application of nonlinear time-fractional partial differential equations to image processing via...

Research ArticleApplication of Nonlinear Time-Fractional PartialDifferential Equations to Image Processing via HybridLaplace Transform Method

B A Jacobs 123 and C Harley 12

1School of Computer Science and Applied Mathematics University of the Witwatersrand Johannesburg Private Bag 3Wits 2050 South Africa2DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) South Africa3School of Mathematics and Statistics UNSW Australia Sydney NSW 2052 Australia

Correspondence should be addressed to B A Jacobs byronjacobswitsacza

Received 6 April 2018 Accepted 30 August 2018 Published 16 September 2018

Academic Editor Morteza Khodabin

Copyright copy 2018 B A Jacobs and C Harley This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This work considers a hybrid solutionmethod for the time-fractional diffusionmodel with a cubic nonlinear source term in one andtwo dimensions Both Dirichlet and Neumann boundary conditions are considered for each dimensional case The hybrid methodinvolves a Laplace transformation in the temporal domain which is numerically inverted and Chebyshev collocation is employedin the spatial domain due to its increased accuracy over a standard finite-difference discretization Due to the fractional-orderderivative we are only able to compare the accuracy of this method with Mathematicarsquos NDSolve in the case of integer derivativeshowever a detailed discussion of the merits and shortcomings of the proposed hybridization is presented An application to imageprocessing via a finite-difference discretization is included in order to substantiate the application of this method

1 Introduction

This work examines the performance of a hybrid LaplacetransformndashChebyshev collocation technique applied to thetime-fractional diffusion equation in two dimensions with anonlinear source term119906119905120572 = 119888119889 (119906119909119909 + 119906119910119910) + 119888119904119906 (1 minus 119906) (119906 minus 119886) (1)

where 119906119905120572 = 120597120572119906120597119905120572 119906119909119909 = 12059721199061205971199092 119906119910119910 = 12059721199061205971199102 (2)

This model is explored subject to both Dirichlet and Neu-mann boundary conditions on the bounded domain Ω =

[minus1 1] times [minus1 1] to satisfy the domain required by the Cheby-shev polynomials with initial condition 119906(119909 119910 0) = 119891(119909 119910)This method benefits from the analyticity of the Laplacetransform and efficient numerical inversion of this trans-form an accurate discretization approach through Cheby-shev collocation and a convergent linearization techniquewhich results in a robust method for solving nonlineartime-fractional partial differential equations on a boundeddomain Following the method detailed by Jacobs and Harleyin [1 2] we also employ a finite-difference discretization inapplying this method to imagesThis is elaborated on furtherin Section 5

Recently fractional derivatives and fractional partial dif-ferential equations (FPDEs) have received great attentionboth in analysis and application (see [3ndash5] and referencestherein) Despite this large effort very little attention hasbeen paid to solving FPDEs on a bounded domain throughtransformation techniques Agrawal [6] makes use of theLaplace transform and the finite sine transform to obtain an

HindawiJournal of MathematicsVolume 2018 Article ID 8924547 9 pageshttpsdoiorg10115520188924547

2 Journal of Mathematics

analytic solution to the fractional diffusion-wave equation ona bounded domain Other techniques such as the variationaliteration method Adomian decomposition method and dif-ferential transformmethod have all been applied to fractionalpartial differential equations but with a focus on unboundeddomains More recently Hashemizadeh and Ebrahimzadeh[7] and Zaky et al [8] present solutions to the linear and non-linear time-fractional Klein-Gordon equations respectivelyTheir solutions were obtained by using an operational matrixapproach and Legendre polynomial interpolation

The linear diffusion model has been applied in manydifferent fields including image processing [9ndash14] Howeverto the best of the authorsrsquo knowledge the application of atime-fractional partial differential equation has not yet beenthoroughly examined within such a context It is from thestandpoint of applying a fractional partial differential equa-tion to an image thatwe examine the time-fractional diffusionequation in two dimensions In a recent paper Jacobs andMomoniat [15 16] show that the diffusion equation with thisnonlinear source term is able to binarize a document imagewith great success In extending this concept to fractional-order derivatives we seek to preserve the symmetry of thediffusion equation Values of 120572 isin (0 1] resolve the diffusionequation to subdiffusion which preserves symmetry andexhibits some interesting dynamics which are discussed lateron in this work Values of 120572 gt 1 introduce a transport effectwhich then breaks symmetry It is for this reason that weimpose the restriction 120572 isin (0 1]

In this work we make use of the Laplace transform whichallows us to handle the fractional-order derivative in analgebraic way and incur no error in doing so Inversion of theLaplace transform is however difficult to obtain analyticallyWe therefore make extensive use of the numerical inversionprocedure described by Weideman and Trefethen in [17]which defines a contour of integration that maps the domainof the Bromwich integral from the entire complex space to thereal space from which we can approximate this integral witha trapezoidal rule

With a robustmethod for inverting the Laplace transformwe may then hybridize the transform with a discretizationtechnique The Laplace transform of the temporal variableavoids the need for a time-marching scheme as well as reduc-ing the fractional-order derivative to an algebraic expressionThe transformed model is then discretized by the use ofChebyshev collocation due to its superior performance tofinite differences as illustrated in [1 2] The resulting systemis solved and the transform inverted to obtain a semianalyticsolution continuous in time and discrete in space

In the following section we present some preliminaryresults which are put to use throughout the paper In Section 3the implemented methods are described including the dif-ferent cases for boundary conditions Section 4 presents thesolutions obtained by the proposedmethod aswell as an errorcomparison with Mathematicarsquos NDSolve for the one- andtwo-dimensional cases of our model as well as both Dirichletand Neumann boundary conditions In Section 5 we providea real-world application of thismethod andmodel in the formof document image binarization illustrating the ability toobtain a useful result with the present method when coupled

with the finite-difference discretization A discussion of theresults and their relationship to work beyond this researchis presented in Section 6 and some concluding remarks aremade in Section 7

2 Preliminaries

In this work we employ Caputorsquos definition of a fractionalderivative over the Riemann-Liouville derivative due to thefact that the Caputo derivative makes use of the physicalboundary conditions whereas the Riemann-Liouville deriva-tive requires fractional-order boundary conditions

Definition 1 The Riemann-Liouville integral of order 120572 gt 0of a function 119906(119905) is119869120572119906 (119905) = 1Γ (120572) int1199050 (119905 minus 120591)120572minus1 119906 (120591) 119889120591 119909 gt 0 (3)

Definition 2 The fractional derivative of 119906(119905) according to theCaputo definition with119898 minus 1 lt 120572 le 119898 119898 isin N is120597120572119906 (119905)120597119905120572 = 119869119898minus120572119863119898119906 (119905) = 119863120572lowast119906 (119905)

fl1Γ (119898 minus 120572) int1199050 (119905 minus 120591)119898minus120572minus1 119906(119898) (120591) 119889120591 (4)

