ, θ
)Yml
(ψ, θ
)sinψ dψ dθ. (7.6)
The coefficients fml
are sometimes referred to as the spherical Fourier transform of f(ψ, θ) [35].
7.3. 3D Fourier Transforms in Spherical Polar Coordinates
To find 3D Fourier transforms in spherical polar coordinates, both the function and its
3D Fourier transform are written in terms of polar coordinates in the spatial and spatial
frequency domains. That is, a function in 3D space is expressed as f(�r) = f(r, ψr , θr) as a
function of spherical polar coordinates and its 3D Fourier transform is written as F( �ω) =F(ρ, ψω, θω) in frequency spherical polar coordinates. Certain relationships can be shown to
hold. The relationship between the function and its transform are summarized here, with the
relevant details omitted for brevity. The function itself can be expanded as a series in terms
of the spherical harmonics as
f(�r) = f(r, ψr , θr
)=
∞∑l=0
l∑k=−l
fkl (r)Y
kl
(ψr, θr
), (7.7)
where Ykl(ψr, θr) are the spherical harmonics and the Fourier coefficients are given by
fkl (r) =
∫2π
0
∫π
0
f(r, ψr , θr
)Ykl
(ψr, θr
)sinψrdψrdθr. (7.8)
The 3D Fourier transform of f can also be written in Fourier space in spherical polar coor-
dinates as
F( �ω) = F(ρ, ψω, θω
)=
∞∑l=0
l∑k=−l
Fkl
(ρ)Ykl
(ψω, θω
), (7.9)
where
Fkl
(ρ)=∫2π
0
∫π
0
F(ρ, ψω, θω
)Ykl
(ψω, θω
)sinψωdψωdθω. (7.10)
We emphasize again that the relationship between fkl(r) and Fk
l(ρ) is not that of a Fourier
transform. In fact, this relationship is given by [29]
Fkl
(ρ)= 4π(−i)l
∫∞
0
fkl (r)jl
(ρr)r2dr = 4π(−i)lSl
{fkl (r)
}, (7.11)
Mathematical Problems in Engineering 11
where Sl denotes a spherical Hankel transform of order l. The inverse relationship is given by
fkl (r) =
1
4π(i)l
2
π
∫∞
0
Fkl
(ρ)jl(ρr)ρ2dρ =
1
4π(i)lS−1
l
{Fkl
(ρ)}
. (7.12)
8. Spatial Transfer Function Relationship in 3D
With this long set of preliminaries, definitions, and conventions established, we can now
begin the derivation of the 3D spatial transfer function relationship between input s(�r, ω)and output u(�r, ω). The foregoing section makes use of several key results in [36] which
will be presented here without proof. The results in [36] address general cases, not all of
which are relevant to the work here. The versions of the theorems required herein are stated
based on the sign conventions presented in Section 5 with regards to the sign conventions
for the complex wavenumbers and also for the chosen definition of the Fourier transform.
These affect the implied time dependence, the Sommerfeld radiation condition, and thus
the physically correct solution that is chosen from several mathematical possibilities. The
versions of the theorems presented here ensure that the chosen waves are outwardly
propagating (Sommerfeld radiation condition) and also decay to zero at infinity for a damped
wavefield, thus ensuring bounded and physically meaningful solutions.
Theorem 8.1. It is shown in [36] that the following result holds true:
I =∫∞
0
φ(x)x2 − k2
jn(xr)x2dx = −πik2h(2)n (kr)φ(k). (8.1)
Here, φ is any analytic function defined on the positive real line that remains bounded as x goes toinfinity, jn(x) is a spherical Bessel function of order n, h(2)
n (x) is a spherical Hankel function of ordern, and k is a wavenumber which may be real or complex, chosen such that the imaginary part of kis negative. Given the definition of the Fourier transform that is being currently used, the presentedresult satisfies the Sommerfeld radiation condition, ensuring an outwardly propagating wave.
8.1. 3D Transfer Function Relationship in Spherical Polar Coordinates
From (4.7) and using the conversion to spherical polar coordinates in 3D, the transfer function
for the Helmholtz equation in the Fourier domain can be written as
G( �ω,ω) =1(
ρ2 − k2) , (8.2)
which depends on the frequency spherical radial variable ρ only.
