boundary value problems for fractional differential...
TRANSCRIPT
Boundary Value Problems for
Fractional Differential Equations:
Existence Theory and Numerical Solutions
Mujeeb ur Rehman
Centre for Advanced Mathematics and Physics
National University of Sciences and Technology
PhD Thesis
2011
Boundary Value Problems for
Fractional Differential Equations:
Existence Theory and Numerical Solutions
by
Mujeeb ur Rehman
Supervised by
Dr. Rahmat Ali Khan
Centre for Advanced Mathematics and Physics,
National University of Sciences and Technology, Islamabad
A thesis submitted for the degree of
Doctor of Philosophy
c⃝ Mujeeb ur Rehman 2011
Abstract
Fractional calculus can be considered as supper set of conventional calculus in the sense that it extends
the concepts of integer order differentiation and integration to an arbitrary (real or complex) order. This
thesis aims at existence theory and numerical solutions to fractional differential equations. Particular focus
of interest are the boundary value problems for fractional order differential equations. This thesis begins
with the introduction to some basic concepts, notations and definitions from fractional calculus, functional
analysis and the theory of wavelets. Existence and uniqueness results are established for boundary value
problems that include, two–point, three–point and multi–point problems. Sufficient conditions for the
existence of positive solutions and multiple positive solutions to scalar and systems of fractional differential
equations are established using the Guo–Krasnoselskii cone expansion and compression theorems.
Owning to the increasing use of fractional differential equations in basic sciences and engineering,
there exists strong motivation to develop efficient, reliable numerical methods. In this work wavelets are
used to develop a numerical scheme for solution of the boundary value problems for fractional ordinary
and partial differential equations. Some new operational matrices are developed and used to reduce the
boundary value problems to system of algebraic equations. Matlab programmes are developed to compute
the operational matrices. The simplicity and efficiency of the wavelet method is demonstrated by aid of
several examples and comparisons are made between exact and numerical solutions.
i
Acknowledgements
First and foremost, I would like to thank National University of Sciences and Technology (NUST) for its
financial support. I would like to express my gratitude to my Ph.D supervisor Dr. Rehmat Ali Khan for
his guidance and encouragements during the whole course of my studies leading to this thesis. I would like
to thank Prof. Asghar Qadir whose cardial, rigorous and elegant lectures on special functions sparkled
my interest in the field of fractional calculus. During my research phase, I was in constant need of getting
some books on the subject of fractional calculus and fractional differential equations. In this context, I am
also very thankful to Prof. Faiz Ahmad, who managed to provide me a number of good books despite the
limited availability of funds. I want to express my thanks to my colleagues for their encouragements and
many useful discussions. I am thankful to Dr. Tyab Kamran, Dr. M. Rafique, Dr. Rashid Farooq and
Dr. Jamil Raza for their moral support and encouragements. I am grateful to Principle CAMP, Professor
Azad Akhtar Siddiqui for providing an impressive research environment at the centre. I am also thankful
to the University of Malakand for its hospitality during my short stay there.
ii
iii
.
Dedicated to My Mother
Contents
1 Introduction 1
2 Preliminaries 7
2.1 Some special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Euler’s gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Mittag–Leffler function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Fractional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 The Riemann–Liouville fractional integration . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 The Riemann–Liouville fractional derivative . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 The Caputo fractional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.1 The Haar scaling function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.2 Multiresolution Analysis (MRA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3 The Haar wavelet function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.4 Orthogonality of the Haar wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.5 Function approximation by the Haar wavelets . . . . . . . . . . . . . . . . . . . . . 28
2.4.6 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Existence and uniqueness of solutions 31
3.1 Two–point boundary value problems (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Two–point boundary value problems (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Three–point boundary value problems (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.2 Uniqueness of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Three–point boundary value problems (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.2 Uniqueness of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Multi–point boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5.1 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.2 Uniqueness of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
iv
v
3.6 Boundary value problems with integral boundary conditions . . . . . . . . . . . . . . . . . 53
3.6.1 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6.2 Uniqueness of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Existence and multiplicity of positive solutions 59
4.1 Positive solutions for three–point boundary value problems (I) . . . . . . . . . . . . . . . . 60
4.1.1 Green’s function and its properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1.2 Existence of at least one positive solution . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.3 Existence of at least two positive solutions . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.4 Existence of at least three positive solutions . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Positive solutions to three–point boundary value problems (II) . . . . . . . . . . . . . . . 67
4.2.1 Green’s function and its properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 Existence of positive solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.3 Uniqueness of positive solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Existence and multiplicity of positive solutions for systems of fractional differential
equations 74
5.1 Positive solutions for a coupled system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.1 Green’s function and its properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1.2 Existence of at least one positive solution . . . . . . . . . . . . . . . . . . . . . . 76
5.1.3 Existence of at least two positive solutions . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Positive solutions to a system of fractional differential equations with three–point boundary
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.1 Greens’s function and its properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2.2 Existence of at least one positive solution . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.3 Existence of at least two positive solutions . . . . . . . . . . . . . . . . . . . . . . . 91
6 Numerical solutions to fractional differential equations by the Haar wavelets 93
6.1 Numerical solutions to fractional ordinary differential equations . . . . . . . . . . . . . . . 93
6.1.1 Linear fractional differential equations with constant coefficients . . . . . . . . . . . 99
6.1.2 Linear fractional differential equations with variable coefficients . . . . . . . . . . . 108
6.2 Numerical solutions to fractional partial differential equations . . . . . . . . . . . . . . . . 114
6.2.1 Fractional partial differential equations with constant coefficients . . . . . . . . . . 114
6.2.2 Fractional partial differential equations with variable coefficients . . . . . . . . . . 118
7 Numerical solutions to fractional differential equations by the Legendre wavelets 125
7.0.3 The Legendre wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.0.4 Function approximations by the Legendre wavelets . . . . . . . . . . . . . . . . . . 126
7.1 An operational matrices of fractional order integration . . . . . . . . . . . . . . . . . . . . 127
7.2 Numerical solutions of fractional differential equations . . . . . . . . . . . . . . . . . . . . 128
vi
8 Conclusions 135
A Matlab and Mathematica programs 137
A.1 Computations of some operational matrices by Matlab . . . . . . . . . . . . . . . . . . . . 137
A.2 Computations by Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B Useful results from Analysis 144
References 146
List of Tables
6.1 Absolute error for different values of m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Maximum absolute error for α = 2, β = 0, 1 and different values of m, ω. . . . . . . . . . . 102
6.3 Absolute error for m = 32 and α = 1.2, 1.4, 1.6, 1.8, 2. . . . . . . . . . . . . . . . . . . . . . 105
6.4 Absolute error for α = 32 and different values of m. . . . . . . . . . . . . . . . . . . . . . . 106
6.5 Absolute error for different values of m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.6 Absolute error for different values of m and α = 114 . . . . . . . . . . . . . . . . . . . . . . 110
6.7 The maximum absolute error for m = 32, and different values of α and β. . . . . . . . . . 111
6.8 For problem (6.1.85), absolute error for m = 32 and different values of α. . . . . . . . . . 114
6.9 Maximum absolute error for different values of J and α. . . . . . . . . . . . . . . . . . . . 118
6.10 The Haar wavelet solutions and solutions obtained in [105], using ADM and VIM. . . . . 121
7.1 Absolute error for M = 3 and different values of k. . . . . . . . . . . . . . . . . . . . . . . 129
7.2 Absolute error for M = 3 and different values of k. . . . . . . . . . . . . . . . . . . . . . . 132
7.3 Numerical results with comparison to Ref. [104] and [12]. . . . . . . . . . . . . . . . . . . . 133
7.4 Maximum absolute error for the Haar wavelet and the Legendre wavelets. . . . . . . . . . 133
7.5 The absolute error for M = 3, k = 3 and different values of α. . . . . . . . . . . . . . . . 134
vii
List of Figures
2.1 Gamma function and its reciprocal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 The Mittag–Leffler function Eα,β(−(3x)2), for β = 1/2 and different values of α. . . . . . . . . . 10
2.3 Fractional order integrals and derivatives of some elementary functions. . . . . . . . . . . . 15
2.4 The Maxican hat ψ(t) = (1− t2)e−12 t
2
and its dilated shifts. . . . . . . . . . . . . . . . . . . . 26
2.5 Haar wavelets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Approximating f(t) = sin(9t) + 2 cos(11t) + 12 sin(50t) by the Haar wavelets. . . . . . . . . . . . 29
6.1 Exact and numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Solutions y(t) of problem (6.1.29), for 1 ≤ α ≤ 2, y0 = cosωπ − 1. . . . . . . . . . . . . . . 103
6.3 Numerical solutions of problem (6.1.38) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.4 Exact and Numerical solutions for problem (6.1.5) . . . . . . . . . . . . . . . . . . . . . . 105
6.5 Exact and Numerical solutions for problem (6.1.48) . . . . . . . . . . . . . . . . . . . . . . 107
6.6 Exact and Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.7 Exact and Numerical solutions for problem (6.1.73), (6.1.74) . . . . . . . . . . . . . . . . . 110
6.8 Numerical and exact solutions for the boundary value problems (6.1.73), (6.1.74) and
(6.1.75), (6.1.76) ((c)-(f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.9 Exact and Numerical solutions for problem (6.1.85). . . . . . . . . . . . . . . . . . . . . . 113
6.10 Numerical and exact solutions for telegraph equation (6.2.12). . . . . . . . . . . . . . . . . 116
6.11 Solutions for (6.2.13) and (6.2.15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.12 Numerical and exact solutions for the equation (6.2.14) for α = 0.3, J = 5.0 . . . . . . . . 117
6.13 The absolute error between Haar wavelet solution and analytic solution. . . . . . . . . . . 120
6.14 Exact and numerical solutions for different values of J , α, β and γ. . . . . . . . . . . . . 123
6.15 The absolute error between exact and numerical solutions for different values of α, β and γ. 124
7.1 The Legendre wavelets for M = 3, k = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2 Numerical and exact solutions of problem (7.2.1), (7.2.2) for 1 ≤ α ≤ 2. . . . . . . . . . . 128
7.3 Solutions y(t) for Example 7.2.2 for ω = 11, y0 = 0, y1 = 1 and different values of α, β. . . 131
viii
List of publications from thesis
1. Mujeeb ur Rehman and R.A. Khan, Existence and uniqueness of solutions for multi–point boundary
value problem for fractional differential equations, Appl. Math. Lett., 23 (2010) 1038–1044.
2. Mujeeb ur Rehman, R.A. Khan and Naseer Ahmad Asif, Three point boundary value problems for
nonlinear fractional differential equations, Acta Math. Sci., 31 (2011).
3. Mujeeb ur Rehman and R.A. Khan, Positive Solutions to Coupled system of fractional differential
equations, Int. J. Nonlin. Sci., 10 (2010) 96–104.
4. Mujeeb ur Rehman and R.A. Khan, Positive solutions to nonlinear higher-order nonlocal boundary
value problems for fractional differential equations, Abs. Appl. Anal., Vol. 2010, Article ID 501230,
15 pages doi:10.1155/2010/501230
5. R.A. Khan, Mujeeb ur Rehman and J. Hendersom, Existence and uniqueness of solutions for non-
linear fractional differential equations with integral boundary conditions, Fract. Diff. Cal., 1 (2011)
29–43.
6. R. A. Khan, Mujeeb ur Rehman, Existence of multiple positive solutions for a general system of
fractional differential equations, Commun. Appl. Nonlin. Anal. 18 (2011) 25–35.
7. Mujeeb ur Rehman and R.A. Khan, The Legendre wavelet method for solving fractional differential
equations, Commun. Nonlin. Sci. Numer. Simulat., 16 (2011) 4163–4173.
8. Mujeeb ur Rehman and R.A. Khan, A numerical method for solving boundary value problems for
fractional differential equations, Appl. Math. Mod., (2011), doi: 10.1016/j.apm.2011.07.045
9. Mujeeb ur Rehman and R.A. Khan, Existence and uniqueness of solutions for fractional order dif-
ferential equations with nonlocal boundary conditions, Int. J. Math. Anal., (Accepted).
10. Mujeeb ur Rehman, R.A. Khan and Paul W. Eloe, Positive solutions to three-point boundary value
problem for higher order fractional differential system, Dyn. Syst. Appl., (Accepted)
11. Mujeeb ur Rehman and R.A. Khan, Numerical solutions to a class of partial fractional differential
equations (submitted)
12. Mujeeb ur Rehman, Numerical solutions to initial and boundary value problems for fractional partial
differential equations with variable coefficients (submitted)
ix
Chapter 1
Introduction
The discovery of differential calculus is attributed to Isaac Newton (1642-1727) and Gottfried Leibniz
(1646-1716) who independently developed the foundations of the subject in the seventeenth century. The
first mention of the fractional calculus can be traced back to a letter exchange between Leibniz and a
French mathematician Marquis de L’Hôpital. Leibniz introduced the notation dnydxn (still used today) for
nth order derivative with the assumption that n ∈ N and repotted this to L’Hôpital. In his letter L’Hôpital
posed the question to Leibniz “What would be the result if n = 12?” Leibniz in his replay, dated 30th
September 1695, writes “... this is an apparent paradox from which, one day, useful consequences will
be drawn. Since there are little paradoxes without usefulness. ... ”. S. F. de Lacroix (1819) for the
first time introduced the fractional derivatives in published text. Subsequent contributions to fractional
calculus were made by many great mathematicians of the time such as J.P.J. Fourier (1822), N.H. Abel
(1823-1826), J. Liouville (1832), B. Riemann (1847), A.K. Grunwald (1867), A.V. Letnikov (1868), J.
Hadamard (1892), O. Heaviside (1892), H. Weyl (1917), A. Erdélyi (1939), H. Kober (1940) and M. Riesz
(1949). Excellent summary of key milestones in the history of fractional calculus can be found in [110,134].
Recently, in a survey report [134], J. T. Machado, V. Kiryakova and F. Mainardi have comprehensively
listed the major documents and key events in this area of mathematics since 1974 up to April 2010.
Fractional calculus has long and rich history, but due to lack of suitable physical and geometrical inter-
pretations, it remained unfamiliar to applied scientists up to recent years and was considered mathematical
curiosities, not useful for solving problems arising from applied sciences. Several attempts have been made
to provide physical and geometric interpretations for fractional operators. However, these interpretations
are limited to only a small collection of selected applications of fractional derivatives and integrals in the
context of hereditary effects and self-similarity. In 2002, I. Podlubny [115] proposed a convincing physical
and geometric interpretation of fractional derivatives and integrals.
There are several competing definitions of fractional derivatives and integrals. Some of them include,
the Riemann, the Liouville, the Riemann–Liouville, the Caputo, the Weyl, the Hadamard, the Marchaud,
the Gränwald-Letnikov, the Erdélyi-Kober and the Riesz-Feller fractional derivatives and integrals. In
general, these definitions are not equivalent except for some special cases. Probably the most frequently
used definition of fractional derivative and integral is due to B. Riemann and J. Liouville, commonly
known as the Riemann-Liouville fractional derivative (integral). But in some situations, this approach is
1
2
not useful due to lack of physical interpretation of initial and boundary conditions involving fractional
derivatives, and also the Riemann-Liouville approach may yield derivative of a constant different from zero.
A useful alternative to Riemann-Liouville derivative is the Caputo fractional derivative, introduced by M.
Caputo in 1967 and adopted by Caputo and Mainardi 1969 in the context of the theory of viscoelasticity.
Fractional derivatives are non–local in nature. Local fractional derivatives have been proposed in [13, 28]
to study properties of irregular functions.
As a first application of fractional calculus, consider the problem of determination of shape of a
frictionless wire lying in a vertical plane such that the time required for a bead placed on the wire to slide
to the lowest point of the wire is the same regardless of its starting position (tautochronous problem). This
problem can be formulated by the integral equation√2gT =
∫ y
η=0(y − n)−
12ds,
where g is gravitational acceleration, (x, y) is initial position. The arc length s may be expressed as a
function of the y, say s = H(η). Thus
T√
2g =
∫ η=y
η=0(y − η)−
12H ′(η)dη, or T
√2g = Γ
(12
)I
120 H
′(y).
In 1823, N.H. Abel solved this problem by applying operator D120 on both sides of the above equation and
obtained
T√
2gD120 1 =
√πH ′(y).
By computing the fractional derivative of constant,
H ′(y) =ds
dy=T√2g
π√y.
The fact that the constant functions may have nonzero fractional derivative, is not always a drawback.
Because the Abel’s solution of the tatochronous problem rests on this fact.
For almost three centuries fractional calculus had been treated as an interesting, but abstract, mathe-
matical concept. It had significantly been developed with in pure mathematics. However the applications
of the fractional calculus just emerged in last few decades in several diverse areas of sciences, such as
physics, bio-sciences, chemistry and engineering. It is realized widely that in many situations fractional
derivative based models are much better than integer order models. Being nonlocal in nature, the frac-
tional derivatives provide an excellent tool for the understanding of memory and hereditary properties
of various materials and processes. This is the main advantage of fractional derivatives in comparison
with classical integer order derivatives. A new application field for fractional calculus is psychological and
life sciences, to characterize the time variation of emotions of people [6, 132]. In addition to the above
mentioned applications, there are several applications of fractional calculus within different fields of math-
ematics itself. For example, the fractional operators are useful for the analytic investigation of various
spacial functions [70, 71]. There are several collections of articles such as [25, 54, 119], which exhibit wide
variety of applications of fractional calculus and present many of the key developments of the theory. The
3
list of applications of fractional calculus is still growing; perhaps “the fractional calculus is the calculus of
twenty–first century”.
There are a number of books and monographs dealing with theory and applications of fractional
calculus. For the first book, the merit is ascribed to K.B. Oldham and J. Spanier [110] which gives a
historical survey and comprehensive overview on the topic of fractional calculus. The text by K.S Miller and
B. Ross [103] provides an easy introduction to the theory of fractional derivatives and fractional differential
equations. The book of I. Podlubny [114] deals particularly with fractional differential equations and their
applications. S. Samko, A. Kilbas and O. Marichev [124] published an encyclopedic type monograph in
Russian in 1987 and in English in 1993. In addition numerous other works have also been appeared. These
include A.A. Kilbas, H.M. Srivastava, J.J. Trujillo [69], A. Carpinteri and F. Mainardi [25], F. Mainardi
[101], R. Hilfer [54], J. Sabatier, O.P. Agrawal and J.A. Tenreiro Machado [119], V. Lakshmikantham, S.
Leela, J. Vasundhara Devi [77] and K. Diethelm [39].
We make a remark about notations used for fractional differentiation and integration. As pointed
out by K.S Miller and B. Ross [103] “...some of the power and elegance of fractional calculus rests in its
simplified notation.” R. Hilfer [57] have summarized the notations used by various authors for fractional
derivatives and integrals. In this work we will denote the Riemann–Liouville fractional integral by Iαa ,
Riemann–Liouville derivative by Dαa and the Caputo derivative by cDα
a .
The thesis is organized as follows: In Chapter 2 we recall some basic results from spacial functions,
fractional calculus, fixed point theory and wavelet analysis that form basis for our further investigations.
The Chapter begins with a brief introduction to the gamma function and the Mittage-Lefller function.
Analytic results of fractional calculus, frequently needed for our intended investigations in the succeeding
chapters, are briefly discussed. For most of the results the concise and rigorous proofs are established using
fundamental properties of fractional operators. Comparisons of Riemann–Liouville and caputo approaches
are occasionally provided. We also state some commonly used fixed point theorems needed to establish
the existence results in the next three chapters, corresponding to boundary value problems for fractional
differential equations.
Due to frequent applications of fractional differential equations in many standard models, there has
been significant interest for obtaining exact analytical and numerical solutions for them. The exact so-
lutions of initial value problems for fractional differential equations have been investigated by classical
integral transform methods, such as the Laplace transform method, the Fourier transform method and the
Mellin transform method [103,114]. Undoubtedly, “boundary value problems” for classical as well as frac-
tional differential equations is one of the fundamental topic and an active area of research. In general there
exist no method to find an exact analytic solution for boundary value problems for fractional differential
equations. Several numerical methods for solving integer order differential equations have been generalized
to solve initial value problems for fractional differential equations. In contrast, boundary value problems
have received much less attention and few results can be found concerning their numerical solutions. So
there exists a strong motivation to develop an efficient numerical technique for the treatment of boundary
value problems for fractional differential equation.
The concept of approximating complicated functions, which are known implicitly via differential or
4
integral equations, with simpler functions plays a decisive role in many areas of modern mathematics and
its applications. In particular, if solution y(x) to some differential equation belongs to certain class of
functions, then we are interested to find basis functions ϕ0(x), ϕ1(x), ϕ2(x), . . . such that y has represen-
tation
y(x) =
∞∑j=0
cjϕj(x), (1.0.1)
for some coefficients cj . If (1.0.1) holds, one might hope that the finite partial sum
y(x) ≈n∑j=0
cjϕj(x), for some n ∈ N, (1.0.2)
approximates y well. This idea is similar to that of power series (or the Fourier series) with ϕj being
polynomials ( or trigonometric functions). The functions ϕ0(x), ϕ1(x), ϕ2(x), . . . are required to have
simple and convenient structure. Power series and the Fourier series fulfill these requirements, but working
with them have certain disadvantages. One of the disadvantage is that they are available for a limited
classes of functions. In Section 2.4, we will introduce wavelets, a comprehensive mathematical tool leading
to representation of type (1.0.1) for relatively a large class of functions. Our focus of interest will be the
Haar wavelets and the Legendre wavelets. Some of the basic properties of these wavelets are discussed
which will be used in Chapter 6 and Chapter 7 dealing with numerical solutions for boundary value
problems of fractional ordinary and partial differential equations.
In Chapter 3, we will establish existence and uniqueness of solutions for different types of fractional
differential equations. In Section 3.1, we will study existence and uniqueness of solutions to a nonlinear
class of fractional differential equations involving Caputo fractional derivative
cDαy(t) = g(t, y(t)), y(0) = y0, y(1) = y1, t ∈ [0, 1],
where, 1 < α ≤ 2, y0, y1 ∈ R. In Section 3.2, we study the existence and uniqueness of solutions to a more
general problem
cDαy(t) = g(t, y(t), cDβy(t)), y(0) = y0, y(1) = y1, 1 < α ≤ 2, 0 ≤ β ≤ 1.
Section 3.3, is concerned with the existence and uniqueness of solutions to the following three–point
boundary value problem for nonlinear fractional differential equations
cDαy(t) = g(t, y(t), cDβy(t)), y(0) = µy(η), y(a) = νy(η),
where 1 < α < 2, 0 < β < 1, µ, ν ∈ R, η ∈ (0, a), µη(1− ν)+ (1−µ)(a− νη) = 0. In Section 3.4 we study
the existence and uniqueness of solution to following three–point boundary value problem
Dαy(t) = g(t, y(t),Dβy(t)), y(0) = 0, Dβy(1) = γDβy(η),
where, 1 < α ≤ 2, 0 < β < 1, α − β > 1, ∆α,β := (1 − γηα−β−1) > 0. We also give some new results
for the uniqueness of solutions. Section 3.5 deals with the existence and uniqueness of solutions to the
following class of multi–point boundary value problem for nonlinear fractional differential equations
Dαy(t) = g(t, y(t),Dβy(t)), t ∈ (0, 1), y(0) = 0, Dβy(1) =m−2∑i=1
aiDβy(ξi) + y0,
5
where 1 < α ≤ 2, 0 < β < 1, 0 < ξi < 1 (i = 1, 2, · · · ,m − 2), with, Λα,β :=∑m−2
i=1 aiξα−β−1i < 1. In
Section, 3.6 we study existence and uniqueness of solutions to nonlinear fractional differential equations
cDαy(t) = f(t, y(t), cDβy(t)), for t ∈ [0, l],
satisfying integral boundary conditions
py(0)− qy′(0) =
∫ l
0g(s, y(s))ds, γy(1) + δy′(1) =
∫ l
0h(s, y(s))ds,
where 0 < β < 1, 1 < α ≤ 2, p, δ > 0, q, γ ≥ 0 (or p, δ ≥ 0, q, γ > 0). The functions f, g and h
are assumed to be continuous. For the existence of solutions, we employ the nonlinear alternative of the
Leray–Schauder and a uniqueness result is established using the Banach fixed point theorem.
In many situations, only positive solutions of boundary value problems, that model important physical
phenomena, are meaningful. This is what we will pursue in Chapter 4 and Chapter 5. In Chapter 4, we
study the existence and multiplicity of positive solutions to three–point boundary value problems. The
tool that will be used to achieve this goal is Gau-Krasnosel’skii’s fixed point theorem on cone expansion
and compression. In Section 4.1, we investigate the existence and multiplicity results for the following
class of nonlinear three–point boundary value problems for fractional differential equations of type
cDαy(t) + a(t)g(t, y(t)) = 0, t ∈ (0, 1), n− 1 < α ≤ n,
y′(0) = y′′(0) = y′′′(0) = · · · = y(n−1)(0) = 0, y(1) = ξy(η),
where ξ, η ∈ (0, 1). Existence theorems for positive and multiple positive solutions are established by
assuming that f satisfies certain growth conditions. An existence theorem for triple positive solutions is
proved using the Leggett-Wiliam fixed point theorem.
In Section 4.2, we study the existence of positive solutions to a nonlinear higher order three–point
boundary value problem
cDδy(t) + f(t, y(t)) = 0, t ∈ (0, 1), 0 < t < 1, n− 1 < δ < n, n(≥ 3) ∈ N,
y(1) = βy(η) + λ2, y′(0) = αy′(η)− λ1, y
′′(0) = 0, y′′′(0) = 0 · · · y(n−1)(0) = 0,
where, 0 < η, α, β < 1. The boundary parameters are assumed to be nonnegative. Sufficient conditions
for the existence and uniqueness of positive solutions are obtained by employing Gau-Krasnosel’skii’s fixed
point theorem. Also, a result for the existence of a unique positive solution is established. Applicability
of the proposed results is demonstrated by including some examples.
In Chapter 5, some existence theory for positive and multiple positive solutions for nonlinear systems
of fractional differential equations is developed. Section 5.1, is concerned with existence results for positive
solutions to following system of fractional differential equationscDαx(t) + λφ(t)f(y(t)) = 0, n− 1 < α ≤ n,
cDβy(t) + λψ(t)g(x(t)) = 0, n− 1 < β ≤ n,(1.0.3)
6
satisfying the two point boundary conditionsx(1) = 0, x′(0) = 0, x′′(0) = 0, · · · , x(n−2)(0) = 0, x(n−1)(0) = 0,
y(1) = 0, y′(0) = 0, y′′(0) = 0, · · · , y(n−2)(0) = 0, y(n−1)(0) = 0,(1.0.4)
where t ∈ [0, 1], λ > 0. The nonlinear functions f, g : [0,∞) → [0,∞) are continuous and trφ(t), tsψ(t) :
[0, 1] → [0,∞) are also assumed to be continuous for r, s ∈ [0, 1) and do not vanish identically on any
subinterval. Several results on the existence of positive solutions are obtained for the above two point
boundary value problem. In section 5.2, we study existence and multiplicity results for a coupled system
of nonlinear three–point boundary value problems for higher order fractional differential equations of the
type cDαx(t) = λφ(t)f(x(t), y(t)), n− 1 < α ≤ n, n ∈ N,cDβy(t) = µψ(t)g(x(t), y(t)), n− 1 < α, β ≤ n
(1.0.5)
satisfying the boundary conditionsx′(0) = x′′(0) = x′′′(0) = · · · = x(n−1)(0) = 0, x(1) = θ1x(µ1),
y′(0) = y′′(0) = y′′′(0) = · · · = y(n−1)(0) = 0, y(1) = θ2x(µ2),(1.0.6)
where λ, µ > 0, for n ∈ N; θi, µi ∈ (0, 1) for i = 1, 2. We will also derive explicit intervals for the
parameters λ, µ, for with the above system has positive solutions.
In Chapter 6 we focus on providing a numerical scheme based on the Haar wavelets for solving different
types of boundary value problems for fractional differential equations. We drive a useful operational matrix,
and use it together with some other operational matrices developed in [27, 73, 80] for solving boundary
value problems. Various types of examples are presented to demonstrate the accuracy and simplicity of
our proposed numerical scheme.
In Chapter 7, an operational matrix of fractional order integration for Legendre wavelets is developed.
A numerical scheme based on this operational matrix is used to solve fractional differential equations. The
numerical solutions obtained by using Legendre wavelet method are compared with exact solutions and
with solutions obtained by some other numerical methods to demonstrate the accuracy, simplicity and
validity of the method.
The thesis includes two Appendices. In Appendix A we developed some Matlab programs to compute
various operational matrices that are used in the numerical solutions to fractional boundary value problems.
In Appendix B, some basic results from functional analysis are reviewed.
Chapter 2
Preliminaries
We review some elements of fractional calculus, fixed point theory and wavelet analysis that will be used
throughout this work. We begin by introducing Euler’s gamma function and using it to give a general
introduction to the idea of differentiation and integration of arbitrary order. The analytic results of
fractional calculus presented in section 2.2 are well known in literature and can be found in [69,103,110,114].
Some fixed point theorems are outlined in section 2.3 which are needed for the analysis of fractional
differential equations. The last section of the chapter is a brief introduction to the theory of wavelets. For
the most part, we use the notations and symbols that are commonly used in the current literature.
2.1 Some special functions
The generalization of integer order derivatives and multiple integrals to the derivatives and integrals of
arbitrary order is associated with the generalization of factorial function to gamma function. Also, the
Mittag–Leffler function, which is generalization of exponential function, plays an important role in the
theory of fractional differential equations and is connected with gamma function. In what follows, we
define and briefly discuss some properties of these special functions that will be frequently used in this
work.
2.1.1 Euler’s gamma function
In 1729, Euler discovered the gamma function while investigating the interpolation problem for the factorial
function. There are several approaches leading to the definition of gamma function. The most preferred
way of defining it, is the use of Euler’s integral
ϕ(x) =
∫ ∞
0txe−tdt, x ∈ N, (2.1.1)
as the starting point. Integration of (2.1.1) by parts, yields
ϕ(x) = [−txe−t]∞0 + x
∫ ∞
0tx−1e−tdt
= xϕ(x− 1), x = 1, 2, 3, . . . .
(2.1.2)
7
8
Since, ϕ(1) = 1, therefore repeated application of (2.1.2) gives ϕ(x) = x!. Thus we have integral represen-
tation of the factorial function x! as
x! =
∫ ∞
0txe−tdt, x ∈ N. (2.1.3)
Legendre introduced the notation Γ(x) for the function (x − 1)!. The integral Γ(x) =∫∞0 tx−1e−tdt
converges for x ∈ R+. From (2.1.2) we have the following fundamental equation
Γ(x+ 1) = xΓ(x). (2.1.4)
For n ∈ N, if some value of the function Γ(x) is known on the interval (n − 1, n], then with the help of
(2.1.4), we can find its value on the interval (n, n+1]. Now, one can extend the domain of Γ(x) to include
the negative real numbers. The repeated application of equation (2.1.4) gives
Γ(x) =Γ(x+ n)
x(x+ 1) . . . (x+ n− 2)(x+ n− 1), x ∈ R\0,−1,−2, . . . . (2.1.5)
The equations (2.1.4), (2.1.5) are valid even for complex values of x, provided x ∈ C\0,−1,−2, . . . . At
this point, we have following definition of gamma function.
Definition 2.1.1. The Euler’s gamma function is defined as
Γ(x) =
∫ ∞
0tx−1e−tdt if R(x) > 0,Γ(x) = Γ(x+ 1)/x. (2.1.6)
Theorem 2.1.2. (Weierstrass infinite product) For any x ∈ C,
1
Γ(x)= xeγx
∞∏n=1
(1 +
x
n
)e−
xn , (2.1.7)
where γ is the Euler’s constant given by
γ = limn→∞
( n∑k=1
1
k− log n
).
From (2.1.5), it follows that the gamma function has poles at x = 0,−1,−2, . . . , but 1Γ(x) is entire
function with zeroes at these points.
Another special function, closely related to the gamma function is the beta function. It has a simple
and useful integral representation.
Definition 2.1.3. The beta function is defined by Euler’s integral of first kind:
B(x, y) =
∫ 1
0sx−1(1− s)y−1ds, ℜ(x) > 0,ℜ(y) > 0.
This function is related to Euler’s gamma function as
B(x, y) =Γ(x)Γ(y)
Γ(x+ y), x, y ∈ C\0,−1,−2, . . . .
Thus the beta function is analytically continued to entire complex plane. Instead of using combinations
of gamma function, it is convenient to use beta function.
9
−5 −4 −3 −2 −1 0 1 2 3 4−20
−15
−10
−5
0
5
x
Γ(x)
1/Γ(x)
Figure 2.1: Gamma function and its reciprocal.
2.1.2 Mittag–Leffler function
In 1902, a Swedish mathematician Gosta Mittag–Leffler introduced the Mittag–Leffler function. It is a
straightforward generalization of exponential function. In recent years, the Mittag–Leffler function have
received much attention from researchers due to its role played in the investigation of fractional differential
equations. It arises frequently in the solutions of fractional differential and integral equations. The one–
parameter Mittag–Leffler function Eα(x) is defined by
Eα(x) =
∞∑k=0
xk
Γ(αk + 1), x ∈ C,ℜ(α) > 0. (2.1.8)
For 0 < α < 1, the one–parameter Mittag–Leffler function interpolates between exponential function ex
and hypergeometric function 1x−1 . Some special cases of the Mittag–Leffler function are [101]
(i) E0(x) =1
1− x, (ii) E1(x) = ex,
(iii) E2(−x2) = cos(x), (iv) E2(x2) = cosh(x),
(v) E3(x) =1
2
[ex
13 + 2e−
12x13 cos
(√3
2x
13
)], (vi) E4(x) =
1
2
[cos(x
14 ) + cosh(x
14 )].
The two–parameter Mittag–Leffler function Eα,β(x), which is defined as
Eα,β(x) =
∞∑k=0
xk
Γ(αk + β), x ∈ C,ℜ(α) > 0, ℜ(β) > 0, (2.1.9)
is a generalization of the one–parameter Mittag–Leffler function, to which it reduces to for β = 1. This
function was originally introduced by Wiman in 1905 and later investigated by Agrawal and Humbert in
1953. The two–parameter Mittag–Leffler function is related to the generalized hyperbolic function of order
n as
Hr(n, x) =
∞∑k=0
xnk+r−1
(nk + r − 1)!= xr−1En,r(x
n), r ∈ N. (2.1.10)
10
0 2 4 6 8 10 12 14 16 18 20−4
−3
−2
−1
0
1
2
3
4
5
x
α=1.3
α=1.5
α=1.7
α=1.9
α=1.95
α=1.99
2.0
Figure 2.2: The Mittag–Leffler function Eα,β(−(3x)2), for β = 1/2 and different values of α.
Also, Eα,β(x) is related to the generalized trigonometric function as
Kr(n, x) =∞∑k=0
(−1)kxnk+r−1
(nk + r − 1)!= xr−1En,r(−xn), r ∈ N. (2.1.11)
Another generalization of the Mittag–Leffler function is discussed by Prabhakar in 1971. He introduced
the function
Eδα,β(x) =
∞∑k=0
(δ)kxk
Γ(αk + β)k!, x, δ ∈ C,ℜ(α) > 0, ℜ(β) > 0 ℜ(δ) > 0. (2.1.12)
were (δ)0 = 0, (δ)k = δ(δ+1) · · · (δ+k−1) (k ∈ N) is the Pochhammer symbol. Many other generalizations
of the Mittag–Leffler functions have been appeared recently [49,68,72].
The Mittag–Leffler function E1α,1(x) = Eα(x) have no zero for α ∈ (0, 1] and for α ∈ (1, 2), it has
odd number of zeros on negative real axis. Moreover, for α ≥ 2, Eα(x) have infinite number of zeros on
negative real axis. Note that Eα(x) have no positive zero. For detailed information describing the zeroes
of the Mittag–Leffler functions we refer to [53,143].
2.2 Fractional calculus
Fractional calculus deals with derivatives and integrals of arbitrary order that are joined under the name
of differintegration. There are several approaches to fractional differentiation and integration that are not
equivalent, except for some special classes of functions. B. Ross [103] provided the following set of criteria
for fractional differintegration.
• Zero property: The operation of zero order must leave the function unchanged.
• Compatibility: The fractional differintegrals must produce the same results as ordinary differentia-
tion and integration when the order of differintegrals is an integer.
• Linearity: The fractional operators must be linear.
11
• Law of exponents: The fractional integrals must satisfy the law of exponents.
The most commonly used definitions of fractional integration and differentiation that fulfill these demands
are due to Riemann, Liouville and Caputo. In what follows, we define fractional derivatives and integral and
discuss some of their most useful properties. We begin with the Riemann–Liouville fractional integration.
2.2.1 The Riemann–Liouville fractional integration
One of the possible approach to define non-integer order differentiation and integration is through the use
of well known Cauchy’s integral formula for n-fold integral
Ina f(t) =∫ t
a
∫ tn−1
a· · ·∫ t1
af(t0)dt0 · · · dtn−2dtn−1
=1
(n− 1)!
∫ t
a(t− s)n−1f(s)ds, n ∈ N,
(2.2.1)
where f ∈ L2[a, b], a, b ∈ R. The generalization of factorial function to gamma function allows us to
replace n in (2.2.1) with an arbitrary real number α (or, even complex number) provided that the integral
on right side converges. Hence, it is natural to define the fractional integral as follows:
Definition 2.2.1. [114] Let f ∈ L1[a, b], α ∈ R+, the Riemann–Liouville fractional integral operator of
order α is defined as
Iαa f(t) =1
Γ(α)
∫ t
a(t− s)α−1f(s)ds (2.2.2)
for all t ∈ [a, b]. In particular, when α = 0, we set I0a := I; the identity operator.
For arbitrary lower limit (2.2.2) is the Riemann version and for infinite lower limit, i.e., for a = −∞,
(2.2.2) is the Liouville version of fractional integral. The case when a = 0, namely Iα0 is called the
Riemann–Liouville fractional integral and is quite convenient for further manipulations. On the other
hand if we keep lower limit arbitrary, take upper limit ∞ and replace kernel in (2.2.2) with (s− t)α−1 then
the resulting integral operator, for a reasonable class of functions, is called the Weyl fractional integral of
order α and is usually denoted by xWα∞. Historically, Abel (1823) used the fractional integral operator Iα0 ,
for α = 1/2 to solve his celebrated integral equation. Later in (1826) he generalized it to order α ∈ (0, 1).
Therefor, some authors prefer to name Iα0 , as the “Abel–Riemann fractional integral”.
Lemma 2.2.2. If α ≥ 0, β > −1, then the Riemann–Liouville fractional integral of the function (x− a)β
is given by
(Iαa (s− a)β)(t) =Γ(β + 1)
Γ(β + α+ 1)(t− a)β+α.
Theorem 2.2.3. [39] Let f ∈ L1[a, b] and α ∈ R+. Then the integral Iαa exists almost every where on
[a, b] and also Iαa is an element of L1[a, b].
Proof. Define a function φ : [a, b]× [a, b] → R by
φ(t, s) =
(t− s)α−1, if s ≤ t,
0, if t < s.
12
Case 1. If α ≥ 1, then φ(t, s) is continuous on [a, b]. Since f ∈ L1[a, b], therefore the product φ(t, s)f(s)
is integrable on [a, b]. Thus Iαa ∈ L1[a, b] in this case.
Case 2. If 0 ≤ α ≤ 1 then∫ ba φ(t, s)dt =
1α(b− s)α and∫ b
a
(∫ b
aφ(t, s)|f(s)|dt
)ds =
∫ b
a|f(s)|
( ∫ b
aφ(t, s)dt
)ds
=
∫ b
a|f(s)|(b− s)
α
α
ds
≤ (b− a)α
α∥f(s)∥1 <∞.
Thus, the product φ(t, s)f(s) is integrable over [a, b] × [a, b]. Therefore by Fubini’s Theorem the func-
tion g(t) =∫ ba φ(t, s)f(s)ds is integrable over [a, b]. Hence the fractional integral Iαa f(t) exists almost
everywhere on [a, b].
Theorem 2.2.4. [39] Let α ≥ 1 and f ∈ L1[a, b]. Then Iαa f ∈ C[a, b].
Before establishing the composition of fractional integral with the Mittag–Leffler function, let us point
out that one of the fundamental property of integer order integral, namely the semi group property, caries
over the Riemann–Liouville fractional order integral.
Lemma 2.2.5. [69] Let α, β ∈ R+ ∪ 0 and f be an element of L1[a, b]. Then
Iαa Iβa f(t) = Iα+βa f(t) = Iβa Iαa f(t) (2.2.3)
is valid almost everywhere on [a, b]. In addition to this, if f ∈ C[a, b] or α + β ≥ 1, then (2.2.3) is
identically true for all t ∈ [a, b].
Proof. Since I0a = I is defined as identity operator. Therefore for α = 0 or β = 0 or α = 0 = β, the
statement of the theorem holds trivially. By the definition of fractional integral, we have
Iαa Iβa f(t) =1
Γ(α)Γ(β)
∫ t
a(t− s)α−1
(∫ s
af(τ)(s− τ)β−1dτ
)ds.
The integral exists and Fubini’s Theorem allows us to interchange the order of integration, that is,
Iαa Iβa f(t) =1
Γ(α)Γ(β)
∫ t
af(τ)
(∫ t
τ(t− s)α−1(s− τ)β−1ds
)dτ.
The substitution s = τ + x(t− τ) yields
Iαa Iβa f(t) =1
Γ(α)Γ(β)
∫ t
af(τ)(t− τ)α+β−1
∫ 1
0xβ−1(1− x)α−1dxdτ
=1
Γ(α+ β)
∫ t
a(t− τ)α+β−1f(τ)dτ = Iα+βa f(t).
Therefore (2.2.3) holds almost everywhere on [a, b].
Now, if f ∈ C[a, b], then by Theorem 2.2.4 Iα+βa f(t) ∈ C[a, b] for α + β ≥ 1 and any t ∈ [a, b]. An
application of Fubini’s Theorem yields Iαa Iβa f(t) ∈ C[a, b] and is equal to Iα+βa f(t) ∈ C[a, b] for every
t ∈ [a, b].
13
The composition relation between fractional integral and the Mittag–Leffler functions are useful for
evaluation of fractional integrals of some of the elementary functions and in solutions of differential equa-
tions of fractional order.
Theorem 2.2.6. [125] For α, β, γ, δ ∈ R+ and λ ∈ R, the following holds:
(Iαa [(s− a)γ−1Eδβ,γ(λ(s− a)β)])(t) = (t− a)α+γ−1Eδβ,α+γ(λ(t− a)β). (2.2.4)
Proof. By definition of the Riemann–Liouville fractional integral and the generalized Mittag–Leffler func-
tion Eδβ,γ , we have
(Iαa [(s− a)γ−1Eδβ,γ(λ(s− a)β)])(t) =
∞∑k=0
(δ)kλk
Γ(βk + γ)k!Iαa (t− a)kβ+γ−1
= (t− s)γ+α−1∞∑k=0
(δ)k(λ(t− a)β)k
Γ(α+ βk + γ)k!
= (t− a)α+γ−1Eδβ,α+γ(λ(t− a)β).
The convergence of series in the definition of Eδβ,γ allows us to interchange the order of integration and
summation. The Lemma 2.2.2 is used to evaluate the fractional integral involved.
2.2.2 The Riemann–Liouville fractional derivative
Having defined the concept of fractional integrals, we intend to develop the notion of fractional order
derivatives and some of their important basic properties.
From now and onwards, Dn will denote the ntn order differential operator with D1 := D. In accordance
with these notations, the fundamental theorem of integer order calculus becomes
DIaf = f. (2.2.5)
A repeated application of above equation yields the following relation
f = DnIna f, n ∈ N. (2.2.6)
Replacing n in (2.2.6) by m− n with n < m , m ∈ N and applying Dn on both sides, we have
Dnf = DnDm−nIm−na f = DmIm−n
a f. (2.2.7)
This relation is still valid and meaningful, for some reasonable class of functions, if n is replaced by α ∈ Rprovided that m − α > 0 (m ≥ ⌈α⌉). Using the semigroup property of fractional integrals together with
the index law of classical derivative D and the fact that D is inverse of I, we have
DnIn−αa f = DαDn−⌈α⌉In−⌈α⌉a I⌈α⌉−αa f = D⌈α⌉I⌈α⌉−αa f = Dα
a f.
It is worthmentioning that the operator defined in this way depends on the choice of lower limit a of
fractional integral operator involved. One can define a fractional order derivative of a function as follows.
14
Definition 2.2.7. [69] Let α ∈ R+, m = ⌈α⌉ and f ∈ ACm[a, b]. The Riemann–Liouville fractional
derivative of order α is defined by
Dαa f := Dm
a Im−αa f, D0
a := I (the identity operator). (2.2.8)
The above definition is valid for arbitrary integer m provided m > α. There is no loss of generality
while considering narrow condition m = ⌈α⌉ or m − 1 ≤ α < m. In view of (2.2.7), the operator Dαa
coincides with classical nth order derivative operator when α is replaced with a positive integer.