If 120572 isin Z the Caputo fractional derivative reduces tothe ordinary derivative or integral Podlubny [4] illustratesthe pleasing property of the Laplace transform of a Caputoderivative as can be seen in (5) In our case where 0 lt 120572 lt 1we have

L120597120572119906 (119909 119910 119905)120597119905120572 = 119904120572119880 (119909 119910) minus 119904120572minus1119906 (119909 119910 0) (5)

This property allows one to treat fractional-order derivativesalgebraically

Definition 3 The Generalized Mittag-Leffler function of theargument 119911 is 119864120572120573 (119911) = infinsum

119896=0

119911119896Γ (119896120572 + 120573) (6)

Theuse of Laplace transformallows one to circumvent theproblems that arise in the time-domain discretization How-ever using the Laplace transform for the fractional-orderderivative presents the problem of inverting the transformto find a solution Analytic inversion of the transform isinfeasible and hence the numerical scheme for evaluating theBromwich integral presented by Weideman and Trefethen in[17] is put to extensive use The Bromwich integral is of theform 119906 (119905) = 12120587119894 int120590+119894infin120590minus119894infin 119890119904119905119880 (119904) 119889119904 120590 gt 1205900 (7)

where 1205900 is the convergence abscissa Using the parabolicconformal map119904 = 120583 (119894119908 + 1)2 minus infin lt 119908 lt infin (8)

(7) becomes

Journal of Mathematics 3

119906 (119905) = 12120587119894 intinfinminusinfin 119890119904(119908)119905119880 (119904 (119908)) 1199041015840 (119908) 119889119908 (9)

which can then be approximated simply by the trapezoidalrule or any other quadrature technique as

119906ℎ119873119905 (119905) = ℎ2120587119894 119873119905sum119896=minus119873119905

119890119904(119908119896)119905119880 (119904 (119908119896)) 1199041015840 (119908119896) 119908119896 = 119896ℎ (10)

On the parabolic contour the exponential factor in (7)forces rapid decay in the integrand making it amenable toquadrature All that is left is to choose the parameters ℎand 120583 optimally which is done by asymptotically balancingthe truncation error and discretization errors in each of thehalf-planes The methodology described by Weideman andTrefethen achieves near optimal results (see [17] and figurestherein) and the interested reader is directed there for athorough description of the implementation of the methodIn this work we make use of the parabolic contour due to theease of use and the hyperbolic contour only exhibits a slightimprovement in performance over the parabolic contour

3 Methods

This section introduces the methodologies used for the two-dimensional model previously presented We may write ourmodel as119888119889119906119909119909 + 119888119889119906119910119910 + 119888119904 (119886 + 1) 1199062 minus 119888119904119886119906 minus 1198881199041199063 minus 119906119905120572= Φ(119906119909119909 119906119910119910 119906 119906119905120572) = 0 (11)

The quasi-linearization technique can be viewed as a gen-eralized Newton-Raphson method in functional space Aniterative scheme is constructed creating a sequence of linearequations that approximate the nonlinear equation (11) andboundary conditions Furthermore this sequence of solu-tions converges quadratically andmonotonically [18ndash20]Thequasi-linear form is given by119888119889119906119899+1119909119909 + 119888119889119906119899+1119910119910 + 1198831198991119906119899+1 + 1198831198992119906119899+1119905120572 = 1198831198993 (12)

where 119899 indicates the index of successive approximationMoreover 119906(119909 119910 119905)119899 is known entirely at the explicit index119899 The coefficients are given by

1198831198991 = 120597Φ120597119906 = 1198881199042 (119886 + 1) 119906119899 minus 119888119904119886 minus 3119888119904 (119906119899)2 (13)

and 1198831198992 = 120597Φ120597119906119905120572 = minus1 (14)

If the elements in 119906 = (119906119909119909 119906119910119910 119906 119906119905120572) are indexed by 119895 thenfor the 119899th iteration in the quasi-linearization we have

1198831198993 = sum119895

119906119899119895 120597Φ119899120597119906119895 = [119906119909119909 120597Φ120597119906119909119909 + 119906119910119910 120597Φ120597119906119910119910 + 119906120597Φ120597119906 + 119906119905120572 120597Φ120597119906119905120572 ]119899 = [119888119889119906119909119909 + 119888119889119906119910119910 + 2 (119886 + 1) 1199062 minus 119886119906 minus 31199063 minus (119906119905120572)]119899 = [119888119889119906119909119909 + 119888119889119906119910119910 + (119886 + 1) 1199062 minus 119886119906 minus 1199063 minus (119906119905120572)]119899+ [(119886 + 1) 1199062 minus 21199063]119899

(15)

Since the first term above satisfies (11) it is replaced with 0giving

1198831198993 = [(119886 + 1) 1199062 minus 21199063]119899 (16)

Equation (12) can now be transformed by the Laplace trans-form a linear operator to obtain

119888119889119880119899+1119909119909 + 119888119889119880119899+1119910119910 + 1198831198991119880119899+1 minus 119904120572119880119899+1= 1198831198993119904 minus 119904120572minus1119891 (119909 119910) (17)

where 119880 (119909 119910 119904) = L 119906 (119909 119910 119905) (18)

Equation (17) may be discretized by Chebyshev collocationdescribed in the following section

31 Chebyshev Collocation Chebyshev polynomials form abasis on [minus1 1] and hence we dictate the domain of our PDEto be Ω We note here however that any domain in R2 canbe trivially deformed to match Ω We discretize our spatialdomain using Chebyshev-Gauss-Lobatto points

119909119894 = cos( 119894120587119873119909) 119894 = 0 1 119873119909119910119895 = cos( 119895120587119873119910) 119895 = 0 1 119873119910 (19)

Given this choice of spatial discretization we have 1199090 = 1119909119873119909 = minus1 1199100 = 1 and 119910119873119910 = minus1 indicating that the domainis in essence reversed and one must exercise caution whenimposing the boundary conditions

In mapping our domain to Ω we may assume that 119873119909 =119873119910 ie we have equal number of collocation points in eachspatial direction We now define a differentiation matrix119863(1) = 119889119896119897


119889119896119897 =

119888119896 (minus1)119896+119897119888119897 (119909119896 minus 119909119897) 119896 = 119897minus 1199091198962 (1 minus 1199092119896) 119896 = 11989716 (21198732119909 + 1) 119896 = 119897 = 0minus16 (21198732119909 + 1) 119896 = 119897 = 119873119909where 119888119896 = 2 119896 = 01198731199091 119896 = 1 119873119909 minus 1

(20)