In general, as the imaginary part of the wavenumber k gets smaller, the transfer
function becomes more frequency selective, in the sense of strongly passing a smaller
bandwidth. As the imaginary part of the wavenumber k gets larger, the transfer function
becomes more low-pass in nature, passing lower frequencies more strongly than the higher
frequencies, with a corresponding larger bandwidth of frequencies being passed. This is
physically meaningful as the imaginary part of k represents the damping inherent in this
12 Mathematical Problems in Engineering
wave modality. Wavenumbers with larger imaginary parts are more heavily damped and
that damping tends to affect the higher frequencies more—that is, the higher frequencies are
attenuated and the lower ones are passed through, implying a low-pass nature to the physical
system.
Recall that ρ is the magnitude of the spatial frequency variables. The case for the
imaginary part of k equal to zero (in other words, a real wavenumber) is easy to visualize.
For the case of a real k, the function is discontinuous at ρ = k, meaning that only those
spatial frequencies where exactly ρ = k are passed and the rest are attenuated. In other
words, the resulting wave must have ρ2 = ω2x + ω2
y + ω2z = k2. While these statements are
fairly mathematical in nature, they explain the nature of the resolution of various imaging
modalities. For example, it is far more difficult to achieve the resolution of acoustic imaging
with thermal imaging and one of the explanations for this is the “low-pass” blurring nature
of the complex wavenumber of a thermal wave versus the highly selective band-pass nature
of the acoustic wavenumber. This is discussed further in other papers, for example, in [18].
8.2. 3D Green Function Coefficients
Let us define a set of functions that will be referred to as the Green function coefficients.
The actual Green function for the system is the full 3D inverse Fourier transform of the
transfer function, which for a spherically symmetric function is equivalent to a spherical
Hankel transform of order zero only [33]. With the help of Theorem 8.1, we define these Green
function coefficients as
g3Dn (r, k) = S−1
n
{1
ρ2 − k2
}=
2
π
∫∞
0
1
ρ2 − k2jn(ρr)ρ2dρ = −ikh(2)
n (kr), (8.3)
where g3Dn (r, k) has been used to denote the nth order Green function coefficient, with
the subscript indicating the order of the coefficient and the wavenumber k included as a
parameter of the function.
8.3. Transfer Function Relationship in Spherical Polar Coordinates
The 3D Fourier transform of the input (source) function is written in polar coordinates and
expanded in terms of a spherical harmonic series as
S( �ω,ω) =∞∑l=0
l∑m=−l
Sml
(ρ,ω
)Yml
(ψω, θω
). (8.4)
Hence, (4.1) for the output wavefield becomes
U( �ω,ω) =∞∑l=0
l∑m=−l
Sml
(ρ,ω
)ρ2 − k2
Yml
(ψω, θω
). (8.5)
Mathematical Problems in Engineering 13
The symbol k is used to denote the wavenumber. The spatial inverse Fourier transform and
thus the output in spatial coordinates is given by
u(r, ψr , θr , ω
)=
∞∑l=0
l∑m=−l
(i)l
4π
{2
π
∫∞
0
Sml
(ρ,ω
)jl(ρr)
ρ2 − k2ρ2dρ
}Yml
(ψr, θr
). (8.6)
The temporal frequency ω is a constant as far as the spatial inverse Fourier transformation
is concerned. We recognize that the quantity within the curly brackets can be evaluated with
the help of Theorem 8.1 and using the definition of the Green function coefficients in (8.3) to
obtain
2
π
∫∞
0
Sml
(ρ,ω
)jl(ρr)
ρ2 − k2ρ2dρ = g3D
l (r, k)Sml (k,ω). (8.7)
It now follows that the wavefield expression becomes
u(�r, ω) =∞∑l=0
l∑m=−l
il
4πg3Dl (r, k)Sm
l (k,ω)Yml
(ψr, θr
). (8.8)
If the measured wavefield itself, namely, the left-hand side of (8.8), is expanded as in a
spherical harmonic series so that
u(�r, ω) = u(r, ψr , θr , ω
)=
∞∑l=0
l∑m=−l
uml (r, ω)Ym
l
(ψr, θr
), (8.9)
then (8.8) gives us the simple input-output transfer function relationship we seek:
uml (r, ω) =
il
4πg3Dl (r, k)Sm
l (k,ω). (8.10)
Note how this relationship is in the spatial domain (not the frequency spatial domain) and
gives a multiplicative transfer function relationship between the input coefficients Sml(k,ω)
and the resulting wavefield uml(r, ω).