Lemma 2.2.8. If α ≥ 0, β > −1, then the Riemann–Liouville fractional derivative of function (x − a)β
is given by
(Dαa (s− a)β)(t) =
Γ(β + 1)
Γ(β − α+ 1)(t− a)β−α.
In particular, when β = α− j, (j = 1, 2, . . . , ⌈α⌉+ 1), we have (Dαa (t− a)α−j = 0.
The composition relation between the Riemann–Liouville fractional derivative and the generalized
Mittag–Leffler function Eδβ,γ is of significant importance for evaluating fractional derivatives of various
functions.
Theorem 2.2.9. [125] For α, β, γ, δ ∈ R+ and λ ∈ R, the following holds:
(Dαa [(s− a)γ−1Eδ
β,γ(λ(s− a)β)])(t) = (t− a)γ−α−1Eδβ,γ−α(λ(t− a)β). (2.2.9)
Proof. By definition of the Riemann–Liouville fractional integral, the generalized Mittag–Leffler functions
Eδβ,γ and Lemma 2.2.8 we have
(Dαa [(s− a)γ−1Eδβ,γ(λ(s− a)β)])(t) =
∞∑k=0
(δ)kλk
Γ(βk + γ)k!Dαa (t− a)kβ+r−1
= (t− s)γ−α−1∞∑k=0
(δ)k(λ(t− a)β)k
Γ(βk + γ − α)k!
= (t− a)γ−α−1Eδβ,γ−α(λ(t− a)β).
The convergence of series in the definition of Eδβ,γ allows us to interchange the order of integration and
summation.
Remark 2.2.10. The fractional derivatives and integrals of exponential function et plotted in Figures 2.3
(a)-(c) are computed by applying Theorem 2.2.6 and Theorem 2.2.9. The fraction derivatives and integrals
of f(t) = t3 are evaluated by the application of Lemma 2.2.2 and Theorem 2.2.8. The fractional derivatives
and integrals of trigonometric and hyperbolic functions can be evaluated using the relation between the
Mittag–Leffler function and generalized trigonometric functions (2.1.11), generalized hyperbolic functions
(2.1.10). But, the numerical evaluation of the Mittag–Leffler functions is itself difficult. We have used
a much simpler method based on the Haar wavelets, (which will be discussed later in this chapter), to
evaluate the fractional integrals of some functions plotted in Figures 2.3(f)-(h). For the classical cases
i.e. α = 1, 2, the obtained results by the Haar wavelets are in good agreement with the exact values. In
particular, for f(t) = e−t cos(7t) and α = 1, 2, (m=32) the maximum absolute error is 5.15 × 10−4 and
8.52× 10−5 respectively.
15
0 0.5 1 1.5 2 2.50
2
4
6
8
10
t
α=0
α=1/3
α=7/10
α=1
(a) Iα0 e
t, α = 0, 13, 73, 1.
0 0.2 0.4 0.6 0.8 10
0.5
1−0.5
0
0.5
1
1.5
2
2.5
3
α
t
(b) Iα0 e
t, 0 ≤ α ≤ 1.
0 0.2 0.4 0.6 0.8 10
0.5
10.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
tα
(c) Dα0 e
t, 0 ≤ α ≤ 1.
0 0.4
0.61.2
1.62
00.5
11.5
22.5
30
2
4
6
8
10
12
14
tα
D0α t2
(d) Dα0 t
3, 0 ≤ α ≤ 3.
−2
−1.2
−0.4
−3−2−10123−20
−10
0
10
20
tα
−2
(e) Differintegration Dα−2t
3,−3 ≤ α ≤ 3.
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
t
α=1α=1(Exact)α=1.3α=1.5α=1.7α=2α=2(Exact)
(f) Iα0 sin(7t), α = 1, 1.3, 1.5, 1.7, 2
00.2
0.40.6
0.81
1
1.2
1.4
1.6
1.8
20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
x
α
(g) Iα0 sin(7t), 1 ≤ α ≤ 2.
0 0.2 0.4 0.6 0.8 1−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
t
α=1α=1(Exact)α=1.3α=1.5α=1.7α=2α=2
(h) Iα0 e
−t cos(7t), 1 ≤ α ≤ 2.
Figure 2.3: Fractional order integrals and derivatives of some elementary functions.
16
The following theorem characterizes the conditions for the existence of the Riemann–Liouville fractional
derivative Dαa defined above.
Theorem 2.2.11. [69, 114] Let f ∈ ACm[a, b] and m − 1 ≤ α < m. Then, the Riemann–Liouville
fractional derivative Dαa exists almost everywhere on [a, b]. Furthermore Dα
a f ∈ Lp[a, b] for 1 ≤ p < 1α and
Dαa f(t) =
m−1∑j=0
Djf(a)
Γ(j − α+ 1)(t− a)j−α + Im−α
a Dmf(t). (2.2.10)
Proof. Since f ∈ ACm[a, b], therefore by equations (2.2.6), (2.2.8), Lemma 2.2.5, Lemma 2.2.8 and Theo-
rem B.0.3, we have
Dαa f(t) = Dα
a
[m−1∑j=0
Djf(a)
Γ(j + 1)(t− a)j + Ima Dmf(t)
]
=m−1∑j=0
Djf(a)
Γ(j − α+ 1)(t− a)j−α +DmIm−αIma Dmf(t)
=
m−1∑j=0
Djf(a)
Γ(j − α+ 1)(t− a)j−α + Im−α
a Dmf(t).
Since, by Theorem B.0.3, Dmf(t) ∈ L[a, b] , therefore the integral Im−αa exists. Also we observe that the
existence of Dαa , α > 0 implies the existence of Dβ
a , 0 < β < α.
Before presenting properties of the Riemann–Liouville fractional derivative, we define a useful function
space
IαaL1[a, b] =f : Iαa f = φ, φ ∈ L1[a, b], α > 0
.
In the following , we establish composition relations between fractional derivative and integrals [114].
Lemma 2.2.12. If α, β ∈ R+, α > β and f ∈ L1[a, b], then DβaIαa f = Iα−βa f holds almost everywhere
on [a, b].
Proof. Using definition of the Riemann–Liouville fractional derivative, and semigroup property of frac-
tional integrals, we have DβaIαa f = D⌈β⌉I⌈β⌉−β
a Iαa f = Iα−βa f.
When α = β it immediately follows from above Lemma that the Riemann–Liouville fractional derivative
is left inverse to fractional integral operator. For some restricted class of functions the Riemann–Liouville
fractional derivative is also right inverse of fractional integral, as shown in the following lemma.
Lemma 2.2.13. Let a > 0 and f ∈ IαaL1[a, b]. Then IαaDαa f = f .
Proof. By same arguments as in the proof of the Lemma 2.2.12, we have
IαaDαa f = IαaD⌈α⌉I⌈α⌉−α
a f = IαaD⌈α⌉I⌈α⌉−αa Iαa φ = IαaD⌈α⌉I⌈α⌉
a φ = f.
In general, Fractional derivatives and integrals do not commute. We have the following result, which is
crucial for the investigations of fractional differential equations involving the Riemann–Liouville derivative.
We use it to convert differential equations into corresponding integral equations.
17
Theorem 2.2.14. [114] Let m ∈ N, m − 1 ≤ α < m and f ∈ L1[a, b]. Then, for Im−αa f ∈ ACm[a, b],
following holds:
IαaDαa f(t) = f(t)−
m∑j=1
(t− a)α−j
Γ(α− j + 1)
[Dα−ja f(t)
]t=a
. (2.2.11)
Proof. By definition of the Riemann–Liouville fractional derivative it follows that
Iα+1a Dα
a f(t) = Iα+1a DmDm−α
a f(t).
Integrating by parts and making use of Lemma 2.2.5 and Lemma 2.2.12, then by induction, we have
Iα+1a Dα
a f(t) =−k∑j=1
(t− a)α−j+1
Γ(α− j + 2)
[Dm−ja Im−α
a f(t)]t=a
+ Iα−m+1a Im−α
a f(t)
=Iaf(t)−k∑j=1
(t− a)α−j+1
Γ(α− j + 2)
[Dα−ja f(t)
]t=a
.
Applying D on both sides, we arrive at (2.2.11).
Having established relation between the Riemann–Liouville fractional derivative and integral, we now
turn to discuss the composition relation between two Riemann–Liouville fractional derivatives, namely
Dαa , (m − 1 ≤ α < m) and Dβ
a , (n − 1 ≤ α < n). The classical derivatives satisfy an unconditional
semigroup property. This is not the case when we are dealing with the fractional differential operators.
More precisely, we state and prove following result.
Theorem 2.2.15. [39] Let m, n ∈ N, m − 1 ≤ α < m, n − 1 ≤ α < n, α + β < m and f ∈ L1[a, b].
Then, for In−αa f ∈ ACn[a, b], following holds:
DαaDβ
af(t) = Dα+βa f(t)−
n∑j=1
Dβ−ja f(a)
Γ(1− α− j)(t− a)−α−j . (2.2.12)
Proof. By definition of the Riemann–Liouville fractional derivative and Lemma 2.2.5, we have
DαaDβ
af(t) = DmIm−αa Dβ
af(t) = DmIm−α−βa
[IβaDβ
af(t)]= Dα+β
a
[IβaDβ
af(t)].
Therefore, by Lemma 2.2.8 and Theorem 2.2.14 we obtain
DαaDβ
af(t) = Dα+βa
[f(t)−
n∑j=1
Dβ−ja f(a)
Γ(β − j + 1)(t− a)β−j
]= Dα+β
a f(t)−n∑
j=1
Dβ−ja f(a)
Γ(1− α− j)(t− a)−α−j .
In order to have a semigroup property for the Riemann–Liouville fractional derivative when α = β, we
must have DαaD
βaf(t) = Dβ
af(t)DβaDα
a f(t), which is possible only if Dα−ja f(a) and Dβ−j
a f(a) vanish simul-
taneously. For this, the function f must have a certain degree of smoothness. The following smoothness
assumptions on f are explored in [114, p. 215] by I. Podlubny.
Theorem 2.2.16. Assume that conditions of the Theorem 2.2.15 are satisfied. Then Dα−ja f(a), (j =
1, 2, . . . ,m) and Dβ−ja f(a), (j = 1, 2, . . . , n) vanish simultaneously, if and only if Djf(a) = 0 for j =
1, 2, . . . , q − 1, where q = maxm,n.
18
At this point, we introduce a useful function space for our purposes.
Lemma 2.2.17. [129] The space Bδ =y : y ∈ C[a, b] and Dδ
ay ∈ C[a, b], 0 < δ < 1
supplied with the
norm ∥y∥δ = max0≤t≤1
|y|+ max0≤t≤1
|Dδay| is a Banach space.
2.2.3 The Caputo fractional derivatives
The Riemann–Liouville fractional differential operators have played a significant role in the development
of the theory of differentiation and integration of arbitrary order. However, there are certain disadvan-
tages of using the Riemann–Liouville fractional derivatives for modeling the real world phenomena. In
Lemma 2.2.8 when β = 0 and f(t) = C(t − a)β , we observe that the fractional derivative of constant,
(Dαa (C))(t) =
C(t−a)−α
Γ(1−α) is a function of t and is never zero except for a = −∞. But for most of the physical
applications the lower limit is required to be a finite number. We also note that the initial value problems
for fractional differential equations with the Riemann–Liouville approach leads to the initial conditions
involving fractional derivative at lower limit. Mathematically, such problems can successfully be solved.
Being familiar with interpretation of real world problems with classical derivatives, we at least at the
present do not have any known physical interpretation of initial conditions involving fractional deriva-
tives. Applied problems, modeled using fractional operators, require an approach to fractional derivatives
which can utilize physically meaningful initial conditions involving classical derivatives. To cop with these
situations, M. Caputo in 1967 introduced another definition of fractional derivative and later in 1969
Caputo and Mainardi used it in the framework of viscoelasticity theory.
In what follows, we give a formal definition of the Caputo derivative and discuss its relations with the
Riemann–Liouville fractional integral and derivative.
Definition 2.2.18. [69] Let α ∈ R+ and f ∈ ACm[a, b], m = ⌈α⌉. Then the Caputo fractional derivative
of order α is defined bycDα
a f(t) = Im−αDmf(t). (2.2.13)
For α = m, the equation (2.2.13) yields cDαa f(t) = Dmf(t). Thus for integer values of α, the Caputo
fractional derivative becomes the conventional derivative. Historically, this concept was known to many
authors, including Rabotnov (1969), Dzherbashyan and Nersesian (1968), Gerasimov (1948), Gross (1947)
and even can traced back to Liouville (1832). Following the most common convention, we will name it
the Caputo derivative.
The definitions of fractional derivative, both in the sense of Riemann–Liouville and Caputo , utilize
the definition of Riemann–Liouville fractional integral but the order of fractional integration with integer
differentiation is interchanged. Also, note that both in the definition of the Riemann–Liouville fractional
derivative and the Caputo fractional derivative we require m = ⌈α⌉. This condition is not strict in the
case of the Riemann–Liouville definition of fractional derivative. We may chose any integer m such that
m ≥ α. However in the case of the Cputo fractional derivative, we may not use the condition m > ⌈α⌉.Another difference between Riemann–Liouville and caputo fractional derivatives is that the Riemann–
Liouville fractional derivative exists for a class of integrable function while the existence of the Caputo
19
fractional derivative requires the integrability of m times differentiable functions. This can be seen from
following Lemma.
Lemma 2.2.19. [39] If α ≥ 0 and f(t) = (t− a)β, m = ⌈α⌉, then
cDαa f(t) =
0, if β ∈ 0, 1, 2, . . . ,m− 1,Γ(β+1)
Γ(β−α+1)(t− a)β−α, if β ∈ N, and β ≥ m or β /∈ N, and β > m− 1.
Using this lemma, we can compute the Caputo fractional derivative of Mittag–Leffler function. More
precisely, we have following result.
Theorem 2.2.20. For α, β, γ, δ ∈ R+ and λ ∈ R, the following holds:
(cDαa [(s− a)γ−1Eδβ,γ(λ(s− a)β)])(t) = (t− a)γ−α−1Eδβ,γ−α(λ(t− a)β). (2.2.14)
Proof. By definition of the generalized Mittag–Leffler function and Lemma 2.2.19, we have
(cDαa [(s− a)γ−1Eδβ,γ(λ(s− a)β)])(t) =
∞∑k=0
(δ)kλk
Γ(βk + γ)k!Dαa (t− a)kβ+r−1
= (t− s)γ−α−1∞∑k=0
(δ)k(λ(t− a)β)k
Γ(βk + γ − α)k!
= (t− a)γ−α−1Eδβ,γ−α(λ(t− a)β).
The convergence of series in the definition of Eδβ,γ allow us to interchange the order of integration.
The following result shows an important connection between the Riemann–Liouville and the Caputo
fractional derivatives and is key to the construction of the Caputo differential operator.
Theorem 2.2.21. For α ≥ 0, assume that f ∈ ACm[a, b], m = ⌈α⌉. Then
cDαa f(t) = Dα
a
[f(t)−
m−1∑j=0
Djf(a)
j!(t− a)j
](2.2.15)
Proof. In view of definition of the Riemann–Liouville fractional derivative, we have
Dαa
[f(t)−
m−1∑j=0
Djf(a)
j!(t− a)j
]= DmIm−α
a
[f(t)−
m−1∑j=0
Djf(a)
j!(t− a)j
].
Repeatedly integrating by parts, we obtain
Im−αa f(t) =
−1
Γ(m− α+ 1)
[(f(t)−
m−1∑j=0
Djf(a)
j!(t− a)j
)(t− a)m−α
]ta+ Im−α+1
a Df(t)
= Im−α+1a Df(t) = Im−α+2
a D2f(t) = · · · = I2m−αa Dmf(t).
Thus, we have DmI2m−αa Dmf(t) = Im−α
a Dmf(t) = cDαa f(t).
Using the above Theorem and Lemma 2.2.8 another relation between the Riemann–Liouville and
Caputo fractional derivatives can be given as.
20
Theorem 2.2.22. For α ≥ 0, m = ⌈α⌉ assume that for some f , both Dαa f(t) and cDα
a f(t) exist. Then
cDαa f(t) = Dα
a f(t)−m−1∑j=0
Djf(a)
Γ(j − α+ 1)(t− a)j−α. (2.2.16)
The above relation between the Riemann–Liouville and the Caputo derivatives says that they are
not equal in general. However, the equality cDαa f(t) = Dα
a f(t) holds if and only if Djf(a) = 0, j =
0, 1, 2, . . . ,m− 1. Having established relationship between the Riemann–Liouville and the Caputo deriva-
tives, we now turn to the composition relations between the Riemann–Liouville fractional integral and the
Caputo fractional derivatives.
Lemma 2.2.23. Let α, β ∈ R+, α ≥ β and f be continuous function. Then cDαaIαa f(t) = f(t).
Proof. First, we consider the case α = β. Since f is continuous on [a, b], therefore it is easy to see that
Iαa f(t) is bounded. Replacing f(t) by Iαa f(t) in (2.2.16), we get the required result.
In the case, α > β, using definition of the Caputo fractional derivative and semigroup property of
fractional integrals, we have
cDβaIαa f(t) = I⌈β⌉−β
a D⌈β⌉Iαa f(t) = I⌈β⌉−βa Iα−⌈β⌉
a f(t) = f(t).
Lemma 2.2.23 shows that the Caputo fractional derivative is left inverse of the Riemann–Liouville
fractional integral but on the other hand the following lemma shows the it is not right inverse of the
Riemann–Liouville fractional integral.
Lemma 2.2.24. Let α > 0, m = ⌈α⌉ and f ∈ ACm[a, b]. Then
Iαa [cDαa f(t)] = f(t)−
m−1∑j=0
Djf(a)
j!(t− a)j . (2.2.17)
Proof. Using (2.2.13), Lemma 2.2.5 and classical Taylor Theorem , we have
Iαa [cDαa f(t)] = Iαa Im−α
a Dmf(t) = Ima Dmf(t) = f(t)−m−1∑j=0
Djf(a)
j!(t− a)j .
The equation (2.2.17) also serves as the fractional Taylor series in term of the Caputo fractional
derivative. It has a significant importance in the theory of fractional differential equations involving the
Caputo fractional derivative for obtaining the corresponding integral representation.
We leave this section by introducing a function space which will be used to discuss matters.
Lemma 2.2.25. [152] The space Bα =y : y ∈ C[a, b] and cDα
a y ∈ C[a, b], α > 0
furnished with the
norm ∥y∥α = max0≤t≤1
|y|+ max0≤t≤1
|cDαa y| is a Banach space.
Remark 2.2.26. The conclusion of above Lemma holds if cDαa is replaced with Dα
a .
21
2.3 Fixed Point Theorems
One of the objective of this work is to provide a numerical technique for the treatment of boundary value
problems for fractional differential equations. While solving equations, or in particular solving differential
equations, the important thing to know is whether a particular equation has a solution or not. The
presence of solutions is guaranteed by so called fixed point theorems. So for in this chapter we have
been mainly concerned with the basic theory of fractional calculus. Now we also provide the terminology,
basic concepts and notations from analysis and fixed point theory. We state some important fixed point
theorems needed to establish existence of solutions for fractional differential equations.
Many of the most important problems of applied mathematical sciences can be described in a unitary
fission as: Find an object which belongs to a given class O of objects and is related to a given object φ
by a certain relation R. More precisely x ∈ O : x R φ
. (2.3.1)
It will be useful to explain this problem by a specific example. Consider the initial value problem for
fractional differential equationscDαa y(t) = f(t, y), α > 0, t ∈ [a, b]
Dky(a) = bk, k = 0, 1, 2, . . . ,m− 1, m = ⌈α⌉.(2.3.2)
Indeed, here we have O = C[a, b], φ = (f, a, bk), where y is a continuous function [a, b] :→ R and R is the
system (2.3.2). If f [a, b] : R is continuous, then by Lemma 2.2.23, the problem (2.3.2) can be transformed
into following Volterra integral equation
y(t) = ga(t) + Iαa f(t, y) (2.3.3)
where ga(t) =∑m−1
j=0 bj(t − a)j , bj = Djy(a)j! for j = 0, 1, 2, . . . ,m − 1. Since f is continuous, therefore
(2.3.2) can be recovered from (2.3.3). Thus (2.3.2) and (2.3.3) are equivalent.
Define an operator A : C[0, 1] → C[0, 1] as
Ay(t) = ga(t) + Iαa f(t, y). (2.3.4)
The continuity of f implies that the operator A is continuous. In view of (2.3.4) the equation (2.3.3) takes
the form
y = Ay(t). (2.3.5)
The first question in the study of this operator equation involves the existence of solutions. The solutions
of equation (2.3.5) are called the fixed points of A, which in turn are the solutions of problem (2.3.2). The
second questions involves the uniqueness of the fixed points of operator A. Perhaps the most simplest and
well known elementary result in fixed point theory employed to answer these questions is due to Banach
(1922)and is commonly known as the Banach Fixed Point Theorem or the Contraction Mapping Principle.
Being based on iterative process, it provides a constructive way of finding fixed points.
22
Theorem 2.3.1. Let M be a closed subset of the Banach space B and A : M → M. Then A has a
unique fixed point y in M and for any initial guess y0 ∈ M, the successive approximations ym+1 = Aymconverges to y, provided that ∥Ay −Ay∥ ≤ q∥Ay −Ay∥ for all y, y ∈ M and q < 1.
The Banach fixed point theorem yields a great deal of information about the solutions of certain
equations, but the class of equations to which it is applicable is very limited. Thus various alternative
have been developed.
Theorem 2.3.2. (The Brouwer Fixed Point Theorem) Suppose that M is a nonempty, convex, compact
subset of a finite dimensional normed vector space and A : M → M is a continuous mapping. Then Ahas a fixed point in M.
Now the question arises wether the condition of finite dimensionality, in the Brouwer fixed point
theorem, can be removed. The first powerful result, which avoids the restriction of this type, is the Schauder
fixed point theorem. This result uses the idea of compactness of operators to bridge the gape between
finite and infinite dimensional normed spaces. The boundary value problems for fractional differential
equations on a finite domain can often be posed in term of nonlinear compact operators using Green’s
functions. For this reason the Schauder fixed point theorem occupies a central position in the existence
theory of differential equations.
Theorem 2.3.3. (The Schauder Fixed Point Theorem) Suppose that M is a nonempty, convex, compact
subset of the Banach space B and A : M → B is a compact operator that maps M into itself. Then A has
a fixed point in M.
One of the possible way of solving a given operator equations is to embed it into continuum of equations
and then starting from one of the solution of simpler equation to obtain the solution of given equation.
More explicitly, for given two nonempty sets Σ and Ω consider a mapping Υ1 : Σ → Ω and U : a proper
subset of Ω. Our goal is to investigate the solvability of the problem
Υ1(x) ∈ U, x ∈ Σ. (2.3.6)
The idea behind the continuation methods is combining the problem (2.3.6) with a simpler one
Υ0(x) ∈ U, x ∈ Σ, (2.3.7)
using homotopy µ : Σ× [0, 1] → Ω such that
µ(., 0) = Υ0, µ(., 1) = Υ1.
Key role is played by conditions in continuation theorem which assure that the solvability of (2.3.7) implies
the solvability of (2.3.6). This important method is called Leray–Schauder continuation principle and is
key to establish the existence results for nonlinear boundary value problems for differential equations.
Historically, the continuation method was introduced by Poincaré (1884) and Bernstein (1912) used this
method to establish existence results by introducing technique of priori estimates. However, the first
rigorous study of continuation principle was carried out by Leray and Schauder (1934). In the following
we recall the statement of a Leray–Schauder continuation principle.
23
Theorem 2.3.4. [144, Theorem 6.A](Leray–Schauder Continuation Principle). Assume that
(i) the operator A : B → B is compact in the Banach space B;
(ii) there is a constant ρ > 0 and y = λAy for λ ∈ (0, 1) implies ∥y∥ ≤ ρ.
Then the equation (2.3.5) has a solution.
Another important fixed point theorem, derived from the Schauder fixed point theorem, commonly
used in the existence theory of differential equations is the nonlinear alternative of Leray–Schauder type.
Theorem 2.3.5. [48, Theorem 2.3] (Nonlinear alternative of Leray–Schauder type) Let B be a Banach
space and M be a nonempty convex subset of B and U be open in M with 0 ∈ U . Let A : U → M be a
compact operator. Then either
(i) A has a fixed point, or
(ii) there exists y ∈ ∂U and λ ∈ (0, 1) such that y = λAy.
In the case when the solutions of the operator equation (2.3.5) are not unique then the theorems
of non-uniqueness, that is the existence theorems for two or more solutions, are of basic interest. The
existence of solutions ensured by pure mathematical reasoning often requires some additional conditions
be satisfied. Most of the mathematical models in biomathematics, economics, physics and engineering
require the nonnegativity of solutions. Usually the existence theorems for positive solutions are established
in specially constricted cones defined as:
Definition 2.3.6. A closed convex subset P of a Banach space B is called cone, if λP ⊂ P, for all λ ≥ 0
and −P ∩ P = 0.
The most frequently used result to prove the existence of multiple positive solutions for differential
equations is the Guo-Krasnosel’skii Fixed Point Theorem, which is stated as:
Theorem 2.3.7. [75] Let B be Banach space and P ⊂ B be a cone. Assume that Ω1,Ω2 are open disks
in B such that 0 ∈ Ω1 ⊂ Ω1 ⊂ Ω2. Let A : P ∩ (Ω2\Ω1) → P be completely continuous such that either
(i) ∥Ay∥ ≤ ∥y∥, for y ∈ P ∩ ∂Ω1 and ∥Ay∥ ≥ ∥y∥, for y ∈ P ∩ ∂Ω2 or
(ii) ∥Ay∥ ≥ ∥y∥, for y ∈ P ∩ ∂Ω1 and ∥Ay∥ ≤ ∥y∥, for y ∈ P ∩ ∂Ω2.
Then, A has a fixed point in P ∩ (Ω2\Ω1).
In the following, we state another fixed point theorems which is commonly used to ensure the existence
of three fixed points.
Theorem 2.3.8. [83] (Leggett-Wiliam’s Fixed Point Theorem) Let θ : B → R+ be a continuous concave
functional and θ(y) ≤ ∥y∥, for all y ∈ Ec, where Ec = y ∈ E : ∥y∥ < c. Let 0 < a < b < d ≤ c be given
and define Eθ(b, d) = y ∈ E : b ≤ θ(y), ∥y∥ ≤ d. Assume that A : Ec → Ec is completely continuous and
satisfies
24
(i) y ∈ Eθ(b, d) : θ(y) > b = ∅ and θ(Ay) > b, for y ∈ Eθ(b, d),
(ii) ∥Ay∥ < a, for ∥y∥ ≤ a and
(iii) θ(Ay) > b for y ∈ Eθ(b, c) with ∥Ay∥ > d.
Then, A has at least three fixed points y1, y2 and y3 such that ∥y1∥ < a, b < θ(y2) and ∥y3∥ > a with
θ(y3) < b.
Finally, we make a remark about notation. In the case when lower limit a = 0, in fractional operators,
we shall use the notation Iα for Riemann–Liouville fractional integral and Dα, cDα will be used for
Riemann–Liouville and Caputo fractional derivatives. The symbol B will denote the Banach space C[a, b]
equipped with the norm ∥y∥ = maxt∈[a,b]
|y(t)|.
2.4 Wavelets
The origan of wavelets goes back to the beginning of 20th century, when Hungarian mathematician Alfred
Haar in 1910 constructed an orthonormal system of functions on the unit interval [0, 1] even though he did
not named it. Historically, the concept of wavelets was formally introduced at the beginning of eighties by
J. Morlet, a French geophysical engineer, as a family of functions constructed by translation and dilation
of single function, called the mother wavelet, for the analysis of seismic signals. The reason behinds the
discovery of wavelets is that Fourier series represents frequency of a signal well, but it does not model its
localized features appropriately. This is because the building blocks of Fourier series, the sine and cosine
functions, are periodic waves which continue forever.
The wavelet theory have drawn great deal of attention from scientists working in various disciplines
because of its comprehensive mathematical power and wide range applications in science and engineering.
Particularly, wavelets are very useful in signal processing, image processing, edge extraction, computer
graphics, approximation theory, biomedical engineering, differential equations, numerical analysis, etc. As
pointed out by Stephane Mallat,
“Wavelet theory is the result of a multidisciplinary effort that brought together mathematicians
physicists and engineers...”.
Wavelets are special kind of functions which exhibit oscillatory behavior for a short period of time and
then become zero. Wavelets are constructed from dilation and translation of single function ψ(t), called
mother wavelet and thus generating a two parameter family of functions ψa,b. It is convenient to define
ψa,b as follows.
ψa,b(t) =1√|a|ψ( t− b
a
), a, b ∈ R, a = 0,
where a is dilation parameter and b is translation parameter. If |a| < 1, then ψa,b is compressed form of
mother wavelet and corresponds to higher frequencies. On the other hand, for |a| > 1 the wavelet ψa,bcorresponds to lower frequencies. More precisely, we have the following definition of wavelets:
25
Definition 2.4.1. [43] A function ψ ∈ L2(R) is admissible as a wavelet if and only if
Cψ =
∫ ∞
−∞
|ψ(ω)|2
|ω|dω <∞,
where ψ is the Fourier transform of ψ, i.e.,
ψ(ω) =1
2√π
∫ ∞
−∞e−iωxψ(x)dx.
The admissibility condition requires that Cψ is finite. This implies that ψ(0) = 0 i.e. the mean value
of ψ should vanish:∫∞−∞ ψ(s)ds = 0. As an example of wavelets, the Maxican hat and its dilated shifts
are shown in Figure 2.4.
The continuous wavelets are not useful for many practical purposes. In particular they do not form
basis. For that reason, discretization is performed by fixing the positive constants a0 > 1, b0 > 0 and
setting a = a−j0 , b = kb0a−j0 , where n, k ∈ N. Thus, we define the following family of discrete wavelets as
ψj,k(t) = (a0)j2ψ(aj0t− kb0
).
Usually a0 is chosen to be 2 and b = 1. Ingrid Daubechies gave solid foundations for wavelet theory. In
1988 she provided major break through by constructing a system of orthonormal wavelets with compact
support. The Haar wavelet is the simplest example of orthogonal wavelets compactly supported on the
interval [0, 1] and was constructed by Haar in 1910 in his Ph.D. dissertation.
2.4.1 The Haar scaling function
In discrete wavelet transform we consider two sets of functions, scaling functions and wavelet functions.
The Haar scaling function φ(t) is defined on [0, 1] as
φ(t) = χ[0,1)(t) =
1, 0 ≤ t < 1,
0, elsewhere,(2.4.1)
which is a characteristics function of the interval [0, 1). The translates of Haar scaling function φ(t−k)k∈Zform an orthonormal set of functions. That is
∫R φ(t−m)φ(t−n) = δmn. The subspace of L2(R) spanned
by translates of the Haar scaling functions is denoted by V0. Scaling translates of φ(t) by 2j , we get the
functions φj,k = 2j2φ(2jt − k) supported on the dyadic subintervals Ij,k = [k2−j , (k + 1)2−j), j, k ∈ Z,
where φ0,0 is abbreviated as φ. For fixed j, the functions φj,k are orthonormal among themselves and
the space spanned by φj,k is denoted by Vj . The Haar scaling function φ satisfies the dilation equation
φ(t) =√2
∞∑k=−∞
ckφ(2t− k) (2.4.2)
where ck are given by
ck =√2
∫ ∞
−∞φ(s)φ(2s− k)ds. (2.4.3)
Evaluating (2.4.3), we have c0 = c1 = 1√2
and ck = 0 for k > 1. Therefore the dilation equation (2.4.2)
becomes
φ(t) = φ(2t) + φ(2t− 1). (2.4.4)
26
−10 −5 0 5 10−0.5
0
0.5
1
t
ψψ
−1,0
ψ−1,−1
ψ−1,1
Figure 2.4: The Maxican hat ψ(t) = (1− t2)e−12 t
2
and its dilated shifts.
The spaces spanned by the scaling function, that we briefly mentioned above, define a multiresolution
representation in L2(R). The idea of multiresolution is to express functions in L2(R) as limit of successive
approximations. These successive approximations use different levels of resolutions. Before proceeding
further, in the following subsection, we define multiresolution analysis.
2.4.2 Multiresolution Analysis (MRA)
The central idea of multiresolution analysis for the constructions of wavelet basis was formulated by S.
Mallat and Y. Meyer in 1986. The MRA is a formal approach for the construction of orthogonal wavelet
basis.
Definition 2.4.2. A multiresolution analysis is a sequence Vjj∈Z of closed subspaces of L2(R), such
that following conditions are satisfied:
(i) The sequence Vjj∈Z is nested: · · · ⊂ V−1 ⊂ V0 ⊂ V1 · · ·Vm ⊂ Vm+1 · · · .
(ii)∪j∈Z
Vj = L2(R) i.e.∪j∈Z
Vj is dense in L2(R).
(iii)∩j∈Z
Vj = 0.
(iv) f(t) ∈ Vm if and only if f(2t) ∈ Vm+1.
(v) There exists a function φ ∈ V0 such that φ0,k = φ(t− k), k ∈ Z is a Riez basis for V0, that is, for
every f ∈ V0, there exists a unique sequence ckk∈Z ∈ l2(Z) such that f(t) =∑
k∈Z ckφ(t− k) with
convergence in L2(R) and there exist two positive constants A, B independent of f ∈ V0 such that
A∑
k∈Z |ck|2 ≤ ∥f∥2 ≤ B∑
k∈Z |ck|2 where 0 < A < B <∞.
The subspaces Vjj∈Z of L2(R) spanned by the sets of functions φj,k is a MRA. Thus the Haar
scaling function generates the MRA.
27
0 1/2 1
(a) ψ0,0
000 1/4 1/21
(b) ψ1,0
0 1/2 13/4
(c) ψ1,1
Figure 2.5: Haar wavelets.
2.4.3 The Haar wavelet function
Based on the relation Vj ⊂ Vj+1 we are interested in the decomposition of Vj+1 as an orthonormal sum of
Vj and its orthogonal complement. For j = 0, the space V0 is spanned by the integer translates of scaling
function φ. This observation motivates for the construction of a function ψ(t) whose translate form the
basis for the orthogonal complement W0 of V0. The function ψ should be member of V1 and orthogonal to
V0. The simplest ψ that fulfills these requirements is ψ(t) = χ[0, 12) −χ[ 1
2,1) and is referred as Haar wavelet
function. The translates of ψ(t) i.e. ψ(t− k)k∈Z form an orthonormal set. Furthermore it is easy to see
that the system ψ(t− k)k∈Z forms basis for the space W0. For each pair of integers j, k ∈ Z, the dilated
translates of ψ are given as
ψj,k(t) = 2j2 (χIlj,k
(t)− χIrj,k(t)) = 2j2ψ(2jt− k). (2.4.5)
The functions ψj,k are compactly supported on the dyadic intervals Ij,k, j, k ∈ Z. For fixed j ∈ Z, we
define Wj as the orthogonal complement of Vj in Vj+1. The system ψj,kk∈Z, forms an orthonormal basis
of the complementary space Wj . In what follows, we show that the Haar wavelet functions are orthogonal
among themselves.
2.4.4 Orthogonality of the Haar wavelets
Theorem 2.4.3. The Haar wavelet system ψj,kj,k∈Z is orthonormal in L2(R).
Proof. Consider the inner product
⟨ψj,k, ψl,m⟩ =∫ ∞
−∞2
j2ψ(2jt− k)2
l2ψ(2lt−m)dt. (2.4.6)
Using change of variables 2jt− k = x, we find
⟨ψj,k, ψl,m⟩ = 212(l−j)
∫ ∞
−∞ψ(x)ψ(2l−j(x+ k)−m)dx. (2.4.7)
We consider the following cases:
Case 1. j = l. If k = m then ⟨ψj,k, ψj,k⟩ = ∥ψ∥2 = 1. On the other hand, if k = m then from (2.4.7) we
have ⟨ψj,k, ψj,k⟩ = ⟨ψ,ψ0,m−k⟩. Since m− k = 0, therefore the supports of ψ, ψ0,m−k are not overlapping,
i.e., I = [0, 1] ∩ I0,m−k = ∅ and the integral in (2.4.7) vanishes. Hence ⟨ψj,k, ψj,k⟩ = δk,m.
28
Case 2. j = l. It is sufficient to consider l > j. By symmetry, the result will hold for l < j. Substituting
r = l − j and n = m− 2rk in (2.4.7), we get.
⟨ψj,k, ψl,m⟩ = 2r2
∫ ∞
−∞ψ(x)ψ(2rx− n)dx. (2.4.8)
In the case of non overlapping diadic intervals, i.e., I ∩Ir,m = ∅, the integral (2.4.8) becomes 0. The diadic
intervals either do not overlap or one is contained in other. If one diadic interval is contained in other,
then it is contained in the left half or in the right half of it. So, if I ∩ Ir,m = ∅, then ψ(t) is constant over
Ir,m. Therefore the integral (2.4.7) vanishes.
In view of orthogonality of the system ψi,kj,k∈Z, the subspace Wj , j ∈ Z of L2(R) have following
important property:
· · ·W−1⊥W0⊥W1⊥W2⊥ · · ·Wj⊥Wj+1 · · · .
Furthermore the space Wj is an orthogonal compliment of Vj
Vj+1 =Wj ⊕ Vj , Wj⊥Vj . (2.4.9)
The repeated application of (2.4.9) yields the following relation
V0 ⊕W0 ⊕ · · · ⊕Wj = Vj+1. (2.4.10)
The nested structure of Vj together with properties (ii) and (iv) of MRA provide a decomposition of the
space L2(R). When m→ ∞, we have the decomposition.
L2(R) = V0 ⊕(⊕∞j=0Wj
). (2.4.11)
Thus MRA enables us to construct an orthonormal basis for L2(R).
2.4.5 Function approximation by the Haar wavelets
The decomposition of L2(R) into scaling space Vj and wavelet space Wj allows us decomposing any
function f(t) ∈ L2(R) such that components of f(t) lie in both subspaces. For each j ∈ Z, the successive
approximations of given function f(t) are defined as orthogonal projections Pj of function f ∈ L2(R) on
the space Vj given by
Pjf =∑k∈Z
⟨f, φj,k⟩φj,k(t), j ∈ Z, (2.4.12)
where ⟨f, φj,k⟩ are scaling coefficients. Since φj,k = 2j2χIj,k(t), therefore
Pjf = 2j∑k∈Z
(∫Ij,k
f(s)ds)χIj,k(t), j ∈ Z. (2.4.13)
Thus for each k ∈ Z, the projection Pjf is the average value of f(t) on the interval Ij,k. Therefore, in
relation to previous approximation Pj−1f , the function approximations determined at resolution level j
29
0 0.2 0.4 0.6 0.8 1-2
-1
0
1
2
Figure 2.6: Approximating f(t) = sin(9t) + 2 cos(11t) + 12 sin(50t) by the Haar wavelets.
lost any details about variation of f of scale smaller then 2−j on Ij,k. The lost details can be recovered
by orthogonal projection Qjf on the orthogonal complement Wj of Vj in Vj+1 so that
Qjf =∑k∈Z
⟨f, ψj,k⟩ψj,k(t), j ∈ Z, (2.4.14)
where ⟨f, ψj,k⟩ are the wavelet coefficients. The projection operator Qjf contains the details of f smaller
the 2−j but larger then 2−j−1. In view of (2.4.9),we have
Pj+1f = Pjf +Qjf. (2.4.15)
Therefore, in accordance with relation (2.4.11), we have multiresolution decomposition of the function
f ∈ L2(R) as
f(t) =∑k∈Z
⟨f, φ0,k⟩φ0,k(t) +∞∑j=0
∑k∈Z
⟨f, ψj,k⟩ψj,k(t) (2.4.16)
For the function f : [0, 1] → R, the Haar wavelet series representation is
f(t) = ⟨f, χ[0,1)⟩χ[0,1)(t) +
∞∑j=0
2j−1∑k=0
⟨f, ψj,k⟩ψj,k(t), (2.4.17)
The infinite series (2.4.17) is a powerful mathematical tool for the representation of large class of functions.
2.4.6 Error analysis
When dealing with series representation in practice, we are only able to deal with finite sums. We are
required to choose some fixed J such that the partial sum
SN (t) = ⟨f, χ[0,1]⟩χ[0,1](t) +
J−1∑j=0
2j−1∑k=0
⟨f, ψj,k⟩ψj,k(t), N = 2J , (2.4.18)
30
approximates f sufficiently well. The Haar wavelet approximation of the function f with bounded first
order derivative on (0, 1), gives the following estimate for error [15].
∥∥∥f(t)− SN (t)∥∥∥2 = ∥∥∥f − ⟨f, χ[0,1]⟩χ[0,1](t)−
J∑j=0
2j−1∑k=0
⟨f, ψj,k⟩ψj,k(t)∥∥∥2
=∥∥∥ ∞∑j=J
2j−1∑k=0
⟨f, ψj,k⟩ψj,k(t)∥∥∥2 = ∞∑
j=J
2j−1∑k=0
|⟨f, ψj,k⟩|2.
Now, by mean value theorem of integral calculus, there exist t1 ∈ I lj,k, t2 ∈ Irj,k, such that
⟨f, ψj,k⟩ = 2j2
(∫Ilj,k
f(t)dt−∫Irj,k
f(t)dt
)= 2−
j2−1(f(t1)− f(t2)
).
By mean value theorem of differential calculus, there exists ξ ∈ (t1, t2), such that
|⟨f, ψj,k⟩|2 = 2−j−2(f(t1)− f(t2)
)2= 2−j−2(t2 − t2)
2(f ′(ξ))2
≤ 2−3j−2L2, for some L > 0.
Thus, we have
∥∥∥f(t)− SN (t)∥∥∥2 ≤ L2
∞∑j=J
2j−1∑k=0
2−3j−2 = L2∞∑j=J
2−2j−2 =L2
N2= O
( 1N
).
Thus, with increasing J the error of approximation decreases.
Chapter 3
Existence and uniqueness of solutions
The objective of this chapter is to study some existence results for solutions of certain classes of nonlin-
ear two–point, three–point and multi–point boundary value problems for differential equations involving
fractional derivatives. The existence theory for initial value problems for fractional differential equations
have received considerable attention in the last two decades. However the existence theory for linear
and nonlinear boundary value problems for differential equations of fractional order is in its initial stages
of development. The theory of boundary value problems for fractional or even integer order differential
equations is much more complicated as compared with that of initial value problems.
This chapter is organized as follows: In section 3.1 and section 3.2, we are concerned with the existence
and uniqueness of solutions to two–point boundary value problems for nonlinear differential equations of
fractional order. In section 3.3, we establish sufficient conditions for the existence and uniqueness of
solutions to a general class of three–point boundary value problems by imposing some growth conditions
on the nonlinear functions involved. Section 3.4 deals with a class of three–point boundary value problems
for fractional order differential equations where the nonlinear term as well as the boundary conditions
involve the derivatives of unknown functions. Section 3.5 concerns with existence and uniqueness results
for a class of multi–point boundary value problems. Finally, in section 3.6, existence and uniqueness results
are established for nonlinear boundary value problems involving integral boundary conditions. Examples
are included to show the applicability of our results.
3.1 Two–point boundary value problems (I)
The question of existence and uniqueness of solution for two–point boundary value problems for fractional
differential equations have been studied in [3,21,126,128] and the references therein. Here we consider the
following nonlinear boundary value problem with the Caputo fractional derivative
cDαy(t) = g(t, y(t)), t ∈ [0, 1], (3.1.1)
y(0) = y0, y(1) = y1, (3.1.2)
where, 1 < α ≤ 2, y0, y1 ∈ R. The nonlinear function g : [0, 1] × R → R is assumed to be continuous.
Existence theory for the problem (3.1.1), (3.1.2) under much stronger hypothesis is developed in [128].
31
32
Different from the technique presented in [128], we prove the existence results not only in much simpler
way but also under weaker hypothesis on the nonlinear function g. Moreover, we also provide sufficient
conditions for the existence of a unique solution to the boundary value problem.
Lemma 3.1.1. A function y ∈ C[0, 1] is a solution of the boundary value problem (3.1.1), (3.1.2) if and
only if it is solution of the integral equation
y(t) = Iαg(t, y(t))− tIαg(1, y(1)) + (y1 − y0)t+ y0 =
∫ 1
0G(t, s)ds+ (y1 − y0)t+ y0, (3.1.3)
where
G(t, s) =
(t−s)α−1−t(1−s)α−1
Γ(α) , if s ≤ t,
− t(1−s)α−1
Γ(α) , if t ≤ s .(3.1.4)
Proof. Assume that y ∈ C[0, 1] is a solution of the boundary value problem (3.1.1), (3.1.2). Let w(t) :=
g(t, y(t)) = cDαy(t) = I2−αD2y(t). Therefore I2−α0 D2y(t) is continuous. Hence y(t) ∈ AC2[0, 1]. By
Lemma 2.2.24, we have
y(t) = Iαw(t) + c0 + c1t, c0, c1 ∈ R. (3.1.5)
Using the boundary conditions (3.1.2), we obtain c0 = y0, c1 = y1 − y0 − Iαw(1). Thus
y(t) = Iαg(t, y(t))− tIαg(1, y(1)) + (y1 − y0)t+ y0 =
∫ 1
0G(t, s)ds+ (y1 − y0)t+ y0.