Bayliss et al [21] describe a method for minimizing theround-off errors incurred in the calculations of higher orderdifferentiation matrices Since we write 119863(2) = 119863(1)119863(1)we implement the method described in [21] in order tominimize propagation of round-off errors for the secondderivative in space

The derivative matrices in the 119909 direction are119863(1)119909 = 119863(1)119863(2)119909 = 119863(2) (21)

where 119863(1) is the Chebyshev differentiation matrix of size(119873119909 + 1) times (119873119910 + 1)Because we have assumed 119873119909 = 119873119910 we derive the

pleasing property that119863(1)119910 = (119863(1)119909 )119879 and119863(2)119910 = (119863(2)119909 )119879Writing the discretization of (17) in matrix form yields119863(2)119909 U119899+1 + U119899+1119863(2)119910 + 1198831198991U119899+1 minus 119904120572U119899+1 = 1198831198993119904 minus F (22)

where 119865119894119895 = 119904120572minus1119891 (119909119894 119910119895) (23)

By expanding (22) in summation notation we have119873119909sum119896=0

119889(2)119894119896 119880119899+1 (119909119896 119910119895) + 119873119910sum119896=0

119889(2)119896119895 119880119899+1 (119909119894 119910119896)+ [1198831]119899119894119895119880119899+1 (119909119894 119910119895) minus 119904120572119880119899+1 (119909119894 119910119895)= [1198833]119899119894119895119904 minus 119904120572minus1119891 (119909119894 119910119895)

(24)

for 119894 = 0 1 119873119909 119895 = 0 1 119873119910 By extracting the firstand last terms in sums we obtain119889(2)1198940 119880119899+1 (1199090 119910119895) + 119889(2)119894119873119909119880119899+1 (119909119873119909 119910119895)+ 119889(2)0119895 119880119899+1 (119909119894 1199100) + 119889(2)119873119910119895119880119899+1 (119909119894 119910119873119910)

+ 119873119909minus1sum119896=1

119889(2)119894119896 119880119899+1 (119909119896 119910119895) + 119873119910minus1sum119896=1


(25)

for 119894 = 1 119873119909 minus 1 119895 = 1 119873119910 minus 1 We use the form of(25) to impose the boundary conditions

The solution U = 119880(1199091 1199101) 119880(1199091 1199102) 119880(119909119873119909minus1119910119873119910minus1) which is the matrix of unknown interior points ofU can be found by solving the system(A minus s120572I) U + Xn

1 ∘ U + UB = Xn3119904 minus F (26)

where A is the matrix of interior points of D(2)119909 and B isthe matrix of interior points of D(2)119910 so that A and B matchthe dimensions of U We also use ∘ to denote the Hadamardproduct between two matrices Also119865119894119895 = 119904120572minus1119891 (119909119894 119910119895) + 119889(2)1198940 119880(1199090 119910119895)+ 119889(2)119894119873119909119880(119909119873119909 119910119895) + 119889(2)0119895 119880(119909119894 1199100)+ 119889(2)119873119910119895119880(119909119894 119910119873119910)

(27)

for 119894 = 1 119873119909 minus 1 119895 = 1 119873119910 minus 1311 Dirichlet Boundary Conditions Boundary conditionsmay be in the form of Dirichlet conditions119906 (minus1 119910 119905) = 119886119906 (1 119910 119905) = 119887 (28)

119906 (119909 minus1 119905) = 119888119906 (119909 1 119905) = 119889 (29)

and hence 119880 (minus1 119910) = L 119886 119880 (1 119910) = L 119887 (30)

119880 (119909 minus1) = L 119888 119880 (119909 1) = L 119889 (31)

The parameters 119886 119887 119888 and 119889 are potentially functions of thetemporal variable and one of the spatial variables ie 119886 =119886(119910 119905) We assume that 119886 119887 119888 and 119889 are constant

Dirichlet boundary conditions can be imposed directlyby substituting (28) and (29) into (25) and collecting all theknown terms in F

312 Neumann Boundary Conditions Alternatively Neu-mann boundary conditions give119906119909 (minus1 119910 119905) = 119886119906119909 (1 119910 119905) = 119887 (32)

119906119910 (119909 minus1 119905) = 119888119906119910 (119909 1 119905) = 119889 (33)

with

Journal of Mathematics 5119880119909 (minus1 119910) = L 119886 119880119910 (1 119910) = L 119887 (34)

119880119910 (119909 minus1) = L 119888 119880119910 (119909 1) = L 119889 (35)

Neumann boundary conditions given by (30) are dis-cretized as120597119880120597119909 (119909119873119909 = minus1 119910119895) asymp 119873119909sum

119896=0

119889119873119909119896119880119899 (119909119896 119910119895) = L 119886 (36)

120597119880120597119909 (1199090 = 1 119910119895) asymp 119873119909sum119896=0

1198890119896119880119899 (119909119896 119910119895) = L 119887 (37)

Similarly for equations (31)120597119880120597119910 (119909119894 119910119873119910 = minus1) asymp 119873119910sum119896=0

119889119896119873119910119880119899 (119909119894 119910119896) = L 119888 (38)

120597119880120597119910 (119909119894 1199100 = 1) asymp 119873119910sum119896=0

1198891198960119880119899 (119909119894 119910119896) = L 119889 (39)

By extracting the first and last terms in the sum thediscretizations can be written as

(1198891198731199090 11988911987311990911987311990911988900 1198890119873119909 )( 119880119899 (1199090 119910119895)119880119899 (119909119873119909 119910119895))= ( L 119887 minus 119873119909minus1sum

119896=1

1198890119896119880119899 (119909119896 119910119895)L 119886 minus 119873119909minus1sum

119896=1

119889119873119909119896119880119899 (119909119896 119910119895))(40)

and

(1198890119873119910 11988911987311991011987311991011988900 1198891198731199100 )( 119880119899 (119909119894 1199100)119880119899 (119909119894 119910119873119910))= (L 119889 minus 119873119910minus1sum

119896=1

1198891198960119880119899 (119909119894 119910119896)L 119888 minus 119873119910minus1sum

119896=1

119889119896119873119910119880119899 (119909119894 119910119896))(41)

and the solutions to these linear systems are then substitutedinto equation (25)

4 Results

In this section we consider only the results obtained byChebyshev collocation due to the enormous increase inaccuracy obtained over the finite-difference method that waspresented by the authors in [1 2]

Table 1 Maximum absolute error in the presented methodrsquossolution of the problem described by Example 1 at time 119905 = 05119873119909 Error vs NDSolve5 002914558401868394510 00001560755247342893215 00001499233624842055720 00001494908109620496425 0000149488995827673530 000014948859870700382

41 Example 1 The first example we consider is119906 (119909 0) = 119890minus1199092 (42)

with Dirichlet boundary conditions consistent with thisinitial condition 119906 (minus1 119905) = 119906 (1 119905) = 119890minus1 (43)