Using the relationships proposed between the coefficients of the function in the spatial
domain and the coefficients in the Fourier domain as given in (7.11), the general relationship
between Hankel and Fourier transforms is given by
Sml
(ρ,ω
)= 4π(−i)mSl
{sml (r, ω)
}= 4π(−i)l
∫∞
0
sml (r, ω)jl(ρr)r2dr. (8.11)
Hence (8.10) can be written as
uml (r, ω) = g3D
l (r, k)∫∞
0
sml (x, ω)jl(kx)x2dx, (8.12)
14 Mathematical Problems in Engineering
so that the value of the measured wavefield is related to the spherical Hankel transform of
the input function, evaluated at the wavenumber of the wavefield in question. Using the
definition of the spherical Hankel transforms, this becomes a very compact yet powerful
expression:
uml (r, ω) = g3D
l (r, k)Sl
{sml (k,ω)
}. (8.13)
The value of defining the Green function coefficients now becomes apparent in that it permits
the relationship between uml(r, ω) and sm
l(k,ω) to have a simple multiplicative transfer
function relationship. In other words, the relationship between output wave coefficients and
input is one of multiplication with the transfer function coefficients instead of a complicated
convolution-type relationship.
If we also assume that the input function is separable in time and space (a fairly
general assumption) so that s(�r, ω) = q(�r)η(ω). This would be the case for a spatial
inhomogeneity and a temporal input. In this case, then sml(r, ω) = qm
l(r)η(ω) and the formula
for the forward problem is given by
uml (r, ω) = g3D
l (r, k)η(ω)∫∞
0
qml (r)jl(kr)r2dr = g3D
l (r, k)η(ω)Sl
{qml (k)
}. (8.14)
8.4. Discussion and Relationship with the Fourier Diffraction Theorem
Equation (8.13) is the key input-output relationship that is sought. It gives the relationship
between input and output and the relationship is a transfer function (multiplication) type
of relationship, even in the spatial domain where the normal relationship to be expected is
one of convolution. This relationship is for the spherical harmonic expansion coefficients, not
between the full functions themselves. However, because it is a direct, proportional type of
relationship between the (l,m)th term of the input and the (l,m)th term of the output, it is
particularly useful and simple to apply.
In particular, (8.13) states that the wavefield (output) coefficients are directly
proportional to the transfer function coefficients and the proportionality factor between
them is the spherical Hankel transform of the input coefficients, evaluated on the sphere
ρ = k. Note that (l,m)th order output coefficients are related to (l,m)th order transfer
function coefficients in proportion to the lth order spherical Hankel transform of the (l,m)thorder input coefficient. This is in fact a generalization of the Fourier diffraction theorem of
tomography [3] which loosely states that the output wavefield is proportional to the Fourier
transform of the input inhomogeneity evaluated somewhere on the ρ = k sphere in Fourier
space. This is still true but we have expressed a more precise version of this theorem in the
sense that we have removed any ambiguity regarding the “somewhere” on the ρ = k sphere
and replaced it with a relationship (8.14) that does not depend on angular location. This was
enabled by the use of the Fourier transform in curvilinear coordinates so that a full Fourier
transform requires an (l,m) set of Fourier coefficients and spherical Hankel transforms.
Furthermore, the idea of proportionality between Fourier transform of the inhomogeneity
and output wavefield is also made more precise by stating that the proportionality between
them is actually the coefficients of the Green function itself. In essence, the output wavefield
can be seen as being the Green function coefficients, with each term weighted by the spherical
Hankel transform of the input inhomogeneity.
Mathematical Problems in Engineering 15
9. 2D Transfer Function Relationship in 2D Polar Coordinates
We proceed to develop the transfer function expressions for the 2D case in polar coordinates.
As for the 3D case, we begin with a necessary theorem which is stated without proof. As
previously mentioned, the results in [36] address general cases, and the versions of these
theorems required herein are stated here based on the sign conventions presented in Section 5
for the complex wavenumbers and also for the Fourier transform which affects the implied
time dependence and thus the Sommerfeld radiation condition.
Theorem 9.1. It is shown in [36] that the following result holds true:
I =∫∞
0
φ(ρ)
ρ2 − k2Jn(ρr)ρ dρ = −πi1
2H
(2)n (kr)φ(k). (9.1)
Here, φ is an analytic function defined on the positive real line that remains bounded as x goes toinfinity, Jn(x) is a Bessel function of order n, H(2)
n (x) is a Hankel function of order n, and k is awavenumber which may be real or complex, chosen such that the imaginary part of k is negative.Given the definition of the Fourier transform that is being currently used, the presented result satisfiesthe Sommerfeld radiation condition, ensuring an outwardly propagating wave.