Conversely suppose that y(t) is a solution of the integral equation (3.1.3). The application of the operatorcDα on both the sides of (3.1.3) and the use of Lemma 2.2.19 and Lemma 2.2.23, implies that y(t) satisfies
the differential equation (3.1.1) and the boundary conditions.
Lemma 3.1.2. The Green function G(t, s) defined in (3.1.4) has the following properties:
(i) |G(t, s)| ≤ 1Γ(α)(1− s)α−1, for all t, s ∈ [0, 1],
(ii)∣∣∣ ∂∂tG(t, s)∣∣∣ ≤ γ
Γ(α−1)(1− s)α−2, where γ := min1, Γ(α−1)Γ(α) .
Proof. For s ≤ t, using (3.1.4) we have G(t, s) = 1Γ(α)(t − s)α−1 − t(1 − s)α−1 ≤ (1 − t)(1 − s)α−1. Thus
G(t, s) ≤ 1Γ(α)(1− s)α−1. For s < t, obviously |G(t, s)| ≤ 1
Γ(α)(1− s)α−1.
Now, differentiating (3.1.4) with respect to t, we get
∂
∂tG(t, s) =
(t−s)α−2
Γ(α−1) − (1−s)α−1
Γ(α) , if s ≤ t,
− (1−s)α−1
Γ(α) , if t ≤ s .(3.1.6)
We observe that (1− s)α−2 ≥ (1− s)α−1 for s ∈ [0, 1] and 1 < α ≤ 2. Now, for s ≤ t we have
∂
∂tG(t, s) ≤ (t− s)α−2
Γ(α− 1)
and for t < s, we have ∣∣∣ ∂∂tG(t, s)
∣∣∣ ≤ (t− s)α−1
Γ(α)≤ (t− s)α−2
Γ(α).
Therefore∣∣∣ ∂∂tG(t, s)∣∣∣ ≤ γ
Γ(α−1)(1− s)α−2, where γ = min1, Γ(α−1)Γ(α) .
33
For the establishment of sufficient conditions for the existence of at least one solution to the fractional
boundary value problem (3.1.1), (3.1.2), we choose the Banach space of all continuous functions B :=
C[0, 1], equipped with Chebyshev norm.
Theorem 3.1.3. Assume that there exists continuous function φ : [0, 1] → R+ such that
|g(t, y)| ≤ φ(t) + c|y|θ, c, θ ∈ R+, θ = 1, (3.1.7)
then the boundary value problem (3.1.1), (3.1.2) has at least one solution.
Proof. Let R ≥ max2(Iαφ(1) + 2|y0| + |y1|,
(2c
Γ(α+1)
) 11−θ
and define a set U := y ∈ B : ∥y∥ ≤ R.Obviously U is nonempty closed convex subset of the Banach space B. Define an operator T : B → B as
T y(t) =∫ 1
0G(t, s)g(t, y(t))ds+ (y1 − y0)t+ y0. (3.1.8)
From (3.1.3) and (3.1.8), we have T y(t) = y(t). Thus in order to prove the existence of solutions to the
boundary value problem (3.1.1), (3.1.2), we have to prove that the operator T has a fixed points.
For the continuity of the operator T , consider a sequence of functions ynn∈N in B such that yn →y ∈ B as n→ ∞. For each t ∈ B, we have
|T yn(t)− T y(t)| ≤∫ 1
0|G(t, s)||g(s, yn(s))− g(s, y(s))|ds.
From the continuity of g, it follows that ∥T yn(t)− T y(t)∥ → 0 as n→ ∞. Hence, T is continuous.
Now to show that T maps U into itself, take y ∈ U and consider
|T y(t)| ≤∫ 1
0|G(t, s)||g(t, y(t))|ds+ |y1 − y0|t+ |y0|
≤∫ 1
0|G(t, s)|φ(s)ds+ cRθ
∫ 1
0|G(t, s)|ds+ 2|y0|+ |y1|
≤ Iαφ(1) + 2|y0|+ |y1|+cRθ
Γ(α+ 1)≤ R
2+R
2= R.
Hence, ∥T y(t)∥ ≤ R. Therefore, the set T (U) :=T y : y ∈ U
is bounded. In order to show that T (U)
is relatively compact, it remains to show that T (U) is equicontinuous.
Let L = maxt∈[0,1]
g(t, y(t)) : y ∈ U
. In view of (3.1.8) and Lemma 3.1.7, we obtain
∣∣∣ ddt(T y(t))
∣∣∣ ≤ ∫ 1
0
∣∣∣ ∂∂tG(t, s)
∣∣∣|g(s, y(s))|ds+ |y1|
≤ γL
Γ(α− 1)
∫ 1
0(1− s)α−2 + |y1| =
γL
Γ(α)+ |y1|.
For t1, t2 ∈ [0, 1] with t1 ≤ t2, using mean value theorem, we have∣∣∣T y(t2)− T y(t1)∣∣∣ = ∣∣∣[ d
dt(T y(t))
]t=ξ
(t2 − t1)∣∣∣ ≤ ( γL
Γ(α)+ |y1|
)(t2 − t1).
where ξ ∈ [t1, t2]. Hence, if for some δ > 0, (t2 − t1) < δ, then ∥T y(t2) − T y(t1)∥ < ε, where ε :=(γLΓ(α) + |y1|
)δ is independent of y, t1, t2. Therefore the set T (U) is equicontinuous. By the Arzela–Ascoli
theorem, T (U) is relatively compact. Hence, by the Schauder fixed point theorem, the boundary value
problem (3.1.1), (3.1.2) has at least one solution.
34
Theorem 3.1.4. Assume that there exists a continuous function φ : [0, 1] → R+, such that
|g(t, y)| ≤ φ(t) + c|y|, c ≤ Γ(α+ 1)
2, (3.1.9)
then the boundary value problem (3.1.1), (3.1.2) has at least one solution.
Proof. Let R ≥ min 2(Iαφ(1) + 2|y0|+ |y1|) and define the set U := y ∈ B : ∥y∥ ≤ R. Then, for y ∈ U ,
we have∣∣∣T y(t)∣∣∣ ≤ Iαφ(1) + Γ(α+1)
2 ≤ R. Thus T : U → U . The rest of the proof is similar to the proof
of the Theorem 3.1.3.
Remark 3.1.5. If φ(t) = 0 in (3.1.7) and (3.1.9), then we can choose R ≤(Γ(α+1)
2c
) 1θ−1 , θ = 1 in the
proof of the Theorem 3.1.3 and R > 0 in Theorem 3.1.4.
Theorem 3.1.6. Assume that there exists a constant k < Γ(α+ 1) such that
|g(t, y)− g(t, z)| ≤ k|y − z|, for each t ∈ [0, 1] and all y, z ∈ R. (3.1.10)
Then the boundary value problem (3.1.1), (3.1.2) has a unique solution.
Proof. Define U :=y ∈ B : ∥y∥ ≤ K
and using arguments similar to used in the Theorem 3.1.4, we
conclude that the operator T defined by (3.1.8) maps the bounded set U into itself. It remains to prove
that T is contraction. For y, z ∈ U , we have
∣∣T y(t)− T z(t)∣∣ ≤∫ 1
0|G(t, s)||g(s, y(s))− g(s, z(s))|ds
≤k∥y − z∥∫ 1
0|G(t, s)|ds ≤ k
Γ(α+ 1)∥y − z∥ ≤ q∥y − z∥,
where q = kΓ(α+1) < 1. Hence by contraction mapping principle the boundary value problem (3.1.1), (3.1.2)
has a unique solution.
3.2 Two–point boundary value problems (II)
In this section, we study the existence and uniqueness of solutions to the following fractional differential
equation
cDαy(t) = g(t, y(t), cDβy(t)), 1 < α ≤ 2, 0 ≤ β ≤ 1. (3.2.1)
subject to the boundary conditions
y(0) = y0, y(1) = y1. (3.2.2)
This problem generalizes the results studied in section 3.1 in the sense that the nonlinear function depends
on fractional derivative of y as well.
35
Lemma 3.2.1. Assume that g : [0, 1]×R×R → R is continuous. Then, a function y ∈ C[0, 1] is a solution
of the boundary value problem (3.2.1), (3.2.2) if and only if y(t) is a solution of the integral equation
y(t) = Iαg(t, y(t), cDβy(t))− tIαg(1, y(1), cDβy(1)) + (y1 − y0)t+ y0
=
∫ 1
0Gα(t, s)g(s, y(s),
cDβy(s))ds+ (y1 − y0)t+ y0.(3.2.3)
where G(t, s) is the Green function, given by
Gα(t, s) =
(t−s)α−1−t(1−s)α−1
Γ(α) , 0 ≤ s ≤ t,
− t(1−s)α−1
Γ(α) , t < s ≤ 1.(3.2.4)
The proof of the Lemma 3.2.1 is similar to the proof of Lemma 3.1.1. Following will be assumed to
establish sufficient conditions for the existence and uniqueness of solutions.
(A1) g : [0, 1]× R× R → R is continuous;
(A2) there exists a nonnegative function φ ∈ C[0, 1] such that |g(t, y, z)| ≤ φ(t) + µ1|y|ν1 + µ2|z|ν2 , where
µ1, µ2 ∈ R+ and 0 ≤ ν1, ν2 < 1;
(A3) |g(t, y, z)| ≤ µ1|y|ν1 + µ2|z|ν2 , where µ1, µ2 ∈ R+ and ν1, ν2 > 1;
(A4) there exists a constant k > 0 such that
|g(t, y, z)− g(t, y, z))| ≤ k(|y − y|+ |z − z|), for each t ∈ [0, 1] and all y, z, y, z ∈ R.
Consider the Banach space Bβ defined in Lemma 2.2.25. Define an operator A : Bβ → Bβ by
T y(t) =∫ 1
0Gα(t, s)g(s, y(s),
cDβy(s))ds+ (y1 − y0)t+ y0 (3.2.5)
In order to show that the boundary value problem (3.2.1), (3.2.2) has a solution, it is sufficient to prove
that the operator T has a fixed point.
For convenience, we define the following constants:
kα = Iα|φ(1)| + 2|y0| + |y1|, kα,β = σα,βIα−βφ(s) + |y1−y0|Γ(2−β) , where σα,β =
(1
Γ(α−β) +1
Γ(α)Γ(2−β)
)and
qα,β = 1Γ(α+1) +
σα,β
Γ(α−β+1) .
Theorem 3.2.2. Assume that (A1) and (A2) hold. Then the fractional boundary value problem (3.2.1),
(3.2.2) has at least one solution.
Proof. Continuity of T follows form the continuity of g. Choose
R ≥ max3(kα + kα,β), (3qα,βµ1)
11−ν1 , (3qα,βµ2)
11−ν2
and define a set M =
y ∈ Bβ : ∥y∥β ≤ R
. Then for an arbitrary y ∈ M, we have the following estimate.
|T y(t)| ≤∫ 1
0|Gα(t, s)|φ(s)ds+ (µ1R
ν1 + µ2Rν2)
∫ 1
0|Gα(t, s)|ds+ 2|y0|+ |y1|
≤ Iαφ(1) + 2|y0|+ |y1|+ (µ1Rν1 + µ2R
ν2)
∫ 1
0
(t− s)α−1
Γ(α)ds
≤ kα +1
Γ(α+ 1)(µ1R
ν1 + µ2Rν2).
36
In view of Lemma 2.2.2 and Lemma 2.2.5, we have
cDβ(T y)(t) = I1−β(cD(T y))(t)
= I1−β(Iα−1g(t, y(t), cDβy(t))− Iαg(1, y(1), cDβy(1)) + y1 − y0
)= Iα−βg(t, y(t), cDβy(t))−
(Iαg(1, y(1), cDβy(1))− y1 + y0
) t1−β
Γ(2− β)
=
∫ 1
0Gα,β(t, s)g(s, y(s),
cDβy(s))ds+(y1 − y0)
Γ(2− β)t1−β
where Gα,β is defined by
Gα,β =
(t−s)α−β−1
Γ(α−β) − t1−β(1−s)α−1
Γ(α)Γ(2−β) , 0 ≤ s ≤ t,
− t1−β(1−s)α−1
Γ(α)Γ(2−β) , t < s ≤ 1.(3.2.6)
For, s ≤ t, we have |Gα,β | ≤ (1−s)α−β−1
Γ(α−β) + (1−s)α−1
Γ(α)Γ(2−β) ≤(
1Γ(α−β) +
1Γ(α)Γ(2−β)
)(1− s)α−β−1 and for t < s,
we have |Gα,β | ≤ (1−s)α−1
Γ(α)Γ(2−β) ≤(1−s)α−β−1
Γ(α)Γ(2−β) . Thus, |Gα,β | ≤σα,β(1−s)α−β−1
Γ(α−β) . Hence, it follows that∣∣cDβ(T y)(t)∣∣ ≤ ∫ 1
0|Gα,β(t, s)||g(s, y(s), cDβy(s))|ds+ |y1 − y0|
Γ(2− β)
≤∫ 1
0|Gα,β(t, s)|φ(s)ds+
|y1 − y0|Γ(2− β)
+ (µ1Rν1 + µ2R
ν2)
∫ 1
0|Gα,β(t, s)|ds
≤ σα,βIα−βφ(s) +|y1 − y0|Γ(2− β)
+ (µ1Rν1 + µ2R
ν2)σα,βt
α−β
Γ(α− β + 1)
≤ kα,β + (µ1Rν1 + µ2R
ν2)σα,β
Γ(α− β + 1).
Therefore,
∥T y)(t)∥β ≤ kα + kα,β + qα,β(µ1Rν1 + µ2R
ν2) ≤ R
3+R
3+R
3= R,
which imply that T maps the bounded set M of the Banach space Bβ into itself.Now, we prove that the set T (M) is equicontinuous. Let Λ = max
t∈[0,1]
∣∣g(t, y(t), cDβy(t))∣∣ : y ∈ M
,
then for 0 ≤ t ≤ τ ≤ 1, we have
|T y(τ)− T y(t)|
≤∣∣Iαg(τ, y(τ), cDβy(τ))− Iαg(t, y(t), cDβy(t))
∣∣+ (∣∣Iαg(1, y(1), cDβy(1))∣∣+ |y1 − y0|
)(τ − t)
≤ Λ
(∫ t
0
(τ − s)α−1 − (t− s)α−1
Γ(α)ds+
∫ τ
t
(τ − s)α−1
Γ(α)ds
)+
(Λ
∫ 1
0
(1− s)α−1
Γ(α)ds+ |y1 − y0|
)(τ − t)
≤ Λ
(−(τ − t)α + τα − tα
αΓ(α)+
(τ − t)α
αΓ(α)
)+
(Λ
αΓ(α)+ |y1 − y0|
)(τ − t)
=Λ(τα − tα)
Γ(α+ 1)+
(Λ
Γ(α+ 1)+ |y1 − y0|
)(τ − t),
∣∣cDβ(T y)(τ)− cDβ(T y)(t)∣∣ ≤ ∣∣Iα−βg(τ, y(τ), cDβy(τ))− Iα−βg(t, y(t), cDβy(t))
∣∣+(∣∣Iαg(1, y(1), cDβy(1))
∣∣+ |y1 − y0|) (τ1−β − t1−β)
Γ(2− β)
≤Λ(τα−β − tα−β
)Γ(α− β + 1)
+
(Λ
Γ(2− β)Γ(α− β + 1)+
|y1 − y0|Γ(2− β)
)(τ1−β − t1−β)
Γ(2− β).
37
Hence, it follows that ∥T y(τ)−T y(t)∥ → 0, as τ − t→ 0. By the Arzela–Ascoli theorem, T : M → M is
compact operator. Thus by Theorem 2.3.3, the boundary value problem (3.2.1), (3.2.2) has a solution.
Theorem 3.2.3. Assume that (A1) and (A3) hold. Then the boundary value problem (3.2.1), (3.2.2) has
at least one solution.
Remark 3.2.4. The proof of the Theorem 3.2.3 is similar to the proof of the Theorem 3.2.2. The bounded
convex subset of Bβ can be defined by chosing
0 < R ≤ min( 1
2qα,βµ1
) 1ν1−1
,( 1
2qα,βµ2
) 1ν2−1
.
Remark 3.2.5. Theorem 3.2.2 and Theorem 3.2.3 do not cover the cases ν1 = ν2 = 1, ν1 = 1, ν2 = 1, (or
ν1 = 1, ν2 = 1). We consider these cases separately.
Theorem 3.2.6. Assume that A1 holds. Furthermore, there exists a nonnegative function φ(t) and
µ1, µ2 ≥ 0 such that
|g(t, y, z)| ≤ φ(t) + µ1|y|+ µ2|z|, µ1 + µ2 ≤1
2qα,β, or
|g(t, y, z)| ≤ φ(t) + µ1|y|+ µ2|z|ν , 0 < ν < 1, µ1 ≤1
3qα,β.
Then, the boundary value problem (3.2.1), (3.2.2) has a solution.
The proof is similar to the proof of Theorem 3.2.2. Therefore, it is omitted.
Theorem 3.2.7. Assume that A1 is satisfied and there exists a constant k <(
1Γ(α−β+1) +
1Γ(2−β)Γ(α+1)
)such that
|g(t, y, z)− g(t, y, z)| ≤ k(|y − y|+ |z − z|), for each t ∈ [0, 1] and all y, y, z, z ∈ R, (3.2.7)
then there exists a unique solution of the boundary value problem (3.2.1), (3.2.2).
Proof. From (3.2.5), we have,∣∣T y(t)− T y(t)∣∣ ≤ ∫ 1
0|G(t, s)||g(s, y(s), cDαy(t))− g(s, y(s), cDαy(t))|ds
≤ ∥y − y∥kΓ(α)
(∫ t
0(t− s)α−1ds+ t
∫ 1
0(1− s)α−1ds
)=
∥y − y∥kΓ(α+ 1)
(tα + t) ≤ 2k
Γ(α+ 1)||y − y||,
∣∣cDα(T y)(t)− cDα(T y)(t)∣∣ ≤ ∫ 1
0|G(t, s)||g(s, y(s), cDαy(t))− g(s, y(s), cDαy(t))|ds
≤ ∥y − y∥k(∫ t
0
(t− s)α−β−1
Γ(α− β)ds+
t1−β
Γ(2− β)
∫ 1
0
(1− s)α−1
Γ(α)ds
)= ∥y − y∥k
(tα−β
Γ(α− β + 1)+
t1−β
Γ(2− β)Γ(α+ 1)
)≤ ∥y − y∥k
(1
Γ(α− β + 1)+
1
Γ(2− β)Γ(α+ 1)
).
38
Thus, we have ∥∥T y(t)− T y(t)∥∥ ≤ ρ∥y − y∥,
where, ρ = k(
1Γ(α−β+1) +
1Γ(2−β)Γ(α+1)
)< 1. Therefore, by the contraction mapping principle, the bound-
ary value problem (3.2.1), (3.2.2) has a unique solution.
3.3 Three–point boundary value problems (I)
Multi–point boundary value problems arise in different areas of physics and mathematics. The most
commonly quoted example in this respect is modeling the vibration of a guy-wire with n parts of different
densities, but having uniform cross-section. Bitsadze and Samarski [22] initiated the study of multi–point
boundary value problems for integer order differential equations. Later II’in and Moreover [58,59] played
the leading role for the development of the existence theory of such problems. Since then, the multi–
point boundary value problems have been investigated by several researchers including [51, 97, 98, 100].
In contrast, the multi–point boundary value problems for fractional differential equations have received
attention quite recently. For details, we refer to [4,19,44,123,153]. In this section, we study existence and
uniqueness of solutions to a class of nonlinear fractional differential equations of the type [87]
cDαy(t) = g(t, y(t), cDβy(t)), t ∈ [0, a], a > 0, (3.3.1)
y(0) = µy(η), y(a) = νy(η), (3.3.2)
where 1 < α < 2, 0 < β ≤ 1, µ, ν ∈ R, η ∈ (0, a), µη(1− ν) + (1− µ)(a− νη) = 0.
We need the following fundamental lemma for our main result.
Lemma 3.3.1. Let h ∈ C[0, a] and 1 < α ≤ 2. Then the linear problem
cDαy(t) = h(t), 0 < t < 1, t ∈ [0, a], (3.3.3)
y(0) = µy(η), y(a) = νy(η), (3.3.4)
has solution
y(t) = Iαh(t) + 1
∆(µa+ (ν − µ)t)Iαh(η)− (µη + (1− µ)t)Iαh(a) =
∫ a
0G(t, s)h(s)ds, (3.3.5)
where ∆ = µη(1− ν) + (1− µ)(a− νη) = 0 and G(t, s) is the Green function
G(t, s) =
(t−s)α−1
Γ(α) + (µa+(ν−µ)t)(η−s)α−1
∆Γ(α) − (µη+(1−µ)t)(a−s)α−1
∆Γ(α) , if 0 ≤ s < t, η ≥ s,
(µa+(ν−µ)t)(η−s)α−1
∆Γ(α) − (µη+(1−µ)t)(a−s)α−1
∆Γ(α) , if 0 ≤ t ≤ s ≤ η,
(t−s)α−1
Γ(α) − (µη+(1−µ)t)(a−s)α−1
∆Γ(α) , if η ≤ s ≤ t ≤ 1,
− (µη+(1−µ)t)(a−s)α−1
∆Γ(α) , if 0 ≤ t ≤ s, s ≥ η .
(3.3.6)
Proof. In view of Theorem 2.2.24, the general solution of (3.3.3), (3.3.4) is
y(t) = Iαh(t) + c0 + c1t, (3.3.7)
39
where c0, c1 ∈ R. Using (3.3.4) and (3.3.7), we have
(1− µ)c0 − µηc1 = µIαh(η), (1− ν)c0 + (a− νη)c1 = νIαh(η)− Iαh(a),
which implies that
c0 =µ
∆(aIαh(η)− ηIαh(a)), c1 =
1
∆((ν − µ)Iαh(η) + (µ− 1)Iαh(a).
Therefore, the solution of (3.3.3), (3.3.4) is
y(t) = Iαh(t) + 1
∆(µa+ (ν − µ)t)Iαh(η)− (µη + (1− µ)t)Iαh(a). (3.3.8)
Now, for t ≤ η, equation (3.3.1) can be written as
y(t) =
∫ t
0
((t− s)α−1
Γ(α)+
(µa+ (ν − µ)t)(η − s)α−1
∆Γ(α)− (µη + (1− µ)t)(a− s)α−1
∆Γ(α)
)h(s)ds
+
∫ η
t
((t− s)α−1
Γ(α)+
(µa+ (ν − µ)t)(η − s)α−1
∆Γ(α)− (µη + (1− µ)t)(a− s)α−1
∆Γ(α)
)h(s)ds
−∫ a
η
(µη + (1− µ)t)(a− s)α−1
∆Γ(α)h(s)ds =
∫ a
0
G(t, s)h(s)ds.
For t > η, we have
y(t) =
∫ η
0
((t− s)α−1
Γ(α)+
(µa+ (ν − µ)t)(η − s)α−1
∆Γ(α)− (µη + (1− µ)t)(a− s)α−1
∆Γ(α)
)h(s)ds
+
∫ t
η
((t− s)α−1
Γ(α)− (µη + (1− µ)t)(a− s)α−1
∆Γ(α)
)h(s)ds
−∫ a
t
(µη + (1− µ)t)(a− s)α−1
∆Γ(α)h(s)ds =
∫ a
0
G(t, s)h(s)ds.
Define an operator T : Bβ → Bβ by
T y(t) =∫ a
0G(t, s)g(s, y(s), cDβy(s))ds. (3.3.9)
In order to find the solutions of the boundary value problem (3.3.1), (3.3.2), we need to find the fixed
points of the operator T .
For convenience, define the constants:
l∗ = max(∫ a
0 | ∂∂tG(t, s)φ(s)|ds), l = max
(∫ a0 |G(t, s)φ(s)|ds
), Nα = |ν − µ|ηα + |1− µ|aα,
Mα,β =[(∆ + |µ|η)aα + |µ|aηα + aβ−1
Γ(2−β)(∆aα−1 +Nα)
], kαµ,ν = 2|µ|+|ν|
∆αΓ(α) , kαµ = 1+|µ|
∆αΓ(α) and kα = aα
∆αΓ(α) .
3.3.1 Existence of solutions
Theorem 3.3.2. Assume that (A1), (A2) or (A1), (A3) are satisfied. Then the boundary value problem
(3.3.1), (3.3.2) has at least one solution.
Proof. Suppose (A2) holds. Choose a constant R ≥ max3(l + l∗a1−β
Γ(2−β)), (3Mα,βµ1)1
1−ν1 , (3Mα,βµ2)1
1−ν2
and define W = y ∈ Bβ : ∥y∥β ≤ R. W is bounded convex subset of the Banach space Bβ . For an
40
arbitrary y ∈ W , using assumption (A2), we have
|T y(t)| =∣∣∣∣∫ a
0
G(t, s)g(s, y(s), cDβy(s))ds
∣∣∣∣ ≤ ∫ a
0
|G(t, s)φ(s)|ds+ (µ1Rν1 + µ2R
ν2)
[∫ t
0
(t− s)α−1
Γ(α)ds
+1
∆(|µ|a+ |ν − µ|t)
∫ η
0
(η − s)α−1
Γ(α)ds+
1
∆(|µ|η + |1− µ|t)
∫ a
0
(a− s)α−1
Γ(α)ds
]≤l + (µ1R
ν1 + µ2Rν2)
(tα
αΓ(α)+
1
∆(|µ|a+ |ν − µ|t) ηα
αΓ(α)+
1
∆(|µ|η + |1− µ|t) aα
Γ(α)
),
which implies that |T y(t)| ≤ l + (µ1Rν1+µ2R
ν2 )α∆Γ(α) (∆aα + |µ|(aηα + ηaα) +NαT ). Also,
|(T y)′(t)| ≤∫ a
0
| ∂∂tG(t, s)||g(s, y(s), cDβy(s))|ds ≤
∫ a
0
| ∂∂tG(t, s)φ(s)|ds+ (µ1R
ν1 + µ2Rν2)
(∫ t
0
(t− s)α−2
Γ(α− 1)ds
+|ν − µ|
∆
∫ η
0
(η − s)α−1
Γ(α)ds+
|1− µ|∆
∫ a
0
(a− s)α−1
Γ(α)ds
)≤l∗ + (µ1R
ν1 + µ2Rν2)
(tα−1
(α− 1)Γ(α− 1)+
Nα
α∆Γ(α)
).
Hence,
|DβT (y)| =∣∣∣∣ 1
Γ(1− β)
∫ t
0
(t− s)−β(T y)′(s)ds∣∣∣∣ ≤ 1
Γ(1− β)
∫ t
0
(t− s)−β |(T u)′(s)|ds
≤ l∗t1−β
(1− β)Γ(1− β)+
(µ1Rν1 + µ2R
ν2)t1−β
(1− β)Γ(1− β)
(tα−1
Γ(α)+
|ν − µ|ηα
∆αΓ(α)+
|1− µ|aα
∆αΓ(α)
),
which gives |DβT (y)| ≤ l∗a1−β
Γ(2−β) +(µ1R
ν1+µ2Rν2 )
α∆Γ(α)Γ(2−β)
(∆aα−1 +Nα
)a1−β . Finally, we have,
∥T y(t)∥β ≤l + l∗a1−β
Γ(2− β)+µ1R
ν1 + µ2Rν2
∆αΓα
[(∆ + |µ|η)aα + |µ|aηα +
aβ−1
Γ(2− β)(∆aα−1 +Nα)
]≤R
3+ (µ1R
ν1 + µ2Rν2)Mα,β ≤ R
3+R
3+R
3= R.
Thus, T : W → W. The continuity of T follows from the continuity of f and G.Now, if (A3) holds, we choose 0 < R ≤ min
3(l + l∗a1−β
Γ(2−β) ), (1
3Mα,βµ1)
11−ν1 , ( 1
3Mα,βµ1)
11−ν1
and by the same
process as above, we obtain ∥T y(t)∥β ≤ R3 + (µ1R
ν1 + µ2Rν2)Mα,β ≤ R, which implies that T : W → W.
Now we show that T is completely continuous operator. Let K = max|g(t, y(t), cDβy(t))| : t ∈ [0, a], y ∈ W. Fort1, t2 ∈ [0, a] such that t1 < t2, we have
|T y(t1)− T y(t2)| =∣∣∣∣∫ a
0
(G(t1, s)−G(t2, s))g(s, y(s),cDβy(s))ds
∣∣∣∣≤K
[∫ t1
0
|G(t1, s)−G(t2, s)|ds+∫ t2
t1
|G(t1, s)−G(t2, s)|ds +
∫ a
t2
|G(t1, s)−G(t2, s)|ds],
which in view of (3.3.6) implies that
|T y(t1)− T y(t2)| ≤ K[ ∫ t2
0
(t2 − s)α−1
Γ(α)ds−
∫ t1
0
(t1 − s)α−1
Γ(α)ds
+|ν − µ|(t2 − t1)
∆Γ(α)
∫ η
0
(η − s)α−1ds+|1− µ|(t2 − t1)
∆Γ(α)
∫ a
0
(η − s)α−1ds]
≤ K
α∆Γ(α)[∆(tα2 − tα1 ) + (|ν − µ|ηα + |1− µ|aα)(t2 − t1)]
=K
αpΓ(α)
[∆(tα2 − tα1 ) +Nα(t2 − t1)
].
41
Also,
|cDβT (y)(t1)− cDβT (y)(t2)| =1
Γ(1− β)
∣∣∣∣∫ t2
0
(t2 − s)−β(T y)′(s)ds−∫ t1
0
(t1 − s)−β(T y)′(s)ds∣∣∣∣
≤ 1
Γ(1− β)
∣∣∣∣∫ t2
0
(t2 − s)−β(T y)′(s)ds−∫ t1
0
(t2 − s)−β(T y)′(s)ds∣∣∣∣
+1
Γ(1− β)
∣∣∣∣∫ t1
0
(t2 − s)−β(T y)′(s)ds−∫ t1
0
(t1 − s)−β(T y)′(s)ds∣∣∣∣
≤ 1
Γ(1− β)
(∫ t2
t1
(t2 − s)−β |(T y)′(s)|ds+∫ t1
0
((t2 − s)−β − (t1 − s)−β)|(T y)′(s)|ds)
≤ 1
Γ(1− β)
[∫ t2
t1
(t2 − s)−β
(∫ a
0
| ∂∂sG(s, t)||g(z, y(t), cDβy(t))|dt
)ds
+
∫ t1
0
((t2 − s)−β − (t1 − s)−β)
(∫ a
0
| ∂∂sG(s, t)||g(t, y(t), cDβy(t))|dt
)ds
]≤ K
α∆Γ(α)Γ(1− β)(∆aα−1 +Nα)
[∫ t2
t1
(t2 − s)−βds+
∫ t1
0
((t2 − s)−β − (t1 − s)−β)ds
]≤K(∆aα−1 +Nα)
α∆Γ(α)Γ(2− β)(t1−β
2 − t1−β1 ).
Hence, ∥T y(t1) − T y(t2)∥β ≤ Kα∆Γ(α)
[∆(tα2 − tα1 ) +Nα(t2 − t1) +
∆aα−1+Nα
Γ(2−β) (t1−β2 − t1−β
1 )], which implies that
∥T y(t1) − T y(t2)∥β → 0 as t1 → t2. Thus, by Arzela–Ascoli theorem, it follows that T is compact operator.Therefore, using the Schauder fixed point theorem, we conclude that T has at least one fixed point which is solutionof the boundary value problem (3.3.1), (3.3.2).
Now, if (A3) holds then, we can choose 0 < R ≤ max
(1
3Mα,βµ1
) 1ν1−1
,(
13Mα,βµ2
) 1ν2−1
. Using the same arguments
as above, we conclude that the problem (3.3.1), (3.3.2) has at least one solution.
Example 3.3.3. Consider the boundary value problem,
cDαy(t) =Γ(α+ 1)
64√π
et +Γ(α+ 1)
120e−κt(y(t))ν1 +
cos te−πt
187(cDβy(t))ν2 , t ∈ [0, 2], (3.3.10)
y(0) =3
7y(32
), y(1) =
9
11y(32
), (3.3.11)
where 1 < α < 2, 0 < β < 1 and κ > 0. Choose µ = 37 , ν = 9
11 , η = 32 , φ(t) =
Γ(α+1)64
√πet, µ1 = Γ(α+1)
120 ,
µ2 =1
187 and g(t, y, z) = Γ(α+1)64
√πet + Γ(α+1)
120 e−κt(y(t))ν1 + e−πt cos t187 (z(t))ν2 , then ∆ = 5847
1078 and for t ∈ [0, 2],
we have |g(t, y, z)| ≤ φ(t) + µ1|y|ν1 + µ1|z|ν2 .For 0 < ν1, ν2 < 1, condition (A2) of the Theorem 3.3.2 is satisfied and for ν1, ν2 > 1, condition (A3) of
the Theorem 3.3.2 is satisfied. Therefore the boundary value problem (3.3.10), (3.3.11) has a solution.
3.3.2 Uniqueness of solution
Theorem 3.3.4. Assume (A1), (A4) are satisfied: Furthermore, if there exists a constant
k <
(kαµ,ν
(aα +
ηαa1−β
Γ(2− β)
)+ (kαµa
α+1 + kα)
(1 +
a−β
Γ(2− β)
))−1
,
then the boundary value problem (3.3.1), (3.3.2) has a unique solution.
42
Proof. From (3.3.9), we have following estimates
|T (y)(t)− T (y)(t)| ≤∫ a
0
|G(t, s)||g(s, y(s), cDβy(s))− g(s, y(s), cDβy(s))|ds
≤k∥y − y∥β∆Γ(α)
[∫ t
0
(t− s)α−1ds+ (|µ|a+ |ν − µ|t)∫ η
0
(η − s)α−1ds+ (|µ|η + |1− µ|t)∫ a
0
(a− s)α−1ds
]≤k∥y − y∥β
∆αΓ(α)(tα + (|µ|a+ |ν − µ|t)ηα + (|µ|η + |1− µ|t)aα)
≤k∥y − y∥β(
aα
∆αΓ(α)+
(2|µ|+ |ν|)ηaα
∆αΓ(α)+
(1 + |µ|)aα+1
∆αΓ(α)
)= k∥y − y∥β
(kα + kαµ,νηa
α + kαµaα+1
),
|cDβ(T u)(t)− cDβ(T y)(t)| =∣∣∣∣ 1
Γ(1− β)
∫ t
0
(t− s)−β((T y)′(s)− (Ay)′(s))ds
∣∣∣∣≤ 1
Γ(1− β)
∫ t
0
(t− s)−β
(∫ 1
0
∣∣∣∣ ∂∂sG(s, z)∣∣∣∣ |g(z, y(z), cDβy(z)− g(z, y(z), cDβy(z)|dz
)ds
≤k∥y − y∥βΓ(1− β)
∫ t
0
(t− s)−β
(∫ 1
0
∣∣∣∣ ∂∂sG(s, z)∣∣∣∣ dz) ds.
Using equation (3.3.6), we obtain∫ 1
0
| ∂∂tG(t, s)|ds ≤
∫ t
0
(t− s)α−2
Γ(α− 1)ds+
|ν − µ|∆Γ(α)
∫ η
0
(η − s)α−1ds+|1− µ|∆Γ(α)
∫ a
0
(a− s)α−1ds
≤ tα−1
αΓ(α)+
|ν − µ|ηα + |1− µ|aα
∆αΓ(α)≤ aα−1
αΓ(α)+
|ν − µ|ηα + |1− µ|aα
∆αΓ(α).
Consequently,
|cDβ(T y)(t)− cDβ(T y)(t)| ≤ k∥y − y∥βt1−β
(1− β)Γ(1− β)
(aα−1
αΓ(α)+
|ν − µ|ηα + |1− µ|aα
∆αΓ(α)
)≤ k∥y − y∥βa1−β
Γ(2− β)
(aα−1
∆αΓ(α)+
(2|µ|+ |ν|)ηα
∆αΓ(α)+
(1 + |µ|)aα
∆αΓ(α)
)=k∥y − y∥βΓ(2− β)
(kαa−β + kαµ,νη
αa1−β + kαµaα−β+1
).
Hence, it follows that ∥T y − T y∥β ≤ q∥y − y∥β , where q = k(kαµ,ν(a
α + ηαa1−β
Γ(2−β) ) + (kαµaα+1 + kα)(1 + a−β
Γ(2−β) )).
Obviously, q < 1. Thus, by the contraction mapping principle guarantees the unique solution of the boundary valueproblem (3.3.1), (3.3.2) has a unique solution.
Example 3.3.5. Consider the following fractional differential equation,
cD32 y(t)
(24
√π + e−πt)(1 + |y|+ |cD
12 y(t)|
)= e−πt(|y|+ |cD
12 y(t)|), t ∈ [0, 1], (3.3.12)
y(0) =5
7y(
1
4), y(1) =
9
7y(
1
4). (3.3.13)
Set g(t, y, z) = e−πt(y(t)+z(t))(24
√π+e−πt)(1+y(t)+z(t))
, t ∈ [0, 1], y, z ∈ [0,∞). For t ∈ [0, 1] and y, y, v, v ∈ [0,∞), we have
|g(t, y, z)− g(t, y, v)| = e−πt
(24√π + e−πt)
∣∣∣∣ y(t) + z(t)
1 + y(t) + z(t)− y(t) + z(t)
1 + y(t) + z(t)
∣∣∣∣≤ e−πt(|y(t)− y(t)|+ |z(t)− z(t)|)
24√π + e−ct
≤ 1
24√π(|y(t)− y(t)|+ |z(t)− z(t)|).
For µ = 57 , ν = 9
7 , η = 14 , α = 3
2 and β = 12 , we have ∆ = 3, kαµ, ν = 76
63√π, kαµ = 16
3√π
and kα = 49√π.
Therefore q = 102261π < 1. Hence by Theorem 3.3.4 the fractional boundary value problem (3.3.12), (3.3.13)
has a unique solution.
43
3.4 Three–point boundary value problems (II)
In this section we study existence and uniqueness of solutions to a class of nonlinear boundary value
problems for fractional differential equations with three–point boundary conditions involving standard
Riemann–Liouville fractional derivative. A particular focus concerns the nonlinear term satisfying the
Caratheodory conditions in Lp[0, 1]. The studies dealing with existence of solutions for boundary value
problems using Caratheodory conditions in Lp[0, 1] space are well studied in [51,74,98,100] and references
therein.
N. Kosmatov [74] established sufficient conditions for the existence of solutions for the following bound-
ary value problem,
Dαy(t) = f(t, y(t), y′(t)), t ∈ [0, 1], y(0) = 0, y(1) = 0, (3.4.1)
where Dα is the Riemann–Liouville fractional derivative. We consider a more general boundary value
problem [90],
Dαy(t) = g(t, y(t),Dβy(t)), t ∈ [0, 1], (3.4.2)
y(0) = 0, Dβy(1) = γDβy(η), (3.4.3)
where, 1 < α ≤ 2, 0 < β ≤ 1, α − β > 1, ∆α,β := (1 − γηα−β−1) > 0. The nonlinear function g and the
boundary condition involves the fractional derivative of an unknown function. We study the existence as
well as uniqueness of the solutions by imposing some growth conditions on g.
Lemma 3.4.1. Let h ∈ C[0, 1], 1 < α ≤ 2, 0 < β ≤ 1, α− β > 1 and ∆α,β > 0. Then the linear problem
Dαy(t) = h(t), t ∈ [0, 1], (3.4.4)
y(0) = 0, Dβy(1) = γDβy(η), (3.4.5)
has solution
y(t) =Iαh(t) + Γ(α− β)tα−1
∆α,βΓ(α)
[γIα−βh(η)− Iα−βh(1)
]=
∫ 1
0G(t, s)h(s)ds, (3.4.6)
where
G(t, s) =
(t−s)α−1
Γ(α) + [γ(η−s)α−β−1−(1−s)α−β−1]tα−1
∆α,βΓ(α), s ≤ t, η ≥ s,
[γ(η−s)α−β−1−(1−s)α−β−1]tα−1
∆α,βΓ(α), t ≤ s ≤ η ≤ 1,
(t−s)α−1
Γ(α) − tα−1(1−s)α−β−1
∆α,βΓ(α), η ≤ s ≤ t ≤ 1,
− tα−1(1−s)α−β−1
∆α,βΓ(α), s ≥ t, η ≤ s,
(3.4.7)
is the Green’s function associated with the problem (3.4.4), (3.4.5).
Proof. From Theorem 2.2.14 and equation (3.4.4), we have
y(t) = Iαh(t) + c0tα−1 + c1t
α−2. (3.4.8)
44
Using boundary condition y(0) = 0 in (3.4.8) we obtain c1 = 0 and in view of Lemma 2.2.8 and Lemma
2.2.12, we have, c0 =Γ(α−β)
Γ(α)(1−γηα−β−1)
[γIα−βh(η)−Iα−βh(1)
]. Therefore the unique solution of the bound-
ary value problem (3.4.3), (3.4.4) is given by
y(t) = Iαh(t) + Γ(α− β)tα−1
∆α,βΓ(α)
[γIα−βh(η)− Iα−βh(1)
]. (3.4.9)
For t ≤ η, using (3.4.9) we have the following estimates
y(t) =
∫ 1
0
(t− s)α−1
Γ(α)h(s)ds+
∫ t
0
γtα−1(η − s)α−β−1
∆α,βΓ(α)h(s)ds+
∫ η
t
γtα−1(η − s)α−β−1
∆α,βΓ(α)h(s)ds
−∫ t
0
tα−1(1− s)α−β−1
∆α,βΓ(α)h(s)ds−
∫ η
t
tα−1(1− s)α−β−1
∆α,βΓ(α)h(s)ds−
∫ 1
η
tα−1(1− s)α−β−1
∆α,βΓ(α)h(s)ds
=
∫ t
0
[ (t− s)α−1
Γ(α)+
[γ(η − s)α−β−1 − (1− s)α−β−1]tα−1
∆α,βΓ(α)
]h(s)ds
+
∫ η
t
[γ(η − s)α−β−1 − (1− s)α−β−1]tα−1
∆α,βΓ(α)h(s)ds−
∫ 1
η
tα−1(1− s)α−β−1
∆α,βΓ(α)h(s)ds =
∫ 1
0
G(t, s)h(s)ds.
For η < t, we have
y(t) =
∫ η
0
(t− s)α−1
Γ(α)h(s)ds+
∫ t
η
(t− s)α−1
Γ(α)h(s)ds+
∫ η
0
γtα−1(η − s)α−β−1
∆α,βΓ(α)h(s)ds
−∫ η
0
tα−1(1− s)α−β−1
∆α,βΓ(α)h(s)ds−
∫ t
η
tα−1(1− s)α−β−1
∆α,βΓ(α)h(s)ds−
∫ 1
t
tα−1(1− s)α−β−1
∆α,βΓ(α)h(s)ds
=
∫ η
0
[ (t− s)α−1
Γ(α)+
[γ(η − s)α−β−1 − (1− s)α−β−1]tα−1
∆α,βΓ(α)
]h(s)ds
+
∫ t
η
[ (t− s)α−1
Γ(α)− (1− s)α−β−1tα−1
∆α,βΓ(α)
]h(s)ds−
∫ 1
t
tα−1(1− s)α−β−1
∆α,βΓ(α)h(s)ds =
∫ 1
0
G(t, s)h(s)ds.
Consider the Banach space Bβ and define an operator T : Bβ → Bβ by
T y(t) =Iαg(s, y(s),Dβy(t)) +Γ(α− β)tα−1
∆α,βΓ(α)
[γIα−βg(η, y(η),Dβy(η))− Iα−βg(1, y(1),Dβy(1))
]=
∫ 1
0
G(t, s)g(s, y(s),Dβy(s))ds.
(3.4.10)
The fixed points of T are the solutions of the boundary value problem (3.4.2), (3.4.3).
Define Ω1,α,β,γ,η,q =1
((α−1)q+1)1q Γ(α)
+ γηα−β−1+1
q +1
((α−β−1)q+1)1q ∆α,βΓ(α)
and Ω2,α,β,γ,η,q =2−γηα−β−1(1−η
1q )
((α−β−1)q+1)1q ∆α,βΓ(α−β)
.
In the following, we prove that the operator T is completely continuous.
Lemma 3.4.2. Assume that g satisfies the Carathéodory conditions. Then T : Bβ → Bβ is completely
continuous.
Proof. Let M ⊂ Bβ be bounded i.e. there exists a positive number r > 0 such that ∥y∥β ≤ r for ally ∈ M . Since g is Carathéodory function. Therefore, there exists a nonnegative function ψr such that|g(t, y(t),Dβy(t))| ≤ ψr(t), for all y ∈ M and a.e. t ∈ [0, 1]. First we prove that T (M) ⊂ Bα is bounded.