Table 1 illustrates the maximum absolute error between thesolution obtained by the present method and the solutionobtained by NDSolve in Mathematica 9 for 120572 = 1 To thebest of the authorsrsquo knowledge no exact solution exists forthe fractional case and hence no comparison can be made

However we present Figure 1 which illustrates thebehaviour of the mid-point of the solution as time evolves forvarious values of 12057242 Example 2 We now consider the one-dimensional casewith initial condition 119906 (119909 0) = 119890minus(119909+1) (44)

and Neumann Boundary conditions119906119909 (minus1 119905) = minus1 (45)119906119909 (1 119905) = minus11198902 (46)

Once again our solution converges with 119873 moving from 5to 10 but does not continue to improve Table 2 presents theerrors in our method when compared to NDSolve for 120572 = 1We present the behaviour of themid-point 119906(119909 = 0 119905) as timeevolves for different values of 120572 in Figure 2

43 Example 3 This example considers the two-dimensionalcase with initial condition119906 (119909 119910 0) = (119909 minus 1) (119909 + 1) (119910 minus 1) (119910 + 1) (47)

and consistent Dirichlet boundary conditions described as119906 (minus1 119910 119905) = 119906 (1 119910 119905) = 119906 (119909 minus1 119905) = 119906 (119909 1 119905) = 0 (48)

Table 3 presents the errors in the present method whencompared to NDSolve Figure 3 describes the evolution of themid-point in two dimensions with time for various 120572 values


=1=08=06

=04=02

002 004 006 008 010000t

093

094

095

096

097

098

099

u(x=

0t)

Figure 1 Plot of 119906(119909 = 0 119905) for various values of 120572

=1=08=06

=04=02

002 004 006 008 010000t

026

028

030

032

034

036

u(x=

0t)


00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t

=1=08=06

=04=02

Figure 3 Plot of 119906(119909 = 0 119910 = 0 119905) for various values of 120572


Table 3 Maximum absolute error in the presented methodrsquossolution of the problem described by Example 3 at time 119905 = 05119873119909 = 119873119910 Error vs NDSolve5 000005058103549999177610 00000647871853557783515 00000631024907378034320 00000647873964241364125 0000064241523463922630 000006478739639844785


44 Example 4 Finally we consider the case of a two-dimensional initial condition with Neumann conditions119906 (119909 119910 0) = cos (119909) cos (119910) (49)

and 119906119909 (minus1 119910 119905) = sin (1) cos (119910) (50)119906119909 (1 119910 119905) = minus sin (1) cos (119910) (51)119906119910 (119909 minus1 119905) = cos (119909) sin (1) (52)119906119910 (119909 1 119905) = minus cos (119909) sin (1) (53)

The errors for this example when compared to NDSolve arepresented in Table 4 Figure 4 describes the evolution of themid-point in two dimensions with time for various 120572 values

5 Image Processing Application

Despite the impressive accuracy obtained by Chebyshevcollocation for small values of 119873119909 the method suffers fromsevere round-off error for values of 119873119909 gt 100 due to finite-precision arithmetic [22 23] In applying this scheme to


=1=08=06

=04=02

00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t

Figure 4 Plot of 119906(119909 = 0 119910 = 0 119905) for various values of 120572an image the dimensions of the input image dictate theresolution of the scheme and in most cases the input imagedimensions will exceed 100 This forces us to employ a finite-difference discretization scheme instead of the Chebyshevscheme as described by Jacobs and Harley in [2] Jacobs andMomoniat [15] present the results of applying the integerorder model to an image for the purpose of documentimage binarization We present here some examples of theresults obtained using the hybrid-transform method Thepower of the hybrid-transform method lies in the abilityto immediately determine the solution at any point in timeas opposed to needing to iterate to a given point in timeFigure 5 illustrates the solutions obtained given an inputimage at varying points in time Figure 6 shows the effectsof the fractional term on the resulting image wherein thesteady state has been altered as well as the transient phasebeing executed much more quickly This mirrors the effectsillustrated in the examples above Figures 1 2 3 and 4 wherethe steady state of the equation is alteredwith changing valuesof 120572 as well as the equation reaching a steady state morerapidly This suggests the order of the derivative may be usedas an extra dimension of control in image processing wherea smaller 120572 value actually brightens the background of theimage it also reduces diffusivity of the model by altering thesteady state away from a uniformly diffused image

If we take for example 119873119909 = 119873119910 = 100 then thestability criteria for a standard time-marching scheme wouldbe 119905 = (1100)2120572 As 120572 decreases from 1 119905 becomes sosmall that the number of iterations required to attain a finaltime of 119905 = 0003 is unfeasibly large This further justifies theuse of a hybrid-transformmethod in the application of imageprocessing

51 Application Example We present here a document imagebinarization application where the input image is corruptedby noise as well as ink bleed-through from the reverse side ofthe page The solution image we seek has black pixels whichrepresent text and white pixels everywhere else Figure 7

illustrates the input image and the resulting binary images fordifferent values of 120572 as well as the ground truth image againstwhichwemay quantitativelymeasure our resultThe1198711 normis used as an error measure and is presented at various pointsin time in Figure 8 Due to the changing 120572 values we mustnormalize time to accurately compare the performance of thepresent method over various 120572 values Due to the acceleratedconvergence to the steady state with varying 120572 as illustratedin the examples in Section 4 we note that 120572 = 05 results inan optimally processed image supporting the application offractional partial differential equations in image processing

6 Discussion

The results above indicate a strong convergence to a solutionwith increasing 119873 The similar results obtained for 119873 gt 5are attributed to the comparison being drawn between twonumerical methods which may be different from the trueexact solution Results obtained in [1] indicate that whenthe present method is compared with an exact solution theaccuracy increases dramatically with increasing 119873 Despitethis the present method attains results similar to that of anindustry standard NDSolve which is encouraging

Figures 1 2 3 and 4 indicate the dynamic behaviour ofthe subdiffusive process Interestingly the diffusive processbecomes increasinglymore aggressive as120572decreases from 1 to0 however the final ldquosteady staterdquo achieved by these processesis typically less severe than the standard diffusion modelIn image processing terms we could achieve a semidiffusedresult extremely quickly by choosing120572 to be small rather thanusing 120572 = 1 and diffusing over a longer period

In both the one- and two-dimensional cases the Neu-mann boundary conditioned problems result in an accuracyof two orders of magnitude worse than that of the Dirichletcases In the linear case where an exact solution exists theaccuracy of the present method increases dramatically withincreasing 119873 [2] however due to the comparison beingdrawn with another numerical method NDSolve the freeends affect the comparative solution rather than reinforcinga condition on the boundary as in the Dirichlet case