9.1. 2D Green Function Coefficients
The overall system transfer function in 2D is given by
G( �ω,ω) =1(
ρ2 − k2) , (9.2)
which depends on the frequency radial variable only.
As for the 3D case, we define the 2D Green function coefficients as the nth-order
inverse Hankel transform of the overall system spatial transfer function. These will be shown
to be the required transfer functions for working in the 2D polar formulation. From the
definition of the inverse Hankel transform, these are given by
g2Dn (r, k) = H−1
n
{1
ρ2 − k2
}=∫∞
0
1
ρ2 − k2Jn(ρr)ρ dρ = −πi
2H
(2)n (kr). (9.3)
Recall that the actual (full) Green function for the system is the full 2D inverse Fourier
transform of the transfer function, which is an inverse Hankel transform of order zero only.
In (9.3), g2Dn (r, k) has been used to denote the nth order Green function coefficients with the
subscript indicating the order.
16 Mathematical Problems in Engineering
9.2. Fourier Theorem for 2D Fourier Transforms in Polar Coordinates
The 2D Fourier transform of the input function is written in polar coordinates as
S(ρ, ψ,ω
)=
∞∑n=−∞
Sn
(ρ,ω
)ejnψ, (9.4)
where Sn(ρ,ω) can be found from
Sn
(ρ,ω
)=
1
2π
∫2π
0
S(ρ, ψ,ω
)e−jnψdψ. (9.5)
The output wavefield u(�r, ω) is similarly expanded as a series so that in the spatial domain
we can write
u(�r, ω) = u(r, θ, ω) =∞∑
n=−∞un(r)ejnθ, (9.6)
while in the spatial Fourier domain this is
U( �ω,ω) = U(ρ, ψ,ω
)=
∞∑n=−∞
Un
(ρ,ω
)ejnψ. (9.7)
The output wavefield is given by U( �ω,ω) = G( �ω,ω)S( �ω,ω) and since G( �ω,ω) is radially
symmetric and does not require a full series, the output wavefield in the Fourier domain is
given by
U(ρ, ψ,ω
)=
∞∑n=−∞
Un
(ρ,ω
)ejnψ =
∞∑n=−∞
Sn
(ρ,ω
)(ρ2 − k2
)ejnψ, (9.8)
or equivalently as
Un
(ρ,ω
)=
Sn
(ρ,ω
)(ρ2 − k2
) . (9.9)
The equivalent expression to (9.8) in the spatial domain is given by
u(�r, ω) =∞∑
n=−∞un(r)ejnθ =
∞∑n=−∞
in
2π
{∫∞
0
Sn
(ρ,ω
)(ρ2 − k2
)Jn(ρr)ρ dρ}ejnθ, (9.10)
or
un(r) =in
2π
∫∞
0
Sn
(ρ,ω
)(ρ2 − k2
)Jn(ρr)ρ dρ. (9.11)
Mathematical Problems in Engineering 17
The temporal frequency ω is a constant as far as the spatial inverse Fourier transformation
is concerned. The quantity within the curly brackets can be evaluated with the help of
Theorem 9.1 as
∫∞
0
Sn
(ρ,ω
)(ρ2 − k2
)Jn(ρr)ρ dρ = −πi2H
(2)n (kr)Sn(k,ω) = g2D
n (r, k)Sn(k,ω). (9.12)
Hence, the output wavefield expression is given by
u(�r, ω) =∞∑
n=−∞un(r)ejnθ =
∞∑n=−∞
in
2πg2Dn (r, k)Sn(k,ω)ejnθ. (9.13)
The definition of forward and inverse Hankel transforms, the general relationship between
Hankel and Fourier transforms is given by
Sn
(ρ)= 2πi−nHn(sn(r)) = 2πi−n
∫∞
0
sn(r)Jn(ρr)r dr, (9.14)
and may be used to simplify (9.13) to yield
u(�r, ω) =∞∑
n=−∞un(r)ejnθ =
∞∑n=−∞
g2Dn (r, k)
∫∞
0
sn(x, ω)Jn(kx)x dxejnθ, (9.15)
or more compactly as
un(r, ω) = g2Dn (r, k)
∫∞
0
sn(x, ω)Jn(kx)x dx. (9.16)
It is in (9.15) that the interpretation of the evaluation of the 2D Fourier transform becomes
apparent, namely through the evaluation of the Bessel function at the (real or complex)wavenumbers. Using the definition of the Hankel transform, (9.16) can be written in the
compact and powerful formulation of
un(r, ω) = g2Dn (r, k)Hn(sn(k,ω)). (9.17)
If it is also further assumed that the inhomogeneity function is separable in time and space
(a fairly general assumption) so that s(�r, ω) = q(�r)η(ω), and
∞∑n=−∞
sn(r, ω)ejnθ = η(ω)∞∑
n=−∞qn(r)ejnθ, (9.18)
then (9.16) can be further reduced to
un(r, ω) = g2Dn (r, k)η(ω)Hn
{qn(k)
}. (9.19)
18 Mathematical Problems in Engineering
It is noted that the value of the measured wavefield at any position r is related to the Hankel
transform of the heterogeneity function evaluated at the wavenumber of the wavefield in
question, with the Green function coefficient acting as the proportionality term.