45
For t ∈ [0, 1], y ∈M using the Hölder’s inequality, we have
|y(t)| ≤∫ t
0
(t− s)α−1
Γ(α)|ψr(s)|ds+
tα−1γ
∆α,βΓ(α)
[ ∫ η
0
(η − s)α−β−1|ψr(s)|ds+∫ 1
0
(1− s)α−β−1|ψr(s)|ds
≤ 1
Γ(α)
(∫ t
0
(t− s)(α−1)qds
) 1q
∥ψ∥p]+
1
∆α,βΓ(α)
[γ
(∫ η
0
(η − s)(α−β−1)qds
) 1q
+
(∫ 1
0
(1− s)(β−α−1)qds
) 1q
]∥ψr∥p ≤ tα−1+ 1
q
((α− 1)q + 1)1q Γ(α)
∥ψ∥p +γηα−β−1+ 1
q + 1
((α− β − 1)q + 1)1q∆α,βΓ(α)
∥ψr∥p
≤Ω1,α,β,γ,η,q∥ψr∥p,
Therefore
∥y(t)∥ ≤ Ω1,α,β,γ,η,q∥ψr∥p. (3.4.11)
Now, in view of Lemma 2.2.8 and Lemma 2.2.12, we have
|Dβy(t)| =∣∣∣Iα−βg(s, y(s),Dβy(t)) +
tα−β−1
∆α,β
[γIα−βg(η, y(η),Dβy(η))− Iα−βg(1, y(1),Dβy(1))
]∣∣∣≤∫ t
0
(t− s)α−β−1
Γ(α− β)|ψr(s)|ds+
tα−β−1
∆α,βΓ(α− β)
[γ
∫ η
0
(η − s)α−β−1|ψr(s)|ds+∫ 1
0
(1− s)α−β−1|ψr(s)|ds]
≤(∫ t
0
(t− s)(α−β−1)q
Γ(α− β)ds
) 1q
∥ψ∥p +1
∆α,βγ
(∫ t
0
((η − s)(α−β−1)
Γ(α− β)
)q
ds
) 1q
∥ψr∥p
+1
∆α,β
(∫ t
0
((1− s)(α−β−1)
Γ(α− β)
)q
ds
) 1q
∥ψr∥p
≤ tα−β−1+ 1q
((α− β − 1)q + 1)1q Γ(α− β)
∥ψ∥p +γηα−β−1+ 1
q + 1
((α− β − 1)q + 1)1q ∆α,βΓ(α− β)
∥ψr∥p
≤ 2− γηα−β−1(1− η1q )
((α− β − 1)q + 1)1q ∆α,βΓ(α− β)
∥ψr∥p = Ω2,α,β,γ,η,q∥ψr∥p,
Therefore
∥Dβy∥β ≤ Ω2,α,β,γ,η,q∥ψr∥p. (3.4.12)
Hence, from (3.4.11) and (3.4.12), we have
∥y∥β ≤ (Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q)∥ψr∥p. (3.4.13)
Next we prove that the operator T is equicontinuous. Let τ, t ∈ [0, 1], τ < t. For any y ∈M ,
|T y(t)− T y(τ)| =∣∣∣Iαh(t)− Iαh(τ) +
Γ(α− β)(tα−1 − τα−1)
∆α,βΓ(α)
[γIα−βh(η)− Iα−βh(1)
]≤ 1
Γ(α)
[∫ τ
0
[(t− s)α−1 − (τ − s)α−1]|ψr(t)|ds+∫ t
τ
(t− s)α−1|ψr(t)|ds
+tα−1 − τα−1
1− γηα−β−1
∣∣∣γ ∫ η
0
(η − s)α−β−1|ψr(t)|ds−∫ 1
0
(1− s)α−β−1|ψr(t)|ds∣∣∣]
≤( (t− τ)(α−1)+ 1
q + tα−1+ 1q − τα−1+ 1
q
((α− 1)q + 1)1qΓ(α)
+(tα−1 − τα−1)γηα−β−1+ 1
q
((α− β − 1)q + 1)1q∆α,βΓ(α)
)∥ψr∥p
<(t− τ)(α−1)+ 1
q + tα−1+ 1q − τα−1+ 1
q + (tα−1 − τα−1)γηα−β−1+ 1q
((α− β − 1)q + 1)1q∆α,βΓ(α)
∥ψr∥p
46
|Dβ(T u)(t)−Dβ(T y)(τ)|(α− β)[ ∫ τ
0
[(t− s)α−β−1 − (τ − s)α−β−1]|ψr(t)|ds+∫ t
τ
(t− s)α−β−1|ψr(t)|ds
+tα−β−1 − τα−β−1
1− γηα−β−1
∣∣∣γ ∫ η
0
(η − s)α−β−1|ψr(t)|ds−∫ 1
0
(1− s)α−β−1|ψr(t)|ds∣∣∣]
≤( (t− τ)α−β−1+ 1
q + tα−β−1+ 1q − τα−β−1+ 1
q
((α− β − 1)q + 1)1qΓ(α− β)
+(tα−β−1 − τα−β−1)γηα−β−1+ 1
q
((α− β − 1)q + 1)1q ∆α,βΓ(α− β)
)∥ψr∥p
<(t− τ)α−β−1+ 1
q + tα−β−1+ 1q − τα−β−1+ 1
q + (tα−β−1 − τα−β−1)γηα−β−1+ 1q
((α− β − 1)q + 1)1q∆α,βΓ(α− β)
∥ψr∥p
Now, it is obvious that ∥T y(t) − T y(τ)∥β → 0 as τ → t. We conclude that T (M) is an equicontinuous
set. Obviously it is uniformly bounded since T (M) ⊂ Bβ . Thus, by Arzela–Ascoli theorem we conclude
that T : Bβ → Bβ is completely continuous.
Corollary 3.4.3. Assume that h ∈ Lp[0, 1], then the solution y(t) of the boundary value problem (3.4.4),
(3.4.5) satisfies
∥y∥β ≤(Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)∥h∥p.
3.4.1 Existence of solutions
Theorem 3.4.4. Assume that the function g : [0, 1]×R2 → R satisfies the Carathéodory conditions B.0.8
and there exist functions a(t), b(t), φ(t) ∈ Lp[0, 1] such that
|g(t, u, v)| ≤ a(t)|y|+ b(t)|z|+ φ(t), (3.4.14)
for a.e. t ∈ [0, 1] and y, z ∈ R.Then boundary value problem (3.4.2), (3.4.3) has at least one solution
y ∈ Bβ provided that (Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)(∥a∥p + ∥b∥p∥
)< 1. (3.4.15)
Proof. By Lemma 3.4.2, the operator T is completely continuous. We apply Theorem 2.3.4 to obtain the
existence of at least one solution for boundary value problem (3.4.2), (3.4.3) in B. Now we prove that for
λ ∈ [0, 1] the possible solution set of family of boundary value problems
Dαy(t) = λg(t, y(t),Dβy(t)), t ∈ (0, 1), (3.4.16)
y(0) = 0, Dβy(1) = γDβy(η), (3.4.17)
is a priori bound in Lp[0, 1]. The domain of Riemann–Liouville linear operator
z = Dα(.) : Bβ → Lp[0, 1] is given by
D(L) :=y ∈ AC[0, 1], Dβy ∈ AC[0, 1], y(0) = 0, Dβy(1) = γDβy(η)
.
For y ∈ D(L), in view of Corollary 3.4.3, we have
∥y∥β ≤(Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)∥g(t, y,Dβy(t))∥p
≤(Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)(∥a∥p∥y∥+ ∥b∥p∥Dβy∥+ ∥φ∥p
)≤(Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)((∥a∥p + ∥b∥p∥)∥y∥β + ∥φ∥p
).
47
Hence,
∥y∥β ≤
(Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)∥φ∥p
1−(Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)(∥a∥p + ∥b∥p∥
) .Therefore solution set of (3.4.16) and (3.4.17) is bounded in Bβ . All conditions of the Theorem 2.3.4 are
satisfied. Thus the problem (3.4.2), (3.4.3) has at least one solution in Bβ .
3.4.2 Uniqueness of solution
Theorem 3.4.5. Let g(t, y, z) satisfies Carathéodory conditions and there exists ϕ(t) ∈ Lp[0, 1] such that
|g(t, y, z)− g(t, y, z)| ≤ ϕ(t)(|y − y|+ |z − z|), for all y, z, y, z ∈ R. (3.4.18)
If 0 <∫ 10 φ(s)ds < ln 2, where φ(s) = max
0≤t≤1|G(t, s)||ϕ(s)|, then the boundary value problem (3.4.2), (3.4.3)
has a unique solution.
Proof. Let ω(t) =∫ t0 φ(s)ds, then ω′(t) = φ(t) for a.e. t ∈ [0, 1]. Define norm
∥y∥ω = max0≤t≤1
e−ω(t)|y(t)|+ max0≤t≤1
e−ω(t)|Dβy(t)|.
This norm is equivalent to the norm ∥.∥, because
e−∥ϱ∥1∥y∥ ≤ ∥y∥ω ≤ ∥y∥, where ∥ϱ∥1 =∫ 1
0|φ(t)|dt.
Consequently (Bω, ∥.∥ω) is a Banach space. Next, we show that T is a contraction on (Bω, ∥.∥ω). Let u,
y ∈ B, then
e−ω(t)|T y(t)− T y(t)| ≤e−ω(t)∫ 1
0|G(t, s)||g(s, y(s),Dβy(s))− g(s, y(s),Dβ y(s))|ds
≤e−ω(t)∫ 1
0|G(t, s)||ϕ(s)|(|y(s)− y(s)|+ |Dβy(s)−Dβ y(s))|)ds
≤e−ω(t)∫ 1
0φ(s)eω(s)
e−ω(s)|y(s)− y(s)|+ e−ω(s)|Dβy(s)−Dβ y(s))|)
ds
≤e−ω(t)(eω(1) − 1)∥y − y∥ω.
Therefore
∥T y(t)− T y(t)∥ω ≤ e−ω(t)(eω(1) − 1)∥y − y∥ω.
Since e−ω(t)(eω(1) − 1) < 1, the Banach contraction principle guarantees the existence of unique solution
for the boundary value problem (3.4.2), (3.4.3).
Theorem 3.4.6. Let g(t, y, z) satisfies Carathéodory conditions and there exists ϕ(t) ∈ Lp[0, 1] such that
(3.4.18) is satisfied. Furthermore, if(Ω1,α,β,γ,η,q + Ω2,α,β,γ,η,q
)∥ϕ∥p < 1 then the boundary value problem
(3.4.2), (3.4.3) has a unique solution.
48
Proof. From (3.4.18) we have
|g(t, y, z)| ≤ |g(t, 0, 0)|+ ϕ(t)(|y|+ |z|). (3.4.19)
By similar arguments used in the Theorem 3.4.4, we conclude that there exists at least one solution for the
boundary value problem (3.4.2), (3.4.3). In order to prove uniqueness of solution, we assume that there
exist two solutions y, y for the the boundary value problem (3.4.2), (3.4.3). Let z = y − y, then we have
Dαz(t) = g(t, y(t),Dβy(t))− g(t, y(t),Dβ y(t)), t ∈ [0, 1], (3.4.20)
z(0) = 0, Dβz(1) = γDβz(η), (3.4.21)
Now, in view of (3.4.18) and Corollary 3.4.3, we have following estimates
∥z∥β ≤(Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)(∥g(t, y(t),Dβy(t))− g(t, y(t),Dβ y(t))∥p
≤(Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)∥ϕ∥p∥y − y∥β
≤(Ω1,α,β,γ,η,q +Ω2,α,β,γ,η,q
)∥ϕ∥p∥z∥β
Which implies that z(t) = 0 for all t ∈ [0, 1]. Hence y(t) = y(t).
3.5 Multi–point boundary value problems
Multi–point boundary value problems for fractional differential equations have received some attention
from researchers quite recently, for example, Moustafa El-Shahed and Juan J. Nieto [44], Hussein A.H.
Salem [123], Z. Bai and Y. Zhang [19], B. Ahmad and S. Sivasundaram [7], W. Zhong, and W. Lin, [153]. In
this section, we establish sufficient conditions for the existence and uniqueness of solutions to the following
class of multi–point boundary value problem for nonlinear fractional differential equations [86]
Dαy(t) = g(t, y(t),Dβy(t)), t ∈ [0, 1] (3.5.1)
y(0) = 0, Dβy(1) =
m−2∑i=1
aiDβy(ξi) + y0, (3.5.2)
where 1 < α ≤ 2, 0 < β ≤ 1, 0 < ξi < 1 (i = 1, 2, · · · ,m − 2), ai ≥ 0 with, Λα,β =∑m−2
i=1 aiξα−β−1i < 1.
The nonlinear function f and boundary conditions involve fractional derivatives of unknown functions. For
the degenerate case (i.e. β = 1), the above problem is general compared to the problem studied in [153]
in the sense that the nonlinearity still depends on the first derivative. Moreover, the boundary conditions
are general too.
Lemma 3.5.1. Let 1 < α ≤ 2 and 0 < β ≤ 1. Then a function y ∈ Bβ is a solution of the boundary valueproblem (3.5.1), (3.5.2) if and only if it is solution of the integral equation
y(t) = Iαg(t, y(t),Dβy(t))+Γ(α− β)
Γ(α) (1− Λα,β)
(m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβy(ξi))−Iα−βg(1, y(1),Dβy(1))+ y0
)tα−1.
(3.5.3)
49
Proof. Assume that y ∈ Bβ is a solution of the boundary value problem (3.5.1), (3.5.2). By Theorem
2.2.15 and equation (3.5.1), we have
y(t) = Iαg(t, y(t),Dβy(t)) + c0tα−1 + c1t
α−2, for c0, c1 ∈ R. (3.5.4)
The boundary condition y(0) = 0 implies that c1 = 0. Using Lemma (2.2.8), equation (3.5.4) reduces to
Dβy(t) = Iα−βg(t, y(t),Dβy(t)) +Γ(α)tα−β−1
Γ(α− β)c0. (3.5.5)
Using the second boundary condition, from equation (3.5.5), we obtain
c0 =Γ(α− β)
Γ(α)(1− Λα,β)
(m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβy(ξi))− Iα−βg(1, y(1),Dβy(1)) + y0
).
Thus we have
y(t) = Iαg(t, y(t),Dβy(t))+Γ(α− β)
Γ(α) (1− Λα,β)
(m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβy(ξi))−Iα−βg(1, y(1),Dβy(1))+ y0
)tα−1.
Conversely suppose that y ∈ Bβ is a solution of the integral equation (3.5.3). Denoting the right handside of (3.5.3) by w(t), that is,
w(t) =Iαg(t, y(t),Dβy(t)) +Γ(α− β)
Γ(α) (1− Λα,β)
(m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβξiy(ξi))− Iα−βg(1, y(1),Dβ
1 y(1)) + y0
)tα−1.
Using the Lemma 2.2.12 we have Dαw(t) = g(t, y(t),Dβy(t)), that is Dαy(t) = g(t, y(t),Dβy(t)). On theother hand it is quite easy to verify that y(0) = 0. Now, in view of Lemma 2.2.8 , we obtain
Dβy(1) =1
1− Λα,β
(m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβξiy(ξi))− Iβ−αg(1, y(1),Dβ
1 y(1))Λα,β + y0
),
m−2∑i=1
aiDβy(ξi) =1
1− Λα,β
(m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβξiy(ξi))− Iα−βg(1, y(1),Dβ
1 y(1))− y0Λα,β
)= Dβy(1) + y0.
Thus, we conclude that y ∈ Bβ is a solution of the boundary value problem (3.5.1), (3.5.2).
3.5.1 Existence of solutions
For convenience, we define:
ρα,β =(Γ(α− β) + Γ(α)
)∑m−2i=1 aiξ
α−βi + (2− Λα,β)Γ(α) + Γ(α− β),
Q1α,β = max
t∈[0,1]Iα|φ(t)|+ Γ(α−β)
Γ(α)(1−Λα,β)
(∣∣∣∑m−2i=1 aiIα−βφ(ξi)− Iα−βφ(1)
∣∣∣+ |y0|
),
Q2α,β = max
t∈[0,1]Iα−β |ϕ(t)|+ 1
1−Λα,β
(∣∣∣∑m−2i=1 aiIα−β |ϕ(ξi)| − Iα−β |ϕ(1)|
∣∣∣+ |y0|
),
Q3α,β = ρα,β +
((Γ(α− β))2 + Γ(α)
)|y0|, and
Kα,β = 1Γ(α+1) +
1(1−Λα,β)
(|∑m−2
i=1 aiξα−βi −1|
Γ(α)(α−β) +|∑m−2
i=1 aiξα−βi −Λα,β |
Γ(α−β+1)
).
50
Theorem 3.5.2. Assume that (A1) and (A2) hold. Then the boundary value problem (3.5.1), (3.5.2) has
a solution.
Proof. Define an operator T : Bβ → Bβ by
T y(t) = Iαg(t, y(t),Dβy(t)) +Γ(α− β)tα−1
Γ(α) (1− Λα,β)
(m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβξiy(ξi))− Iα−βg(1, y(1),Dβ
1 y(1)) + y0
).
(3.5.6)
By Lemma 3.5.1, fixed points of the operator T are solutions of the boundary value problem (3.5.1), (3.5.2).
In view of the continuity of f , the operator T is continuous. Define U =y ∈ Bβ : ∥y∥Bβ
≤ R, t ∈ [0, 1].
Now we show that T : U → U . Choose
R ≥ max3(Q1
α,β +Q2α,β), (3µ1Kα,β)
11−ν1 , (3µ2Kα,β)
11−ν2
then for every y ∈ U , we have
|T y(t)| ≤ Iα|g(t, y(t),Dβy(t))|
+Γ(α− β)
Γ(α) (1− Λα,β)
(∣∣∣m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβξiy(ξi))− Iα−βg(1, y(1),Dβ
1 y(1))∣∣∣+ |y0|
)
≤Iα|φ(t)|+ Γ(α− β)
Γ(α) (1− Λα,β)
∣∣∣m−2∑i=1
aiIα−βφ(ξi)− Iα−βφ(1)∣∣∣+ Γ(α− β)|y0|
Γ(α) (1− Λα,β)
+ (µ1Rν1 + µ2R
ν2)
(∫ t
0
(t− s)α−1
Γ(α)ds+
Γ(α− β)
Γ(α) (1− Λα,β)
∣∣∣m−2∑i=1
ai
∫ ξi
0
(ξi − s)α−β−1
Γ(α− β)ds−
∫ 1
0
(1− s)α−β−1
Γ(α− β)ds∣∣∣)
≤Q1α,β + (µ1R
ν1 + µ2Rν2)
(1
Γ(α+ 1)+
|∑m−2
i=1 aiξα−βi − 1|
Γ(α)(α− β)(1− Λα,β)
).
In view of Lemma 2.2.8 and equation (3.5.6), we have the following estimate:
|Dβ(T y)(t)| =∣∣∣Iα−βg(t, y(t),Dβy(t)) +
Γ(α− β)
Γ(α) (1− Λα,β)
×
(m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβy(ξi))− Iα−βg(1, y(1),Dβy(1)) + y0
)Dβtα−1
∣∣∣≤Iα−β |ϕ(t)|+ 1
1− Λα,β
(∣∣∣m−2∑i=1
aiIα−β |ϕ(ξi)| − Iα−β |ϕ(1)|∣∣∣+ |y0|
)
+ (µ1Rν1 + µ2R
ν2)∣∣∣ ∫ t
0
(t− s)α−β−1
Γ(α− β)ds+
1
1− Λα,β
(m−2∑i=1
ai
∫ ξi
0
(ξi − s)α−β−1
Γ(α− β)ds−
∫ 1
0
(1− s)α−β−11
Γ(α− β)
)∣∣∣≤Q2
α,β + (µ1Rν1 + µ2R
ν2)|∑m−2
i=1 aiξα−βi − Λα,β |
(1− Λα,β)Γ(α− β + 1).
Hence, ∥T y(t)∥Bβ≤ Q1
α,β +Q2α,β + (µ1R
ν1 + µ2Rν2)Kα,β ≤ R
3 + R3 + R
3 = R. Consequently, T : U → U .In the following, we show that the operator T is completely continuous.
Let N = maxt∈[0,1]
|g(t, y(t),Dβy(t))|, for y ∈ U . Taking t, τ ∈ [0, 1] such that t < τ . Then, for given ε > 0,
51
choose δ = εN
(αηα−1
Γ(α) +Q3
α,β(α−1)ηα−2
(1−Λα,β)Γ(α)Γ(α−β−1)
)−1
such that τ − t < δ.
|T y(τ)− T y(t)| ≤ |Iαg(τ, y(τ),Dβy(τ))− Iαg(t, y(t),Dβy(t))|
+Γ(α− β)
Γ(α) (1− Λα,β)
(∣∣∣m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβξiy(ξi))− Iα−βg(1, y(1),Dβ
1 y(1))∣∣∣+ |y0|
)(τα−1 − tα−1)
≤N
∣∣∣∣∣∫ τ
0
(τ − s)α−1 − (t− s)α−1
Γ(α)ds
∣∣∣∣∣+
Γ(α− β)
Γ(α)(1− Λα,β)
(N(m−2∑
i=1
ai
∫ ξi
0
(ξi − s)α−β−1
Γ(α− β)ds+
∫ 1
0
(1− s)α−β−1
Γ(α− β)ds)+ |y0|
)(τα−1 − tα−1)
=N
Γ(α+ 1)(τα − tα) +
N(∑m−2
i=1 aiξα−βi + Γ(α− β)|y0|+ 1
)Γ(α)(1− Λα,β)(α− β)
(τα−1 − tα−1),
|(DβT y)(τ)− (DβT y)(t)| ≤∣∣Iα−βg(τ, y(τ),Dβy(τ))− Iα−βg(t, y(t),Dβ
t y(t))∣∣
+N
1− Λα,β
(m−2∑i=1
aiIα−βg(ξi, y(ξi),Dβy(ξi)) + Iα−βg(1, y(1),Dβy(1)) + |y0|)(τα−β − tα−β)
≤N
∣∣∣∣∣∫ τ
0
(τ − s)α−β−1 − (t− s)α−β−1
Γ(α− β)ds
∣∣∣∣∣+
N1− Λα,β
(m−2∑i=1
ai
∫ ξi
0
(ξi − s)α−β−1
Γ(α− β)ds+
∫ 1
0
(1− s)α−β−1
Γ(α− β)ds)+ |y0|
)(τα−β − tα−β)
≤ NΓ(α− β + 1)
(τα−β − tα−β) +N(∑m−2
i=1 aiξα−βi + |y0|+ 1
)(1− Λα,β)Γ(α− β + 1)
(τα−β−1 − tα−β−1)
≤N(2− Λα,β +
∑m−2i=1 aiξ
α−βi + |y0|
)(1− Λα,β)Γ(α− β + 1)
(τα−β−1 − tα−β−1).
By Mean Value Theorem, there exists η ∈ (0, 1) such that
τα − tα = αηα−1(τ − t), τα−1 − tα−1 = (α− 1)ηα−2(τ − t) and
τα−β−1 − tα−β−1 = (α − 1)ηα−β−2(τ − t) ≤ (α − β − 1)ηα−β−2(τ − t). Obviously ηα−β−2 ≥ ηα−2. This
gives τα−1 − tα−1 ≤ (α− 1)ηα−β−2(τ − t). Therefore, we have
∥T y(τ)− T y(t)∥ ≤ N
(αηα−1
Γ(α)+
Q3α,β(α− 1)ηα−β−2
(1− Λα,β)Γ(α)Γ(α− β − 1)
)(τ − t) < ε.
Thus, we conclude that T U is equicontinuous set. Also T U is a uniformly bounded set. We have T U ⊂ U .By the Arzela–Ascoli theorem, T : U → U is completely continuous. Hence by the Schauder fixed point
theorem the boundary value problem (3.5.1), (3.5.2) has at least one solution.
Theorem 3.5.3. Assume that (A1) and (A3) hold. Then the boundary value problem (3.5.1), (3.5.2) has
a solution.
Example 3.5.4. Consider the fractional boundary value problem
Dαy(t) =ω1t
νeγt
1 + t2+ω2 sinπt√π + |y|
|y|ν1 + ω3e−υt√
2 + |Dαy||Dβy|ν2 , t ∈ (0, 1), (3.5.7)
y(0) = 0, Dβy(1) =1
2Dβy
(14
)+
1
4Dβy
(12
)+
1
4Dβy
(34) + eπ, (3.5.8)
52
where 1 < α < 2, 0 < β < 1, α−β > 1 and ν, γ, ωi ∈ R+ (i = 1, 2, 3), υ ≥ 0. Choose ai =(i−1)!2i
and ξi = i4
(i = 1, 2, 3). Then Λα,β =∑3
i=1(i−1)!2i
( i4)α−β−1 < (34)
α−β−1 < 1. For g(t, y, z) = ω1tνeγt
1+t2+ ω2 sinπt√
π+|y||y|ν1 +
ω3e−υt√2+|Dαy|
|Dβy|ν2 , t ∈ (0, 1). Thus, we have |g(t, y, z)| < φ(t) + µ1|y|ν1 + µ2|z|ν2 , where φ(t) = ω1tνeγt
1+t2,
µ1 =ω2√π, µ2 = ω3√
2. For 0 < ν1, ν2 < 1, the assumption (A2) is satisfied and for ν1, ν2 > 1, the assumption
(A3) holds. Therefore, by Theorems 3.5.2 and 3.5.5, the boundary value problem (3.5.1), (3.5.2) has a
solution.
3.5.2 Uniqueness of solution
Theorem 3.5.5. Assume that (A1) and (A4) hold. If there exists k ∈ R+, such that
k <( 1
Γ(α+ 1)+
ρα,β(1− Λα,β)Γ(α)Γ(α− β + 1)
)−1.
Then the boundary value problem (3.5.1), (3.5.2) has a unique solution.
Proof. By assumption (A4), we have
|T y(t)− T y(t)| = |Iα(g(t, y(t),Dβy(t))− g(t, y(t),Dβy(t)))|+ Γ(α− β)
Γ(α) (1− Λα,β)
×
(∣∣∣m−2∑i=1
aiIα−β(g(ξi, y(ξi),Dβξiy(ξi))− g(ξi, y(ξi),Dβ
ξiy(ξi)))− Iα−β(g(1, y(1),Dβy(1))− g(1, y((1),Dβy((1)))
∣∣∣)
≤k∥y − y∥β
(∫ t
0
(t− s)α−1
Γ(α)ds+
Γ(α− β)
Γ(α)(1− Λα,β)
(m−2∑i=1
aiIα−β(1)
∫ ξi
0
(ξi − s)α−β−1
Γ(α− β)ds+
∫ 1
0
(1− s)α−β−1
Γ(α− β)ds))
<k( 1
Γ(α+ 1)+
(∑m−2i=1 aiξ
α−βi + 1
)Γ(α)(1− Λα,β)(α− β)
)∥y − y∥β ,
|(DβT y(t)− (Dβ(T y)(t)| ≤
(Iα−β |g(t, y(t),Dβy(t))− g(t, y(t),Dβy(t))|
+1
1− Λα,β
(m−2∑i=1
aiIα−β |g(ξi, y(ξi),Dβy(ξi))− g(ξi, y(ξi),Dβy(ξi))|
+ Iα−β |g(1, y(1),Dβ1 y(1))− g(1, y(1),Dβy(1))|
))tα−β−1
≤k∥y − y∥β
(∫ t
0
(t− s)α−β−1
Γ(α− β)ds+
1
1− Λα,β
(m−2∑i=1
ai
∫ ξi
0
(ξi − s)α−β−1
Γ(α− β)ds+
∫ 1
0
(1− s)α−β−1
Γ(α− β)ds
))
≤k∥y − y∥β
(tα−β
Γ(α− β + 1)+
∑m−2i=1 aiξ
α−βi + 1
(1− Λα,β)Γ(α− β + 1)
)<k(2 +
∑m−2i=1 aiξ
α−βi − Λα,β
)(1− Λα,β)Γ(α− β + 1)
∥y − y∥β .
Thus, we have ∥T y(t)− (T y)(t)∥β < L∥y−y∥β, where L = k(
1Γ(α+1) +
ρα,β
(1−Λα,β)Γ(α)Γ(α−β+1)
)< 1. Hence,
by the Banach fixed point theorem, the multi–point boundary value problem problem (3.5.1), (3.5.2) has
a unique solution.
53
Example 3.5.6. Consider the five–point fractional boundary value problem
D32 y(t)(|D
12 y|+ |y|+ 1) =
(|y|+ |D12 y(t)|)e−κt
(42√π + 85e−κt)
, t ∈ (0, 1), (3.5.9)
y(0) = 0, D12 y(1) =
5
7D
12 y(25
)+
2
3D
12 y(35
)+
8
21D
12 y(45
), (3.5.10)
where κ > 0. Let g(t, y, z) = e−γt(y+z)(42
√π+85e−κt)(1+y+z)
, t ∈ (0, 1), y, z ∈ [0,∞), α = 32 , β = 1
2 , a1 = 57 , a2 = 2
3 ,
a3 = 821 , and ξk = k+1
5 (k = 1, 2, 3). Also Λα,β = 3435 < 1. For y, y, z, z we have |g(t, y, z) − g(t, y, z)| <
142
√π+85
(|y + z| + |y + z|
). Hence the condition (A4) is satisfied. By calculations, L = 146
105√π< 1. Thus
by Theorem 3.5.2 the boundary value problem (3.5.1), (3.5.2) has a unique solution.
3.6 Boundary value problems with integral boundary conditions
In [24], Cannon initiated the study of nonlocal boundary value problems with integral boundary conditions.
Since then the subject has been addressed by many authors [5,34,64,148]. Boundary value problems with
integral boundary conditions arises in thermoplasticity, papulation dynamics, underground water flow and
blood flow problems. For details,we refer to [29, 32, 47, 131]. This interesting class of boundary value
problem includes two-point, three–point and multi–point boundary value problems as special cases.
In this section, we study existence and uniqueness of solutions to a nonlinear fractional differential
equations with integral boundary conditions in an ordered Banach space. We use the Caputo fractional
differential operator and the nonlinearity depends on the fractional derivative of an unknown function. For
the existence of solutions, we employ the nonlinear alternative of the Leray–Schauder and a uniqueness
result is established using the Banach fixed point theorem. We are concerned with existence and uniqueness
of solutions to nonlinear fractional differential equations of the type [89]
cDαy(t) = f(t, y(t), cDβy(t)), for t ∈ [0, l], (3.6.1)
subject to the integral boundary conditions
py(0)− qy′(0) =
∫ l
0g(s, y(s))ds, γy(1) + δy′(1) =
∫ l
0h(s, y(s))ds, (3.6.2)
where 0 < β ≤ 1, 1 < α ≤ 2, p, δ > 0, q, γ ≥ 0 (or p, δ ≥ 0, q, γ > 0). The functions f, g and h are assumed
to be continuous. For particular case, when β = 0, p = q = γ = δ = 1, the existence of solutions for the
boundary value problem (3.6.1), (3.6.2) is studied in [3]. The boundary value problem (3.6.1), (3.6.2) is
more general than that considered in [3] in the sense that the nonlinear function g depends on fractional
derivative of y as well.
Lemma 3.6.1. A function y ∈ Bβ is solution of fractional boundary value problem (3.6.1), (3.6.2) if and
only if y ∈ Bβ is solution of the fractional integral equation
y(t) =
∫ l
0G(t, s)f(s, y(s), cDβy(s))ds+ φ(t), (3.6.3)
54
where, G(t, s) is the Green function given as
G(t, s) =
(t−s)α−1
Γ(α) − (q+pt)γ(l−s)α−1
∆p,qΓ(α)− (q+pt)δ(α−1)(l−s)α−2
∆p,qΓ(α), 0 ≤ s ≤ t,
− (q+pt)γ(l−s)α−1
∆p,qΓ(α)− (q+pt)δ(α−1)(l−s)α−2
∆p,qΓ(α), 0 ≤ t ≤ s,
(3.6.4)
φ(t) =δ + γ(l − t)
∆p,q
∫ l
0g(s, y(s))ds+
q(1 + t)
∆p,q
∫ l
0h(s, y(s))ds and ∆p,q = p(δ + γl) + qγ.
Proof. Taking into account the equation (3.6.1) and Lemma 2.2.24, we infer that the general solution is
y(t) = Iαf(t, y(t), cDβy(t)) + c0 + c1t, c0, c1 ∈ R. (3.6.5)
Using boundary condition (3.6.2), we obtain
c0 = − q
∆p,q(γIαf(l, y(l), cDβy(l))+ δIα−1f(l, y(l), cDβy(l)))+
1
∆p,q
∫ l
0((δ+γl)g(s, y(s))+ qh(s, y(s)))ds,
(3.6.6)
c1 = − p
∆p,q(γIαy(l) + δIα−1y(l)) +
1
∆p,q
∫ l
0(ph(s, y(s))− γg(s, y(s)))ds. (3.6.7)
Therefore, we have the integral equation
y(t) =Iαf(s, y(s), cDβy(t))− 1
∆p,q(γIαf(l, y(l), cDβy(l)) + δIα−1y(l))(q + pt)
+δ + γ(l − t)
∆p,q
∫ l
0
g(s, y(s))ds+(q + pt)
∆p,q
∫ l
0
h(s, y(s))ds,
which can be written as
y(t) =
∫ l
0G(t, s)f(s, y(s), cDβy(s))ds+ φ(t).
Conversely, let y ∈ Bβ satisfy (3.6.3) and denote the right hand side of equation (3.6.3) by w(t). Then, byLemmas 2.2.5 and Lemma 2.2.23, we obtain
w(t) =
∫ l
0
G(t, s)f(s, y(s), cDβy(s))ds+ φ(t)
=Iαf(t, y(t), cDβy(t))− γ
∆p,qIαf(l, y(l), cDβy(l))(q + pt)− δ
∆p,qIα−1f(l, y(l), cDβy(l))(q + pt) + φ(t),
which implies that
cDαw(t) =cDαIαf(t, y(t), cDβy(t))− γ
∆p,q
cDαIαf(l, y(l), cDβy(l))(q + pt)
− δ
∆p,q
cDαIα−1f(l, y(l), cDβy(l))(q + pt) = f(t, y(t), cDβy(t)).
Hence, y(t) is a solution of the fractional differential equation cDαy(t) = f(t, y(t), cDβy(t)). Also, one can
easily verify that py(0)− qy′(0) =∫ l0 g(s, y(s))ds, γy(t) + δy′(l) =
∫ l0 h(s, y(s))ds.
Remark 3.6.2. The Green function (3.6.4) satisfies the following properties:
(i)∫ 10 |G(t, s)| ≤ lα
Γ(α+1) +q+plα+1
∆p,qΓ(α)
(γα + δ
l
), t ∈ (0, 1).
(ii)∫ l0 |
∂∂tG(t, s)|ds ≤
lα−1
Γ(α) +q+plα
∆p,qΓ(α−1)
(γ
α−1 + δl
).
55
Proof. (i) By definition of G(t, s),∫ l
0
|G(t, s)|ds ≤∫ t
0
(t− s)α−1
Γ(α)ds+
∫ l
0
(q + pt)
(γ(l − s)α−1
∆p,qΓ(α)+δ(α− 1)(l − s)α−2
∆p,qΓ(α)
)ds
≤ tα
αΓ(α)+
q + pt
∆p,qαΓ(α)(γlα + δlα−1) ≤ lα
Γ(α+ 1)+q + plα+1
∆p,qΓ(α)
(γα+δ
l
).
(ii) Also,∫ l
0
| ∂∂tG(t, s)|ds ≤
∫ t
0
(t− s)α−2
Γ(α− 1)ds+
p
∆p,qΓ(α)
∫ l
0
(γ(l − s)α−1 + δ(α− 1)(l − s)α−2)ds
≤ tα−1
Γ(α)+
γplα
∆p,q(α− 1)Γ(α− 1)+
δplα−1
∆p,qΓ(α− 1)≤ lα−1
Γ(α)+
q + plα
∆p,qΓ(α− 1)
( γ
α− 1+δ
l
).
Define an operator T : Bβ → Bβ by
T y(t) =∫ l
0G(t, s)f(s, y(s), cDαy(s))ds+ φ(t). (3.6.8)
Then, the boundary value problem (3.6.1), (3.6.2) is equivalent to the fixed point problem T y = y.
In what follows, we establish an existence result using the nonlinear alternative of Leray–Schauder type
by imposing some growth conditions on f , g and h.
(A5) There exist continuous functions ψif : [0,∞) → (0,∞), ϕif ∈ L1[0, l], (i = 1, 2) such that |f(t, y, z)| ≤ϕ1f (t)ψ
1f (|y|) + ϕ2f (t)ψ
2f (|z|).
(A6) There exist continuous and nondecreasing function ψg : [0,∞) → (0,∞) and a function ϕg ∈ L1[0, l]
such that |g(t, y)| ≤ ϕg(t)ψg(|y|), for t ∈ [0, l],
(A7) There exist continuous and nondecreasing function ψh : [0,∞) → (0,∞) and a function ϕh ∈ L1[0, l]
such that |h(t, y)| ≤ ϕh(t)ψh(|y|), for t ∈ [0, l].
Define a =∫ l0 ϕg(s)ds, b =
∫ l0 ϕh(s)ds, Nα := lα
Γ(α+1) + q+plα+1
∆p,qΓ(α)
(γα + δ
l
), kη,α :=
(ψ1f (η)∥ϕ1f∥L1 +
ψ2f (η)∥ϕ2f∥L1
)Nα and Mβ,η =
l1−β
Γ(2−β)
(kη,α−1 +
aγψg(η)∆p,q
+ bpψh(η)∆p,q
)kη,α.
(A8) There exists r > 0 such that rMβ,r
> 1,
(A9) There exists constants k1, k2 > 0 such that
|g(t, y)− g(t, y)| ≤ k1|y − y| and |h(t, y)− h(t, y)| ≤ k2|y − y|, for each t ∈ [0, l].
3.6.1 Existence of solutions
Theorem 3.6.3. Under the assumptions (A5) − (A8), the fractional boundary value problem (3.6.1),
(3.6.2) has at least one solution on [0, l].
Proof. In view of the continuity of f , g and h, the operator T is continuous. To show that T mapsbounded sets into bounded sets in Bβ , choose η > 0 (fixed).Let ρ ≥ maxη,Mβ,η. Define U = y ∈ B : ∥y∥β < η and V = y ∈ B : ∥y∥β < ρ. For t ∈ [0, l] andy ∈ U , we have
|T y(t)| ≤∫ l
0
|G(t, s)||f(s, y(s), cDβy(s))|ds+ δ + γ(l − t)
∆p,q
∫ l
0
|g(s, y)|ds+ q + pt
∆p,q
∫ l
0
|h(s, y)|ds.
56
Using (A5)− (A7) and Remark 3.6.2, we obtain
|T y(t)| ≤
(lα
Γ(α+ 1)+q + plα+1
∆p,qΓ(α)
( γα+δ
l
))(|ϕ1f (s)|ψ1
f (∥y∥β) + |ϕ2f (s)|ψ2f (∥y∥β))ds
+δ + γ(l − t)
∆p,q
∫ l
0
|ϕg(s)|ψg(∥y∥β)ds+q + pt
∆p,q
∫ l
0
|ϕh(s)|ψh(∥y∥β)ds
≤(ψ1f (η)∥ϕ1f∥L1 + ψ2
f (η)∥ϕ2f∥L1
)Nα
+(δ + 2γl)ψg(η)
∆p,q
∫ l
0
ϕg(s)ds+(q + pl)ψh(η)
∆p,q
∫ l
0
ϕh(s)ds
≤kη, α +a
∆p,q(δ + 2γl)ψg(η) +
b
∆p,q(q + pl)ψh(η) = kη, α +Mη,
|(T y)′(t)| ≤∫ l
0
| ∂∂tG(t, s)||f(s, y(s), cDβy(s))|ds+ γ
∆p,q
∫ l
0
|g(s, y)|ds+ p
∆p,q
∫ l
0
|h(s, y)|ds
≤
(lα−1
Γ(α)+
q + plα
∆p,qΓ(α− 1)
( γ
α− 1+δ
l
))(|ϕ1f (s)|ψ1
f (∥y∥β) + |ϕ2f (s)|ψ2f (∥y∥β)
)ds
+γ
∆p,q
∫ l
0
|ϕg(s)|ψg(∥y∥β)ds+p
∆p,q
∫ l
0
|ϕh(s)|ψh(∥y∥β)ds
≤(ψ1f (η)∥ϕ1f∥L1 + ψ2
f (η)∥ϕ2f∥L1
)Nα−1 +
γψg(η)
∆p,q
∫ l
0
ϕg(s)d+pψh(η)
∆p,q
∫ l
0
ϕh(s)ds
≤kη, α−1 +aγψg(η)
∆p,q+bpψh(η)
∆p,q.
Hence, it follows that
|cDβT (y)(t)| ≤ 1
Γ(1− β)
∫ t
0
(t− s)−β |(T y)′(s)|ds ≤ l1−β
Γ(2− β)
(kη,α−1 +
aγψg(η)
∆p,q+bpψh(η)
∆p,q
).
Therefore, ∥T y∥β ≤ ρ, which implies that T y ∈ V . Hence, T maps bounded sets into bounded setsin Bβ . Now, we show that T maps bounded sets into equicontinuous sets of Bβ . For this, we takeK = max|f(t, y(t), cDβy(t))| : y ∈ U, t ∈ [0, l], L1 = max|g(t, y(t)| : y ∈ U, t ∈ [0, l] and L2 =
max|h(t, y(t)| : y ∈ U, t ∈ [0, l]. Taking t, τ ∈ (0, l] such that t < τ and y ∈ U . Then,
|T y(τ)− T y(t)| ≤∫ l
0
|G(τ, s)−G(t, s)||f(s, y(s), cDβy(s))|ds+ |φ(τ − t)|
≤K
[∫ t
0
|G(τ, s)−G(t, s)|ds+∫ τ
t
|G(τ, s)−G(t, s)|ds
+
∫ l
τ
|G(τ, s)−G(t, s)|ds
]+
l
∆p,q(γL1 + pL2)(τ − t)
≤K[∫ t
0
((τ − s)α−1 − (t− s)α−1
Γ(α)+
(pγ(l − s)α−1
Γ(α)+pδ(α− 1)(l − s)α−2)
∆p,qΓ(α)
)(τ − t)
)ds
+
∫ τ
t
((τ − s)α−1
Γ(α)+
(pγ(l − s)α−1
Γ(α)+pδ(α− 1)(l − s)α−2)
∆p,qΓ(α)
)(τ − t)
)ds
+
∫ l
τ
(pγ(l − s)α−1
Γ(α)+pδ(α− 1)(l − s)α−2)
∆p,qΓ(α)
)(τ − t)ds
]+
l
∆p,q(γL1 + pL2)(τ − t)
≤K
[∫ l
0
(pγ(l − s)α−1
Γ(α)+pδ(α− 1)(l − s)α−2)
∆p,qΓ(α)
)(τ − t)ds
]+
l
∆p,q(γL1 + pL2)(τ − t)
=Kp
∆p,qΓ(α)
(γlα
α+ δlα−1
)(τ − t) +
K(τα − tα)
qΓ(α)+
l
∆p,q(γL1 + pL2)(τ − t),
57
|cDβT y(τ)− cDβT y(t)| = 1
Γ(1− β)
∣∣∣∣∫ τ
0
(τ − s)−β(T y)′(s)ds−∫ t
0
(t− s)−β(T y)′(s)ds∣∣∣∣
≤ 1
Γ(1− β)
(∫ τ
t
(τ − s)−β |(T y)′(s)|ds+∫ t
0
((τ − s)−β − (t− s)−β)|(T y)′(s)|ds)
≤ 1
Γ(1− β)
[∫ τ
t
(τ − s)−β
(∫ l
0
| ∂∂sG(s, z)||f(z, y(z), cDβy(z))|dz + φ′(s)
)ds
+
∫ t
0
((τ − s)−β − (t− s)−β)
(∫ l
0
| ∂∂sG(s, z)||f(z, y(z), cDβy(z))|dz + φ′(s)
)ds
]
≤ K
∆p,qΓ(α)Γ(1− β)((pδ + p)lα−1 + pγlα)
[∫ τ
t
(τ − s)−βds+
∫ t
0
((τ − s)−β − (t− s)−β)ds
]+
(γL1 + pL2)l
∆p,qΓ(1− β)
[∫ τ
t
(τ − s)−βds+
∫ t
0
((τ − s)−β − (t− s)−β)ds
]≤ K
∆p,qΓ(α)Γ(2− β)((pδ + p)lα−1 + pγlα)(τ1−β − t1−β) +
(γL1 + pL2)l
∆p,qΓ(2− β)(τ1−β − t1−β)
≤(
K
∆p,qΓ(α)Γ(2− β)((pδ + p)lα−1 + pγlα) +
(γL1 + pL2)l
∆p,qΓ(2− β)
)(τ1−β − t1−β).