7 Conclusions

In addition to obtaining a high accuracy the present methodis robust for 0 lt 120572 le 1 and is in fact exact in the trans-formation from a time-fractional partial differential equa-tion to partial differential equation or differential equationdepending on the dimensionality The errors incurred arethen discretization error and numerical inversion of theBromwich integral which is O(102minus119873) [17]

The application of this method to images introducesa computational hurdle since the resolution of the imagedictates the spatial discretization resolution After the trans-formation one must solve a linear system that is of thesame dimension as the input image This is exacerbatedby the use of Chebyshev collocation since the derivativematrices are full and not tridiagonal or banded as theyare when using a standard finite-difference scheme wherethe Thomas algorithm can be put to use as in [18ndash20] In


(a) (b) (c) (d) (e)

Figure 5 Figures depicting results obtained at increasing time using the current method and 120572 = 1 (a) The original image (b) binarizedoriginal image at 119905 asymp 0 (c) processed binary image at 119905 = 0003 (d) processed binary image at 119905 = 0008 (e) overdiffused image at 119905 = 001

(a) (b) (c) (d)

Figure 6 Figures depicting results obtained at increasing time using the current method and 120572 = 02 (a) The original image (b) processedimage at 119905 = 001 (c) processed binary image at 119905 = 0001 (d) processed binary image at 119905 = 001

(a) (b) (c)

(d) (e) (f)

Figure 7 Figures illustrating processed binary images obtained for various values of 120572 (a) The original image (b) 120572 = 1 (c) 120572 = 075 (d)120572 = 05 (e) 120572 = 025 (f) ground truth image

02 04 06 08 1000Normalized Time

=1=08=075

=06=05=025

0

5

10

15

20

25

30

35

L1

Erro

r

Figure 8 Plot 1198711 errors against number of iterations for variousvalues of 120572

such a case a numerical method should be implemented toimprove the computational time required for large input dataThe Chebyshev collocation method is also known to havelarge round-off errors for large problems 119873119909 gt 100 dueto finite-precision [22 23] Alternatively for large systemsa finite-difference discretization may be used to speed upcomputational time at the cost of accuracy in the solutionWehave shown for example the images obtained by employingthe hybrid-transform method to illustrate that the methoddoes produce the expected results in a real-world applica-tion

We have presented a method that is robust for a time-fractional diffusion equation with a nonlinear source termfor one- and two-dimensional cases with both Dirichlet andNeumann boundary conditions We have shown throughnumerical experiments that in the case of 120572 = 1 our solutionobtains a result similar to that of NDSolve in Mathematica 9and extends trivially to fractional-order temporal derivativeof order 120572 isin (0 1] The application to image processing


substantiates this method for real-world problems highlight-ing the effectiveness of the fractional ordered derivative asa further dimension of control which has a dramatic effectboth on the transition time of the model and on the finalstate attained Within the realm of image processing thisopens new avenues of research allowing more control overphysically derived processes to construct methods that arephysically substantial

To the best of the authorsrsquo knowledge this hybridizationof the Laplace transform Chebyshev collocation and quasi-linearization schemehas not yet been applied to FPDEs in oneor two dimensionsThis collaboration yields a method that isaccurate and robust for time-fractional derivatives

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

Opinions expressed and conclusions arrived at are those ofthe authors and are not necessarily to be attributed to theCoE-MaSSThe work contained in this paper also forms partof B A Jacobsrsquo Doctoral thesis

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

B A Jacobs acknowledges the support of the NRF underThuthuka grant 94005 The DST-NRF Centre of Excellencein Mathematical and Statistical Sciences (CoE-MaSS) isacknowledged for funding

References

[1] B A Jacobs and C Harley ldquoA comparison of two hybridmethods for applying the time-fractional heat equation to a twodimensional functionrdquo in Proceedings of the 11th InternationalConference of Numerical Analysis and Applied Mathematics2013 ICNAAM 2013 pp 2119ndash2122 Greece September 2013

[2] B A Jacobs and C Harley ldquoTwo Hybrid Methods for SolvingTwo-Dimensional Linear Time-Fractional Partial DifferentialEquationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[3] G-h Gao Z-z Sun and Y-n Zhang ldquoA finite differencescheme for fractional sub-diffusion equations on an unboundeddomain using artificial boundary conditionsrdquo Journal of Com-putational Physics vol 231 no 7 pp 2865ndash2879 2012

[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[5] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[6] O PAgrawal ldquoSolution for a fractional diffusion-wave equationdefined in a bounded domainrdquoNonlinear Dynamics vol 29 no1ndash4 pp 145ndash155 2002

[7] E Hashemizadeh and A Ebrahimzadeh ldquoAn efficient numer-ical scheme to solve fractional diffusion-wave and fractionalKleinndashGordon equations in fluid mechanicsrdquo Physica A Sta-tistical Mechanics and its Applications vol 503 pp 1189ndash12032018

[8] M A Zaky D Baleanu J F Alzaidy and E Hashem-izadeh ldquoOperational matrix approach for solving the variable-order nonlinear Galilei invariant advection-diffusion equationrdquoAdvances in Difference Equations Paper No 102 11 pages 2018

[9] F Catte P-L Lions J-M Morel and T Coll ldquoImage selectivesmoothing and edge detection by nonlinear diffusionrdquo SIAMJournal on Numerical Analysis vol 29 no 1 pp 182ndash193 1992

[10] F-C Liu and Y Ha ldquoNoise reduction based on reaction-diffusion systemrdquo in Proceedings of the 2007 International Con-ference on Wavelet Analysis and Pattern Recognition ICWAPRrsquo07 pp 831ndash834 China November 2007

[11] G-H Cottet and L Germain ldquoImage processing throughreaction combined with nonlinear diffusionrdquo Mathematics ofComputation vol 61 no 204 pp 659ndash673 1993

[12] J Kacur and K Mikula ldquoSolution of nonlinear diffusionappearing in image smoothing and edge detectionrdquo AppliedNumerical Mathematics vol 17 no 1 pp 47ndash59 1995

[13] L Alvarez P-L Lions and J-M Morel ldquoImage selectivesmoothing and edge detection by nonlinear diffusion IIrdquo SIAMJournal on Numerical Analysis vol 29 no 3 pp 845ndash866 1992

[14] S Morfu ldquoOn some applications of diffusion processes forimage processingrdquo Physics Letters A vol 373 no 29 pp 2438ndash2444 2009

[15] B A Jacobs and E Momoniat ldquoA novel approach to textbinarization via a diffusion-based modelrdquo Applied Mathematicsand Computation vol 225 pp 446ndash460 2013

[16] B A Jacobs and E Momoniat ldquoA locally adaptive diffusionbased text binarization techniquerdquo Applied Mathematics andComputation vol 269 pp 464ndash472 2015