Equation (9.19) is the 2D equivalent of (8.14). The same comments can be made
as those made in the discussion after the 3D version of this problem, as the relationship
is identical, save for translating 3D tools such as the spherical harmonic expansions and
spherical Hankel transforms into 2D tools involving Fourier series and Hankel transforms.
10. Transfer Function Relationship in 1D
We proceed to develop the equivalent relationship for the 1D case. We have seen from (4.7)that the 1D transfer function in the Fourier domain is given by
G(ωx, ω) =1(
ω2x − k2
) , (10.1)
which depends on only a single spatial frequency. Equation (4.2) in 1D then becomes
u(r, ω) =1
(2π)
∫∞
−∞
S(ωx, ω)(ω2
x − k2)eiωxrdωx. (10.2)
The temporal frequency ω is a constant as far as the spatial inverse Fourier transformation is
concerned. We are thus interested in calculating integrals of the form
I =1
2π
∫∞
0
φ(x)x2 − k2
eixrdx, (10.3)
where φ is an analytic function defined on the positive real line that approaches zero as x goes
to infinity. The required theorem is given in [36] and the relevant result is presented below.
Theorem 10.1. It is shown in [36] that the following is true
1
2π
∫∞
−∞
φ(ρ)
ρ2 − k2eiρrdρ =
⎧⎪⎪⎨⎪⎪⎩1
2ikφ(−k)e−ikr , r > 0,
1
2ikφ(k)eikr , r < 0.
(10.4)
Here, φ is an analytic function defined on the positive real line that remains bounded as x goes toinfinity and k is a wavenumber which may be real or complex, chosen such that the imaginary part ofk is negative. Given the definition of the Fourier transform that is being currently used, the presentedresult satisfies the Sommerfeld radiation condition, ensuring an outwardly propagating wave.
Mathematical Problems in Engineering 19
10.1. 1D Green’s Functions via Inverse Fourier Transformation of the SpatialTransfer Functions
The Green’s function for the Helmholtz equation is 1D and is given by inverse Fourier
transform of the spatial transfer function as
g1D(r, k) =1
2π
∫∞
−∞
eiωxr
ω2x − k2
dωx, (10.5)
and can be evaluated with the help of Theorem 10.1 to give
g1D(r, k) =
⎧⎪⎪⎨⎪⎪⎩1
2ike−ikr r > 0
1
2ikeikr r < 0
⎫⎪⎪⎬⎪⎪⎭ =1
2ike−ik|r|. (10.6)
10.2. Transfer Function Relationship in 1D
Equation (10.2) can now be evaluated with the help of Theorem 10.1 as well as the result in
(10.6) as
u(r, ω) =1
(2π)
∫∞
−∞
S(ωx, ω)(ω2
x − k2)eiωxrdωx =
⎧⎪⎨⎪⎩1
2ikS(−k,ω)e−ikr r > 0
1
2ikS(k,ω)eikr r < 0
= g1D(r, k)S(− sgn(r)k,ω
),
(10.7)
where g1D(r, k) = (1/2ik)e−ik|r|. This is the 1D version of the transfer function relationship,
with the wavefield being directly proportional to the Fourier transform of the object function
evaluated at k. This is similar to the results for the 2D and 3D cases. For a real wavenumber,
a “sphere” in 1D becomes the two points on the real line at ±k and the proportionality term
is the Green’s function for the space in question.
As before, we assume that the inhomogeneity function is separable in time and space
(a fairly general assumption) so that s(r, ω) = q(r)η(ω) → S(ωx, ω) = Q(ωx)η(ω), where
Q(ωx) is the 1D Fourier transform of q(r). The relevant result now reads
u(r, ω) = g1D(r, k)η(ω)Q(− sgn(r)k
). (10.8)
This result is directly applicable for computational purposes.