Obviously, |T y(τ)−T y(t)| → 0 and |cDβT y(τ)−cDβT y(t)| → 0 as t→ τ . Therefore, ∥T y(τ)−T y(t)∥ →0, as t→ τ . By the Arzela–Ascoli theorem it follows that T : Bβ → Bβ is completely continuous.Define U1 = y ∈ Bβ : ∥y∥β < r and assume that there exists y ∈ ∂U1 such that y = λT y for someλ ∈ (0, 1). Therefore, y(t) = λ
(∫ l0 G(t, s)f(s, y(s),
cDβy(s))ds+ φ(t)).
In view (A5)− (A7) and Remark 3.6.2, we have
|y(t)| <∫ 1
0
|G(t, s)|ds+ |φ(t)|
≤Nα(|ϕ1f (s)|ψ1f (∥y∥β) + |ϕ2f (s)|ψ2
f (∥y∥β))
+δ + γ(l − t)
∆p,q
∫ l
0
|ϕg(s)|ψg(∥y∥β)ds+q + pt
∆p,q
∫ l
0
|ϕh(s)|ψh(∥y∥β)ds
≤k∥y∥β , α +a
∆p,q(δ + 2γl)ψg(∥y∥β) +
b
∆p,q(q + pl)ψh(∥y∥β).
Similarly,
|y′(t)| < k∥y∥β ,α−1 +aγψg(∥y∥β)
∆p,q+bpψh(∥y∥β)
∆p,q.
Hence,
|cDβy(t)| < 1
Γ(1− β)
∫ t
0
(t− s)−β |y′(s)|ds ≤ l1−β
Γ(2− β)
(k∥y∥β , α−1 +
aγψg(∥y∥β)∆p,q
+bpψh(∥y∥β)
∆p,q
).
Therefore, ∥y∥βMβ,∥y∥β
< 1, a contradiction to (A8). Hence, y = λT y for y ∈ ∂U1, λ ∈ [0, 1]. By Theorem
2.3.5, the boundary value problem (3.6.1), (3.6.2) has at least one solution.
3.6.2 Uniqueness of solution
The uniqueness result is based on the Banach contraction principal.
Theorem 3.6.4. Assume that (A4), (A9) hold. Furthermore, if k < 13
(Nα + l1−β
Γ(2−β)Nα−1
)−1, k1 <
∆p,q
3l
(δ + 2γl + pl2−β
Γ(2−β)
)−1and k2 <
∆p,q
3l
(q + pl + pl1−β
Γ(2−β)
)−1. Then the boundary value problem (3.6.1),
(3.6.2) has a unique solution.
58
Proof. Let y, y ∈ Bβ , then for each t ∈ [0, l] we have
|T y(t)− T y(t)| ≤∫ l
0
G(t, s)|f(s, y(s), cDβy(s))− f(s, y(s), cDβy(s))|ds
+δ + γ(l − t)
∆p,q
∫ l
0
|g(s, y(s))− g(s, y(s))|ds+ (q + pt)
∆p,q
∫ l
0
|g(s, y(s))− g(s, y(s))|ds
≤∥y − y∥β
(k
(lα
Γ(α+ 1)+q + plα+1
∆p,qΓ(α)
(γα+δ
l
))+
l
∆p,q(k1(δ + γ(l − t)) + k2(q + pt)
)
≤∥y − y∥β(kNα +
l
∆p,q(k1(δ + 2γl) + k2(q + pl))
).
Also,
|cDβT y(t)− cDβT y(t)| =∣∣∣∣ 1
Γ(1− β)
∫ t
0
(t− s)−β((T y)′(s)− (T y)′(s))ds∣∣∣∣
≤ 1
Γ(1− β)
∫ t
0
(t− s)−β
(∫ l
0
∣∣∣∣ ∂∂sG(s, z)∣∣∣∣ |f(z, y(z), cDβy(z)− f(z, y(z), cDβy(z)|dz
+γ
∆p,q
∫ l
0
|g(z, y(z))|dz + p
∆p,q
∫ l
0
|h(z, y(z))|dz
)ds
≤ l1−β
Γ(2− β)
(kNα−1 +
(γk1 + pk2)l
∆p,q
)∥y − y∥β .
Therefore, ∥T y − T y∥β ≤ L∥y − y∥β , where
L =
(Nα +
l1−β
Γ(2− β)Nα−1
)k +
lk1∆p,q
(δ + 2γl +
pl2−β
Γ(2− β)
)+
lk2∆p,q
(q + pl +
pl1−β
Γ(2− β)
)< 1.
By the contraction mapping principle, the boundary value problem (3.6.1), (3.6.2) has a unique solution.
Example 3.6.5. Consider the following fractional differential equation,
cD 32 y(t)(e−λt + 124
√π)(|y|+ |cD 1
2 y(t)|+ 1) = e−λt(|y|+ |cD 12 y(t)|), t ∈ [0, 1], (3.6.9)
y(0)− y′(0) =1
9
∫ 1
0
|y(s)|e−s
(1 + |y(s)|)ds, y(1) + y′(1) =
1
36
∫ 1
0
|y(s)| sin sds. (3.6.10)
Set f(t, y, z) = e−λt(y(t)+z(t))(24
√π+e−λt)(1+y(t)+z(t))
, g(t, y) = y(t)e−t
9(1+y(t)) and h(t, y) = y(t) sin t36 for t ∈ [0, 1] and y, z ∈
[0,∞). Let y, y, z, z ∈ [0,∞), then we have
|f(t, y, z)− f(t, y, z)| = e−λt
(124√π + e−λt)
∣∣∣∣ y(t) + z(t)
1 + y(t) + z(t)− y(t) + z(t)
1 + y(t) + z(t)
∣∣∣∣≤ e−λt(|y(t)− y(t)|+ |z(t)− z(t)|)
(124√π + e−λt)(1 + y(t) + z(t))(1 + y(t) + z(t))
≤ e−λt(|y(t)− y(t)|+ |z(t)− z(t)|)124
√π + e−λt
≤ 1
124√π(|y(t)− y(t)|+ |z(t)− z(t)|).
By some calculations, ∆p,q = 3, Nα = 143√π, 13Nα
(1 + l−β
Γ(2−β)
)−1= π
14(√π+2)
. Here
k = 1124
√π< π
14(√π+2)
. Furthermore, |g(t, y)− g(t, y)| ≤ e−t|y(t)−y(t)|9(1+y(t))(1+y(t)) ≤
19 |y(t)− y(t)|, where k1 = 1
9 <∆p,q
3l(δ+2γl) = 13 . Also, note that |h(t, y) − h(t, y)| = sin t
36 |y(t) − y(t)| ≤ 136 |y(t) − y(t)|, where k2 = 1
36 <∆p,q
3l(q+pl) = 13 . All conditions of the Theorem 3.6.4 are satisfied. Therefore the boundary value problem
(3.6.9), (3.6.10) has a unique solution.
Chapter 4
Existence and multiplicity of positive
solutions
In many situations, we require positive solutions of boundary value problems that arise in the modeling of
problems from natural and social sciences such as papulation dynamics, engineering physics and finance.
One of the most useful tool which have been used effectively for proving existence of positive solutions for
boundary value problems, is Krasnosel’skii’s fixed point theorem on cone expansion and compression and
its norm type version due to Gau [50]. Most early work related to the application of Krasnosel’skii’s fixed
point theorem to eigne value problems to establish the interval of parameter for which there exists at least
one positive solution, was carried out by Wang [137]. Since this pioneering work, notable contributions
to the existence theory of positive solutions have been carried out [63, 99, 139]. The attention drawn to
the theory of existence and multiplicity of positive solutions for fractional differential equations is quite
evident from the increasing number of recent publications. To identify few, we refer to [20,81,145,146] and
references cited therein. However, few results can be found in the literature concerning existence of positive
solutions to nonlinear three–point boundary value problems for fractional differential equations [20,81].
In this chapter, we are concerned with the existence, multiplicity and uniqueness of positive solutions
to boundary value problems of fractional differential equations. A particular focus concerns boundary
value problems with three–point boundary conditions. In section 4.1, we investigate sufficient conditions
for the existence and multiplicity of positive solutions to nonlinear three–point boundary value problems
for fractional differential equations of type
cDαy(t) + a(t)g(t, y(t)) = 0, t ∈ [0, 1], n− 1 < α ≤ n,
satisfying boundary conditions
y′(0) = y′′(0) = y′′′(0) = · · · = y(n−1)(0) = 0, y(1) = ξy(η),
We apply superlinear, sublinear type growth conditions on nonlinearity which will enable us to apply
the Guo-Krasnoselskii and the Leggett-William fixed point theorems to establish several existence and
multiplicity results for positive solutions. In section 4.2, we study the existence and uniqueness of positive
solutions for three–point boundary value problems for fractional differential equations of the type [85]:
59
60
cDαy(t) + g(t, y(t)) = 0, t ∈ [0, 1], n − 1 < α ≤ n, n ∈ N, satisfying the boundary conditions y′(0) =
µy′(η) − θ1, y′′(0) = 0, y′′′(0) = 0, . . . , y(n−1)(0) = 0, y(1) = νy(η) + θ2, where, 0 < η, µ, ν < 1, n > 2,
the boundary parameters θ1, θ2 ∈ R+. Examples are included to show the applicability of some of our
results.
4.1 Positive solutions for three–point boundary value problems (I)
In this section, we study existence and multiplicity results for a class of nonlinear three–point boundary
value problems for fractional differential equations of the type
cDαy(t) + a(t)g(t, y(t)) = 0, t ∈ [0, 1], n− 1 < α ≤ n, (4.1.1)
y′(0) = y′′(0) = y′′′(0) = · · · = y(n−1)(0) = 0, y(1) = ξy(η), (4.1.2)
where ξ, η ∈ (0, 1).
We require the following assumptions:
(C1) (i) g : [0, 1]× [0,∞) → [0,∞) is continuous; (ii) a : [0, 1] → (0,∞) is continuous;
(C2) there exist 0 < µ1, µ2 ≤ 1 such that limy→0
g(t,y(t))yµ1 = ∞, lim
y→∞g(t,y(t))yµ2 = 0, for all t ∈ (0, 1);
(C3) there exist ν1, ν2 ≥ 1 such that limy→0
g(t,y(t))yν1 = 0, lim
y→∞g(t,y(t))yν2 = ∞, for all t ∈ (0, 1);
(C4) there exist positive constants r∗1 < r∗2 and κ satisfying κ∫ 10 Φα(s)a(s)ds = 1, such that
(i) g(t, y(t)) ≤ κ−1r∗1 , for y ∈ [0, r∗1],
(ii) g(t, y(t)) ≥ κ−1r∗2 , for y ∈ [γαr∗2, r
∗2];
(C5) there exist 0 < µ1 ≤ 1, µ2 ≥ 1, such that limy→0
g(t,y(t))yµ1 = lim
y→∞g(t,y(t))yµ2 = ∞, for all t ∈ (0, 1);
(C6) there exist ψ1(y) ∈ C([0,∞), [0,∞)) and φ1(t) ∈ C([0, 1], [0,∞)) such that
g(t, y(t)) ≤ φ1(t)ψ1(y) and there exists ρ > 0 such that ψ1(y) ≤ ερ, for y ∈ [0, ρ] and 0 < ε <
(∫ 10 Φα(s)a(s)φ1(t)ds)
−1;
(C7) there exist µ1 ≥ 1, 0 < µ2 ≤ 1 such that limy→0
g(t,y(t))yµ1 = lim
y→∞g(t,y(t))yµ2 = 0, for all t ∈ (0, 1);
(C8) there exists ψ2(y) ∈ C([0,∞), [0,∞)), φ2(t) ∈ C([0, 1], [0,∞)) such that g(t, y(t)) ≥ φ2(t)ψ2(y) and
there exists ρ∗ > 0 such that ψ2(y) ≥ ε∗ρ∗, for y ∈ [γαρ∗, ρ∗].
4.1.1 Green’s function and its properties
In this subsection, we derive Green’s function for the boundary value problem (4.1.1), (4.1.1) and list some
of its properties useful for the proof of our existence and multiplicity results.
61
Lemma 4.1.1. [93] Let h ∈ C(0, 1]. Then the linear three–point boundary value problem
cDαy(t) + h(t) = 0, t ∈ (0, 1), n− 1 < α ≤ n, (4.1.3)
y′(0) = y′′(0) = y′′′(0) = · · · = y(n−1)(0) = 0, y(1) = ξy(η), (4.1.4)
has solution given by y(t) =∫ 10 G(t, s)h(s)ds, where
G(t, s) =1
(1− ξ)Γ(α)
(1− s)α−1 − (1− ξ)(t− s)α−1 − ξ(η − s)α−1, s ≤ t, η ≥ s,
(1− s)α−1 − (1− ξ)(t− s)α−1, η ≤ s ≤ t ≤ 1,
(1− s)α−1 − ξ(η − s)α−1, 0 ≤ t ≤ s ≤ η,
(1− s)α−1, t ≤ s, s ≥ η.
(4.1.5)
Proof. In view of Lemma 2.2.24, the general solution of differential equation (4.1.3) is given by
y(t) = −Iαh(t) +n∑i=1
citi−1, ci ∈ R, i = 1, 2, . . . , n. (4.1.6)
By Lemma 2.2.23 and Lemma 2.2.19, we obtain
y(m)(t) = −Iα−mh(t) +n∑
i=m+1
(i− 1)!ci(i−m− 1)!
ti−m−1, (4.1.7)
where m = 1, 2, . . . , n− 1. Using the boundary conditions we have
ci = 0, i = 2, 3, . . . , n and c1 =
∫ 1
0
(1− s)α−1
Γ(α)(1− ξ)h(s)ds−
∫ η
0
ξ(η − s)α−1
Γ(α)(1− ξ)h(s)ds.
Substituting values of ci in (4.1.18), we get
y(t) =
∫ 1
0
(1− s)α−1
Γ(α)(1− ξ)h(s)ds−
∫ η
0
ξ(η − s)α−1
Γ(α)(1− ξ)h(s)ds−
∫ t
0
(t− s)α−1
Γ(α)h(s)ds. (4.1.8)
Thus, the solution of the boundary value problem (4.1.3),(4.1.4) is given by y(t) =∫ 10 G(t, s)h(s)ds.
Lemma 4.1.2. [93] The Green’s function G(t, s) defined by (4.1.5) satisfies the following properties:
(P1) G(t, s) > 0 for all t, s ∈ (0, 1);
(P2) For each s ∈ [0, 1], G(t, s) is nonincreasing in t;
(P3) For ℓ ∈ (0, 1), Φα(s) ≥ G(t, s) ≥ minℓ≤t≤1
G(t, s) ≥ γαΦα(s) where
γα = ξ(1− ηα−1), Φα(s) =(1− s)α−1
(1− ξ)Γ(α).
Proof. (P1): For η ≥ s,
G(t, s) =(1− s)α−1 − (1− ξ)(t− s)α−1 − ξ(η − s)α−1
(1− ξ)Γ(α).
For t < η,
G(t, s) >(1− s)α−1 − (η − s)α−1
(1− ξ)Γ(α)> 0
62
and for t ≥ η,
G(t, s) ≥ (1− s)α−1 − (t− s)α−1
(1− ξ)Γ(α)> 0.
For η ≤ s ≤ t ≤ 1,
G(t, s) =(1− s)α−1 − (1− ξ)(t− s)α−1
(1− ξ)Γ(α)≥ (1− s)α−1 − (t− s)α−1
(1− ξ)Γ(α)> 0.
Thus, for each case, η ≤ s ≤ t ≤ 1 and 0 ≤ t ≤ s, s ≥ η, G(t, s) > 0.
(P2) : From (4.1.5), if t > s,
∂
∂tG(t, s) = −(α− 1)
Γ(α)(t− s)α−2 = −(t− s)α−2
Γ(α− 1)≤ 0.
If t ≤ s, then∂
∂tG(t, s) ≡ 0.
Thus, G(t, s) is nonincreasing in t.
(P3): Clearly, G(t, s) ≤ (1−s)α−1
Γ(α)(1−ξ) = Φα(s).
Case(i): s ≤ η, using (P2), we have
G(t, s) ≥ G(1, s) =ξ(1− ηα−1)(1− s)α−1
(1− ξ)Γ(α)= γαΦα(s), for t ≥ s.
For t ≤ s, we have
G(t, s) ≥ (1− s)α−1 − ξηα−1(1− s)α−1
(1− ξ)Γ(α)≥ ξ(1− ηα−1)Φα(s) = γαΦα(s).
Case(ii): s > η: for t ≥ s,
G(t, s) ≥ G(1, s) =ξ(1− s)α−1
(1− ξ)Γ(α)≥ γαΦα(s).
For t < s, we have G(t, s) = (1−s)α−1
(1−ξ)Γ(α) = Φα(s) ≥ γαΦα(s).
Hence, minℓ≤t≤1
G(t, s) ≥ γαΦα(s) where ℓ ∈ (0, 1), γα = ξ(1− ηα−1) and Φα(s) =(1−s)α−1
(1−ξ)Γ(α) .
We write the three–point boundary value problem (4.1.1),(4.1.2) as an equivalent integral equation
y(t) =
∫ 1
0G(t, s)a(s)g(s, y(s))ds. (4.1.9)
By a solution of the boundary value problem (4.1.1),(4.1.2), we mean a solution of the integral equation
(4.1.9). Define a cone U ⊂ B = (C[0, 1], ∥.∥) by
U = y ∈ B : y(t) ≥ 0, minℓ≤t≤1
y(t) ≥ γα∥y∥
and an operator T : U → U by
T y(t) =∫ 1
0G(t, s)a(s)g(s, y(s))ds, 0 ≤ t ≤ 1. (4.1.10)
By a solution of the integral equation (4.1.9), we mean a fixed point of the operator T .
63
Lemma 4.1.3. Under the assumption (C1), the operator T : U → U is completely continuous.
Proof. The operator T is continuous and T y(t) ≥ 0. Also, for all y ∈ U ,
minℓ≤t≤1
T y(t) = minℓ≤t≤1
(∫ 1
0G(t, s)a(s)g(s, y(s))ds
)≥ γα
(∫ 1
0Φα(s)a(s)g(s, y(s))ds
)≥ γα∥T y(t)∥,
which implies that T (U) ⊂ U . For fixed R > 0, consider a bounded subset M of U defined by
M = y ∈ U : ∥y∥ < R.
Define L = max0≤y≤R
g(t, y(t)) + 1 and K = max0≤t≤1
a(t) + 1, then for y ∈ M, we have
|T y(t)| =∣∣∣∣∫ 1
0G(t, s)a(s)g(s, y(s))ds
∣∣∣∣ ≤ KL
∫ 1
0Φα(s)ds =
KL
αΓ(α)(1− ξ).
Hence ∥T y(t)∥ ≤ KLαΓ(α)(1−ξ) , which implies that T (M) is uniformly bounded. Since g is continuous, it
follows that T is completely continuous.
4.1.2 Existence of at least one positive solution
Theorem 4.1.4. Assume that (C1)-(C3) hold. Then the boundary value problem (4.1.1), (4.1.2) has at
least one positive solution.
Proof. The proof is similar to the proof of the Theorem 3.6 in [127].
Now, we study existence of at least one positive solution under a weaker hypothesis on g, that is,
limy→0
g(t,y(t))yµ , lim
y→∞g(t,y(t))yν ∈ 0,∞ for µ, ν > 0.
Theorem 4.1.5. Assume that (C1) and (C4) are satisfied, then, the boundary-value problem (4.1.1),
(4.1.2) has at least one positive solution y∗ such that r∗1 < ∥y∗∥ < r∗2.
Proof. Define Er∗1 = y ∈ B : ∥y∥ < r∗1 and Er∗2 = y ∈ B : ∥y∥ < r∗2. For y ∈ U ∩ ∂Er∗1 , using Lemma
4.1.2, we obtain
|T y(t)| ≤ κ−1r∗1
∫ 1
0Φα(s)a(s)ds < r∗1 = ∥y∥,
which implies that
∥T y(t)∥ ≤ ∥y∥, for y ∈ U ∩ ∂Er∗1 . (4.1.11)
For y ∈ U ∩ ∂Er∗2 , we have, y ≥ minℓ≤t≤1
y(t) ≥ γαr∗2. Using Lemma 4.1.2, we get
|T y(t)| ≥ κ−1r∗2
∫ 1
0Φα(s)a(s)ds > r2 = ∥y∥,
which yields
∥T y(t)∥ ≥ ∥y∥, for y ∈ U ∩ ∂Er∗2 . (4.1.12)
Hence, in view of (4.1.11), (4.1.12) and Theorem 2.3.7, it follows that T has at least one fixed point y∗ in
U ∩ (Er∗2\Er∗1 ) satisfying r∗1 < ∥y∗∥ < r∗2.
64
4.1.3 Existence of at least two positive solutions
Theorem 4.1.6. Assume that (C1), (C5) and (C6) are satisfied, then the boundary value problem (4.1.1),
(4.1.2) has at least two positive solution y1, y2 such that 0 < ∥y1∥ < ρ < ∥y2∥.
Proof. In view of (C5), there exists m∗1 ∈ (0, ρ) such that for all ε∗1 > γ−2
α κ,
g(t, y(t)) ≥ ε∗1yµ1 , for y ≥ m∗
1.
Define Er1 = y ∈ B : ∥y∥ < r1, where 0 < r1 < m∗1 and using Lemma 4.1.2, we have
|T y(t)| ≥ ε∗1γ2α
∫ 1
0Φα(s)a(s)∥y∥µ1ds > r1 = ∥y∥ for y ∈ U ∩ ∂Er1 .
Hence,
∥T y∥ ≥ ∥y∥, for y ∈ U ∩ ∂Er1 . (4.1.13)
Now, for µ2 ≥ 1, using limu→∞g(t,y(t))yµ2 = ∞, it follows that there exists m∗
2 > ργα
such that for all
0 < ε∗2 < κγ−(µ2+1)α , we have
g(t, y(t)) ≥ ε∗2yµ2 , for y ≥ γαm
∗2.
Let Er2 = y ∈ B : ∥y∥ < r2, where r2 ≥ m∗2. Since y ≥ min
ℓ≤t≤1y(t) ≥ γα∥y∥ for y ∈ U ∩ ∂Er2 , it follows
that
|T y(t)| ≥ ε∗2γ1+µ2α
∫ 1
0Φα(s)a(s)ds > r2 = ∥y∥,
which implies that
∥T y(t)∥ ≥ ∥y∥, for y ∈ U ∩ ∂Er2 . (4.1.14)
Using (C6), we obtain
g(t, y(t)) ≤ ερφ1(t), for 0 ≤ y ≤ ρ.
Hence, for y ∈ U ∩ ∂Eρ, where Eρ = y ∈ B : ∥y∥ < ρ, we have
|T y(t)| ≤ ερ
∫ 1
0Φα(s)a(s)φ1(t)ds < ρ = ∥y∥,
which implies that
∥T y(t)∥ ≤ ∥y∥, for y ∈ U ∩ ∂Eρ. (4.1.15)
Since r1 < ρ < r2, by (4.1.13), (4.1.14), (4.1.15) and Theorem 2.3.7, it follows that T has at least two
fixed points y1 ∈ U ∩ (∂Eρ\∂Er1) and y2 ∈ U ∩ (∂Er2\∂Eρ) such that 0 < ∥y1∥ < ρ < ∥y2∥.
Theorem 4.1.7. Assume that (C1), (C7), (C8) hold, then the boundary value problem (4.1.1), (4.1.2)
has at least two positive solution y∗1 and y∗2 satisfying 0 < ∥y∗1∥ < ρ∗ < ∥y∗2∥.
65
Proof. In view of limy→0g(t,y(t))yµ1 = 0, for any ε ∈ (0, κ] there exists m ∈ [0, ρ∗] such that
g(t, y(t)) ≤ εyµ1 , for u ∈ (0,m].
Let Er = y ∈ B : ∥y∥ < r, for 0 < r ≤ m. For any y ∈ U ∩ ∂Er we have
|T y(t)| ≤ εr
∫ 1
0Φα(s)a(s)ds < r = ∥y∥,
which implies
∥T y(t)∥ ≤ ∥y∥, for y ∈ U ∩ ∂Er. (4.1.16)
Also, in view of limu→∞
g(t,y(t))yµ2 = 0, for any ε∗ ∈ (0, κ2 ], there exists m∗ > ρ∗ such that
g(t, y(t)) ≤ ε∗yµ2 , for y ≥ m∗.
Hence, g(t, y(t)) ≤ ε∗yµ2 + L, for y ∈ [0,∞) where L = max
0≤t≤1,0≤y≤m∗|g(t, y(t))| + 1. Choose r∗ >
max2m∗, 2Lκ−1 and define Er∗ = y ∈ B : ∥y∥ < r∗. For y ∈ U ∩ ∂Er∗ , using Lemma 4.1.2, we obtain
|T y(t)| ≤ ε∗rµ2∗
∫ 1
0Φα(s)a(s)ds+ L
∫ 1
0Φα(s)a(s)ds <
r∗2
+r∗2
= r∗ = ∥y∥,
which implies that
∥T y(t)∥ ≤ ∥y∥ for y ∈ U ∩ ∂Er∗ . (4.1.17)
Finally, for y ∈ U ∩ ∂Eρ∗ , where Eρ∗ = y ∈ B : ∥y∥ < ρ∗, using lemma ?? and (C8), we obtain
|T y(t)| ≥ ε∗ρ∗∫ 1
0Φα(s)a(s)φ2(t)ds > ρ∗ = ∥y∥,
which implies
∥T y(t)∥ ≥ ∥y∥, for y ∈ U ∩ ∂Eρ∗ . (4.1.18)
Since r < ρ∗ < r∗, by (4.1.16), (4.1.17), (4.1.18) and Theorem 2.3.7, it follows that T has at least two
fixed points, y∗1 ∈ U ∩ (∂Eρ∗\∂Er) and y∗2 ∈ U ∩ (∂Er∗\∂Eρ∗) such that 0 < ∥y∗1∥ < ρ < ∥y∗2∥.
4.1.4 Existence of at least three positive solutions
Theorem 4.1.8. Suppose there exist constants 0 < a < b < b/γα ≤ c such that
(i) g(t, y) < κa, for 0 ≤ t ≤ 1, 0 ≤ y ≤ a,
(ii) g(t, y) ≥ κ bγα
, for 0 6 t ≤ 1, b ≤ y ≤ b/γα,
(iii) g(t, y) ≤ κc, for 0 ≤ t ≤ 1, 0 6 y ≤ c.
Then the boundary value problem (4.1.16), (4.1.16) has at least three positive solutions y1, y2 and y3 such
that ∥y1∥ < a, b < θ(y2) and ∥y3∥ > a with θ(y3) < b.
66
Proof. Let y ∈ Uc, then ∥y∥ 6 c and by (iii), we have
∥T y(t)∥ = max0≤t≤1
∣∣∣∣∫ 1
0G(t, s)a(s)g(t, s)ds
∣∣∣∣ ≤ ∫ 1
0Φα(s)a(s)κcds = c,
which implies that T : Uc → Uc.Similarly if y ∈ Ua then in view of (i), we have ∥T y(t)∥ < a . Furthermore, T : Uc → Uc is completely
continuous operator. Hence, condition (ii) of Theorem 2.3.8 is satisfied.
Now choose y(t) = b/γα, 0 ≤ t ≤ 1. Obviously y(t) = bγα
∈ Uθ(b, bγα), which implies that y ∈
Uθ(b, bγα) : θ(y) > b = ∅. Hence, if y ∈ Uθ(b, c), then b ≤ y(t) ≤ c for 0 ≤ t ≤ 1. Also, in view of (ii) and
Lemma 4.1.2, we have
θ(T y(t)) = min0≤t≤1
|T y(t)| ≥∫ 1
0γαΦα(s)a(s)g(t, s)ds >
∫ 1
0Φα(s)a(s)κcds = b
Therefore ∥T y(t)∥ > b, for y ∈ Uθ(b, c) which is condition (i) of Theorem 2.3.8.
Finally to check condition (iii) of Theorem 2.3.8 choose y ∈ Uθ(b, c) with ∥T y∥ > bγα
then,
θ(T y(t)) = min0≤t≤1
|T y(t)| ≥ γα∥T y∥ > γαb
γα= b.
Therefore condition (iii) of Theorem 2.3.8 is satisfied. Thus, the boundary value problem (4.1.2), (4.1.2)
has at least three positive solutions y1, y2 and y3 such that ∥y1∥ < a, b < θ(y2) and ∥y3∥ > a with
θ(y3) < b.
Example 4.1.9. Consider the problem
cD5/3y(t) + a(t)g(t, y) =0, 0 < t < 1, (4.1.19)
y′(0) = 0,1
2y(
1
8) =y(1), (4.1.20)
where
a(t) = e−t, g(t, y) =
Γ(53)u
23 + e−t
12 , 0 ≤ y ≤ 10,
Γ(53)u23 + y2
√u−10
5√3
+ e−t
12 , 10 < y ≤ 30,
2367375000 + e−t
12 , y ≥ 30.
Where κ = 1.59923, a = 10, b = 15, c = 350; consequently we have
g(t, y) ≤ 4.27351 < κa ≈ 15.92228, for 0 ≤ y ≤ 10,
g(t, y) ≥ 63.616 > κb
γα≈ 47.768, for 15 ≤ y ≤ 30,
g(t, y) ≤ 473.557 < κc ≈ 557.28, for 0 ≤ y ≤ 350.
Thus all the assumptions of Theorem 4.1.8 are satisfied. Therefore the problem (4.1.19), (4.1.20) has at
least three positive solutions y1, y2 and y3 such that ∥y1∥ < 10, 15 < θ(y2) and ∥y3∥ > 10 with θ(y3) < 15.
67
4.2 Positive solutions to three–point boundary value problems (II)
In this section, we study existence of positive solutions to nonlinear higher order nonlocal boundary value
problems corresponding to fractional differential equations of the type [85]
cDαy(t) + g(t, y(t)) = 0, t ∈ [0, 1], n− 1 < α < n, n ∈ N, (4.2.1)
y′(0) = µy′(η)− θ1, y′′(0) = 0, y′′′(0) = 0, . . . , y(n−1)(0) = 0, y(1) = νy(η) + θ2, (4.2.2)
where, 0 < η, µ, ν < 1, n > 2, the boundary parameters θ1, θ2 ∈ R+. Sufficient conditions are obtained
for the existence of positive solutions using the Guo–Krasnosel’skii fixed point theorem. The conditions
for the existence of unique positive solution are also obtained.
4.2.1 Green’s function and its properties
Lemma 4.2.1. Let h ∈ C[0, 1], then, the linear problem
cDαy(t) + ψ(t) = 0, t ∈ (0, 1), n− 1 < α < n, (4.2.3)
y′(0) = µy′(η)− θ1, y′′(0) = 0, y′′′(0) = 0, . . . , y(n−1)(0) = 0, y(1) = νy(η) + θ2, (4.2.4)
has solution
y(t) =
∫ 1
0Gα(t, s)ψ(s)ds+
∫ 1
0Gα,β(t; η, s)ψ(s)ds+ φ(t), (4.2.5)
where,
Gα(t, s) =
(1−s)α−1−(t−s)α−1
Γ(α) , s ≤ t,
(1−s)α−1
Γ(α) , t ≤ s,Gα,β(t; η, s) =
ν[(1−s)α−1−(η−s)α−1]
(1−ν)Γ(α)
+µ[1−νη−(1−ν)t](η−s)α−2
(1−µ)(1−ν)Γ(α−1) , s ≤ η,
ν(1−s)α(1−ν)Γ(α−1) , η ≤ s
and φ(t) =(1−νη−(1−ν)t(1−µ)(1−ν)
)θ1 +
θ21−ν .
Proof. In view of Lemma 2.2.24, the equation (4.2.3) is equivalent to the integral equation
y(t) = −Iαψ(t) + c1 + c2t+ c3t2+, . . . ,+cnt
n−1.
Using Lemma 2.2.23, we obtain
y′(t) = −Iα−1ψ(t) + c2 + 2c3t+, . . . ,+(n− 1)cntn−2,
y′′(t) = −Iα−2ψ(t) + 2c3+, . . . ,+(n− 1)(n− 2)cntn−3,
. . . ,
y(n−1)(t) = −Iα−(n−1)ψ(t) + (n− 1)!cn,
where c1, c2, . . . , cn ∈ R. Using the boundary conditions, y′′(0) = 0, y′′′(0) = 0, . . . , y(n−1)(0) = 0, we get
c3 = 0, c4 = 0,· · · , cn = 0. By the boundary conditions y′(0) = µy′(η)− θ1, y(1) = νy(η) + θ2, we have
(1− µ)c2 = −µIα−1ψ(η)− θ1, (1− ν)c1 + (1− νη)c2 = Iαψ(1)− νIαψ(η) + θ2,
68
which implies that c2 = − µ1−µI
α−1ψ(η)− θ11−µ and
c1 =1
1− νIαψ(1)− ν
1− νIαψ(η) + µ(1− νη)
(1− µ)(1− ν)Iα−1ψ(η) +
(1− νη)θ1(1− µ)(1− ν)
+θ2
(1− ν).
Therefore, the solution of the linear boundary value problem (4.2.1), (4.2.2) is given by
y(t) =− Iαψ(t) +1
1− νIαψ(1)− ν
1− νIαψ(η) +
µ(1− νη)− µ(1− ν)t
(1− µ)(1− ν)Iα−1ψ(η) + φ(t)
=− Iαψ(t) + Iαψ(1) +ν
1− νIαψ(1)− ν
1− νIαψ(η) +
µ(1− νη)− µ(1− ν)t
(1− µ)(1− ν)Iα−1ψ(η) + φ(t)
=
∫ t
0
(1− s)α−1 − (t− s)α−1
Γ(α)ψ(s)ds+
∫ 1
t
(1− s)α−1
Γ(α)ψ(s)ds+
∫ η
0
[ν(1− s)α−1 − ν(η − s)α−1
(1− ν)Γ(α)
+((µ(1− νη)− µ(1− ν)t)(η − s)α−2
(1− µ)(1− ν)Γ(α− 1)
]ψ(s)ds+
∫ 1
η
ν(1− s)α−1
(1− ν)Γ(α)ψ(s)ds+ φ(t)
=
∫ 1
0
Gα(t, s)ψ(s)ds+
∫ 1
0
Gα,β(t; η, s)ψ(s)ds+ φ(t).
Lemma 4.2.2. The functions Gα(t, s) and Gα,β(t; η, s) satisfy the following properties:
(i) Gα(t, s) ≥ 0 , Gα,β(t; η, s) ≥ 0 and Gα(t, s) ≤ Gα(s, s) for all 0 ≤ s, t ≤ 1;
(ii) minξ≤t≤τ
Gα(t, s) ≥ (1− τα−1) max0≤t≤1
Gα(t, s) = (1− τα−1)Gα(s, s), for s ∈ (0, 1), 0 < ξ < τ < 1,
(iii) ν(1− ηα−1)(1− s)α−1 ≤ (1− ν)Γ(α)Gα,β(t; η, s) < 2(α− 1)(1− s)α−2.
(iv) minξ≤t≤τ
Gα,β(t; η, s) ≥ (1− τ) max0≤t≤1
Gα,β(t; η, s), for s ∈ (0, 1), 0 < ξ < τ < 1.
Proof. (i) : For α > 1, by definition of Gα(t, s), it follows that Gα(t, s) ≥ 0 and Gα(t, s) ≤ Gα(s, s) for all
0 ≤ s, t ≤ 1.
For s ≤ η and 0 < ν < 1, we have 1− νη − (1− ν)t > (1− νη)(1− t). Hence, Gα,β(t, η, s) > 0. Also we
note that Gα,β(t, η, s) > 0 for s ≥ η.
(ii) : Obviously, maxt∈[0,1]Gα(t, s) = (1−s)α−1
Γ(α) . Now, for s ≤ t, t ∈ [ξ, τ ] we have following estimate for
Gα(t, s):
Gα(t, s) =1
Γ(α)[(1− s)α−1 − (t− s)α−1] = max
t∈[0,1]Gα(t, s)−
(t− s)α−1
Γ(α)≥ max
t∈[0,1]Gα(t, s)− τα−1 (1−
st )α−1
Γ(α).
Since (1− st )α−1 ≤ (1− s)α−1. Which implies that
mint∈[ξ,τ ]
Gα(t, s) ≥ (1− τα−1)maxt∈[0,1]
Gα(t, s), for s ∈ (0, 1).
Now for s ≤ η, we have
Gα,β(t; η, s) =ν[(1− s)α−1 − (η − s)α−1]
(1− ν)Γ(α)+µ[1− νη − (1− ν)t](η − s)α−2
(1− µ)(1− ν)Γ(α− 1)
≤ν(1− s)α−1
(1− ν)Γ(α)+
µ(η − s)α−2
(1− µ)(1− ν)Γ(α)<
2(α− 1)(1− s)α−2
(1− µ)(1− ν)Γ(α).
69
For s ≥ η, obviously Gα,β(η, s) <2(α−1)(1−s)α−2
(1−µ)(1−ν)Γ(α) . From the definition of Gα,β(η, s), it clearly follows that
Gα,β(t; η, s) ≥ν(1− ηα−1)(1− s)α−1
(1− ν)Γ(α).
(iii) : From the definition of Gα,β(t; η, s), we have
∂
∂t(Gα,β(t; η, s)) =
−(η − s)α−2
(1− µ)Γ(α− 1)≤ 0.
Therefore Gα,β(t; η, s) is non–increasing in t, so its minimum value occurs at t = τ for t ∈ [ξ, τ ] and its
maximum value occurs at t = 0 for t ∈ [0, 1]. That is
minξ≤t≤τ
Gα,β(t; η, s) =ν[(1− s)α−1 − (η − s)α−1]
(1− ν)Γ(α)+µ[1− νη − (1− ν)τ ](η − s)α−2
(1− ν)Γ(α)(4.2.6)
and
max0≤t≤1
Gα,β(t; η, s) =ν[(1− s)α−1 − (η − s)α−1]
(1− ν)Γ(α)+µ(1− νη)(η − s)α−2
(1− ν)Γ(α). (4.2.7)
Since (1− s)α−1 − (η − s)α−1 ≥ 0 and 1− τ < 1, therefore
(1− s)α−1 − (η − s)α−1 ≥ (1− τ)((1− s)α−1 − (η − s)α−1). (4.2.8)
Also, as 1− ν ≤ 1− νη. Thus,
1− νη − (1− ν)τ ≥1− νη − (1− νη)τ
≥(1− τ)(1− νη).(4.2.9)
Subletting (4.2.8) and (4.2.9) in (4.2.6) and using (4.2.7), we have
minξ≤t≤τ
Gα,β(t; η, s) ≥ (1− τ)
ν[(1− s)α−1 − (η − s)α−1]
(1− ν)Γ(α)+µ(1− νη)(η − s)α−2
(1− ν)Γ(α)
= (1− τ) max
0≤t≤1Gα,β(t; η, s).
Remark 4.2.3. For θ1, θ2 > 0, minξ≤t≤τ
φ(t) ≥ (1− τ) max0≤t≤1
φ(t).
Proof. As ddtφ(t) = − θ1
(1−µ) < 0, therefor φ(t) is a decreasing function. Hence
max0≤t≤1
φ(t) = φ(0) =(1− νη)θ1
(1− µ)(1− ν)+
θ2(1− ν)
and
minξ≤t≤τ
φ(t) = φ(τ) =[1− νη − (1− ν)τ ]θ1
(1− µ)(1− ν)+
θ2(1− ν)
≥ (1− τ)(1− νη)θ1(1− µ)(1− ν)
+(1− τ)θ2(1− ν)
= (1− τ) max0≤t≤1
φ(t).
Thus, we have minξ≤t≤τ
φ(t) ≥ (1− τ) max0≤t≤1
φ(t).
70
Define a cone V in B by V = y ∈ B : min y(t)ξ≤t≤τ
≥ (1− τ)∥y∥. Also, define T : B → B as
T y(t) =∫ 1
0Gα(t, s)g(s, y(s))ds+
∫ 1
0Gα,β(t; η, s)g(s, y(s))ds+ φ(t). (4.2.10)
The boundary value problem (4.2.1), (4.2.2) has a solution if and only if T has a fixed point.
Lemma 4.2.4. Assume that (C1)(i) is satisfied. Then operator T : V → V is completely continuous.
Proof. First we prove that T (V) ⊂ V. From (4.2.10), Lemma 4.2.2 and Remark 4.2.3, we have
minξ≤t≤τ
(T y(t)) ≥ (1− τα−1)
∫ 1
0
Gα(s, s)g(s, y(s))ds+ (1− τ)
∫ 1
0
maxt∈[0,1]
Gα,β(t; η, s)g(s, y(s))ds+ (1− τ) maxt∈[0,1]
φ(t)
≥ (1− τ)
∫ 1
0
Gα(s, s)g(s, y(s))ds+
∫ 1
0
maxt∈[0,1]
Gα,β(t; η, s)g(s, y(s))ds+ maxt∈[0,1]
φ(t)
≥ (1− τ)∥T y∥.
Hence, T (V) ⊂ V.
Consider a bounded subset M of V defined by M = y ∈ V : ∥y∥ ≤ ℓ, ℓ ∈ R+. Let Λ = max0≤y≤ℓ
g(t, y(t))+1.
Then, for y ∈ M, we have
|T y(t)| ≤∫ 1
0Gα(t, s)
∣∣g(s, y(s))∣∣ds+ ∫ 1
0Gα,β(t; η, s)
∣∣g(s, y(s))∣∣ds+ maxt∈[0,1]
φ(t)
≤ Λ
Γ(α)
∫ 1
0(1− s)α−1ds+
2Λ(α− 1)
(1− µ)(1− ν)Γ(α)
∫ 1
0(1− s)α−2ds+
(1− νη)θ1 + (1− µ)θ2(1− µ)(1− ν)
≤ Λ
(1− µ)(1− ν)
(Iα(1) + 2Iα(1)
)+
(1− νη)θ1 + (1− µ)θ2(1− µ)(1− ν)
=(3α− 1)Λ
α(1− µ)(1− ν)(α− 1)+
(1− νη)θ1 + (1− µ)θ2(1− µ)(1− ν)
.
Hence T (M) is a bounded set.Finally we show that T is compact operator. Define δ = α(1−µ)Γ(α)ε
Λ[α(1−µ)+µηµ−1+θ1]and choose t > τ such that
t− τ < δ. Then, for all ε > 0 and y ∈ M, we have
|T y(t)− T y(τ)| =∣∣∣∣∫ 1
0
(Gα(t, s)−Gα(τ, s))g(s, y(s))ds+
∫ 1
0
(Gα,β(t; η, s)−Gα,β(t; η, s))g(s, y(s))ds+ φ(t− τ)
∣∣∣∣≤ Λ
[∫ 1
0
∣∣Gα(t, s)−Gα(τ, s)∣∣ds+ ∫ 1
0
∣∣Gα,β(t; η, s)−Gα,β(t; η, s)∣∣ds+ θ1
1− µ(t− τ)
]= Λ
[1
Γ(α)
∫ t
0
((t− s)α−1 − (τ − s)α−1)ds+µ(t− τ)
(1− µ)Γ(α− 1)
∫ η
0
(η − s)α−2ds+θ1
1− µ(t− τ)
]=
Λ
αΓ(α)
[tα − τα +
µηα−1 + θ11− µ
(t− τ)].
An application of the Mean Value theorem yields, tα − τα ≤ α(t− τ) < αδ. Hence, it follows that
|T y(t)− T y(τ)| < Λδ[α(1− µ) + µηµ−1 + θ1]
α(1− µ)Γ(α)< ε.
The Arzela–Ascoli theorem guarantees that the operator T : V → V is completely continuous.
For simplicity, we introduce following notations:
Kµ,ν =(3Iαψ1(1) +
6Iα−1ψ1(1)
(1− µ)(1− ν)
)−1, Kν =
([(1− τ) +
ν(1− η)
1− ν
]Iαξ+ψ2(τ)
)−1
where, ψ1, ψ2 ∈ C([0, 1] → R+), and
cµ =1
3(1− µ), cν =
1
3(1− ν), γα = 1− τα−1.
71
4.2.2 Existence of positive solutions
In the following theorem, we establish sufficient conditions for the existence of at least one positive solution
for the boundary value problem (4.2.1), (4.2.2).
Theorem 4.2.5. Suppose that (C1)(i) is satisfied. Furthermore, there exist constants r1, r2 ∈ R+ such
that r2 > r1 and functions ψ1, ψ2 ∈ C([0, 1] → R+) such that
(i) g(t, u) ≥ r2Kνψ2(t), for (t, y) ∈ [0, 1]× [γαr2, r2],
(ii) g(t, u) ≤ r1K1µ,νψ1(t), for (t, y) ∈ [0, 1]× [0, r1],
then, the boundary value problem (4.2.1), (4.2.2) has at least one positive solution for θ1, θ2 small enough
and has no positive solution for θ1, θ2 such that θ1 ≤ cµr1 and θ2 ≤ cνr1.