[17] J A Weideman and L N Trefethen ldquoParabolic and hyperboliccontours for computing the Bromwich integralrdquoMathematics ofComputation vol 76 no 259 pp 1341ndash1356 2007

[18] P J Singh S Roy and I Pop ldquoUnsteady mixed convectionfrom a rotating vertical slender cylinder in an axial flowrdquoInternational Journal of Heat and Mass Transfer vol 51 no 5-6pp 1423ndash1430 2008

[19] P M Patil S Roy and I Pop ldquoUnsteady mixed convection flowover a vertical stretching sheet in a parallel free stream withvariable wall temperaturerdquo International Journal of Heat andMass Transfer vol 53 no 21-22 pp 4741ndash4748 2010

[20] P M Patil and S Roy ldquoUnsteady mixed convection flow froma moving vertical plate in a parallel free stream Influence ofheat generation or absorptionrdquo International Journal of Heatand Mass Transfer vol 53 no 21-22 pp 4749ndash4756 2010

[21] A Bayliss A Class and B J Matkowsky ldquoRoundoff Error inComputing Derivatives Using the Chebyshev DifferentiationMatrixrdquo Journal of Computational Physics vol 116 no 2 pp380ndash383 1995

[22] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995

[23] K S Breuer and R M Everson ldquoOn the errors incurredcalculating derivatives using Chebyshev polynomialsrdquo Journalof Computational Physics vol 99 no 1 pp 56ndash67 1992

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analytic solution to the fractional diffusion-wave equation ona bounded domain Other techniques such as the variationaliteration method Adomian decomposition method and dif-ferential transformmethod have all been applied to fractionalpartial differential equations but with a focus on unboundeddomains More recently Hashemizadeh and Ebrahimzadeh[7] and Zaky et al [8] present solutions to the linear and non-linear time-fractional Klein-Gordon equations respectivelyTheir solutions were obtained by using an operational matrixapproach and Legendre polynomial interpolation

The linear diffusion model has been applied in manydifferent fields including image processing [9ndash14] Howeverto the best of the authorsrsquo knowledge the application of atime-fractional partial differential equation has not yet beenthoroughly examined within such a context It is from thestandpoint of applying a fractional partial differential equa-tion to an image thatwe examine the time-fractional diffusionequation in two dimensions In a recent paper Jacobs andMomoniat [15 16] show that the diffusion equation with thisnonlinear source term is able to binarize a document imagewith great success In extending this concept to fractional-order derivatives we seek to preserve the symmetry of thediffusion equation Values of 120572 isin (0 1] resolve the diffusionequation to subdiffusion which preserves symmetry andexhibits some interesting dynamics which are discussed lateron in this work Values of 120572 gt 1 introduce a transport effectwhich then breaks symmetry It is for this reason that weimpose the restriction 120572 isin (0 1]

In this work we make use of the Laplace transform whichallows us to handle the fractional-order derivative in analgebraic way and incur no error in doing so Inversion of theLaplace transform is however difficult to obtain analyticallyWe therefore make extensive use of the numerical inversionprocedure described by Weideman and Trefethen in [17]which defines a contour of integration that maps the domainof the Bromwich integral from the entire complex space to thereal space from which we can approximate this integral witha trapezoidal rule

With a robustmethod for inverting the Laplace transformwe may then hybridize the transform with a discretizationtechnique The Laplace transform of the temporal variableavoids the need for a time-marching scheme as well as reduc-ing the fractional-order derivative to an algebraic expressionThe transformed model is then discretized by the use ofChebyshev collocation due to its superior performance tofinite differences as illustrated in [1 2] The resulting systemis solved and the transform inverted to obtain a semianalyticsolution continuous in time and discrete in space

In the following section we present some preliminaryresults which are put to use throughout the paper In Section 3the implemented methods are described including the dif-ferent cases for boundary conditions Section 4 presents thesolutions obtained by the proposedmethod aswell as an errorcomparison with Mathematicarsquos NDSolve for the one- andtwo-dimensional cases of our model as well as both Dirichletand Neumann boundary conditions In Section 5 we providea real-world application of thismethod andmodel in the formof document image binarization illustrating the ability toobtain a useful result with the present method when coupled

with the finite-difference discretization A discussion of theresults and their relationship to work beyond this researchis presented in Section 6 and some concluding remarks aremade in Section 7

2 Preliminaries

In this work we employ Caputorsquos definition of a fractionalderivative over the Riemann-Liouville derivative due to thefact that the Caputo derivative makes use of the physicalboundary conditions whereas the Riemann-Liouville deriva-tive requires fractional-order boundary conditions

Definition 1 The Riemann-Liouville integral of order 120572 gt 0of a function 119906(119905) is119869120572119906 (119905) = 1Γ (120572) int1199050 (119905 minus 120591)120572minus1 119906 (120591) 119889120591 119909 gt 0 (3)

Definition 2 The fractional derivative of 119906(119905) according to theCaputo definition with119898 minus 1 lt 120572 le 119898 119898 isin N is120597120572119906 (119905)120597119905120572 = 119869119898minus120572119863119898119906 (119905) = 119863120572lowast119906 (119905)

fl1Γ (119898 minus 120572) int1199050 (119905 minus 120591)119898minus120572minus1 119906(119898) (120591) 119889120591 (4)

If 120572 isin Z the Caputo fractional derivative reduces tothe ordinary derivative or integral Podlubny [4] illustratesthe pleasing property of the Laplace transform of a Caputoderivative as can be seen in (5) In our case where 0 lt 120572 lt 1we have

L120597120572119906 (119909 119910 119905)120597119905120572 = 119904120572119880 (119909 119910) minus 119904120572minus1119906 (119909 119910 0) (5)

This property allows one to treat fractional-order derivativesalgebraically

Definition 3 The Generalized Mittag-Leffler function of theargument 119911 is 119864120572120573 (119911) = infinsum

119896=0

119911119896Γ (119896120572 + 120573) (6)

Theuse of Laplace transformallows one to circumvent theproblems that arise in the time-domain discretization How-ever using the Laplace transform for the fractional-orderderivative presents the problem of inverting the transformto find a solution Analytic inversion of the transform isinfeasible and hence the numerical scheme for evaluating theBromwich integral presented by Weideman and Trefethen in[17] is put to extensive use The Bromwich integral is of theform 119906 (119905) = 12120587119894 int120590+119894infin120590minus119894infin 119890119904119905119880 (119904) 119889119904 120590 gt 1205900 (7)

where 1205900 is the convergence abscissa Using the parabolicconformal map119904 = 120583 (119894119908 + 1)2 minus infin lt 119908 lt infin (8)

(7) becomes




119906ℎ119873119905 (119905) = ℎ2120587119894 119873119905sum119896=minus119873119905

119890119904(119908119896)119905119880 (119904 (119908119896)) 1199041015840 (119908119896) 119908119896 = 119896ℎ (10)