11. Summary of Results
The relationships in the previous sections are summarized in Table 1.
20 Mathematical Problems in Engineering
Table 1
Dimension TransformsPseudo-Green’s (gn)
functionInput/output relationship
3Spherical Hankel and
spherical harmonicg3Dn (r, k) = −ikh(2)
n (kr)ukl(r, ω) =
g3Dl
(r, k)η(ω)Sl{qkl (k)}2 Hankel and Fourier Series g2D
n (r, k) = −πi2H
(2)n (kr) un(r, ω) =
g2Dn (r, k)η(ω)Hn{qn(k)}
1 1D Fourier g1D(x) =−i2k
e−ik|x| u(x, ω) =g1D(x)η(ω)Q(− sgn(x)k)
12. Applications of the Transfer Function Relationships tothe Wave Equation
The transfer function relationships can be used to find time domain Green’s functions where
the temporal Fourier integral can be easily inverted. In particular, we consider the case where
the input to the standard wave equation is a Dirac-delta function at the origin
∇2u(�r, t) − 1
c2
∂2
∂t2u(�r, t) = −δ(�r)δ(t). (12.1)
This corresponds to the Helmholtz equation above with k = ω/c. The temporal and
spatial Fourier transform of δ(�r)δ(t) is 1, which is spherically symmetric so that the transfer
function relationships given above only need the zeroth-order component and simplify
considerably. In 3D, the transfer function relationship is given by
u(r, ω) = g3D0 (r, k) · 1 = −ikh(2)
0 (kr). (12.2)
In 2D, the relationship is given by
u(r, ω) = g2D0 (r, k) · 1 = −πi
2H0(kr). (12.3)
In 1D, the relationship is
u(r, ω) = g1D(r, k) · 1 =1
2ike−ik|r|. (12.4)
These three relationships can now be inverse Fourier transformed in time.
For the 3D case, we use the fact that h(2)0 (kr) = j0(kr)− iy0(kr) = i exp(−ikr)/kr so that
the inverse Fourier transform of (12.2) gives
u(r, t) =1
2π
∫∞
−∞
1
re−iωr/ceiωtdω
=1
rδ
(t − r
c
).
(12.5)
Mathematical Problems in Engineering 21
The 1D time response can be found from the inverse Fourier transform of (12.4)
u(r, t) =1
2π
∫∞
−∞
c
2iωe−iω|r|/ceiωtdω
=c
4
[2H
(t − |r|
c
)− 1
]=
c
4sgn
(t − |r|
c
),
(12.6)
where H(x) is the Heaviside unit step function.
The 2D case is a little more complicated but can nevertheless be evaluated in closed
form by an inverse Fourier transform of (12.3). To do this, we write the zeroth-order Hankel
function in its integral form as [37]
H(2)0 (x) = J0(x) − iY0(x) =
2i
π
∫∞
1
e−ixτ√τ2 − 1
dτ. (12.7)
Hence from (12.7) and (12.3), the inverse Fourier transform of (12.3) gives
u(r, t) =1
2π
∫∞
−∞
∫∞
1
e−iωrτ/c
√τ2 − 1
dτeiωtdω. (12.8)
Changing the order of integration gives
u(r, t) =∫∞
1
1√τ2 − 1
1
2π
∫∞
−∞eiω[t−rτ/c]dωdτ. (12.9)
But
1
2π
∫∞
−∞eiω[t−rτ/c]dω = δ
(t − rτ
c
), (12.10)
so that (12.9) becomes
u(r, t) =∫∞
1
1√τ2 − 1
δ
(t − rτ
c
)dτ =
∫∞
−∞
1√τ2 − 1
H(τ − 1)δ(t − rτ
c
)dτ. (12.11)
Changing variables so that x = rτ/c gives
u(r, t) =c
r
∫∞
−∞H
(cx
r− 1
)1√
(cx/r)2 − 1
δ(t − x)dx = H
(t − r
c
)1√
t2 − r2/c2, (12.12)
which yields the desired result in closed form.