Proof. By Lemma 4.2.4, the operator T is completely continuous. We show that the problem (4.2.1),
(4.2.2) has at least one positive solution in V. Define an open subset of the Banach space B as Er1 = y ∈B : ∥y∥ < r1. Then, for any y ∈ V ∩ ∂Er1 , we have ∥y∥ = r1 and in view of Lemma 4.2.2 and equation
(4.2.10), it follows that
|T y(t)| =∣∣∣ ∫ 1
0Gα(t, s)g(s, y(s))ds+
∫ 1
0Gα,β(t; η, s)g(s, y(s))ds+
(1− νη − (1− ν)t
(1− µ)(1− ν)
)θ1 +
θ21− ν
∣∣∣≤ Iαg(1, y(1)) + 2
(1− µ)(1− ν)Iα−1g(1, y(1)) +
(1− νη)θ1 + (1− µ)θ2(1− µ)(1− ν)
≤ Kµ,νr1
[Iαψ1(1) +
2Iα−1ψ1(1)
(1− µ)(1− ν)
]+
θ11− µ
+θ2
1− ν≤ r1
3+r13
+r13
= ∥y∥.
Therefore, we have ∥T y∥ ≤ ∥y∥, for y ∈ V ∩ ∂Er1 .Define Er2 = y ∈ V : ∥y∥ < r2. For any t ∈ [ξ, τ ] and y ∈ V ∩ ∂Er2 , using Lemma 4.2.1, we have
minξ≤t<τ
y(t) ≥ (1− τ)∥y∥. Therefore, by Remark 4.2.3 and equation (4.2.10), we have the following estimate
|T y(t)| =∣∣∣ ∫ 1
0Gα(t, s)g(s, y(s))ds+
∫ 1
0Gα,β(t; η, s)g(s, y(s))ds+ φ(t)
∣∣∣≥[(1− τα−1) +
ν(1− ηα−1)
1− ν
]Iαξ+g(τ, y(τ)) + φ(t)
≥ r2Kν
[(1− τ) +
ν(1− η)
1− ν
]Iαξ+ψ2(τ) = r2 = ∥y∥.
Thus ∥T y∥ ≥ ∥y∥ for y ∈ V ∩ ∂Er2 .Hence, by Theorem 2.3.7, it follows that T has a fixed point y in V ∩ (Er2\Er1), which is a solution to the
boundary value problem (4.2.1), (4.2.2).
Example 4.2.6. Consider the boundary value problem
cDαy(t) =t2e
32t + 1
8(1 + t2)+t2y2(1 + sin t)
16π, t ∈ [0, 1], (4.2.11)
y(1) = νy(η) + θ2, y′(0) = µy′(η)− θ1, y
′′(0) = 0, y′′′(0) = 0, (4.2.12)
72
where, α = 72 , η = 1
2 , µ = 23 and ν = 1
3 . Let g(t, y) = t2e32 t+1
8(1+t2)+ t2y2(1+sin t)
16π , (t, y) ∈ [0, 1] × [0,∞]. For
r1 = 1, r2 = 550, we observe that
g(t, y) ≤ 1
8
(e+
1
π
)t2, for (t, y) ∈ [0, 1]× [0, 1]
and
g(t, y) ≥(
1√e
)t2
1 + t2for (t, y) ∈ [0, 1]× [283, 550].
Choose ψ1(t) = t2 and ψ2(t) =t2
1+t2and ξ = 1
4 , τ = 34 . By computations we obtain cµ = 1
9 , cν = 29 ,
Iαξ+ψ2(τ) =1
60√π
(49
√7 tan−1
(√27
)− tan−1
(√2)− 1132
√2
21
)≈ 0.001291, also Iαψ1(1) =
12810395
√π. There-
fore Kµ,ν ≈ 0.583566 and Kν ≈ 0.000993874. Hence g(t, y) ≤ 0.583566t2, for (t, y) ∈ [0, 1] × [0, 1], and
g(t, y) ≥ 0.546631(
t2
1+t2
), for (t, y) ∈ [0, 1] × [283, 550]. The assumptions (i) and (ii) of the Theorem
4.2.5 are satisfied. Therefore the boundary value problem (4.2.11), (4.2.12) has a positive solution for
θ1 ∈ [0, 19 ], θ2 ∈ [0, 29 ].
4.2.3 Uniqueness of positive solution
Theorem 4.2.7. Assume that there exists ω(t) ∈ C([0, 1] → R+) such that
|g(t, y)− g(t, z)| ≤ ω(t)|y − z|, for t ∈ [0, 1], y, z ∈ [0,∞),
and
λ := Iα|ω(1)|+ 2
(1− µ)(1− ν)Iα−1|ω(1)| < 1,
then, the boundary value problem (4.2.1), (4.2.2) has a unique positive solution.
Proof. For y, z ∈ V, using (4.2.10) and Lemma 4.2.2, we obtain
|T y(t)− T z(t)| ≤∫ 1
0Gα(t, s)|g(s, y(s))− g(s, z(s))|ds+
∫ 1
0Gα,β(t; η, s)|g(s, y(s))− g(s, z(s))|ds
< ∥y − z∥[∫ 1
0
(1− s)α−1
Γ(α)|ω(s)|ds+
∫ 1
0
2(α− 1)(1− s)α−2
(1− µ)(1− ν)Γ(α)|ω(s)|ds
]= ∥y − z∥
[Iα|ω(1)|+ 2
(1− µ)(1− ν)Iα−1|ω(1)|
]= λ∥y − z∥.
Therefore, ∥T y(t) − T z(t)∥ ≤ λ∥y − z∥. Hence, it follows by Banach Contraction principle that the
boundary value problem (4.2.1), (4.2.2) has a unique positive solution.
Example 4.2.8. Consider the boundary value problem
(14√π + 425e
18t)cDαy(t)(1 + y(t)) = e−33t(11
√πe33t cos(25t) + 500 sin2 t)y2, (4.2.13)
y(1) = νy(η)+θ2 , y′(0) = µy′(η)− θ1, y
′′(0) = 0, y′′′(0) = 0, (4.2.14)
73
where α = 72 , ν = 3
5 , µ = 23 , η = 1
2 and θ1, θ2 ∈ R.
Let g(t, y) = (11√πe33t cos(25t)+500 sin2 t)y2e−33t
(14√π+425e
18 t)(1+y)
, (t, y) ∈ [0, 1]× [0,∞]. For u, v ∈ V, t ∈ [0, 1], we have
|g(t, y)− g(t, z)| ≤ e−33t(11√πe33t + 500)
(14√π + 425e
18t)
(∣∣∣∣ y2
1 + y+
v2
1 + z
∣∣∣∣)≤ e−33t(11
√πe33t + 500)
(14√π + 425e
18t)
(|y − z|(y + z + yz)
(1 + y)(1 + z)
)≤ e−33t(11
√πe33t + 500)
(14√π + 425e
18t)
(|y − z|) .
Here, ω(t) = e−33t(11√πe33t+500)
(14√π+425e
18 t)
. After some calculations, we have Iα−1ω(1) ≈ 0.013469, λ ≈ 0.215504 < 1.
All the conditions of Theorem 4.2.7 are satisfied. Therefore, by Theorem 4.2.7, the boundary value problem
(4.2.13), (4.2.14) has a unique positive solution.
Chapter 5
Existence and multiplicity of positive
solutions for non linear systems of
fractional differential equations
In this chapter we are concerned with the existence and multiplicity of positive solutions for systems of
fractional order differential equations with two-point and three–point boundary conditions. Systems of
fractional differential equations are used to model the problems in physics, biosciences and engineering.
For example, in [11] Atanackovic and Stankovic have described the motion of an elastic column fixed at
one end and loaded at other by modeling it as a system of fractional differential equations. The existence
theory for coupled systems of fractional differential equations is well established in [8,95,129,150]. There
are some recent works dealing with the existence and multiplicity of positive solutions for systems of
fractional differential equations,for example, [14, 18,142].
In section 5.1, we study a coupled system of fractional differential equation, which is an extension
of scalar fractional differential equation considered in [145]. Some growth conditions are imposed on the
nonlinear functions (f and g) involved to obtain the conditions for the existence of positive solutions,
and also the explicit interval are obtained for the parameter for which there exists positive or multiple
positive solutions for the problem. In section 5.2, we present some existence results for positive and
multiple positive solutions for more general systems differential equations satisfying three–point boundary
conditions.
5.1 Positive solutions for a coupled system
In this section, we study existence results for positive solutions to a coupled system of fractional differential
equations [88] cDαx(t) + λφ(t)f(y(t)) = 0, n− 1 < α ≤ n,
cDβy(t) + λψ(t)g(x(t)) = 0, n− 1 < β ≤ n,(5.1.1)
74
75
satisfying the two point boundary conditionsx(1) = 0, x′(0) = 0, x′′(0) = 0, · · · , x(n−2)(0) = 0, x(n−1)(0) = 0,
y(1) = 0, y′(0) = 0, y′′(0) = 0, · · · , y(n−2)(0) = 0, y(n−1)(0) = 0,(5.1.2)
where t ∈ [0, 1], λ > 0. It is assumed that f, g : [0,∞) → [0,∞) are continuous and trφ(t), tsψ(t) : [0, 1] →[0,∞) are also assumed to be continuous for r, s ∈ [0, 1) and do not vanish identically on any subinterval.
We introduce following notations:
f0 = limx→0
f(x)
x, f∞ = lim
x→∞
f(x)
x, g0 = lim
x→0
g(x)
x, g∞ = lim
x→∞
g(x)
x.
The following assumptions will be used in the proof of main results.
(H1) f0 = 0, g0 = 0 and f∞ = ∞ , g∞ = ∞;
(H2) f0 = ∞, g0 = ∞ and f∞ = 0 , g∞ = 0;
(H3) f0 = ∞, g0 = ∞, f∞ = ∞, g∞ = ∞;
(H4) f0 = 0, g0 = 0 and f∞ = 0, g∞ = 0;
(H5) g(0) = 0 and f is an increasing function.
5.1.1 Green’s function and its properties
Lemma 5.1.1. [145] Let h ∈ C[0, 1], h ∈ C[0, 1], 0 ≤ r < 1, then the boundary value problem
Dαx(t) + h(t) = 0, t ∈ [0, 1], n− 1 < α ≤ n, (5.1.3)
x(1) = x′(0) =x′′(0) = · · · = x(n−2)(0) = x(n−1)(0) = 0, (5.1.4)
has a unique solution x(t) =∫ 10 Gα(t, s)h(s)ds, where
Gα(t, s) =
(1−s)α−1−(t−s)α−1
Γ(α) , 0 ≤ s ≤ t ≤ 1,
(1−s)α−1
Γ(α) , 0 ≤ t ≤ s ≤ 1.(5.1.5)
Green’s function Gα(t, s) defined by (5.1.5) has the following properties:
(i) Gα(t, s) ≥ 0 for all t, s∈ [0, 1] and Gα(t, s) > 0 for all t, s∈ (0, 1) .
(ii) max0≤t≤1
Gα(t, s) = Gα(s, s), s ∈ [0, 1].
(iii) min1/4≤t≤3/4
Gα(t, s) ≥ γαGα(s, s), where γα :=(1−
(34
)α−1).
The system of boundary value problems (5.1.1), (5.1.2) is equivalent to the following system of integral
equations x(t) = λ∫ 10 Gα(t, s)φ(s)f(y(s))ds, 0 ≤ t ≤ 1,
y(t) = λ∫ 10 Gβ(t, s)ψ(s)g(x(s))ds, 0 ≤ t ≤ 1.
(5.1.6)
76
A pair (x, y) ∈ C[0, 1]×C[0, 1] is a solution of boundary value problem (5.1.1), (5.1.2) if and only if (x, y)
is a solution of the system of integral equations (5.1.6)
Define Υ : B → B by
Υx(t) = λ
∫ 1
0Gβ(t, s)ψ(s)g(x(s))ds, (5.1.7)
then the system of integral equations (5.1.1) takes the formx(t) = λ∫ 10 Gα(t, s)φ(s)f(Υx(s))ds, 0 ≤ t ≤ 1,
y(t) = Υx(t), 0 ≤ t ≤ 1,(5.1.8)
Define T : B → B by
T x(t) = λ
∫ 1
0Gα(t, s)φ(s)f (Υx(s)) ds. (5.1.9)
If x is a fixed point of T , then (x(t),Υx(t)) is a solution of the system (5.1.8).
We use the following notations:
µ := max
∫ 1
0Gα(s, s)φ(s)ds,
∫ 1
0Gβ(s, s)ψ(s)ds
,
ν := min
∫ 3/4
1/4γαGα(s, s)φ(s)ds,
∫ 3/4
1/4γβGβ(s, s)ψ(s)ds
.
Define a cone V ⊂ C[0, 1] by V =
x ∈ C[0, 1] : x(t) ≥ 0, min
1/4≤t≤3/4x(t) ≥ γ∥x∥
, where γ := minγα, γβ.
The cone V ⊂ C[0, 1] induces a partial ordering in the Banach space C[0, 1], i.e, x ≤ y if x− y ∈ V.
Lemma 5.1.2. The operator T : V → V is completely continuous.
Proof. Since f , g, Gα and Gβ are positive functions. Therefore, for any x ∈ V, Υx(t) ≥ 0 and T x(t) ≥ 0.
Using (5.1.7), (5.1.9) and property (iii) of Green’s function, we obtain
min1/4≤t≤3/4
Υx(t) ≥ γβλ
∫ 1
0Gβ(s, s)ψ(s)g(x(s))ds
≥ γ max0≤t≤1
λ
∫ 1
0Gβ(t, s)ψ(s)g(x(s))ds = γ∥Υx∥,
which implies that Υ(V) ⊂ P. Further, for x ∈ V, we have
min1/4≤t≤3/4
T x(t) ≥ γαλ
∫ 1
0Gα(s, s)φ(s)f(Υ(x(s)))ds
≥ γ max0≤t≤1
λ
∫ 1
0Gα(t, s)φ(s)f(Υ(x(s)))ds = γ∥T x∥.
It follows T (V) ⊂ V. By Arzela–Ascoli’s theorem, T : V → V is completely continuous operator.
5.1.2 Existence of at least one positive solution
In this section, we study existence of at least one positive solution of the system (5.1.1), (5.1.2).
77
Theorem 5.1.3. Assume that µmaxf0, g0 < νminf∞, g∞ then for
λ ∈((νminf∞, g∞)−1, (µmaxf0, g0)−1
), (5.1.10)
the boundary value problem (5.1.1), (5.1.2) has at least one positive solution. Moreover, if (H1) holds,
then for any λ ∈ (0,∞), the boundary value problem (5.1.1), (5.1.2) has at least one positive solution.
Proof. In view of (5.1.10), choose sufficiently small ε > 0 such that
(νminf∞ − ε, g∞ − ε)−1 ≤ λ ≤ (µmaxf0 + ε, g0 + ε)−1. (5.1.11)
Furthermore, by the definitions of f0 and g0, there exists a constant r > 0 such that
f(x) ≤ (f0 + ε)x, g(x) ≤ (g0 + ε)x, for x ≤ r. (5.1.12)
Define Er = x ∈ C[0, 1] : ∥x∥ < r. For any x ∈ V ∩ ∂Er, by (5.1.7), (5.1.11) and (5.1.12), we have
Υx(t) = λ
∫ 1
0Gβ(t, z)ψ(z)g(x(z))dz ≤ λ
∫ 1
0Gβ(z, z)ψ(z)(g0 + ε)rdz ≤ λµ(g0 + ε)r ≤ r.
Using (5.1.12), we get
f(Υx(t)) ≤ (f0 + ε)Υx(t), for any x ∈ V ∩ ∂Er. (5.1.13)
In view of (5.1.9) and (5.1.13), we obtain
T x(t) = λ
∫ 1
0Gα(t, s)φ(s)f (Υx(s)) ds
≤ λ
∫ 1
0Gα(s, s)φ(s)(f0 + ε)Υx(t)ds ≤ λµ(f0 + ε)r ≤ r = ∥x∥.
Hence,
∥T x∥ ≤ ∥x∥, for all x ∈ V ∩ ∂Er. (5.1.14)
Next, we consider two cases:
Case 1. f∞, g∞ are finite. Let ε∗ > 0 such that
0 < (νmin(f∞ − ε∗, g∞ − ε∗)−1 ≤ λ. (5.1.15)
It follows from the definitions of f∞, g∞ that there exists constant r > r such that
f(x) ≥ (f∞ − ε∗)x, g(x) ≥ (g∞ − ε∗)x, for x ≥ γr. (5.1.16)
For any x ∈ V ∩ ∂Er, where Er = x ∈ C[0, 1] : ∥x∥ < r, we have
x(t) ≥ min1/4≤t≤3/4
x(t) ≥ ∥x∥ ≥ γr, t ∈ [1/4, 3/4].
Using, (5.1.7) and (5.1.16), we have
Υx(t) ≥λ∫ 3/4
1/4γβGβ(z, z)ψ(z)(g∞ − ε∗)x(z)dz ≥ λr
∫ 3/4
1/4γ22Gβ(z, z)ψ(z)(g∞ − ε∗)dz
≥ rλν(g∞ − ε∗) ≥ γr.
78
Which in view of (5.1.16) gives
f(Υx(t)) ≤ (f∞ − ε∗)Υx(t), for any x ∈ V ∩ ∂Er. (5.1.17)
Using (5.1.9) and (5.1.17), we get
T x(t) ≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)f(Υx(s))ds ≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)(f∞ − ε∗)A(x(s))ds
≥ λ
∫ 3/4
1/4γ2Gα(s, s)φ(s)(f∞ − ε∗)rds ≥ λν(f∞ − ε∗)r ≥ r = ∥x∥.
Hence
∥T x∥ ≥ ∥x∥, for all x ∈ V ∩ ∂Er. (5.1.18)
By the Theorem 2.3.7 and inequalities (5.1.14), (5.1.18) it follows that the operator T has a fixed point
x ∈ V ∩ E r\Er such that r < ∥x∥ < r.
Case 2. f∞ = ∞ and g∞ = ∞. Choose a constant N ∈ R+ satisfying N ≥ (λν)−1. There exists r > r
such that
f(x) ≥ Nx, g(x) ≥ Nx, for x ≥ γr. (5.1.19)
For any x ∈ V ∩ ∂Er, we have
x(t) ≥ min1/4≤t≤3/4
x(t) ≥ ∥x∥ ≥ γr, t ∈ [1/4, 3/4].
Using (5.1.19), we obtain
g(x) ≥ Nx, for any x ∈ V ∩ ∂Er. (5.1.20)
In view of (5.1.7), (5.1.19) and (5.1.20), it follows that
Υx(t) ≥ λ
∫ 3/4
1/4γβGβ(z, z)ψ(z)g(x(z))dz ≥ λ
∫ 3/4
1/4γβGβ(z, z)ψ(z)Nx(z)dz ≥ λνNγr ≥ γr,
This gives
f(Υx(t)) ≥ Nx(t), x ∈ V ∩ ∂Er.
Thus, by (5.1.9), we have
T x(t) ≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)f(Υx(s))ds ≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)NA(x(s))ds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)Nrds ≥ λνNr ≥ r = ∥x∥.
Hence,
∥T x∥ ≥ ∥x∥, for all x ∈ V ∩ ∂Er. (5.1.21)
By Theorem 2.3.7, and the inequalities (5.1.14), (5.1.21) T has at least one fixed point x ∈ V ∩E r\Er such
that r < ∥x∥ < r.
79
Theorem 5.1.4. Assume that (H5) is satisfied and µmaxf∞, g∞ < νminf0, g0, then for
λ ∈((νminf0, g0)−1, (µmaxf∞, g∞)−1
), (5.1.22)
the boundary value problem (5.1.1), (5.1.2) has at least one positive solution. Moreover, if (H5) holds,
then for all λ ∈ (0,∞), the boundary value problem (5.1.1), (5.1.2) has at least one positive solution.
Proof. We consider two cases:
Case 1. f0, g0 are finite. Choose ε > 0 such that 0 < (νminf0 − ε, g0 − ε)−1 ≤ λ. There exists r > 0
such that
f(x) ≥ (f0 − ε)x, g(x) ≥ (g0 − ε)x, for x ∈ (0, r). (5.1.23)
By (H5), we choose 0 < ρ < rmin1, (λµ(g0 − ε))−1 such that
g(x) ≤ (λµ)−1r for x ∈ [0, ρ].
Using (5.1.7), we have
Υx(t) ≤ λ
∫ 1
0Gβ(z, z)ψ(z)g(x(z))dz ≤ λ
∫ 1
0Gβ(z, z)ψ(z)(λµ)
−1rdz ≤ r.
Which in view of (5.1.23) implies
f(Υx(t)) ≥ (f0 − ε)Υx(t) for x ∈ [0, ρ].
For any x ∈ V ∩ ∂Eρ, where Eρ = x ∈ C[0, 1] : ∥x∥ < ρ, we have x(t) ≥ min1/4≤t≤3/4
x(t) ≥ γ∥x∥ = γρ, t ∈
[1/4, 3/4]. Since, νλ(g0 − ε) ≥ 1. Therefore, using (5.1.9) we obtain
T x(t) ≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)(f0 − ε)Υx(s)ds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)(f0 − ε)λ
∫ 3/4
1/4γGβ(z, z)ψ(z)g(x(z))dzds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)(f0 − ε)λ
∫ 3/4
1/4γ2Gβ(z, z)ψ(z)(g0 − ε)∥x∥dzds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)(f0 − ε)νλ(g0 − ε)∥x∥ds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)(f0 − ε)∥x∥ds ≥ λν(f0 − ε)∥x∥ ≥ ∥x∥.
Which yields
∥T x∥ ≥ ∥x∥, for all x ∈ V ∩ ∂Eρ. (5.1.24)
Case 2. f0 = ∞ and g0 = ∞. Choose a constant N > 0 satisfying λνN ≥ 1. There exists r > 0 such that
f(x) ≥ Nx, g(x) ≥ Nx, for x ∈ (0, r). (5.1.25)
80
By (H5), there exists 0 < ρ < r such that g(x) ≤ (λµ)−1r, for x ∈ [0, ρ]. Therefore,
Υx(t) ≤ λ
∫ 1
0γGβ(s, s)ψ(s)g(x(s))ds ≤ λ
∫ 1
0γGβ(s, s)ψ(s)(λµ)
−1rds ≤ r, x ∈ [0, r1].
Which in view of (5.1.25) implies that
f(Υx(t)) ≤ NΥx(t), x ∈ [0, r1].
For any x ∈ V ∩ ∂Eρ, where Eρ = x ∈ C[0, 1] : ∥x∥ < ρ, using (5.1.9), (5.1.25), we have
T x(t) ≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)NΥx(s)ds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)Nλ
∫ 3/4
1/4γGβ(z, z)ψ(z)g(x(z))dzds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)Nλ
∫ 3/4
1/4γ2Gβ(z, z)ψ(z)N∥x∥dzds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)NνλN∥x∥ds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)N∥x∥ds ≥ λνN∥x∥ ≥ ∥x∥.
Thus
∥T x∥ ≥ ∥x∥, for all x ∈ V ∩ ∂Eρ. (5.1.26)
Now, let λ be as given in (5.1.22) and choose ε∗ > 0 small enough such that
(νminf0 − ε∗, g0 − ε∗)−1 ≤ λ ≤ (µmaxf∞ + ε∗, g∞ + ε∗)−1. (5.1.27)
The right hand side of (5.1.27), gives µλ(f0 − ε∗) ≤ 1. Now, by the definitions of f∞, g∞, there exists a
constant r > ρ such that
f(x) ≤ (f∞ + ε∗)x, g(x) ≤ (g∞ + ε∗)x for x ≥ r. (5.1.28)
We, again consider two cases:
Case 1. g is bounded. There exists a constant l > 0, such that
g(x) ≤ l for x ∈ (0,∞).
Let l∗ = maxf(x) : x ∈ [0, λµl] and choose r2 ≥ max2r, λl∗µ. Then for any x ∈ V ∩ ∂Er2 , where
Er2 = x ∈ C[0, 1] : ∥x∥ < r2, using (5.1.7), we get
Υx(t) ≤∫ 1
0λGβ(z, z)ψ(z)ldz ≤ λµl.
Which implies that f(Υx(t)) ≤ l∗. Thus, we have,
T x(t) ≤ λ
∫ 1
0Gα(s, s)φ(s)l
∗ds ≤ λµl∗ ≤ ∥x∥.
81
Hence,
∥T x∥ ≤ ∥x∥, for all x ∈ V ∩ ∂Er2 . (5.1.29)
Case 2. g is unbounded. There exists a constant r∗0 ≥ max2ρ, r such that
g(x) ≤ g(r∗0) for x ∈ (0, r∗0].
Since the function f is increasing, therefore there exists a constant ρ∗ ≥ maxr∗0, µg(r∗0) such that
f(x) ≤ f(ρ∗), for x ∈ (0, ρ∗].
Let r2 ≥ max2r∗0, λµf(ρ∗) and define Er2 = x ∈ C[0, 1] : ∥x∥ < r2. Then for x ∈ V ∩ ∂Er2 , we obtain
Υx(t) ≤ λ
∫ 1
0Gβ(s, s)ψ(s)g(r
∗0)ds ≤ λµg(r∗0) ≤ ρ∗.
Hence,
T x(t) ≤ λ
∫ 1
0Gα(s, s)φ(s)f(Υx(s))ds ≤ λµf(ρ∗) ≤ ∥x∥.
Which implies that
∥T x∥ ≤ ∥x∥, for all x ∈ V ∩ ∂Er2 . (5.1.30)
Therefore, by Theorem 2.3.7, the inequalities (5.1.24) and (5.1.29) or (5.1.24) and (5.1.30) or (5.1.26) and
(5.1.29) or (5.1.26) and (5.1.30) implies that T has a fixed point x ∈ V ∩ Er2\Eρ.
Example 5.1.5. Consider the following system of nonlinear fractional differential equationscD2.5x(t) + λ (56754y2+160)yt72
1+y2= 0, 0 ≤ t ≤ 1,
cD2.7y(t) + λx(x+1)(23471e−x+120)t92
2+x = 0,(5.1.31)
satisfying the boundary conditions x(1) = 0, x′(0) = 0, x′′(0) = 0,
y(1) = 0, y′(0) = 0, y′′(0) = 0,(5.1.32)
Here, α = 2.5, β = 2.7, φ(t) = t72 , ψ(t) = t
92 , f(y) = (56754y2+160)
1+y2and g(x) = (x+1)(23471e−x+120)
2+x . By
direct calculations γ = 0.512860, µ = 0.016155 and ν = 0.001530. By calculations f0 = 160, f∞ = 56754,
g0 = 39295 and g∞ = 120. All conditions of the Theorem 5.1.3 are satisfied. Therefore, the boundary
value problem (5.1.31) and (5.1.32) has a positive solution for each λ ∈ (0.01151, 0.38687).
5.1.3 Existence of at least two positive solutions
Theorem 5.1.6. Assume that (H3), (H5) holds and for λ > 0, there exists r ∈ R+ such that
f(x) < (λµ)−1r, g(x) < (λµ)−1r, for x ∈ [0, r], (5.1.33)
then, the boundary value problem (5.1.1), (5.1.2) has at least two positive solutions.
82
Proof. Set Er = x ∈ C[0, 1] : ∥x∥ < r. For any x ∈ V ∩ ∂Er, using (5.1.33), we obtain
Υ(x(t)) ≤ λ
∫ 1
0Gβ(s, s)ψ(s)g(x(s))ds < λ
∫ 1
0Gβ(s, s)ψ(s)(λµ)
−1rds ≤ r.
Which gives f(Υx(t)) ≤ (λµ)−1r, x ∈ V ∩ ∂Er. Hence,
T (x(t)) ≤ λ
∫ 1
0Gα(s, s)φ(s)f(Υx(s))ds < λ
∫ 1
0Gα(s, s)φ(s)(λµ)
−1rds ≤ r, x ∈ V ∩ ∂Er.
Thus,
∥T x∥ ≤ ∥x∥, for x ∈ V ∩ ∂Er. (5.1.34)
Now, choose N ∈ R+ satisfying N ≥ (λν)−1. From H3, f0 = ∞, g0 = ∞. Therefore, there exists
0 < ρ0 < r such that
f(x) ≥ Nx, g(x) ≥ Nx for x ∈ (0, ρ0). (5.1.35)
By assumption (H5) and (5.1.33), there exists 0 < r∗ < ρ0 such that
g(x) < (λµ)−1ρ0, for x ∈ [0, r∗].
For any x ∈ V ∩ ∂Er∗ , where Er∗ = x ∈ C[0, 1] : ∥x∥ < r∗, we have
Υx(t) ≤ λ
∫ 1
0Gβ(z, z)ψ(z)g(x(z))dz ≤ λ
∫ 1
0Gβ(z, z)ψ(z)(λµ)
−1ρ0dz ≤ ρ0.
Which in view of (5.1.35) implies that f(Υx) ≥ Nx, x ∈ V ∩ ∂Er∗ . Hence,
T x(t) ≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)NΥx(s)ds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)Nλ
∫ 3/4
1/4γGβ(z, z)ψ(z)g(x(z))dzds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)N(λνN)∥x∥ds
≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)N∥x∥ds ≥ λνN∥x∥ ≥ ∥x∥, x ∈ V ∩ ∂Er∗ .
Consequently,
∥T x∥ ≥ ∥x∥ for x ∈ V ∩ ∂Er∗ . (5.1.36)
Again, from H3, we have f∞ = ∞, g∞ = ∞. Let M ∈ R+, such that M ≥ (λν)−1. Then there exists
ρ∗0 > r such that
f(x(t)) ≥Mx, g(x(t)) ≥Mx, for x ≥ ρ∗0. (5.1.37)
Let r = max2r∗,ρ∗0γ and define Er = x ∈ C[0, 1] : ∥x∥ < r. Then for x ∈ V ∩ ∂Er, we have
min1/4≤t≤3/4
x(t) ≥ γ∥x∥ = γr ≥ ρ∗0. Therefore,
Υx(t) ≥ λ
∫ 3/4
1/4γGβ(s, s)ψ(s)g(x(s))ds ≥ λ
∫ 3/4
1/4γGβ(s, s)ψ(s)Mx(s)ds ≥ λνMρ∗0 ≥ ρ∗0.
83
Thus, in view of (5.1.37) we have f(Υx) ≥Mu, x ∈ V ∩ ∂Er. Hence,
T (x) ≥ λM
∫ 3/4
1/4γGα(s, s)φ(s)A(x(s))ds
≥ λM
∫ 3/4
1/4γGα(s, s)φ(s)
(λ
∫ 3/4
1/4γ2Gβ(z, z)ψ(z)M∥x∥dz
)ds
≥ λM
∫ 3/4
1/4γGα(s, s)φ(s)(λνM)rds ≥ λνMr ≥ r = ∥x∥, x ∈ V ∩ ∂Er.
This implies that,
∥T x∥ ≥ ∥x∥ for x ∈ V ∩ ∂Er. (5.1.38)
From (5.1.34), (5.1.36) and (5.1.38), using Theorem 2.3.7, it follows that the operator T has two fixed
points x1 ∈ V ∩ (Er\Er∗) and x2 ∈ V ∩ (E r\Er) such that 0 < ∥x1∥ < r < ∥x2∥.
Example 5.1.7. Consider the system of nonlinear fractional differential equationscD3.8x(t) + λ√1 + t2(
√y + y3) = 0, 0 < t < 1,
cD3.5y(t) + λ√t(x
34 + x
72 ) = 0, 0 < t < 1,
(5.1.39)
satisfying the boundary conditionsx(1) = 0, x′(0) = 0, · · · , xiii(0) = 0,
y(1) = 0, y′(0) = 0, · · · , yiii(0) = 0.(5.1.40)
Here, α = 3.8, β = 3.5, φ(t) =√1 + t2, ψ(t) =
√t, f(y) =
√y + y3 and g(x) = x
34 + x7. Obviously,
f0 = ∞, g0 = ∞ and f∞ = ∞, g∞ = ∞. By some calculations we have γ = 0.634645, µ = 0.057976 and
ν = 0.012333. Also, for 0 < λ < 0.07657 and x ∈ (0, 15], we have f(y) < 3378 and g(x) < 3378. Hence,
by Theorem 5.1.6, the system of fractional differential equations (5.1.39), (5.1.40) has at least two positive
solutions.
Theorem 5.1.8. Assume that (H4), (H5) are satisfied and for λ > 0, there is r ∈ R+ such that
f(x) ≥ (λν)−1r, g(x) ≥ (λν)−1r, for x ∈ [γr, r], (5.1.41)
then, the boundary value problem (5.1.1), (5.1.2) has at least two positive solutions.
Proof. Choose N ∈ R+ satisfying N ≤ (λµ)−1. Since, from H4, we have f0 = 0, g0 = 0. Thus, we may
choose 0 < ρ < r such that
f(x) < Nx and g(x) < Nx for x ∈ (0, ρ), (5.1.42)
For any x ∈ V ∩ ∂Eρ, where Eρ = x ∈ C[0, 1] : ∥x∥ < ρ, we have
Υ(x(t)) ≤ λ
∫ 1
0Gβ(s, s)ψ(s)g(x(s))ds < λ
∫ 1
0Gβ(s, s)ψ(s)N∥x∥ds ≤ λµN∥x∥ ≤ ρ. (5.1.43)
84
Using (5.1.42) and (5.1.43), we have
T (x(t)) ≤ λ
∫ 1
0Gα(s, s)φ(s)f(Υx(s))ds
≤ λ
∫ 1
0Gα(s, s)φ(s)NΥx(s)ds
< λ
∫ 1
0Gα(s, s)φ(s)N∥x∥ds ≤ λµN∥x∥ ≤ ∥x∥.
Which gives,
∥T x∥ ≤ ∥x∥, for x ∈ V ∩ ∂Eρ. (5.1.44)
Also, from H4, we have f∞ = 0, g∞ = 0. Thus, here exists constant r∗2 > r, such that for M satisfying
M ≤ (λµ)−1, we have
f(x) < Mx, g(x) < Mx for x ∈ [r∗2,∞). (5.1.45)
We consider following two cases:
Case 1. g is bounded. There exists a constant ω > 0 such that g(x) ≤ ω for all x ∈ (0,∞). Therefore, we
have
Υx(t) ≤ λω
∫ 1
0Gβ(s, s)ψ(s)ds ≤ ωλµ.
Let τ = maxf(x) : 0 ≤ u ≤ ωλµ. For r2 ≥ max2r, τλµ, define Er2 = x ∈ C[0, 1] : ∥x∥ < r2. For
any x ∈ V ∩ ∂Er2 , we get
T x(t) ≤∫ 1
0λGα(s, s)φ(s)f(A(x(s)))ds ≤
∫ 1
0λGα(s, s)φ(s)τds ≤ τλµ ≤ ∥x∥.
Hence,
∥T x∥ ≤ ∥x∥ for all x ∈ V ∩ ∂Er2 .
Case 2. g is unbounded. There exists r2 > r∗2 such that g(x) ≤ g(r2) ≤ Mr2, for x ∈ (0, r2]. Therefore,
Υx(t) ≤ λ
∫ 1
0Gβ(s, s)ψ(s)Mr2ds ≤ λMµr2 ≤ r2.
Since f is increasing function, therefore f(x) ≤ f(r2) ≤ Mr2, for x ∈ (0, r2]. It follows that for x ∈ V∩∂Er2 ,we have
T x(t) ≤ λ
∫ 1
0Gα(s, s)φ(s)f(Υx(s))ds ≤ (λMµ)r2 ≤ ∥x∥.
Therefore,
∥T x∥ ≤ ∥x∥, for all x ∈ V ∩ ∂Er2 .
Consequently, in both the cases, we have
∥T x∥ ≤ ∥x∥, for all x ∈ V ∩ ∂Er2 . (5.1.46)
Now, for any x ∈ V ∩ ∂Er, where Er = x ∈ C[0, 1] : ∥x∥ < r, we have
x(t) ≥ min1/4≤t≤3/4
x(t) ≥ ∥x∥ ≥ γr, t ∈ [1
4,3
4],
85
which in view of (5.1.41) implies that
Υx(t) ≥λ∫ 3/4
1/4γGβ(s, s)ψ(s)(g(x(s))ds ≥ γr.
Also, using (5.1.43), we obtain Υx(t) ≤ ρ for any x ∈ V ∩ ∂Eρ. Since, ρ < r, we have γr ≤ Υx(t) ≤ r, for
x ∈ V ∩ ∂Er. Consequently, f(Υx) ≥ (λν)−1r. Hence,
T x(t) ≥ λ
∫ 3/4
1/4γGα(t, s)φ(s)f(Υx)ds ≥ λ
∫ 3/4
1/4γGα(s, s)φ(s)(λν)
−1rds ≥ ∥x∥.
Thus,
∥T x∥ ≥ ∥x∥ for x ∈ V ∩ ∂Er. (5.1.47)
From (5.1.44), (5.1.46) and (5.1.47), using Theorem 2.3.7, we conclude that T has two fixed points x1 ∈Er\Eρ and x2 ∈ Er2\Er.
Example 5.1.9. cD5.4x(t) + λ(1 + t2)y2e−πy = 0, 0 < t < 1,
cD5.8y(t) + λt2x52 e−πx = 0, 0 < t < 1,
(5.1.48)
satisfying the boundary conditionsx(1) = 0, x′(0) = 0, · · · , x5(0) = 0,
y(1) = 0, y′(0) = 0, · · · , y5(0) = 0.(5.1.49)
Set α = 5.4, β = 5.8, φ(t) = (1+ t2), ψ(t) = t2, f(y) = y2e−πy and g(x) = x52 e−πx. By direct calculations,
we get γ = 0.78849004, µ = 0.00432759 and ν = 0.00007918.
Furthermore, for λ > 190716, we have f(y) > 0.05221499, g(x) > 0.05221499, for x ∈ (0.78849004, 1).
Also, f0 = 0, f∞ = 0, g0 = 0 and g∞ = 0. All the conditions of Theorem 5.1.8 are satisfied. Hence, the
system of boundary value problems (5.1.48), (5.1.49) has at least two positive solution.
5.2 Positive solutions to a system of fractional differential equations
with three–point boundary conditions
In this section, we study existence and multiplicity results for a coupled system of nonlinear three–point
boundary value problems for higher order fractional differential equations of the type [93]cDαx(t) = λφ(t)f(x(t), y(t)), n− 1 < α ≤ n, n ∈ N,cDβy(t) = µψ(t)g(x(t), y(t)), n− 1 < α, β ≤ n
(5.2.1)
satisfying the boundary conditionsx′(0) = x′′(0) = x′′′(0) = · · · = x(n−1)(0) = 0, x(1) = θ1x(µ1),
y′(0) = y′′(0) = y′′′(0) = · · · = y(n−1)(0) = 0, y(1) = θ2x(µ2),(5.2.2)
where λ, µ > 0, for n ∈ N; θi, µi ∈ (0, 1) for i = 1, 2. We use Guo-Krasnosel’skii fixed point theorem
to establish existence and multiplicity results for positive solutions. We drive explicit intervals for the
parameters λ and µ for which the system possess the positive solutions or multiple positive solutions.
86
5.2.1 Greens’s function and its properties
Lemma 5.2.1. Let h ∈ C[0, 1], then the linear three–point boundary value problem
cDαx(t) + h(t) = 0, t ∈ (0, 1), n− 1 < α ≤ n,
x′(0) = x′′(0) = x′′′(0) = · · · = x(n−1)(0) = 0, x(1) = θ1x(µ1),(5.2.3)
has a solution given by
x(t) =
∫ 1
0Hα(t, s)h(s)ds,
where,
Hα(t, s) =
(1−s)α−1−(1−θ1)(t−s)α−1−θ1(µ1−s)α−1
(1−θ1)Γ(α) , s ≤ t, µ1 ≥ s,
(1−s)α−1−(1−θ1)(t−s)α−1
(1−θ1)Γ(α) , µ1 ≤ s ≤ t ≤ 1,
(1−s)α−1−θ1(µ1−s)α−1
(1−θ1)Γ(α) , 0 ≤ t ≤ s ≤ µ1,
(1−s)α−1
(1−θ1)Γ(α) , t ≤ s, s ≥ µ1.
(5.2.4)
Proof. The proof follows by slight modifications in the proof of Lemma 4.1.1.
Lemma 5.2.2. The Green’s function Hα(t, s) defined by (5.2.4) satisfies the following properties:
(i) Hα(t, s) > 0 for all t, s ∈ (0, 1),
(ii) For t ∈ [0, 1], Hα(t, s) is decreasing.
(iii) For ℓ ∈ (0, 1), minℓ≤t≤1
Hα(t, s) ≥ ξαΦα(s) where ξα = θ1(1 − µα−11 ), Hα(t, s) ≤ Φα(s) and Φα(s) =
(1−s)α−1
(1−θ1)Γ(α) .
Proof. The proof follows by slight modifications in the proof of Lemma 4.1.2.
Now we write the system of boundary value problem (5.2.1), (5.2.2) as an equivalent system of integral
equations x(t) = λ∫ 10 Hα(t, s)φ(s)f(x(s), y(s))ds
y(t) = µ∫ 10 Hβ(t, s)ψ(s)g(x(s), y(s))ds.
(5.2.5)
Starting from now, we will work in the Banach space B2 = C[0, 1]× C[0, 1] furnished with the norm
∥(x, y)∥2 = ∥x∥+ ∥v∥ = max0≤t≤1
|x(t)|+ max0≤t≤1
|y(t)|.
We define an operators Tλ, Tµ : B2 → B2 by Tλ(x, y)(t) = λ∫ 10 Hα(t, s)φ(s)f(x(s), y(s))ds and
Tµ(x, y)(t) = µ∫ 10 Hβ(t, s)ψ(s)g(x(s), y(s))ds. Define an other operator T : B2 → B2 by
T (x, y) = (Tλ(x, y), Tµ(x, y)). The fixed points of T are the solutions of (5.2.1), (5.2.2).
Let ξ = minξα, ξβ, and define a cone in B2 by
W =(x, y) ∈ B2 : x(t) ≥ 0, y(t) ≥ 0, min
l≤t≤1(x(t) + y(t)) ≥ ξ∥(x, y)∥2
, where ξ := min(ξα, ξβ).
In this section for the system (5.2.1), (5.2.2) we admit following assumptions:
87
(H6) f, g ∈ C(R+ × R+,R), and the limits
f0 = limx+y→0
f(x, y)
x+ y, f∞ = lim
x+y→∞
f(x, y)
x+ y,
g0 = limx+y→0
g(x, y)
x+ y, g∞ = lim
x+y→∞
g(x, y)
x+ y
exist and f0, f∞, g0, g∞ ∈ [0,∞);
(H7) φ,ψ ∈ C([0, 1], (0,∞)) such that Iαφ(1) and Iβψ(1) exist and are finite;
(H8) (1− θ1)Iαφ(1)f0 < ξαξIαl φ(1)f∞, (1− θ2)Iβψ(1)g0 < (1− ξ)ξβIβl ψ(1)g∞;
(H9) (1− θ1)Iαφ(1)f∞ < ξξαIαl φ(1)f0, (1− θ2)Iβψ(1)g∞ < (1− ξ)ξβIβl ψ(1)g0;
(H10) there exist constants r, γ, ζ with
(1− θ1)ζIαφ(1) < ξ2γIαl φ(1) and (1− θ2)ζIβψ(1) < ξ(1− ξ)γIβl ψ(1) such that
(i) f0 = 0, g0 = 0, f∞ = 0, g∞ = 0;
(ii) f(x, y) ≥ γr, g(x, y) ≥ γr, for ∥(x, y)∥ ∈ [ξr, r];
(H11) there exist constants r, γ, ζ with
(1− θ1)γIαφ(1) < ξ2ζIαl φ(1) and (1− θ2)γIβψ(1) < ξ(1− ξ)ζIβl ψ(1) such that
(i) f0 = ∞, g0 = ∞, f∞ = ∞, g∞ = ∞;
(ii) f(x, y) ≤ γr, g(x, y) ≤ γr, for ∥(x, y)∥ ∈ [0, r].
(H12) (i) f0 = 0, g0 = 0, f∞ = ∞, g∞ = ∞; (ii) f0 = ∞, g0 = ∞, f∞ = 0, g∞ = 0.
Lemma 5.2.3. Assume that H6, holds. Then, the operator T : W → W is completely continuous.
Proof. First, we prove that T (W) ⊂ W. For any (t, s) ∈ [l, 1]× [0, 1], by Lemma 5.2.2, we have
minl≤t≤1
(Tλ(x, v)(t) + Tµ(x, v)(t))
= minl≤t≤1
(λ
∫ 1
0Hα(t, s)φ(s)f(x(s), y(s))ds,+µ
∫ 1
0Hβ(t, s)ψ(s)g(x(s), y(s))ds
)≥λξα
∫ 1
0Φα(s)φ(s)f(x(s), y(s))ds+ µξβ
∫ 1
0Φβ(s)ψ(s)g(x(s), y(s))ds
≥max0≤t≤1
(λξα
∫ 1
0Hα(t, s)φ(s)f(x(s), y(s))ds+ µξβ
∫ 1
0Hβ(t, s)ψ(s)g(x(s), y(s))ds
)=ξα∥Tλ(x, y)∥2 + ξβ∥Tµ(x, y)∥2 ≥ ξ∥T (x, y)∥2.
Therefore minl≤t≤1
(Tλ(x, v)(t) + Tµ(x, v)(t)) ≥ ξ∥T (x, y)∥2. Hence T (W) ⊂ W.