3 Methods




1198831198991 = 120597Φ120597119906 = 1198881199042 (119886 + 1) 119906119899 minus 119888119904119886 minus 3119888119904 (119906119899)2 (13)

and 1198831198992 = 120597Φ120597119906119905120572 = minus1 (14)


1198831198993 = sum119895


(15)


1198831198993 = [(119886 + 1) 1199062 minus 21199063]119899 (16)



where 119880 (119909 119910 119904) = L 119906 (119909 119910 119905) (18)



119909119894 = cos( 119894120587119873119909) 119894 = 0 1 119873119909119910119895 = cos( 119895120587119873119910) 119895 = 0 1 119873119910 (19)




119889119896119897 =


(20)





where 119865119894119895 = 119904120572minus1119891 (119909119894 119910119895) (23)


119889(2)119894119896 119880119899+1 (119909119896 119910119895) + 119873119910sum119896=0


(24)


+ 119873119909minus1sum119896=1

119889(2)119894119896 119880119899+1 (119909119896 119910119895) + 119873119910minus1sum119896=1


(25)



1 ∘ U + UB = Xn3119904 minus F (26)


(27)


119906 (119909 minus1 119905) = 119888119906 (119909 1 119905) = 119889 (29)


119880 (119909 minus1) = L 119888 119880 (119909 1) = L 119889 (31)




119906119910 (119909 minus1 119905) = 119888119906119910 (119909 1 119905) = 119889 (33)

with


119880119910 (119909 minus1) = L 119888 119880119910 (119909 1) = L 119889 (35)


119896=0

119889119873119909119896119880119899 (119909119896 119910119895) = L 119886 (36)

120597119880120597119909 (1199090 = 1 119910119895) asymp 119873119909sum119896=0

1198890119896119880119899 (119909119896 119910119895) = L 119887 (37)


119889119896119873119910119880119899 (119909119894 119910119896) = L 119888 (38)

120597119880120597119910 (119909119894 1199100 = 1) asymp 119873119910sum119896=0

1198891198960119880119899 (119909119894 119910119896) = L 119889 (39)


(1198891198731199090 11988911987311990911987311990911988900 1198890119873119909 )( 119880119899 (1199090 119910119895)119880119899 (119909119873119909 119910119895))= ( L 119887 minus 119873119909minus1sum

119896=1

1198890119896119880119899 (119909119896 119910119895)L 119886 minus 119873119909minus1sum

119896=1

119889119873119909119896119880119899 (119909119896 119910119895))(40)

and

(1198890119873119910 11988911987311991011987311991011988900 1198891198731199100 )( 119880119899 (119909119894 1199100)119880119899 (119909119894 119910119873119910))= (L 119889 minus 119873119910minus1sum

119896=1

1198891198960119880119899 (119909119894 119910119896)L 119888 minus 119873119910minus1sum

119896=1

119889119896119873119910119880119899 (119909119894 119910119896))(41)


4 Results













=1=08=06

=04=02

002 004 006 008 010000t

093

094

095

096

097

098

099

u(x=

0t)


=1=08=06

=04=02

002 004 006 008 010000t

026

028

030

032

034

036

u(x=

0t)


00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t

=1=08=06

=04=02











=1=08=06

=04=02

00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t





6 Discussion




7 Conclusions




(a) (b) (c) (d) (e)


(a) (b) (c) (d)


(a) (b) (c)

(d) (e) (f)



=1=08=075

=06=05=025

0

5

10

15

20

25

30

35

L1

Erro

r







Data Availability


Disclosure




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018











119906ℎ119873119905 (119905) = ℎ2120587119894 119873119905sum119896=minus119873119905

119890119904(119908119896)119905119880 (119904 (119908119896)) 1199041015840 (119908119896) 119908119896 = 119896ℎ (10)


3 Methods




1198831198991 = 120597Φ120597119906 = 1198881199042 (119886 + 1) 119906119899 minus 119888119904119886 minus 3119888119904 (119906119899)2 (13)

and 1198831198992 = 120597Φ120597119906119905120572 = minus1 (14)


1198831198993 = sum119895


(15)


1198831198993 = [(119886 + 1) 1199062 minus 21199063]119899 (16)



where 119880 (119909 119910 119904) = L 119906 (119909 119910 119905) (18)



119909119894 = cos( 119894120587119873119909) 119894 = 0 1 119873119909119910119895 = cos( 119895120587119873119910) 119895 = 0 1 119873119910 (19)




119889119896119897 =


(20)





where 119865119894119895 = 119904120572minus1119891 (119909119894 119910119895) (23)


119889(2)119894119896 119880119899+1 (119909119896 119910119895) + 119873119910sum119896=0


(24)


+ 119873119909minus1sum119896=1

119889(2)119894119896 119880119899+1 (119909119896 119910119895) + 119873119910minus1sum119896=1


(25)



1 ∘ U + UB = Xn3119904 minus F (26)


(27)


119906 (119909 minus1 119905) = 119888119906 (119909 1 119905) = 119889 (29)


119880 (119909 minus1) = L 119888 119880 (119909 1) = L 119889 (31)




119906119910 (119909 minus1 119905) = 119888119906119910 (119909 1 119905) = 119889 (33)

with


119880119910 (119909 minus1) = L 119888 119880119910 (119909 1) = L 119889 (35)


119896=0

119889119873119909119896119880119899 (119909119896 119910119895) = L 119886 (36)

120597119880120597119909 (1199090 = 1 119910119895) asymp 119873119909sum119896=0

1198890119896119880119899 (119909119896 119910119895) = L 119887 (37)


119889119896119873119910119880119899 (119909119894 119910119896) = L 119888 (38)

120597119880120597119910 (119909119894 1199100 = 1) asymp 119873119910sum119896=0

1198891198960119880119899 (119909119894 119910119896) = L 119889 (39)


(1198891198731199090 11988911987311990911987311990911988900 1198890119873119909 )( 119880119899 (1199090 119910119895)119880119899 (119909119873119909 119910119895))= ( L 119887 minus 119873119909minus1sum

119896=1

1198890119896119880119899 (119909119896 119910119895)L 119886 minus 119873119909minus1sum

119896=1

119889119873119909119896119880119899 (119909119896 119910119895))(40)

and

(1198890119873119910 11988911987311991011987311991011988900 1198891198731199100 )( 119880119899 (119909119894 1199100)119880119899 (119909119894 119910119873119910))= (L 119889 minus 119873119910minus1sum

119896=1

1198891198960119880119899 (119909119894 119910119896)L 119888 minus 119873119910minus1sum

119896=1

119889119896119873119910119880119899 (119909119894 119910119896))(41)


4 Results













=1=08=06

=04=02

002 004 006 008 010000t

093

094

095

096

097

098

099

u(x=

0t)