22 Mathematical Problems in Engineering
13. Application: Inversion Formulas for Tomographic Applicationswith Real Wavenumbers
In the case of a real wavenumber k, an immediate application of the above formulas is
for closed-form inversion formulas which are useful for tomographic applications. These
inversion formulas are applicable to applications where the wavefield (e.g., acoustic field)is measured and the goal is to reconstruct the source (input) that led to that measured
wavefield (output). As we have seen, formulas above give the output (wavefield) in terms
of the transform of the input evaluated at k. Thus, if the output wavefield is measured, this
implies knowledge of the transform of the input. An inverse transform then leads to the input
itself.
13.1. Inversion Formula in 3D
Equation (8.14) admits an inversion useful for tomographic applications. It can be written as
iukl(r0, ω)
ksh(2)l(ksr0)η(ω)
=∫∞
0
φkl (x)jl(ksx)x
2dx, (13.1)
where x has been used as a dummy integration variable in order to avoid possible confusion
and r0 has been used as the radial variable and indicates the position where a measurement
of the wavefield is made. Multiplying both sides by jl(ksr)k2s and integrating over ks gives
∫∞
0
iukl(r0, ω)
ksh(2)l(ksr0)η(ω)
jl(ksr)k2sdks =
∫∞
0
∫∞
0
φkl (x)jl(ksx)x
2dxjl(ksr)k2sdks
=∫∞
0
∫∞
0
φkl (x)jl(ksx)jl(ksr)k
2sdksx
2dx.
(13.2)
Using the orthogonality of the spherical Bessel functions, it follows that (13.2) becomes
∫∞
0
iukl(r0, ω)
ksh(2)l(ksr0)η(ω)
jl(ksr)k2sdks =
∫∞
0
φkl (x)
π
2x2δ(x − r)x2dx =
π
2φkl (r). (13.3)
Recalling that ks = ω/cs, the inversion formula for the spatial source becomes
φkl (r) =
2i
πc2s
∫∞
0
ukl(r0, ω)
h(2)l(ωr0/cs)η(ω)
jl
(ωr
cs
)ωdω. (13.4)
If the measured wavefield is also separable in space and time so that ukl(r0, ω) = χk
l(r0)T(ω),
then (13.4) can be further simplified to
φkl (r) =
2iχkl(r0)
πc2s
∫∞
0
T(ω)
h(2)l(ωr0/cs)η(ω)
jl
(ωr
cs
)ωdω. (13.5)
Mathematical Problems in Engineering 23
Equation (13.5) gives a simple form for finding the spatial source function at any position
r once having measured the temporal response of the wavefield at a position r0. However,
since the spherical harmonic transform ukl(r0, ω) of the wavefield is required, this implies
that sufficient angular information about the wavefield u must be obtained in order to permit
this spherical harmonic calculation.
13.2. Inversion Formula in 2D
Equation (9.19) admits an inversion leading to the input function. We assume that the source
function is separable in time and space (a fairly general assumption) so that s(�r, ω) =φ(�r)η(ω). In this case, then sk
l(r, ω) = φk
l(r)η(ω) and (8.12) becomes
2iun(r0, ω)
πH(2)n (ksr0)η(ω)
=∫∞
0
φn(x)Jn(ksx)x dx, (13.6)
where x has been used as a dummy integration variable in order to avoid possible confusion
and r0 indicates the radial position where the wavefield is measured. Multiplying both sides
of (13.6) by Jn(ksr)ks and integrating over ks gives
∫∞
0
2iun(r0, ω)
πH(2)n (ksr0)η(ω)
Jn(ksr)ksdks =∫∞
0
∫∞
0
φn(x)Jn(ksx)x dxJn(ksr)ksdks,
=∫∞
0
φn(x)∫∞
0
Jn(ksx)Jn(ksr)ksdks x dx.
(13.7)
Using the orthogonality of the spherical Bessel functions, it follows that (13.7) becomes
∫∞
0
2iun(r0, ω)
πH(2)n (ksr0)η(ω)
Jn(ksr)ksdks =∫∞
0
φn(x)1
xδ(x − r)x dx = φn(r). (13.8)
Recalling that ks = ω/cs, the inversion formula for the inhomogeneity becomes
φn(r) =2i
πc2s
∫∞
0
un(r0, ω)
H(2)n (ωr0/cs)η(ω)
Jn
(ωr
cs
)ωdω. (13.9)
If the measured wavefield is also separable in space and time so that un(r0, ω) = χn(r0)T(ω),then (13.4) can be further simplified to
φn(r) =2iχn(r0)
πc2s
∫∞
0
T(ω)
H(2)n (ωr0/cs)η(ω)
Jn
(ωr
cs
)ωdω. (13.10)
Equation (13.10) gives a simple form for finding the spatial source function at any position
r once having measured the wavefield response at a radial position r0. Comparing (13.10)and (13.5), it is noted that they have identical forms with the exception of the replacement
24 Mathematical Problems in Engineering
of Hankel and Bessel functions for the 2D case with spherical Hankel and Bessel functions
for the 3D case.