Now we prove that the operator T maps bonded sets into uniformly bonded sets. For fixed M ∈ R+,
consider a bounded subset M of W defined by M =(x, y) ∈ W : ∥(x, y)∥2 ≤ M
. Also, define
88
L1 = maxf(x(t), y(t)) : (x, y) ∈ M
, L2 = max
g(x(t), y(t)) : (x, y) ∈ M
, then for (x, y) ∈ M, using
(H7) and Lemma 5.2.2, we have
|Tλ(x, y)| =∣∣∣λ ∫ 1
0Hα(t, s)φ(s)f(x(s), y(s))ds
∣∣∣≤ λL1
∫ 1
0
(1− s)α−1φ(s)
Γ(α)(1− θ1)ds =
L1λ
1− θ1Iαφ(1) < +∞.
Similarly, one can show that |Tµ(x, y)| < +∞. Therefore |Tλ(x, y)|+ |Tµ(x, y)| < +∞. Which implies that
T (M) is bounded.
Finally, it remains to show that T (M) is equicontinuous. By (ii) of lemma 5.2.2, we have
∣∣ ddt
(Tλ(x, y)(t))∣∣ ≤λ ∫ 1
0
(t− s)α−2
Γ(α− 1)φ(s)|f(x(s), y(s))|ds
≤L1λ
∫ 1
0
(1− s)α−2
Γ(α− 1)φ(s)ds = L1λIα−1φ(1) < +∞.
Define δ =(λL1Iα−1φ(1) + µL2Iβ−1ψ(1)
)−1, and chose t, τ ∈ [0, 1] such that t < τ and τ − t < δ. Then
for all ε > 0 and (x, y) ∈ M, we obtain
∣∣Tλ(x, y)(τ)− Tλ(x, y)(t)∣∣ =∣∣∣ ∫ τ
t
d
ds(Tλ(x, y)(s)) ds
∣∣∣≤L1λIα−1φ(1)(τ − t).
Similarly, ∣∣Tµ(x, y)(τ)− Tµ(x, y)(t)∣∣ = ∣∣∣ ∫ τ
t
d
ds(Tµ(x, y)(s)) ds
∣∣∣ ≤ L2µIβ−1ψ(1)(τ − t).
Hence, it follows that∥∥T (x, y)(τ)− T (x, y)(t)∥∥2≤(L1λIα−1φ(1) + L2µIβ−1ψ(1)
)(τ − t) < ε.
Therefore, by the Arzela–Ascoli Theorem, T : W → W is completely continuous.
5.2.2 Existence of at least one positive solution
Theorem 5.2.4. Assume that (H6)− (H8) hold, then for every
λ ∈(
1−θ1ξαIα
l φ(1)f∞, ξ
Iαφ(1)f0
), and µ ∈
(0, 1−ξ
Iβψ(1)g0
)or λ ∈
(0, ξ
Iαφ(1)f0
)and µ ∈
(1−θ2
ξβIβl ψ(1)g∞
, 1−ξIβψ(1)g0
)the boundary value problem (5.2.1), (5.2.2) has at least one positive solution. Moreover, if (H12)(i) holds
then for each λ, µ ∈ (0,∞) the boundary value problem (5.2.1), (5.2.2) has at least one positive solution.
Proof. Choose ε > 0 such that following hold:
1− θ1ξαIαl φ(1)(f∞ − ε)
≤ λ ≤ ξ
Iαφ(1)(f0 + ε), 0 < µ ≤ 1− ξ
Iβψ(1)(g0 + ε).
By definitions of f0 and g0 there exists a constant r > 0 such that
f(x, y) ≤ (f0 + ε)(x+ y), g(x, y) ≤ (g0 + ε)(x+ y), for x+ y ∈ [0, r].
89
Define Er = (x, y) ∈ W : ∥(x, y)∥2 ≤ r . For any (x, y) ∈ W ∩ ∂Er, by Lemma 5.2.2, we have
Tλ(x, y)(t) =λ∫ 1
0Hα(t, s)φ(s)f(x(s), y(s))ds
≤λ∫ 1
0
(1− s)α−1
Γ(α)φ(s)(f0 + ε)(x+ y)ds
≤λ(f0 + ε)Iαφ(1)∥(x, y)∥2 ≤ ξ∥(x, y)∥2,
Tµ(x, y)(t) =µ∫ 1
0Hβ(t, s)ψ(s)f(x(s), y(s))ds
≤µ∫ 1
0
(1− s)β−1
Γ(β)ψ(s)(g0 + ε)(x+ y)ds
≤µ(g0 + ε)Iβψ(1)∥(x, y)∥2 ≤ (1− ξ)∥(x, y)∥2.
Hence,
∥T (x, y)∥2 ≤ξ∥(x, y)∥2 + (1− ξ)∥(x, y)∥2
=∥(x, y)∥2, for all (x, y) ∈ W ∪ ∂Er.(5.2.6)
Next, we consider two cases:
Case 1. f∞, g∞ are finite.
Let ε1 > 0 such that 0 < 1−θ1ξαIα
l φ(1)(f∞−ε1) ≤ λ. By the definition of f∞, g∞ there exists a constant r∗ > r,
such that
f(x, y) ≥ (f∞ − ε1)(x+ y), g(x, y) ≥ (g∞ − ε1)(x+ y), for x+ y ≥ ξr∗.
Define Er∗ = (x, y) ∈ W : ∥(x, y)∥2 ≤ r∗ . Then for (x, y) ∈ W ∩ ∂Er∗ , we have
x(t) + y(t) ≥ minl≤t≤0
(x(t) + y(t)) ≥ ξ∥(x, y)∥2 = ξr∗.
Therefore, by Lemma 5.2.2, we obtain
Tλ(x, y)(t) =λ∫ 1
0Hα(t, s)φ(s)f(x(s), y(s))ds
≥λξα∫ 1
l
(1− s)α−1
(1− θ1)Γ(α)φ(s)(f∞ − ε)(x+ y)ds
≥ λξα1− θ1
Iαl φ(1)(f∞ − ε)∥(x, y)∥2 ≥ ∥(x, y)∥2.
Thus
∥T (x, y)∥2 ≥ ∥Tλ(x, y)∥2 ≥ ∥(x, y)∥2, for all (x, y) ∈ W ∩ ∂Er∗ . (5.2.7)
By Theorem 2.3.7 and inequalities (5.2.6), (5.2.7), the operator T has a fixed point (x, y) ∈ W ∩ Er∗\Ersuch that r ≤ ∥(x, y)∥2 ≤ r∗.
Case 2. f∞ = ∞, g∞ = ∞.
For λ > 0, choose a constant ζ > 0 such that ζ ≥ min
1−θ1λξαIα
l φ(1), 1−θ2µξβIβ
l ψ(1)
. By (H12)(i), f∞ = ∞,
g∞ = ∞. Therefore, there exists r∗ > r, such that
f(x, y) ≥ ζ(x+ y), for x+ y ≥ ξr∗.
90
For (x, y) ∈ W ∩ ∂Er∗ , we have x(t) + y(t) ≥ minl≤t≤0
(x(t) + y(t)) ≥ ξ∥(x, y)∥2 = ξr∗. Hence,
f(x, y) ≥ ζ(x+ y), for any (x, y) ∈ W ∩ ∂Er∗ .
By Lemma 5.2.2, we have
Tλ(x, y)(t) =λ∫ 1
0Hα(t, s)φ(s)f(x(s), y(s))ds
≥λξαζ∫ 1
l
(1− s)α−1
(1− θ1)Γ(α)φ(s)(x+ y)ds
≥ λξαζ
1− θ1Iαl φ(1)∥(x, y)∥2 ≥ ∥(x, y)∥2.
Thus, we have
∥T (x, y)∥2 ≥ ∥Tλ(x, y)∥2 ≥ ∥(x, y)∥2, for all (x, y) ∈ W ∩ ∂Er∗ . (5.2.8)
Hence, by Theorem 2.3.7 and inequalities (5.2.6), (5.2.8), the operator T has a fixed point (x, y) ∈W ∩ Er∗\Er such that r ≤ ∥(x, y)∥2 ≤ r∗.
Theorem 5.2.5. Assume that (H6), (H7) and (H9) hold, then for every
λ ∈(
1−θ1ξαIα
l φ(1)f0, ξ
Iαφ(1)f∞
)and µ ∈
(0, 1−ξ
Iβψ(1)g∞
)or λ ∈
(0, ξ
Iαφ(1)f∞
)and µ ∈
(1−θ2
ξβIβψ(1)g0, 1−ξIβl ψ(1)g∞
),
the boundary value problem (5.2.1), (5.2.2) has at least one positive solution. Moreover, if (H12)(ii) holds,
then for each λ, µ ∈ (0,∞) the boundary value problem (5.2.1), (5.2.2) has at least one positive solution.
Proof. The proof is similar to the Theorem 5.2.4. Therefore it is omitted.
Example 5.2.6. Consider the system of fractional differential equationscD
52x(t) = λ
(π − 6(x+ y + π
2 )− 3
2
)(243578 + eπt),
cD2710 y(t) = µ
((1 + 23√
x+y
)− 52 + π
257
)(sin t+ 256eπt),
(5.2.9)
satisfying the boundary conditionsx′(0) = x′′(0) = 0, x(1) = 12x(
23),
y′(0) = y′′(0) = 0, y(1) = 34x(
35),
(5.2.10)
Set f(x, y) = π − 6(x + y + π2 )
− 32 , g(x, y) =
(1 + 23√
x+y
)− 52+ π
257 , θ1 = 12 , θ2 = 3
4 , µ1 = 23 , µ2 = 3
5 .
Clearly f0 = π − 6( 2π )32 , f∞ = π, g0 = π
257 and g∞ = 1 + π257 . Let l = 1
4 . By calculations ξα ≈ 0.227834,
ξβ ≈ 0.4352, ξ ≈ 0.2278; Iαφ(1) ≈ 1.0794, Iαl φ(1) ≈ 0.7865, Iβψ(1) ≈ 182.2013 and Iβl ψ(1) ≈ 133.3916.
Since, (1 − θ1)Iαφ(1)f0 ≈ 0.050678 < ξαξIαl φ(1)f∞ ≈ 0.128268 and (1 − θ2)Iβψ(1)g0 ≈ 0.556812 <
(1 − ξ)ξβIβl ψ(1)g∞ ≈ 45.382788. We conclude that for λ, µ satisfying 0.888114 < λ < 2.247865 and
0 < µ < 0.346692 or 0 < λ < 2.247865 and 0.004254µ < 0.346692, the system of fractional differential
equations (5.2.9), (5.2.10) has at least one positive solution.
91
5.2.3 Existence of at least two positive solutions
Theorem 5.2.7. Assume that (H6), (H7), (H10) hold, then for any
λ ∈[
1−θ1ξγIα
l φ(1), ξζIαφ(1)
]and µ ∈
(0, 1−ξ
ζIβψ(1)
], or λ ∈
(0, ξ
ζIαφ(1)
]and µ ∈
[1−θ2
ξγIβl ψ(1)
, 1−ξζIβψ(1)
], the boundary
value problem (5.2.1), (5.2.2) has at least two positive solutions (x1, y1), (x2, y2) such that 0 < ∥(x1, y1)∥2 <r < ∥(x2, y2)∥2 for some r > 0.
Proof. By (H10)(i), f0 = 0, g0 = 0, there exists r1 ∈ (0, r) such that
f(x, y) ≤ ζ(x+ y), g(x, y) ≤ ζ(x+ y), for x+ y ∈ (0, r1).
Define Er1 = (x, y) ∈ W : ∥(x, y)∥2 < r1. For any (x, y) ∈ W ∩ ∂Er1 ,by Lemma 5.2.2 we have
Tλ(x, y)(t) =λ∫ 1
0Hα(t, s)φ(s)f(x(s), y(s))ds
≤λζ∫ 1
0
(1− s)α−1
Γ(α)φ(s)(x+ y)ds
≤λζIαφ(1)∥(x, y)∥2 ≤ θ∥(x, y)∥2,
Tµ(x, y)(t) =λ∫ 1
0Hβ(t, s)ψ(s)g(x(s), y(s))ds
≤µζ∫ 1
0
(1− s)β−1
Γ(β)ψ(s)(x+ y)ds
≤µζIβψ(1)∥(x, y)∥2 ≤ (1− θ)∥(x, y)∥2.
Which implies that
∥T (x, y)∥2 ≤ ∥(x, y)∥2, for all (x, y) ∈ W ∩ ∂Er1 . (5.2.11)
Also, from (H10)(i), we have f∞ = 0, g∞ = 0. Therefore, there exists r2 > r such that for some positive
constant ζ, we have
f(x, y) ≤ ζ(x+ y), g(x, y) ≤ ζ(x+ y), for x+ y ≥ r2. (5.2.12)
Set Er2 = (x, y) ∈ W : ∥(x, y)∥2 < r2. For any (x, y) ∈ W ∩ ∂Er2 , by Lemma 5.2.2 we have
∥T (x, y)∥2 ≤ ∥(x, y)∥2, for all (x, y) ∈ W ∩ ∂Er2 . (5.2.13)
Next define Er = (x, y) ∈ W : ∥(x, y)∥2 < r. For any (x, y) ∈ W ∩ ∂Er, by Lemma 5.2.2 we have
x(t) + y(t) ≥ ∥(x, y)∥2 ≥ ξr, for all t ∈ [l, 1].
Therefore
Tλ(x, y)(t) =λ∫ 1
0Hα(t, s)φ(s)f(x(s), y(s))ds
≥λξ∫ 1
l
(1− s)α−1
(1− θ1)Γ(α)φ(s)γrds
=λξIαl φ(1)γr = r = ∥(x, y)∥2.
92
Finally, we have
∥T (x, y)∥2 ≥ ∥(x, y)∥2, for all (x, y) ∈ W ∩ ∂Er. (5.2.14)
Hence, by (5.2.11), (5.2.13), (5.2.11) and by Theorem 2.3.7, it follows that the operator T has two fixed
points x1 ∈ Er\Er1 and x2 ∈ Er2\Er.
Theorem 5.2.8. Assume that (H6), (H7) and (H11) hold, then for any
λ ∈[
1−θ1ξζIα
l φ(1), ξγIαφ(1)
]and µ ∈
(0, 1−ξ
γIβψ(1)
]or λ ∈
(0, ξ
γIαφ(1)
], µ ∈
[1−θ2
ξζIβl ψ(1)
, 1−ξγIβψ(1)
], the boundary
value problem (5.2.1), (5.2.2) has at least two positive solutions (x1, y1), (x2, y2) such that 0 < ∥(x1, y1)∥2 <r < ∥(x2, y2)∥2.
Proof. The proof is similar to the Theorem 5.2.7, so we omit it.
Example 5.2.9. Consider the system of fractional differential equationscD
145 x(t) = λ
(194t2
1+t
)(12π
x+y+12π − e−π(x+y)),
cD125 y(t) = µ
(78125t2
1+t2
)(1√x+y
− e−π(x+y)),
(5.2.15)
satisfying the boundary conditionsx′(0) = x′′(0) = 0, x(1) = 3750x(
1125),
y′(0) = y′′(0) = 0, y(1) = 1120x(
1950).
(5.2.16)
Set f(x, y) = 194(
12πx+y+12π − e−π(x+y)
), g(x, y) = 78125
(12π
x+y+12π − e−π(x+y)). One can easily verify that
f0 = 0, f∞ = 0, g0 = 0 and g∞ = 0. Choose constants l = 12 , r = 5, ζ = 12, γ = 2449. By computations
Iαφ(1) ≈ 0.015652, Iαl φ(1) ≈ 0.007547, Iβψ(1) ≈ 0.034126, Iβl ψ(1) ≈ 0.018676,
(1− θ1)ζIαφ(1) ≈ 0.326817 < ξ2γIαl φ(1) ≈ 0.959355
and (1 − θ2)ζIβψ(1) ≈ 0.184282 < ξ(1 − ξ)γIβl ψ(1) ≈ 8.046569. Also f(x, y) > 12245, for ∥(x, y)∥2 ∈(1.1397, 5) and g(x, y) > 12245, for ∥(x, y)∥2 ∈ (1.1397, 5). Therefore, for any
λ ∈ [0.0617, 1.2130] and µ ∈ (0, 1.8855] or λ ∈ (0, 1.2130] and µ ∈ [0.0431, 1.8855], by Theorem 5.2.7 the
boundary value problem (5.2.15), (5.2.16) has at least two positive solutions.
Chapter 6
Numerical solutions to fractional
differential equations by the Haar wavelets
Fractional differential equations is rapidly growing field of mathematics both in theory and in applica-
tions to real word problems. Numerous problems in physics, engineering and other applied sciences can
be modeled as differential equations of fractional order. However, in spite of a large number of recent
applications of fractional differential equations in applied problems, the state of art is far less developed
for boundary value problems. Exact analytic solutions to most of the differential equations of fractional
order are not available in general. Therefore various numerical schemes for integer order differential equa-
tions are generalized to approximate solutions of fractional differential equations (differential equations
of arbitrary order). Some of these are listed as Adomian decomposition method [42], homotype analy-
sis method [52], homotype perturbation method [1], finite difference method [149], variational iteration
method [140], fractional linear multi-step method [84], generalized differential transform method [109], ex-
trapolation method [36] and predictor-corrector method [38]. Most of these numerical schemes are applied
to deal with initial value problems and so far there has been no considerable advancement in extending
these methods for the boundary value problems of fractional order differential equations. In the literature,
one can find only few papers [9, 33, 60] that deal with numerical solutions of boundary value problems
for fractional ordinary and partial differential equations. Therefore, it appears to be very important to
develop efficient numerical techniques to solve the boundary value problems for differential equations of
fractional order. In this chapter and in chapter 7, we will focus on providing numerical techniques based
on wavelets for the solutions of boundary value problems for fractional differential equations.
6.1 Numerical solutions to fractional ordinary differential equations
In 1997 Chen and Hsiao [27] first proposed the idea of Haar operational matrix for the integration of Haar
function vectors and used it for solving differential equations. This remarkable idea deals with a general
formalism of construction of an operational matrix of integration for Haar wavelets. Since then numerous
operational matrices based on different orthogonal functions, such as the Legendre [10, 67, 117, 147], the
Chebyscev [16,133], the Fourier [112], sine–cosine [62,116] and block–pulse [102] have been established and
93
94
used to solve various problems in engineering. J. Wu, C. Chen and Chih-Fan Chen [141] have developed a
unified approach for deriving the operational matrices of several orthogonal functions including the square
wave group and sinusoidal group of orthogonal functions. In the following we review some operational
matrices of integration and derive a new operational matrix that will be used together existing operational
matrices to solve the boundary value problems for fractional differential equations. In the literature, as
for as we know, the boundary value problems are not solved by operational matrices approach even for
integer order differential equations.
Definition 6.1.1. [102] For l > 0, a set of block-pulse functions is defined on [0, l) as
bi(t) = χIi(t), where Ii :=[(i− 1)l
m,il
m
), i = 1, 2, . . . ,m.
The block-pulse functions are disjoint. That is, for t ∈ [0, 1) and i, j = 1, 2, . . . ,m, we have
bi(t)bj(t) = biδij .
The block-pulse functions are orthogonal among themselves. That is ⟨bi, bj⟩ = lmδij . The orthogonality
property of block-pulse function is obtained from the disjointness property. Any function f ∈ L2[0, l) can
be approximated by block-pulse series as follows
f(t) ≈m−1∑i=0
kibi(t) = kTmBm(t),
where the coefficients ki are given by ki = ml ⟨f, bi⟩. Furthermore, km = [k0, k1, . . . , km−1]
T , is the block-
pulse coefficient vector and Bm(t) = [b0(t), b1(t), . . . , bm−1(t)]T is the block-pulse function vector. For
ti =2i−1m , i = 1, 2, . . . ,m− 1, the block-pulse matrix is simply m×m identity matrix. By Integrating the
block-pulse function vector and expanding it into block-pulse series, we have∫ t
0Bm(s)ds ≈ Bm×mBm(t), (6.1.1)
where Bm×m is the operational matrix of integration for the block-pulse functions [102] and it is given by
Bm×m =1
m
12 1 1 · · · 1
0 12 1 · · · 1
0 0 12 · · · 1
......
.... . .
...0 0 0 · · · 1
2
.
Our aim is to develop a numerical scheme to solve two–point and multi–point boundary value problems
for fractional order differential equations. We derive a matrix and use it together with the existing Haar
wavelet operational matrices and the operational matrix of fractional order integration to solve boundary
value problems for fractional order differential equations. The work is accepted for publication in [94].
Usually the Haar scaling function φ and the mother wavelet ψ are defined on unit interval [0, 1). In
many situations, we need the Haar wavelet system defined on an interval [0, η), η > 0. In particular,
whenever we are interested to solve three–point (or multi–point) boundary value problems on [0, 1], we
95
choose η ∈ (0, 1]. Let I = I00 = [0, η) and Ij,k =[2−jkη, 2−j(k + 1)η
), then the Haar scaling and wavelet
functions on [0, η) are defined as follows:
φ(t) =1√ηχI(t), ψj,k =
2j/2√η
(χIlj,k
(t)− χIrj,k(t)), J ≥ 0, j ≤ J − 1, k ≤ 2j − 1,
where ψ0,0(t) = ψ1(t) = 1√η
(χ[0, η
2)(t) − χ[ η
2, η)(t)
)is the mother wavelet function for the Haar system
ψj,k = 2j/2ψ1(2jt− k. An arbitrary function y ∈ L2[0, η] can be expanded into the Haar wavelet series
as follows:
y(t) = ⟨y, φ⟩φ(t) +J−1∑j=0
2j−1∑k=0
⟨y, ψj,k⟩cj,kψj,k(t)
= cφ(t) +
J−1∑j=0
2j−1∑k=0
cj,kψj,k(t) = CTmΨm(t),
(6.1.2)
where m = 2J , for some fixed J ∈ N. The Haar coefficient vector Cm and the Haar wavelet vectors Ψm(t)
are given as
Cm =[c, c0,0, c1,0, c1,1, c2,0, c2,1, c2,2, c2,3, . . . , cJ−1,0, cJ−1,1, cJ−1,2, . . . , cJ−1,2J−1]T ,
Ψm(t) =[φ(t), ψ0,0(t), ψ1,0(t), ψ1,1(t), ψ2,0(t), ψ2,1(t), ψ2,2(t), ψ2,3(t), . . . ,
ψJ−1,0(t), ψJ−1,1(t), ψJ−1,2(t), . . . , ψJ−1,2J−1(t)]T .
For ti =(2i−1)η
2m , i = 1, 2, . . . ,m, the m×m Haar matrix is defined as
Ψm×m =[Ψm
( η
2m
)Ψm
( 3η
2m
)· · · Ψm
((2m− 1)η
2m
)]. (6.1.3)
For instance, when m = 8, the Haar matrix is given by
Ψ8×8 =1√η
1 1 1 1 1 1 1 1
1 1 1 1 −1 −1 −1 −1√2
√2 −
√2 −
√2 0 0 0 0
0 0 0 0√2
√2 −
√2 −
√2
2 −2 0 0 0 0 0 0
0 0 2 −2 0 0 0 0
0 0 0 0 2 −2 0 0
0 0 0 0 0 0 2 −2
.
The fractional order integral of the Haar wavelets can be approximated by the Haar wavelet series
Iαψj′,k′(t) =⟨Iαψj′,k′ , φ⟩φ(t) +J−1∑j=0
2j−1∑k=0
⟨Iαψj′,k′ , ψj,k⟩ψj,k(t).
Therefore,
IαΨm(t) = Pα,ηm×mΨm(t), (6.1.4)
where Pα,ηm×m = [pα,ηil ]m×m, pα,ηil = ⟨Iαψj′,k′ , ψj,k⟩, i, l ∈ N. The indices, i, l are determined as i = 2j
′+ k′,
l = 2j + k (j, j′ ≤ J, J − 1 ∈ N; k ≤ 2j − 1, k′ ≤ 2j′ − 1). In particular pα,ηi0 := ⟨Iαψj′,k′ , φ⟩ and
pα,η00 := ⟨Iαφ,φ⟩.
96
The Haar wavelets can be approximated by the block-pulse series as
ψj,k(t) =
m−1∑i=0
⟨ψj,k, bi⟩bi(t).
Consequently,
Ψm(t) = Ψm×mBm(t). (6.1.5)
Applying fractional integral of order α > 0 on both side of (6.1.5), we have
IαΨm(t) = Ψm×mIαBm(t)
= Ψm×mFα,ηm×mBm(t),
(6.1.6)
where Fα,ηm×m is the operational matrix of integration for the block-pulse functions [73] and is given by
Fα,ηm×m =( ηm
)α 1
Γ(α+ 2)
1 ζ1 ζ2 · · · ζm−1
0 1 ζ1 · · · ζm−2
0 0 1 · · · ζm−3
0 0 0. . .
...
0 0 0 · · · 1
,
with ζ0 = 1, ζj = (j + 1)α+1 − 2jα+1 + (j − 1)α+1 j = 1, 2, . . . ,m − i + 1 for i = 0, 1, 2, . . . ,m + 1.
Substituting (6.1.5), (6.1.8), into (6.1.4) we have
Pα,ηm×mΨm×m = Ψm×mFα,ηm×m,
Since Ψm×m is non-singular, therefore
Pα,ηm×m =
1
mΨm×mFα,ηm×mΨ
Tm×m, (6.1.7)
where Ψ−1m×m = 1
mΨTm×m. The matrix Pα,η
m×m is called the operational matrix of fractional order integration
for the Haar wavelets supported on [0, η]. In particular when η = 1 then Pα,ηm×m = Pα
m×m is operational
matrix of fractional order integration for the Haar wavelets supported on unit interval [80]. One can easily
verify that
Pα,ηm×m =
(1η
)1/2Pαm×m. (6.1.8)
For the special case, when α = 1, η = 1, the matrix Pα,ηm×m reduces to the well known operational matrix
Pm×m given in [27] and expressed by
Pm×m =1
2m
2mPm2×m
2−1
2Ψm2×m
2
12Ψ
−1m2×m
20
, (6.1.9)
with P1×1 =12 , Ψ
−1m×m = η
mΨTm×m.
Next, we derive a useful matrix for the Haar wavelets which is crucial for developing a numerical scheme
for solutions of boundary value problems to fractional ordinary and partial differential equations.
97
Consider functions Jn ∈ L2[0, η], n = 1, 2, . . . , l. Then for m-dimensional vector Jnm = [Jn1 , Jn2 , . . . , J
nm]
(Jni = Jn(2i−12m ), i = 1, 2, . . . ,m) and for any m×m matrix Am×m = [Ai,j ]m×m, define the product
Jnm ⊗Am×m = [Jnj Ai,j ]m×m.
In general Jnm⊗Am×m = Am×m⊗Jnm and also Jnm⊗(Am×mBm×m
)=(Jnm⊗Am×m
)Bm×m. By equation
(6.1.7), we have
Jnm ⊗(Pα,ηm×mΨm×m
)= Jnm ⊗
(Ψm×mF
α,ηm×m
). (6.1.10)
A function of two variables K(t, s) ∈ (L2[0, η))2 can be approximated by the Haar wavelets as
K(t, s) = ⟨φ(t), ⟨K(t, s), φ(s)⟩⟩φ(t)φ(s) +J−1∑j=0
2j−1∑k=0
⟨ψj,k(t), ⟨K(t, s), φ(s)⟩⟩ψj,k(t)φ(s)
+J−1∑j′=0
2j′−1∑
k′=0
⟨φ(t), ⟨K(t, s), ψj′,k′(s)⟩⟩φ(t)ψj′,k′(s)
+
J−1∑j=0
2j−1∑k=0
J−1∑j′=0
2j′−1∑
k′=0
⟨ψj,k(t), ⟨K(t, s), ψj′,k′(s)⟩⟩ψj,k(t)ψj′,k′(s)
= ΨTm×m(t)Km×mΨm×m(s).
Consider continuous functions ϕn : [0, η] → R, n = 1, 2, . . . , then
ϕn(t)Iαψj1,k1(η) =∫ η
0
(η − s)α−1
Γ(α)ϕn(t)ψj1,k1(s)ds =
∫ η
0Qnα(t, s)ψj1,k1(s)ds
=(⟨φ(t), ⟨Qnα(t, s), φ(s)⟩⟩φ(t) +
J−1∑j=0
2j−1∑k=0
⟨ψj,k(t), ⟨Qnα(t, s), φ(s)⟩⟩ψj,k(t))⟨φ(s), ψj1,k1(s)⟩
+
(J−1∑j′=0
2j′−1∑
k′=0
⟨φ(t), ⟨Qnα(t, s), ψj′,k′(s)⟩⟩φ(t)
+J−1∑j=0
2j−1∑k=0
J−1∑j′=0
2j′−1∑
k′=0
⟨ψj,k(t), ⟨Qnα(t, s), ψj′,k′(s)⟩⟩ψj,k(t)
)⟨ψj′,k′(s), ψj1,k1(s)⟩,
=(Cφ(t) +
J−1∑j=0
2j−1∑k=0
Cj,kψj,k(t))⟨φ(s), ψj1,k1(s)⟩
+
(J−1∑j′=0
2j′−1∑
k′=0
Cj′,k′φ(t) +
J−1∑j=0
2j−1∑k=0
J−1∑j′=0
2j′−1∑
k′=0
Cj,kj′,k′ψj,k(t)
)⟨ψj′,k′(s), ψj1,k1(s)⟩,
where Qnα(t, s) =(η−s)α−1
Γ(α) ϕn(t). Since the Haar wavelets are orthonormal on [0, η], i.e.
⟨ψj′,k′(s), ψj1,k1(s)⟩ = δj′,k1δk′,k1 .
98
Therefore,
ϕn(t)IαΨm(η) = ϕn(t)[Iαφ(t), Iαψ0,0(t), Iαψ1,0(t), Iαψ1,1(t), Iαψ2,0(t), Iαψ2,1(t),
Iαψ2,2(t), Iαψ2,3(t), . . . , IαψJ,0(t), IαψJ−1,1(t), IαψJ−1,2(t), . . . , IαψJ−1,2J−1(t)]T
=
Cφ(t) +∑J−1
j=0
∑2j−1j=0 Cj,kψj,k(t)
C0,0φ(t) +∑J−1
j=0
∑2j−1j=0 Cj,k0,0ψj,k(t)
C1,0φ(t) +∑J−1
j=0
∑2j−1j=0 Cj,k1,0ψj,k(t)
C1,1φ(t) +∑J−1
j=0
∑2j−1j=0 Cj,k1,1ψj,k(t)
...
CJ−1,0φ(t) +∑J−1
j=0
∑2j−1j=0 Cj,kJ−1,0ψj,k(t)
CJ−1,1φ(t) +∑J−1
j=0
∑2j−1j=0 Cj,kJ−1,1ψj,k(t)
...
CJ−1,2Jφ(t) +∑J−1
j=0
∑2j−1j=0 Cj,k
J−1,2Jψj,k(t)
.
Hence,
ϕn(t)IαΨm(η) = Qα,η,nm×mΨm(t). (6.1.11)
Since the Haar wavelets ψj,k, j = 0, 1, . . . , J , 0 ≤ k < 2j − 1 are supported on dyadic subintervals
Ij,k =[ηk2j, η(k+1)
2j
)of [0, η]. Therefore, we have,∫ η
0
(η − s)α−1
Γ(α)ψj,k(s)ds =
∫Ij,k
(η − s)α−1
Γ(α)ψj,k(s)ds
=
∫Ilj,k
(η − s)α−1
Γ(α)ψj,k(s)ds−
∫Irj,k
(η − s)α−1
Γ(α)ψj,k(s)ds,
where, I lj,k =[ηk2j, η(2k+1)
2j+1
)and Irj,k =
(η(2k+1)2j+1 , η(k+1)
2j
]are the left and right halves of the interval Ij,k.
Thus, we have
Λα,ηφ :=
∫ η
0
(η − s)α−1
Γ(α)φ(s)ds =
ηα
Γ(α+ 1)
and
Λα,ηj,k :=
∫ η
0
(η − s)α−1
Γ(α)ψj,k(s)ds =
ηα
Γ(α+ 1)
[(1− k
2j
)α+(1− k + 1
2j
)α− 2(1− 2k + 1
2j+1
)α].
Hence,
IαΨm(η) = [Λα,ηφ ,Λα,η0,0 ,Λα,η1,0 ,Λ
α,η1,1 ,Λ
α,η2,0 , . . . ,Λ
α,η2,3 , . . . ,Λ
α,ηJ−1,0, . . .Λ
α,ηJ−1,2J−1
]T
= [Λα,η0 , Λα,η1 , . . . ,Λα,ηm−1]T . (Λα,η0 := Λα,ηφ , Λα,ηi := Λα,ηj,k , i = 2j + k)
For a continuous function ϕn : [0, η] → R, we define a matrix at collocation points ti = 2i+12J+1 , i =
0, 1, . . . ,m− 1 as follows
Λα,nm×m =
ϕn(t0)Λ
α,η0 ϕn(t1)Λ
α,η0 . . . ϕn(tm−1)Λ
α,η0
ϕn(t0)Λα,η1 ϕn(t1)Λ
α,η1 . . . ϕn(tm−1)Λ
α,η1
......
. . ....
ϕn(t0)Λα,ηm−1 ϕn(t1)Λ
α,ηm−1 . . . ϕn(tm−1)Λ
α,ηm−1
. (6.1.12)
99
Therefore
ϕn(t)IαΨm(η) =η
mΛα,nm×mΨ
Tm×mΨm(t) (6.1.13)
From equation (6.1.11) and (6.1.13), we have
Qα,η,nm×m =
η
mΛα,nm×mΨ
Tm×m. (6.1.14)
In particular, when η = 1, the matrix Qα,η,nm×m will be denoted by Qα,n
m×m. It is worth mentioning that Qα,η,nm×m,
is very useful for solving multi–point boundary value problems for differential equations of arbitrary order.In particular, for m = 8, α = 1.5, η = 0.98 and ϕ1(t) = tα−1eπt, the matrix Qα,η,n
m×m is given by
Qα,η,nm×m =
0.00214 0.0217 0.0788 0.2103 0.4835 1.0174 2.0185 3.8400
0.00151 0.0152 0.0553 0.1478 0.3398 0.7149 1.4183 2.6981
0.00073 0.0074 0.0269 0.0719 0.1654 0.3481 0.6907 1.3141
0.00031 0.0032 0.0116 0.0310 0.0714 0.1503 0.2981 0.5672
0.00029 0.0029 0.0107 0.0286 0.0657 0.1383 0.2745 0.5223
0.00022 0.0022 0.0083 0.0222 0.0510 0.1074 0.2132 0.4056
0.00015 0.0015 0.0056 0.0151 0.0347 0.0731 0.1452 0.2762
0.00006 0.0006 0.0024 0.0065 0.0150 0.0315 0.0626 0.1192
.
Before applying the Haar wavelets to solve the boundary value problems for fractional differential
equations, we need to give a note of caution. The direct use of the Haar wavelets for the solutions of
fractional order differential equations my not suitable in some situations, for example when derivative
involved are of integer order. This is because the Haar wavelets are discontinuous. The obvious way to
overcome this difficulty is to convert the underlying differential equation into an integral equation and
approximating the solution by truncated orthogonal series and using operational matrices of integration
to eliminate the integral operators. An other possibility is that, one can expand the term involving highest
derivative, appearing in the underlying differential equation. The main characteristics of the method is to
convert the fractional order differential equation into system of algebraic equations.
6.1.1 Linear fractional differential equations with constant coefficients
In this section the numerical solutions for boundary value problems for linear fractional differential equa-
tions with constant coefficients are discussed. The linear problems can be treated in two ways.
Let us consider the following boundary value problem:
aDαy(t)+bDβy(t) + cy(t) = g(t), (6.1.15)
y(0) = 0, y(1) = y0, (6.1.16)
where t ∈ [0, 1], 1 ≤ α ≤ 2, 0 ≤ β ≤ 1, α ≥ β + 1 and a, b, c, y0 ∈ R.
Method.1: We approximate the fractional derivative and integral as follows
Dαy(t) = CTmΨm(t), (6.1.17)
IαΨm(t) = CTmPα
m×mΨm(t). (6.1.18)
100
Using properties of fractional derivatives and integrals, together with boundary conditions, from equation
(6.1.17), we have
y(t) = CTmIαΨm(t)− CT
mϕ1(t)IαΨm(1) + y0t
α−1, (6.1.19)
Dβy(t) = CTmIα−βΨm(t)− CT
mϕ2(t)IαΨm(1) +
Γ(α)y0tα−1
Γ(α− β)(6.1.20)
where ϕ1(t) = tα−1 and ϕ2(t) = Γ(α)tα−β−1
Γ(α−β) . In view of (6.1.11) and (6.1.18) the equations (6.1.19) and
(6.1.20) become
y(t) = CTmPα
m×mΨm(t)− CTmQα,1
m×mΨm(t) + y0tα−1, (6.1.21)
Dβy(t) = CTmPα−β
m×mΨm(t)− CTmQα,2
m×mΨm(t) +Γ(α)y0t
α−1
Γ(α− β), (6.1.22)
Substituting (6.1.17) (6.1.21) and (6.1.22) into (6.1.15), we have following linear system of algebraic
equations
CTm
aIm×m + b
(Pα−βm×m − Qα,2
m×m)+ c(Pαm×m − Qα,1
m×m)
Ψm(t) = FTmΨm(t), (6.1.23)
where we have approximated the function f(t;α, β) = g(t)−(
bΓ(α)Γ(α−β) + c
)y0t
α−1 as
f(t;α, β) = FTmΨm(t).
The Bagley–Torvik equation
The Bagley–Torvik equation [17] arises in the modeling of the motion of a rigid plate in a Newtonian fluid
and a gas in a fluid. Several authors have investigated this equation. In [114], I. Podlubny solved this
equation numerically with homogenous initial conditions and he compared the numerical solutions with
exact solutions obtained with the help of fractional Green’s function. S. Saha Ray and R.K. Bera, [122]
solved the Bagley–Torvik equation using the Adomian decomposition method. K. Diethelm and N.J.
Ford [37, 38] have applied fractional linear multistep method and predictor-corrector method of Adams
type to solve the Bagley–Torvik equation. Z.H. Wang, X. Wang [136] provided a general solution of the
Bagley–Torvik equation. In the present work, we solve this equation with two–point boundary conditions
by using Haar wavelets.
Example 6.1.2. Consider the Bagley–Torvik equation with boundary conditions,
y′′(t) +(1817
)Dαy(t) +
(1351
)y(t) =
t−12
89250√π
(48p(t) + 7
√tq(t)
), t ∈ [0, 1],
y(0) = 0, y(1) = 0,
(6.1.24)
where p(t) = 16000t4−32480t3+21280t2−4746t+189 and q(t) = 3250t5−9425t4+264880t3−448107t2+
233262t−34578. The equation (6.1.24) is a prototype fractional differential equation with two derivatives.
Let
y′′(t) = CTmΨm(t). (6.1.25)
101
0 0.2 0.4 0.6 0.8 1
0
0.002
0.004
0.006
0.008
0.01
0.012
Figure 6.1: m = 8 −−−, m = 16 −−−, m = 64 −−−, Exact Solution . . .Then, by Theorem 2.2.14 and boundary conditions in (6.1.24), the equation (6.1.25) reduces to
y(t) = CTmP2
m×mΨm(t)− CTmQ
2,1m×mΨm(t), (6.1.26)
where ϕ1(t) = t. Also,
Dαy(t) = CTmP2−α
m×mΨm(t)− CTmQ
2,2m×mΨm(t), (6.1.27)
where ϕ2(t) = t1−α
Γ(2−α) . Substituting equation (6.1.25), (6.1.26) and (6.1.27) in equation (6.1.24), we have
CTm
Im×m +
(1817
)(P2−αm×m −Q2,2
m×m)+(1351
)(Pαm×m −Q2,1
m×m)
Ψm(t) = FTmΨm(t), (6.1.28)
The function f(t) = t−12
89250√π
(48p(t) + 7
√tq(t)
)is approximated as f(t) = FTmΨm(t). The exact solution
for the boundary value problem (6.1.24) is y(t) = t5 − 29t4
10 + 76t3
25 − 339t2
250 + 27t125 . The numerical and exact
solutions are plotted in Figure 6.1. The absolute error for different values of m is shown in the Table 6.1.
Obviously, the absolute error decreases with increasing m.
t m = 8 m = 16 m = 32 m = 64 m = 128 m = 256
0.1 3.597(−4) 9.128(−5) 9.128(−5) 3.157(−5) 1.107(−5) 3.899(−6)
0.2 1.582(−3) 1.328(−4) 1.328(−4) 4.072(−5) 1.286(−5) 4.171(−6)
0.3 1.786(−3) 1.215(−4) 1.390(−4) 4.110(−5) 1.253(−5) 3.942(−6)
0.4 1.634(−3) 8.992(−4) 1.215(−4) 3.560(−5) 1.079(−5) 3.373(−6)
0.5 1.157(−3) 1.766(−5) 8.992(−5) 2.677(−5) 8.238(−6) 2.609(−6)
0.6 5.835(−4) 5.361(−5) 5.361(−5) 1.683(−5) 5.425(−6) 1.788(−6)
0.7 1.271(−4) 2.153(−5) 2.153(−5) 7.930(−6) 2.888(−6) 1.040(−6)
0.8 1.196(−4) 1.648(−6) 1.648(−6) 2.108(−6) 1.128(−6) 4.920(−7)
0.9 5.540(−4) 1.420(−6) 1.420(−6) 1.327(−6) 6.274(−7) 2.610(−7)
Table 6.1: Absolute error for different values of m.
Example 6.1.3. Consider the boundary value problem
Dαy(t) +Dβy(t) + (ωπ)2y(t) = −ωπ(sin(ωπt) + ωπ
), y(0) = 0, y(1) = −2. (6.1.29)
102
When α = 2, β = 1 and ω = 1, 3, 5, . . . , the boundary value problem (6.1.29) has exact solution y(t) =
cos(ωπ)− 1. Also, when α = 2 and β = 0, the exact solution of (6.1.29) is given by
y(t) = a(t cos√π2ω2 + 1− 1)− ωπ sinπωt+ bt sin
√π2ω2 + 1, (6.1.30)
where a = π2ω2
π2ω2+1and b = 1
sin√π2ω2+1
a(1 − cos
√π2ω2 + 1) + πω sinπω + y0
. The solutions of the
problem (6.1.29) for different values of α, β and ω are plotted in Figure 6.2. Solutions obtained for α = 2,
β = 0 and α = 2, β = 1 are in good agreement with the exact solutions. As α = 2 and β approaches 0 or 1
the solutions of fractional differential equation approach the solution of integer order ordinary differential
equations. The maximum absolute error is tabulated in Table 6.2. The error grows as ω increases. The
results in the Table 6.2 and Figures 6.2(d), 6.2(e) and 6.2(f) show that increasing ω, while not increasing
m produces a solution whose error is large. For large ω, the maximum absolute error can be reduced by
increasing m.
m = 32 m = 128 m = 512
ω β = 0 β = 1 β = 0 β = 1 β = 0 β = 1
0.5 2.966(−4) 1.25(−4) 1.85(−5) 7.85(−5) 1.15(−6) 4.91(−7)
1.5 0.00491 0.00380 3.08(−4) 2.38(−4) 1.93(−5) 1.49(−5)
2.0 0.02108 0.01918 0.00133 0.001209 8.33(−5) 7.56(−5)
3.5 0.05795 0.05552 0.00371 0.003543 2.32(−4) 2.22(−4)
Table 6.2: Maximum absolute error for α = 2, β = 0, 1 and different values of m, ω.
Method 2: The corresponding integral equation for equation (6.1.17), (6.1.18) is
y(t) = −1
a
(bIα−βy(t) + cIαy(t)
)+tα−1
a
(bIα−βy(1) + cIαy(1)
)+ f(t), (6.1.31)
where f(t) = 1a
(Iαg(t)− tα−1Iαg(1)
)+ y0t
α−1,
Set
y(t) = CTmΨm(t), (6.1.32)
then
Iαy(t) = CTmPα
m×mΨm(t), (6.1.33)
Iα−βy(t) = CTmPα−β
m×mΨm(t), (6.1.34)
ϕ1(t)Iαy(1) = CTmQ
α,1m×mΨm(t) (6.1.35)
ϕ1(t)Iα−βy(1) = CTmQ
α−β,1m×mΨm(t), (6.1.36)
where ϕ1(t) = tα−1. Inserting (6.1.32)- (6.1.36), in equation (6.1.31), we have following linear algebraic
system
CTm
Im×m +
( ba
)(Pα−βm×m −Qα−β,1
m×m
)+( ca
)(Pαm×m −Qα,1
m×m
)Ψm(t) = FTmΨm(t). (6.1.37)
103
0 0.2 0.4 0.6 0.8 1−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
t
y
β=1
β=1 (Exact)
β=0
β=0 (Exact)
β=0.2
β=0.4
β=0.6
β=0.8
(a) ω = 1, J = 6
0 0.2 0.4 0.6 0.8 10
0.5
1
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
β
t
y
(b) ω = 1, J = 6
0 0.2 0.4 0.6 0.8 1−2.5
−2
−1.5
−1
−0.5
0
0.5
t
y
β=1 (Exact)
β=1
β=0 (Exact)
β=0
β=0.2
β=0.4
β=0.6
β=0.8
(c) ω = 1.7, J = 6, α = 2
0 0.2 0.4 0.6 0.8 1−8
−6
−4
−2
0
2
4
6
t
y
β=1(Exact)
β=1
β=0 (Exact)
β=0
β=0.2
β=0.4
β=0.6
β=0.8
(d) ω = 2, J = 6, α = 2
0 0.2 0.4 0.6 0.8 1−15
−10
−5
0
5
10
t
y
β=1 (Exact)
β=1
β=0 (Exact)
β=0
β=0.2
β=0.4
β=0.6
β=0.8
(e) ω = 3, α = 2, J = 6
0 0.2 0.4 0.6 0.8 1−15
−10
−5
0
5
10
t
y
β=1 (Exact)
β=1
β=0 (Exact)
β=0
β=0.2
β=0.4
β=0.6
β=0.8
(f) α = 2, ω = 3, J = 8
0
0.5
1
00.20.40.60.81−30
−20
−10
0
10
20
30
tβ
y
(g) α = 2, ω = 3, J = 8, 0 ≤ β ≤ 1.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
−4
−2
0
2
4
6
β
t
y
(h) α = 1.5, w = 1, j = 7, 0 ≤ β ≤ 1
Figure 6.2: Solutions y(t) of problem (6.1.29), for 1 ≤ α ≤ 2, y0 = cosωπ − 1.