=1=08=06

=04=02

002 004 006 008 010000t

026

028

030

032

034

036

u(x=

0t)


00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t

=1=08=06

=04=02











=1=08=06

=04=02

00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t





6 Discussion




7 Conclusions




(a) (b) (c) (d) (e)


(a) (b) (c) (d)


(a) (b) (c)

(d) (e) (f)



=1=08=075

=06=05=025

0

5

10

15

20

25

30

35

L1

Erro

r







Data Availability


Disclosure




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









119889119896119897 =


(20)





where 119865119894119895 = 119904120572minus1119891 (119909119894 119910119895) (23)


119889(2)119894119896 119880119899+1 (119909119896 119910119895) + 119873119910sum119896=0


(24)


+ 119873119909minus1sum119896=1

119889(2)119894119896 119880119899+1 (119909119896 119910119895) + 119873119910minus1sum119896=1


(25)



1 ∘ U + UB = Xn3119904 minus F (26)


(27)


119906 (119909 minus1 119905) = 119888119906 (119909 1 119905) = 119889 (29)


119880 (119909 minus1) = L 119888 119880 (119909 1) = L 119889 (31)




119906119910 (119909 minus1 119905) = 119888119906119910 (119909 1 119905) = 119889 (33)

with


119880119910 (119909 minus1) = L 119888 119880119910 (119909 1) = L 119889 (35)


119896=0

119889119873119909119896119880119899 (119909119896 119910119895) = L 119886 (36)

120597119880120597119909 (1199090 = 1 119910119895) asymp 119873119909sum119896=0

1198890119896119880119899 (119909119896 119910119895) = L 119887 (37)


119889119896119873119910119880119899 (119909119894 119910119896) = L 119888 (38)

120597119880120597119910 (119909119894 1199100 = 1) asymp 119873119910sum119896=0

1198891198960119880119899 (119909119894 119910119896) = L 119889 (39)


(1198891198731199090 11988911987311990911987311990911988900 1198890119873119909 )( 119880119899 (1199090 119910119895)119880119899 (119909119873119909 119910119895))= ( L 119887 minus 119873119909minus1sum

119896=1

1198890119896119880119899 (119909119896 119910119895)L 119886 minus 119873119909minus1sum

119896=1

119889119873119909119896119880119899 (119909119896 119910119895))(40)

and

(1198890119873119910 11988911987311991011987311991011988900 1198891198731199100 )( 119880119899 (119909119894 1199100)119880119899 (119909119894 119910119873119910))= (L 119889 minus 119873119910minus1sum

119896=1

1198891198960119880119899 (119909119894 119910119896)L 119888 minus 119873119910minus1sum

119896=1

119889119896119873119910119880119899 (119909119894 119910119896))(41)


4 Results













=1=08=06

=04=02

002 004 006 008 010000t

093

094

095

096

097

098

099

u(x=

0t)


=1=08=06

=04=02

002 004 006 008 010000t

026

028

030

032

034

036

u(x=

0t)


00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t

=1=08=06

=04=02











=1=08=06

=04=02

00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t





6 Discussion




7 Conclusions




(a) (b) (c) (d) (e)


(a) (b) (c) (d)


(a) (b) (c)

(d) (e) (f)



=1=08=075

=06=05=025

0

5

10

15

20

25

30

35

L1

Erro

r







Data Availability


Disclosure




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









119880119910 (119909 minus1) = L 119888 119880119910 (119909 1) = L 119889 (35)


119896=0

119889119873119909119896119880119899 (119909119896 119910119895) = L 119886 (36)

120597119880120597119909 (1199090 = 1 119910119895) asymp 119873119909sum119896=0

1198890119896119880119899 (119909119896 119910119895) = L 119887 (37)


119889119896119873119910119880119899 (119909119894 119910119896) = L 119888 (38)

120597119880120597119910 (119909119894 1199100 = 1) asymp 119873119910sum119896=0

1198891198960119880119899 (119909119894 119910119896) = L 119889 (39)


(1198891198731199090 11988911987311990911987311990911988900 1198890119873119909 )( 119880119899 (1199090 119910119895)119880119899 (119909119873119909 119910119895))= ( L 119887 minus 119873119909minus1sum

119896=1

1198890119896119880119899 (119909119896 119910119895)L 119886 minus 119873119909minus1sum

119896=1

119889119873119909119896119880119899 (119909119896 119910119895))(40)

and

(1198890119873119910 11988911987311991011987311991011988900 1198891198731199100 )( 119880119899 (119909119894 1199100)119880119899 (119909119894 119910119873119910))= (L 119889 minus 119873119910minus1sum

119896=1

1198891198960119880119899 (119909119894 119910119896)L 119888 minus 119873119910minus1sum

119896=1

119889119896119873119910119880119899 (119909119894 119910119896))(41)


4 Results













=1=08=06

=04=02

002 004 006 008 010000t

093

094

095

096

097

098

099

u(x=

0t)


=1=08=06

=04=02

002 004 006 008 010000t

026

028

030

032

034

036

u(x=

0t)


00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t

=1=08=06

=04=02











=1=08=06

=04=02

00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t





6 Discussion




7 Conclusions




(a) (b) (c) (d) (e)


(a) (b) (c) (d)


(a) (b) (c)

(d) (e) (f)



=1=08=075

=06=05=025

0

5

10

15

20

25

30

35

L1

Erro

r







Data Availability


Disclosure




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









=1=08=06

=04=02

002 004 006 008 010000t

093

094

095

096

097

098

099

u(x=

0t)


=1=08=06

=04=02

002 004 006 008 010000t

026

028

030

032

034

036

u(x=

0t)


00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t

=1=08=06

=04=02











=1=08=06

=04=02

00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t





6 Discussion




7 Conclusions




(a) (b) (c) (d) (e)


(a) (b) (c) (d)


(a) (b) (c)

(d) (e) (f)



=1=08=075

=06=05=025

0

5

10

15

20

25

30

35

L1

Erro

r







Data Availability


Disclosure




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









=1=08=06

=04=02

00

02

04

06

08

u(x=

0y=

0t)

02 04 06 08 1000t





6 Discussion




7 Conclusions




(a) (b) (c) (d) (e)


(a) (b) (c) (d)


(a) (b) (c)

(d) (e) (f)



=1=08=075

=06=05=025

0

5

10

15

20

25

30

35

L1

Erro

r







Data Availability


Disclosure




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









(a) (b) (c) (d) (e)


(a) (b) (c) (d)


(a) (b) (c)

(d) (e) (f)



=1=08=075

=06=05=025

0

5

10

15

20

25

30

35

L1

Erro

r







Data Availability


Disclosure




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018











Data Availability


Disclosure




Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








application of nonlinear time-fractional partial differential equations to image processing via...

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