13.3. Inversion Formula in 1D
We can also further assume that the wavefield is also separable in time and space so that
u(r, ω) = χ(r)T(ω), which leads to to
φ(−ks) =χ(r)T(ω)η(ω)
2ikseiksr =
∫∞
−∞φ(y)eiyksdy r > 0,
φ(ks) =χ(r)T(ω)η(ω)
2ikse−iksr =
∫∞
−∞φ(y)e−iyksdy r < 0.
(13.11)
Using the orthogonality of the Fourier kernel, both sides are multiplied by e±ixks and
integrated over all ks to give
χ(r)∫∞
−∞
T(ω)η(ω)
2ikseiksre−ixksdks
=∫∞
−∞φ(y) ∫∞
−∞eiykse−ixksdksdy = 2π
∫∞
−∞φ(y)δ(y − x
)dy r > 0
=⇒ φ(x) =iχ(r)
πc2s
∫∞
−∞
T(ω)η(ω)
eiω((r−x)/cs)ωdω r > 0,
(13.12)
and similarly
χ(r)∫∞
−∞
T(ω)η(ω)
2ikse−iksreixksdks
=∫∞
−∞φ(y) ∫∞
−∞e−iykseixksdksdy = 2π
∫∞
−∞φ(y)δ(y − x
)dy r < 0
=⇒ φ(x) =iχ(r)
πc2s
∫∞
−∞
T(ω)η(ω)
eiω((x−r)/cs)ωdω r < 0.
(13.13)
Several points need to be made regarding (13.12) and (13.13). First, they both give the source
function as a function of position x along the real line. The interpretation of the variable r
is that of the position at which the measurement is made and is considered to be a fixed
quantity. Both (13.12) and (13.13) are inverse Fourier transforms of (T(ω)/η(ω))ω, evaluated
at (x − r)/cs or (r − x)/cs, depending on whether measurements are made in transmission or
reflection. Clearly |x − r|/cs is the time taken for a wave to travel the distance |x − r|.The 2D and 3D equivalent to (13.12) and (13.13) are (13.5) and (13.10) which also
similarly involve an inverse transform of (T(ω)/η(ω))ω but have a different kernel for the
integration. The reason for the differences in the nature of the kernels from the 2D/3D cases
to the 1D case is that in the 2D and 3D cases the kernels used for the Fourier transforms are
the Bessel and spherical Bessel functions, respectively, which are the standing wave solutions
Mathematical Problems in Engineering 25
in those dimensions. However, Green’s functions in those dimensions are the Hankel and
spherical Hankel functions which are the travelling wave solutions. We note that the kernels
for the inversion formulas of (13.5) and (13.10) involve inverse Hankel and spherical Hankel
functions, which represent the inverse of Green’s function. In contrast, the travelling wave
solutions in 1D are the complex exponential and those are also the kernel of 1D Fourier
transform. The ratio of the two complex exponentials (one represents the Fourier kernel, the
other represents the inverse of 1D Green’s function) finally give another complex exponential
which represents the shift |x − r|/cs.
14. Summary and Conclusions
This paper presents the derivation of spatial transfer function relationships for the Helmholtz
equation. The focus has been on deriving forward transfer function relationships so that
once given a particular input function, the resulting output wavefield can be calculated
at any point in space. These are termed “transfer function” relationships because the
relationship between the input and output quantities is one of multiplication and no
convolution is involved, even in the spatial domain where normally a convolution would
be required. Interestingly, instead of Green’s function itself, Green’s function coefficients
for the space are required. These Green function coefficients form the kernel of the output
response. Each element of the kernel is then weighted (multiplicatively) by the relevant
Fourier/Hankel/Spherical Hankel transform of the input, evaluated on the n-dimensional
sphere of radius k, where k is the wavenumber and n is the dimension of the space.
Combined together, these form the final output response. This is true for dimensions n = 1,
2, 3. These simple but powerful results serve to cast the entire problem as an input-output
problem with a transfer function relating input to output. In this view of the problem, the
input (inhomogeneity) and output (resulting wavefield) in space are related by a simple
multiplicative transfer function relationship and not via a convolution. Some applications of
these results were shown in the manuscript.
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