104
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
Figure 6.3: Numerical solutions of problem (6.1.38)
(α = 1.2 —, α = 1.4 —, α = 1.6 —, α = 1.8 —, α = 2 —).
Example 6.1.4. Consider the boundary value problem for inhomogeneous linear fractional differential
equation
Dαy(t) +( 3
57
)y(t) = t+
3tα+1
57Γ(α+ 2), t ∈ [0, 1], y(0) = 0, y(1) =
1
Γ(α+ 2), (6.1.38)
where 1 < α ≤ 2, a ∈ R. The exact solution of boundary value problem is y(t) = tα+1
Γ(α+1) . The equivalent
integral equation for the boundary value problem (6.1.38) is given by
y(t) = − 3
57
Iαy(t)− tα−1Iαy(1)
+ f(t;α), (6.1.39)
where f(t;α) = Iαg(t)− tα−1Iαg(1)+ tα−1
Γ(α+2) . Inserting (6.1.32), (6.1.33) and (6.1.35) in equation (6.1.39)
we have following system of algebraic equations
CTm
Im×m +
3
57
(Pαm×m −Qα,1
m×m
)Ψm(t) = FTmΨm(t). (6.1.40)
where, f is approximated as f(t;α) = FTmΨm(t).
We solve (6.1.40) for Haar coefficient vector, for m = 32 and α = 1.2, 1.4, 1.6, 1.8, 2. The absolute error
is given in the Table 6.3. Also, numerical solutions are plotted in Figure 6.3.
Three–point boundary value problems
Usually the physical systems are modeled as initial (or terminal) value problems or, as two–point boundary
value problems. There are some applications in science and engineering where multi–point boundary value
problems arise. For example, multi–point boundary value problems appear in wave propagation and in
elastic stability. In the following we solve a three–point boundary value problem numerically using the
Haar wavelets.
Example 6.1.5. Consider boundary value problems for a class of fractional differential equations with
three–point boundary conditions,
Dαy(t) + ay(t) = g(t), 1 ≤ α < 2, y(0) = 0, y(1) = ξy(η), (6.1.41)
105
t α = 1.2 α = 1.4 α = 1.6 α = 1.8 α = 2,
0.1 1.53063(−6) 1.45713(−6) 1.56047(−7) 1.49112(−7) 3.50621(−8)
0.2 1.52699(−7) 6.98702(−9) 3.89064(−8) 8.90900(−7) 6.58227(−8)
0.3 8.07661(−7) 3.38714(−8) 5.73057(−7) 1.14669(−7) 8.79828(−8)
0.4 6.31371(−7) 3.13234(−7) 3.89575(−7) 1.86018(−7) 9.72476(−8)
0.5 5.19845(−7) 7.84210(−7) 3.64839(−7) 2.66526(−7) 8.93295(−8)
0.6 1.82879(−6) 1.58534(−6) 2.33905(−7) 6.05455(−7) 5.99495(−8)
0.7 2.43150(−6) 4.81312(−7) 3.65764(−8) 1.05672(−6) 4.84022(−9)
0.8 3.11752(−6) 7.98561(−7) 2.27054(−7) 1.85259(−7) 8.02523(−8)
0.9 3.96605(−6) 1.17157(−6) 5.58515(−7) 3.54806(−7) 1.99566(−7)
Table 6.3: Absolute error for m = 32 and α = 1.2, 1.4, 1.6, 1.8, 2.
0 0.2 0.4 0.6 0.8 1
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Figure 6.4: Exact and Numerical solutions for problem (6.1.5)
(Exact solution —, Numerical solution . . . ).
where η ∈ (0, 1), a, ξ ∈ R and ∆ := 1− ξη = 0.
We transform the differential equation in (6.1.41), by incorporating the boundary conditions, to an
equivalent integral equation
y(t) = −aIαy(t)− atα−1
∆
(ξIαy(η)− Iαy(1)
)+ f(t;α), (6.1.42)
where f(t;α) := Iαg(t) + atα−1
∆
(ξIαg(η) − Iαg(1)
)may be expanded into the Haar series as f(t;α) =
FTmΨm(t). Let
y(t) = CTmΨm(t), (6.1.43)
then
Iαy(t) = CTmPα
m×mΨm(t), (6.1.44)
ϕ1(t)Iαy(η) = CTmQ
α,η,1m×mΨm(t), (6.1.45)
ϕ1(t)Iαy(1) = CTmQ
α,1m×mΨm(t), (6.1.46)
106
where ϕ1(t) = tα−1. Substituting (6.1.43)-(6.1.46) into equation (6.1.42), we obtain following system
CTm
Im×m + aPα
m×m +a
∆
(ξQα,η
m×m +Qαm×m
)Ψm(t) = FTmΨm(t). (6.1.47)
In particular, for α = 32 , a = e−3π
√π
, g(t) = e−3π
40√π
(t2(40t2 − 74t+ 33) + 4e3π
√t(128t2 − 148t+ 33)
), ξ =
−125196 , η = 2
5 and y(1) = − 140 the exact solution is y(t) = t2(t3 − 37t
20 + 3340). The numerical and exact
solutions for m = 32 are shown in Figure 6.4. The absolute error for different values of m is shown in the
Table 6.4
t m = 8 m = 16 m = 32 m = 64 m = 128, m = 256
0.1 6.973(−5) 2.871(−6) 4.426(−7) 1.076(−8) 1.737(−9) 2.417(−10)
0.2 4.592(−5) 7.082(−6) 1.786(−7) 2.729(−8) 8.407(−10) 3.595(−10)
0.3 4.592(−5) 7.082(−6) 1.800(−7) 2.734(−8) 4.338(−10) 8.063(−10)
0.4 1.133(−4) 2.869(−6) 4.422(−7) 1.089(−8) 3.148(−9) 6.640(−10)
0.5 1.373(−4) 8.583(−6) 5.358(−7) 3.289(−8) 2.368(−9) 6.698(−10)
0.6 1.133(−4) 2.870(−6) 4.418(−7) 9.466(−9) 1.439(−9) 1.415(−9)
0.7 5.800(−5) 7.080(−6) 1.782(−7) 2.677(−8) 5.683(−10) 7.945(−10)
0.8 6.679(−5) 7.079(−6) 1.781(−7) 2.629(−8) 7.103(−10) 8.230(−10)
0.9 6.275(−4) 4.172(−6) 4.408(−7) 1.045(−8) 1.469(−9) 5.191(−10)
Table 6.4: Absolute error for α = 32 and different values of m.
Fractionally damped mechanical oscillator
In [113] Attila Pálfalvi have applied the Adomian decomposition method on a fractionally damped mechan-
ical oscillator equation for a sine excitation. We solve this equation with two–point boundary conditions
using the Haar wavelets.
Example 6.1.6. Consider the fractionally damped mechanical oscillator equation with boundary condi-
tions
Dαy(t) + λDβy(t) + νy(t) = g(t), t ∈ [0, 1], y(0) = 0, y(1) = 0, (6.1.48)
where 1 < α ≤ 2, 0 < β ≤ 1, α − β > 1, λ, ν are prescribed constants and g(t) is the forcing function.
If α = 2, β = 1 then (6.1.48) reduces to the usual differential equation of harmonic oscillator. The
corresponding integral equation for equation (6.1.48) is
y(t) = −λIα−βy(t)− νIαy(t) + tα−1λIα−βy(1) + νIαy(1)
+ f(t;α), (6.1.49)
where f(t;α) = Iαg(t)− tα−1Iαg(1),Inserting (6.1.32)-(6.1.36) into equation (6.1.49), we have
CTm
Im×m + λ
(Pα−βm×m −Qα−β
m×m
)+ ν(Pαm×m −Qα
m×m
)Ψm(t) = FTmΨm(t). (6.1.50)
107
0 0.2 0.4 0.6 0.8 1-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
Figure 6.5: Exact and Numerical solutions for problem (6.1.48)
(Exact solution —, Numerical solution . . . ).
In particular, for α = 74 , β = 1
2 and λ = 1, ν = − 1√π
and g(t) = 1√π
(16t
32 p(t)
45045 + 24t14 q(t)
9945Γ( 54)− t2(5t− 3)2r(t)
),
where p(t), q(t) and r(t) are polynomials given by
p(t) = 28028000t3 − 14620320t2 − 21527571t− 270270,
q(t) = 6400000t5 − 15360000t4 + 13328000t3 − 5021120t2 + 757809t− 29835,
r(t) = 25t3 − 50t2 + 29t− 4.
One can easily verify that the exact solution is y(t) = 625t7 − 2000t6 + 2450t5 − 1420t4 + 381t3 − 36t2.
For m = 64 we solve the system (6.1.50) for the vector Cm. The numerical results for the approximate
solution for the problem (6.1.48) are shown in Figure 6.5. The absolute error for different values of m is
shown in the Table 6.5.
t m = 8 m = 16 m = 32 m = 64 m = 128, m = 256
0.1 0.040484 1.018(−3) 2.271(−4) 8.777(−6) 2.823(−6) 6.499(−7)
0.2 0.017310 1.815(−4) 7.112(−5) 1.361(−5) 3.352(−6) 6.356(−7)
0.3 0.010302 1.455(−3) 9.529(−5) 1.446(−5) 2.577(−6) 3.715(−7)
0.4 0.003190 5.008(−4) 4.983(−5) 2.315(−6) 2.387(−6) 9.482(−7)
0.5 0.026157 7.389(−4) 2.966(−5) 1.007(−5) 5.020(−6) 1.595(−6)
0.6 0.039802 5.940(−5) 2.196(−5) 1.663(−6) 2.822(−6) 1.054(−6)
0.7 0.020226 1.014(−4) 5.453(−5) 4.072(−6) 7.854(−7) 6.346(−7)
0.8 0.018284 3.448(−3) 8.472(−5) 1.099(−7) 5.572(−6) 1.886(−6)
0.9 0.295680 4.654(−3) 5.406(−4) 7.195(−6) 8.658(−6) 3.139(−6)
Table 6.5: Absolute error for different values of m.
108
6.1.2 Linear fractional differential equations with variable coefficients
Consider the following boundary value problem with variable coefficients
Dαy(t) +A(t)Dβy(t) +B(t)y(t) = g(t), (6.1.51)
y(0) = 0, y(1) = y0, 1 < α ≤ 2, 0 < β ≤ 1 (6.1.52)
Approximating the fractional derivative and integral as follows
Dαy(t) = CTmΨm(t), (6.1.53)
IαΨm(t) = CTmPα
m×mΨm(t). (6.1.54)
Using properties of fractional derivatives and integrals, together with boundary conditions, from equation
(6.1.53), we have
y(t) = CTm
(IαΨm(t)− tα−1IαΨm(1)
)+ y0t
α−1, (6.1.55)
Dβy(t) = CTm
(Iα−βΨm(t)−
Γ(α)tα−β−1
Γ(α− β)IαΨm(1)
)+
Γ(α)y0tα−1
Γ(α− β). (6.1.56)
Let ϕ1(t) = tα−1, ϕ2(t) = B(t)ϕ1(t), ϕ3(t) = A(t)Γ(α)tα−β−1
Γ(α−β) , then from (6.1.55) and (6.1.56) we have
y(t) = CTm
(IαΨm(t)− ϕ1(t)IαΨm(1)
)+ y0t
α−1, (6.1.57)
B(t)y(t) = CTm
(B(t)IαΨm(t)− ϕ2(t)IαΨm(1)
)+ y0B(t)tα−1, (6.1.58)
A(t)Dβy(t) = CTm
(A(t)Iα−βΨm(t)− ϕ3(t)IαΨm(1)
)+
Γ(α)y0A(t)tα−1
Γ(α− β). (6.1.59)
Substituting (6.1.4) and (6.1.11) in (6.1.57)-(6.1.59), we obtain
y(t) = CTm
(Pαm×mΨm(t)−Qα,1
m×mΨm(t))+ y0t
α−1, (6.1.60)
B(t)y(t) = CTm
(B(t)Pα
m×mΨm(t)−Qα,2m×mΨm(t)
)+ y0B(t)tα−1, (6.1.61)
A(t)Dβy(t) = CTm
(A(t)Pα−β
m×mΨm(t)−Qα,3m×mΨm(t)
)+
Γ(α)y0A(t)tα−1
Γ(α− β). (6.1.62)
Inserting (6.1.60)-(6.1.62) into equation (6.1.51) we have following system of algebraic equations
CTm
Im×m +B(t)Pα
m×m +A(t)Pα−βm×m −Qα,2
m×m −Qα,3m×m
Ψm(t) = FTmΨm(t), (6.1.63)
where the function f(t;α, β) = g(t)− Γ(α)Γ(α−β)y0B(t)tα−1 − y0A(t)t
α−1 is approximated by FTmΨm(t).
Let ti = 2i−12m (i = 1, 2, . . . ,m), then from (6.1.63) we have
CTm
Im×m +BT
m ⊗(Pαm×mΨm×m
)+AT
m ⊗(Pα−βm×mΨm×m
)−(Qα,2m×m +Qα,3
m×m)Ψm×m
. = FTmΨm×m,
(6.1.64)
Using (6.1.10) in (6.1.64), we get
CTm
Im×m +BT
m ⊗(Ψm×mFαm×m
)+AT
m ⊗(Ψm×mFα−βm×m
)−(Qα,2m×m +Qα,3
m×m)Ψm×m
= FTmΨm×m.
(6.1.65)
109
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
t
α=2.00
.
α=2.25
.
α=2.50
.
α=2.75
.
α=3.00
.
(a)
00.2
0.40.6
0.81
22.5
33.5
4−0.1
0
0.1
0.2
0.3
t
α
y
(b)
Figure 6.6: Numerical (solid circles) and exact (solid lines) solutions for problem (6.1.66), (6.1.67).
Example 6.1.7. Consider the boundary value problem for fractional differential equation with variable
coefficients
Dαy(t) + e−9πty(t) = e−9πt(tα−1 − tα
)− Γ(α+ 1), 2 ≤ α < 4, (6.1.66)
y(0) = 0, y′(0) = 0, y′′(0) = 0, y(1) = 0. (6.1.67)
Let B(t) = e−9πt, g(t;α) = e−9πt(tα−1 − tα
)− Γ(α + 1), ϕ1(t) = tα−1 and ϕ2(t) = B(t)tα−1. We
approximate the fractional derivative (of order α) of y(t) by the Haar wavelets as follows
Dαy(t) = CTmΨm(t). (6.1.68)
Together with boundary conditions, we have
y(t) = CTm
(Pαm −Qα,1
m×m)Ψm(t), (6.1.69)
B(t)y(t) = CTm
(B(t)Pα
m −Qα,2m×m
)Ψm(t). (6.1.70)
Substituting (6.1.68) and (6.1.70) into (6.1.66), we have
CTm
Im×m +B(t)Pα
m×m −Qα,1m×m
Ψm(t) = GT
mΨm(t), (6.1.71)
where we have used the approximation g(t;α) = GTmΨm(t). Let ti = 2i−1
2m (i = 1, 2, . . . ,m), then
(6.1.71) reduces to system of algebraic equations
CTm
(Im×m −Qα,1
m×m)Ψm×m +BT
m ⊗(Ψm×mFαm×m
= GT
mΨm×m. (6.1.72)
The numerical solution of the boundary value problem (6.1.66), (6.1.67) is obtained form (6.1.69) by
solving the linear system of algebraic equations (6.1.72) for the vector Cm. One can verify that the exact
solution of the boundary value problem (6.1.66), (6.1.67) is y(t) =(1t − 1
)tα. The numerical solutions for
m = 8 and different values of α are shown in Figure 6.6. The error analysis is given in Table 6.6.
110
t m = 8 m = 16 m = 32 m = 64 m = 128, m = 256
0.1 3.666(−4) 1.452(−4) 3.442(−5) 8.709(−6) 2.176(−6) 5.437(−7)
0.2 9.604(−4) 2.328(−4) 5.858(−5) 1.464(−5) 3.659(−6) 9.139(−7)
0.3 1.280(−3) 3.175(−4) 7.938(−5) 1.984(−5) 4.959(−6) 1.237(−6)
0.4 1.570(−3) 3.940(−4) 9.850(−5) 2.462(−5) 6.152(−6) 1.535(−6)
0.5 1.868(−3) 4.659(−4) 1.164(−4) 2.910(−5) 7.271(−6) 1.813(−6)
0.6 2.135(−3) 5.340(−4) 1.335(−4) 3.336(−5) 8.335(−6) 2.077(−6)
0.7 2.398(−3) 5.995(−4) 1.498(−4) 3.745(−5) 9.355(−6) 2.331(−6)
0.8 2.652(−3) 6.626(−4) 1.656(−4) 4.139(−5) 1.033(−5) 2.574(−6)
0.9 2.882(−3) 7.238(−4) 1.809(−4) 4.522(−5) 1.129(−5) 2.810(−6)
Table 6.6: Absolute error for different values of m and α = 114 .
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
Figure 6.7: Exact and Numerical solutions for problem (6.1.73), (6.1.74)
(Exact solution —, Numerical solution . . . ).
Example 6.1.8. Consider the fractional boundary value problem
Dαy(t) + sin(t)Dβy(t) + e−ty(t) =e−t
2tα(tβ − 1)(2t− 1) + q1(t;α, β) + q2(t;α, β), (6.1.73)
y(0) = 0, y(1) = 0, (6.1.74)
where, 1 < α ≤ 2, 0 < β ≤ 1, q1(t;α, β) = Γ(α + 1)(12 − t(1 + α)
)− Γ(α+β+1)tβ
Γ(β+1)
(12 − (α+β+1)t
β+1
)and
q2(t;α, β) = sin(t)tα−βΓ(α+1)Γ(α−β+1)
(12 − (α+1)t
α−β+1
)− tαΓ(α+β+1)
Γ(α+1)
(12 − (α+β+1)t
α+1
). Furthermore y0 = 0, ϕ1(t) =
tα−1, ϕ2(t) = sin(t)tα−1 and ϕ3(t) = e−t Γ(α)tα−β−1
Γ(α−β) . Also, f(t;α, β) = g(t;α, β) = e−t
2 tα(tβ − 1)(2t− 1) +
q1(t;α, β)+q2(t;α, β). The exact solution is y(t) = 12 tα(tβ−1)(2t−1). Solving the corresponding system of
algebraic equations (6.1.65) we obtain the numerical solution at points ti =(2i−1)2m (i = 1, 2, . . . ,m). Exact
and numerical solutions are plotted in Figures 6.8(a)-6.8(b). The maximum absolute error for different
values of α and β is displayed in Table 6.7.
Example 6.1.9. Now we consider following boundary value problem for the Caputo fractional order
111
β α = 1.3 α = 1.6 α = 1.8 α = 2.0
0.2 3.53(−4) 1.58(−4) 6.51(−5) 3.76(−5)
0.4 5.41(−4) 2.25(−4) 8.74(−5) 6.39(−5)
0.6 6.06(−4) 2.44(−4) 8.92(−5) 8.62(−5)
0.8 6.01(−4) 2.41(−4) 9.94(−5) 1.04(−4)
1.0 5.69(−4) 2.30(−4) 9.54(−5) 1.14(−4)
Table 6.7: The maximum absolute error for m = 32, and different values of α and β.
differential equations with variable coefficients
cDα0+y(t) +A(t)cDβ
0+y(t) +B(t)y(t) = g(t), t ∈ [0, 1], (6.1.75)
y(0) = y0 y(1) = y1, 1 < α ≤ 2, 0 < β ≤ 1. (6.1.76)
Taking ϕ1(t) = t, ϕ2(t) = A(t)ϕ1(t), ϕ3(t) = B(t)Γ(α)t−β
Γ(2−β) . Using the same method as used for bound-
ary value problem (6.1.51) involving the Riemann–Liouville fractional derivative, we arrive at system of
algebraic equations (6.1.65). In particular, for the choice α = 2, β = 1, A(t) = − 2t1+t2
, B(t) = 21+t2
and g(t) = (1 + t2)et, the exact solution of the boundary value problem (6.1.75), (6.1.76) is y(t) =
y1t+ (1− t)2((1− t)et − (1− y0)(t+ 1)
). The exact and numerical solutions for different values of α, β
are plotted in Figures 6.8(c)-6.8(f).
A nonlinear fractional differential equation
Since the method based on Haar wavelets converts the underlying nonlinear problem to a system of
nonlinear algebraic equations. A Nonlinear system having large dimension is itself difficult to solve due
to increased computational complexity. However, for some simple cases one can apply wavelet method to
solve the boundary value problems numerically.
Consider the following class of boundary value problem for nonlinear fractional order differential equa-
tion
Dαy(t) + a(y(t))n = g(t), (6.1.77)
y(0) = 0, y(1) = y0, (6.1.78)
where 1 < α ≤ 2, y0 ∈ R, n ∈ N and g(t) is a given function. The properties of fractional derivatives
and integrals allow us to reduce the the boundary value problem (6.1.77), (6.1.78) into following integral
equation
y(t) = −aIα(y(t))n − atα−1Iα(y(1))n + f(t;α), (6.1.79)
where,
f(t;α) = Iαg(t)− Iαg(1)tα−1 + y0tα−1. (6.1.80)
112
0 0.2 0.4 0.6 0.8 1−0.025
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
t
y
(a) α = 1.5, β = 0.5, m = 32.
00.2
0.40.6
0.81
0
0.5
1
−0.04
−0.02
0
0.02
t
β
y
(b) α = 2, 0 ≤ β ≤ 1.
0 0.2 0.4 0.6 0.8 10.9
1
1.1
1.2
1.3
1.4
1.5
t
y(t)
ExactNumerical
(c) α = 2, β = 1, m = 64
0 0.2 0.4 0.6 0.8 1 00.2
0.40.6
0.81
0.9
1
1.1
1.2
1.3
1.4
1.5
β
y(t;
β)
t
(d) α = 2, 0 < β ≤ 1, y0 = 1.5, y1 = 1.
0 0.2 0.4 0.6 0.8 1 1
1.5
20.2
0.4
0.6
0.8
1
1.2
1.4
1.6
α
y(t;α
)
t
(e) 1 < α ≤ 2, β = 0.5, y0 = 1.5, y1 = 1
00.2
0.40.6
0.81 1
1.2
1.4
1.6
1.8
2
1
1.5
2
2.5
y(t;α
)
αt
(f) 1 < α ≤ 2, β = 0.5, y0 = 2, y1 = 2.
Figure 6.8: Numerical and exact solutions for the boundary value problems (6.1.73), (6.1.74) and (6.1.75),
(6.1.76) ((c)-(f)).
113
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
y
ExactNumerical
Figure 6.9: Exact and Numerical solutions for problem (6.1.85).
We approximate y(t) as
y(t) = CTmΨm(t)
= CTmΨm×mBm(t),
(6.1.81)
Let CTmΨm×m = [d0, d1, . . . , dm−1], then using properties of block-pulse function, we have
(y(t))2 =([d0, d1, . . . , dm−1]Bm(t)
)2= d20b0(t) + d21b1(t) + · · ·+ d2m−1bm−1(t)
By induction, we have
(y(t))n = dn0b0(t) + dn1 b1(t) + · · ·+ dnm−1bm−1(t)
= [dn0 , dn1 , . . . , d
nm−1]Bm(t)
= DTmBm(t)
where Dm = [dn0 , dn1 , . . . , d
nm−1]
T . Now
Iα(y(t))n = DTmIαBm(t)
= DTmFαm×mBm(t).
(6.1.82)
Let φ(t) ∈ L2[0, 1] be a given function, then
ν(t)Iα(y(1))n =
∫ 1
0
ν(t)(1− s)α−1
Γ(α)(y(s))nds
=
∫ 1
0kα(s, t)(y(s))
nds
= DTm
∫ 1
0Bm(s)Ψ
T (s)Kαm×mΨm(t)ds
= DTm
(∫ 1
0Bm(s)BT
m(s)ds
)Ψm×mK
αm×mΨm(t)
=1
mDTmΨ
Tm×mK
αm×mΨm×mBm(t).
(6.1.83)
114
Defining ν := atα−1 and substituting (6.1.81)-(6.1.83) into equation (6.1.79), we have following system of
nonlinear algebraic equations.
CTmΨm(t) = −aDT
mFαm×mBm(t) +1
mDTmΨm×mK
αm×mΨm×mBm(t) + FTmΨm(t), (6.1.84)
where the function f(t) is approximated by f(t) = FTmΨm(t). This nonlinear system can be solved using
Matlab built in function fsolve.
Example 6.1.10. Consider the following boundary value problem for nonlinear fractional order differential
equation
Dαy(t) + e−5π(y(t))2 =Γ(α+ 3)
2t2 + e−5πt2α+4, y(0) = 0, y(1) = 1, (6.1.85)
where 1 < α ≤ 2. The exact solution is y(t) = tα+2. We compare the numerical solution with exact
solution for m = 32 and different values of α and the results are given below in the Table 6.8.
t α = 1.2 α = 1.5 α = 1.7 α = 1.9 α = 2
0.1 0.0056 0.0028 0.0017 0.0011 0.0009
0.3 0.0495 0.0345 0.0271 0.0213 0.0188
0.5 0.1088 0.0884 0.0769 0.0669 0.0625
0.7 0.1368 0.1229 0.1145 0.1066 0.1028
0.9 0.0793 0.0768 0.0752 0.0736 0.0728
Table 6.8: For problem (6.1.85), absolute error for m = 32 and different values of α.
6.2 Numerical solutions to fractional partial differential equations
In this section we apply the numerical scheme developed in previous section to linear fractional partial
differential equations. Our focus of interest are the two classes of boundary value problems for fractional
partial differential equations. Throughout this section, the fractional partial derivative will be considered
in the Caputo sense.
6.2.1 Fractional partial differential equations with constant coefficients
This subsection considers following class of linear fractional partial differential equations with variable
coefficients
∂αu(x, t)
∂tα+ λ
∂βu(x, t)
∂tβ+ µu(x, t) = η
∂γu(x, t)
∂xγ+ f(t, x), 0 < α ≤ 2, 0 ≤ β ≤ 1, 1 ≤ γ ≤ 2, (6.2.1)
satisfying non–homogenous initial and boundary conditions conditions
u(x, 0) = ρ(x),∂u(x, t)
∂t|t=0 = σ(x), u(0, t) = ξ(t), u(1, t) = ζ(t). (6.2.2)
115
In particular, for 1 < α ≤ 2, λ, µ, η > 0, the equation (6.2.1) reduces to the fractional telegraph
equation. It also includes heat, wave and the Poisson equations as special cases. Many researchers have
recently studied the equation using various numerical techniques. In [35] S. Das et al. have employed the
homotopy analysis method for numerical solutions of the equation satisfying initial conditions u(x, t)|t = 0,∂u(x,t)∂t |t=0 = 0. Some other contributions have been mad by authors, such as S. Momani [106], F. Huang
[55], R. C. Cascaval et al. [26], R. F. Camargo et al. [23], J. Chen et al. [30], and E. Orsingher, L.
Beghin [111]. We use the Haar wavelet method developed in Section 6.1 to provide numerical solutions.
Approximating ∂αu(x,t)∂tα by two dimensional Haar wavelet series as
∂αu(x, t)
∂tα= ΨT
m(x)Km×mΨm(t). (6.2.3)
Operating on both sides of (6.2.3) by fractional integral Iαt in variable t, and using Lemma 2.2.24 we have
u(x, t) = ΨTm(x)Km×m
(∫ t
0
(t− s)β
Γ(α)Ψm(s)ds
)+ p(x)t+ q(x). (6.2.4)
Using the initial conditions u(x, 0) = ρ(x), ∂u(x,t)∂t |t=0 = σ(x), from (6.2.4) we have q(x) = ρ(x) and
p(x) = σ(x). Thus, the equation (6.2.4) takes the form
u(x, t) = ΨTm(x)Km×mP
αm×mΨm(t) + σ(x)t+ ρ(x). (6.2.5)
Applying ∂β
∂tβon both sides of (6.2.5) and using Lemma 2.2.19, we have
∂βu(x, t)
∂tβ= ΨT
m(x)Km×mPα−βm×mΨm(t) + σ(x)
t1−β
Γ(2− β). (6.2.6)
Substituting (6.2.3), (6.2.5) and (6.2.6) in (6.2.1), we get
η∂γu(x, t)
∂xγ=ΨT
m(x)Km×mΨm(t) + λΨTm(x)Km×mP
α−βm×mΨm(t)
+ µΨTm(x)Km×mP
αm×mΨm(t) + g(x, t)
=ΨTm(x)
Km×m(I + λPα−β
m×m + µPαm×m) +Gm×m
Ψm(t),
(6.2.7)
where
g(x, t) = σ(x)( λt1−β
Γ(2− β)+ µt
)+ µρ(x)− f(x, t) = ΨT
m(x)Gm×mΨm(t).
Applying Iγx on both sides of (6.2.7), we have
ηu(x, t) = IγxΨTm(x)
Km×m(I + λPα−β
m×m + µPαm×m) +Gm×m
Ψm(t) + xφ1(t) + φ2(t). (6.2.8)
Now, applying the boundary conditions u(0, t) = ξ(t), we have φ2(t) = ξ(t), and u(1, t) = ζ(t) implies
φ1(t) = −IγxΨTm(1)
Km×m(I + λPα−β
m×m + µPαm×m) +Gm×m
Ψm(t) + ζ(t)− ξ(t). (6.2.9)
Substituting (6.2.9) in (6.2.8) we have
ηu(x, t) =ΨTm(x)
((Pγ
m×m)T − (Qγ
m×m)T)
Km×m(I + λPα−βm×m + µPα
m×m)
+Gm×mΨm(t) + x(ζ(t)− ξ(t)) + ξ(t).
(6.2.10)
116
0 0.2 0.4 0.6 0.8 10
0.5
1−1
−0.5
0
0.5
1
t
x
u
HaarExact
Figure 6.10: Numerical and exact solutions for telegraph equation (6.2.12).
where, IγxΨm(x) = Pγm×mΨm(x) = ΨT
m(x)(Pγm×m)
T and xIγxΨTm(1) = Qγ
m×mΨm(x). From (6.2.5) and
(6.2.10) we get Sylvester matrix equation
((Pγm×m)
T − (Qγm×m)
T )Km×m(I + λPα−βm×m + µPα
m×m)− ηKm×mPαm×m
= Sm×m − ((Pγm×m)
T − (Qγm×m)
T )Gm×m,(6.2.11)
where, s(x, t) := x(ζ(t)− ξ(t)) + ξ(t)− η(σ(x)t+ ϱ(t)) = ΨTm(x)Sm×mΨm(t). Solving (6.2.11) for Km×m
and using (6.2.5) or (6.2.10) we can get the solution of the problem (6.2.1).
Example 6.2.1. We consider the time–fractional telegraph equation
∂αu(x, t)
∂tα+∂α−1u(x, t)
∂tα−1+ u(x, t) =
∂2u(x, t)
∂x2+
(Γ(2α+ 1)
Γ(α+ 1)(1 +
t
α+ 1)− 49tα
)t2α cos (7x), (6.2.12)
satisfying initial and boundary conditions u(x, 0) = 0, ∂u(x,t)∂t |t=0 = 0, u(0, t) = t2α, u(1, t) = 0.7539022t2α.
One can easily verify that the exact solution for the problem is u(x, t) = t2α cos (7x). The numerical so-
lutions are obtained for the points (xi, ti) = (2i−12n , 2i−1
2n ), i = 1, 2, . . . , n, n = 2J . Computer plots for
numerical and exact solutions for J = 4, α = 1.3 are shown in Figure 6.10.
Example 6.2.2. Consider the time–fractional heat equation
∂αu(x, t)
∂tα= λ
∂2u(x, t)
∂x2, 0 < α ≤ 1, (6.2.13)
satisfying the initial condition, u(x, 0) = xα(1− x) and the boundary conditions u(0, t) = 0, u(1, t) = 0.
For integer case, i.e., α = 1 and λ = 0.2 the problem is solved numerically in [79]. The numerical
solutions for fractional case, α = 0.15, 0.35, 0.50, 1.0 are obtained by the method discussed above. The
numerical solutions are plotted in Figure 6.11a. It is observed that the solutions for the fractional cases
are approaching to the solution of classical heat equation.
117
00.2
0.40.6
0.81
00.2
0.40.6
0.81
0
0.05
0.1
0.15
0.2
0.25
tx
u
α=0.15
α=0.35
α=0.50
α=1.00
(a) Numerical solutions of heat equation for J = 5, α = 0.15, α =
0.35, α = 0.50 and α = 1.00.
0
0.5
1
00.20.40.60.81
−0.04
−0.02
0
0.02
0.04
t
x
u(x,
t)
HaarExact
(b) Numerical and exact solutions ( J = 5.0,α = 0.3).
Figure 6.11: Solutions for (6.2.13) and (6.2.15)
Example 6.2.3. Consider the fractional partial differential equation
∂αu(x, t)
∂xα+ λ
∂2u(x, t)
∂t2= xα−1et, 1 ≤ α < 2, x, t ∈ [0, 1], (6.2.14)
with boundary conditions u(0, t) = 0, u(1, t) = et, u(x, 0) = xα−1, u(x, 1) = exα−1.
For α = 2, the equation reduces to Poisson equation having exact solution u(x, t) = xey. Numerical
solutions obtained for α = 1.2, α = 2 are plotted in Figure 6.12.
0 0.2 0.4 0.6 0.8 1 0
0.5
1−1
−0.5
0
0.5
1
1.5
2
2.5
3
xt
u
α=2.0
α=1.2
Figure 6.12: Numerical and exact solutions for the equation (6.2.14) for α = 0.3, J = 5.0
118
Example 6.2.4. Consider partial fractional differential equation
∂αu(x, t)
∂tα−λ∂
2u(x, t)
∂x2=
(t1−α
Γ(2− α)− Γ(3α+ 1)t2α
Γ(2α+ 1)
)+144λt(1−t3α−1) sin(12x), 0 ≤ α ≤ 1, (6.2.15)
satisfying initial and boundary conditions u(x, 0) = 0, u(0, t) = 0, u(1, t) = −0.536573t(1 − t3α−1). The
exact solution of the problem is u(x, t) = t(1−t3α−1) sin(12x). In order to show the accuracy and simplicity
of the method, we compare numerical solutions for different values of α and J by computing the maximum
absolute differences between solutions at collocation points. The numerical results are shown in Table 6.9.
Also, for α = 0.3 and J = 5, the plots for solutions are shown in Figure 6.11b.
J α = 0.1 α = 0.3 α = 0.5 α = 0.7 α = 0.9 α = 1.0
4 5.2008(−4) 7.2442(−5) 3.5490(−4) 7.6899(−4) 1.1825(−3) 1.3946(−3)
5 5.2278(−5) 7.3221(−6) 3.6103(−5) 7.8885(−5) 1.2262(−4) 1.4573(−4)
6 5.4590(−6) 7.6600(−7) 3.7860(−6) 8.3071(−6) 1.2998(−5) 1.5500(−5)
7 6.0733(−7) 8.5326(−8) 4.2210(−7) 9.2925(−7) 1.4640(−6) 1.7403(−6)
8 7.0972(−8) 9.9905(−9) 4.9472(−8) 1.0927(−7) 1.7236(−7) 2.0439(−7)
Table 6.9: Maximum absolute error for different values of J and α.
6.2.2 Fractional partial differential equations with variable coefficients
This subsection concerns with the numerical solutions for the for the class of fractional partial differential
equations∂γu(x, t)
∂tγ− a(x)
∂αu(x, t)
∂xα+ b(x)
∂βu(x, t)
∂xβ+ d(x)u(x, t) = f(x, t), (6.2.16)
( 0 < γ ≤ 2, 1 < α ≤ 2, 0 < β ≤ 1) with initial and boundary conditions,
(i) u(x, 0) = ϕ1(x),∂u(x, t)
∂t|t=0 = ψ1(x), or (ii) u(x, 0) = ϕ1(x), u(x, 1) = ψ2(x), (6.2.17)
and
u(0, t) = µ(t), u(1, t) = ν(t), (6.2.18)
For a(x) > 0, b(x) > 0, d(x) ≥ 0 and 0 < α < 1, 1 < β < 2, the problem reduces to fractional convection–
diffusion equations. In [151] Y. Zhang have used stable difference scheme for the numerical solutions of the
convection–diffusion equation supplemented with initial condition u(x, 0) = 0 and boundary conditions
u(0) = u(1) = 0. For a, b > 0, d = 0 and f(x, t) = 0, the problem reduces to space–time fractional
advection–dispersion equation. Many authors have discussed numerical methods for convection–diffusion,
including [31, 40, 118, 130, 138]. Motivated by the work mentioned above, we give a numerical algorithm
based on operational matrices of integration and the matrix derived in Section 6.1. We approximate∂αu(x,t)∂xα by the Haar wavelets as
∂αu(x, t)
∂xα= ΨT
m(x)Km×mΨm(t). (6.2.19)
119
Applying the fractional integral operator Iαx on both sides of (6.2.19), we have
u(x, t) = IαxΨTm(x)Km×mΨm(t) + p(t)x+ q(t). (6.2.20)
Using the boundary condition (6.2.18), from (6.2.20), we have
q(t) = µ(t), p(t) = −IαxΨTm(1)Km×mΨm(t) + ν(t)− µ(t).
Equation (6.2.20) takes the form
u(x, t) = IαxΨTm(x)Km×mΨm(t)− xIαxΨ
Tm(1)Km×mΨm(t) + x(ν(t)− µ(t)) + µ(t), (6.2.21)
Since IαxΨm(x) = Pαm×mΨm(x) and ϕ1(x)IαxΨm(1) = Qα,1
m×mΨm(x), where ϕ1(x) = x. Therefore, (6.2.21)
takes the form
u(x, t) =ΨTm(x)(P
αm×m)
TKm×mΨm(t)−ΨTm(x)(Q
α,1m×m)
TKm×mΨm(t)
+ x(ν(t)− µ(t)) + µ(t).(6.2.22)
Applying the caputo operator ∂β
∂xβon (6.2.21), using the Lemma 2.2.19 and Lemma 2.2.23, we have
∂βu(x, t)
∂xβ= Iα−βx ΨT
m(x)Km×mΨm(t)−x1−β
Γ(2− β)IαΨT
m(1)Km×mΨm(t) +(ν(t)− µ(t))x1−β
Γ(2− β). (6.2.23)
For simplicity, we introduced some convenient notations.
ϕ2(x) =b(x)x1−β
Γ(2− β), ϕ3(x) = xd(x), r(x, t) =
(ν(t)− µ(t))b(x)
Γ(2− β)x1−β + xd(x)(ν(t)− µ(t)) + d(x)q(t),
s(x, t) = x(ν(t)− µ(t)) + µ(t) + ψ1(x)t+ ϕ1(x),
g(x, t) = −x(ν(t)− µ(t))− µ(t) + (ψ2(x)− ϕ1(x))t+ ϕ1(x)
d(x)IαxΨm(x) = Pαm×mΨm(x), b(x)IαxΨm(x) = Pα
m×mΨm(x).
Substituting (6.2.19), (6.2.21) and (6.2.23) in (6.2.16), we have
∂γu(x, t)
∂tγ=(a(x)ΨT
m(x)−ΨTm(x)(P
α−βm×m)
T +ΨTm(x)(Q
α,2m×m)
T −ΨTm(x)(P
αm×m)
T
+ΨTm(x)(Q
α,3m×m)
T)Km×mΨm(t) +ΨT
m(x)Rm×mΨm(t)
Applying Iγt on both sides of above equation
u(x, t) =(a(x)ΨT
m(x)−ΨTm(x)(P
α−βm×m)
T +ΨTm(x)(Q
α,2m×m)
T −ΨTm(x)(P
αm×m)
T
+ΨTm(x)(Q
α,3m×m)
T)Km×mΨm(t) +ΨT
m(x)Rm×mIγt Ψm(t) + w(x)t+ ω(x).
By initial conditions (6.2.17)(i) we have w(x) = ϕ1(x) and v(x) = ψ1(x). Therefore
u(x, t) =[(a(x)ΨT
m(x)−ΨTm(x)(P
α−βm×m)
T +ΨTm(x)(Q
α,2m×m)
T −ΨTm(x)(P
αm×m)
T
+ΨTm(x)(Q
α,3m×m)
T)Km×m +ΨT
m(x)Rm×m
]Pγm×mΨm(t) + ψ1(x)t+ ϕ1(x).
(6.2.24)
120
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
t
x
u(x,
t)
Haar
Exact
(a) Exact and numerical solutions for J = 5.
00.2
0.40.6
0.81
0
0.5
10
1
2
x 10−4
tx
Err
or
(b) For J = 6, the error between exact and numerical solutions.
Figure 6.13: The absolute error between Haar wavelet solution and analytic solution.
Now, taking into account the boundary conditions (6.2.17)(ii), we have w(x) = ϕ1(x) and
υ(x) =[(a(x)ΨT
m(x)−ΨTm(x)(P
α−βm×m)
T +ΨTm(x)(Q
α,2m×m)
T −ΨTm(x)(P
αm×m)
T
+ΨTm(x)(Q
α,3m×m)
T)Km×m +ΨT
m(x)Rm×m
]Iγt Ψm(1) + (ψ2(x)− ϕ1(x).
(6.2.25)
Therefore, (6.2.24) becomes
u(x, t) =[(a(x)ΨT
m(x)−ΨTm(x)(P
α−βm×m)
T +ΨTm(x)(Q
α,2m×m)
T
−ΨTm(x)(P
αm×m)
T +ΨTm(x)(Q
α,3m×m)
T)Km×m +ΨT
m(x)Rm×m
](Pγm×m −Qγ
m×m
)Ψm(t) + (ψ2(x)− ϕ1(x))t+ ϕ1(x).
(6.2.26)
From 6.2.22 and (6.2.24) we have the Sylvester equation
((Pαm×m)
T − (Qα,1m×m)
T )Km×m −( ηmΨm×mAm×mΨ
Tm×m
− (Pα−βm×m)
T − (Pαm×m)
T + (Qα,2m×m)
T + (Qα,3m×m)
T)Km×mP
γm×m = Rm×mP
γm×m + Sm×m,
(6.2.27)
where Am×m := diag[a(xi)], xi = 2i−12m , i = 1, 2, . . . ,m. Also, from (6.2.22) and (6.2.26), we get following
matrix equation
((Pαm×m)T − (Qα,1
m×m)T )Km×m −( ηmΨm×mAm×mΨT
m×m − (Pα−βm×m)T − (Pα
m×m)T + (Qα,2m×m)T
+ (Qα,3m×m)T
)Km×m
(Pγ
m×m −Qγm×m
)= Rm×m
(Pγ
m×m −Qγm×m
)+Gm×m.
(6.2.28)
Solving equations (6.2.32) and (6.2.28), for Gm×m and then substituting in (6.2.22), we get the approxi-
mate solution of the problem (6.2.16).
Now, we implement the Haar wavelet method to solve different types of fractional partial differential
equations. Also, we compare the results obtained with exact solutions and solutions obtained in literature
by other methods.
121
Example 6.2.5. Consider the partial differential equation
∂αu(x, t)
∂tα=
1
2x2∂2u(x, t)
∂x2, 1 < α ≤ 2, (6.2.29)
subject to initial and boundary conditions
u(x, 0) = x,∂u(x, t)
∂t|t=0 = 0, (6.2.30)
u(0, t) = 0, u(1, t) = 1 + tEα,2(tα). (6.2.31)
The series solution of the problem is [105] u(x, t) = x+x2∑∞
k=0tkα+1
Γ(kα+2) . The numerical solutions by Haar
wavelets for J = 7 and different values of α are shown in Table 6.10. Also we compare the results with
the solutions obtained in reference [105] using Adomian decomposition method and variational iteration
method. Solutions by the Haar wavelets agree well with the solutions by ADM and VIM.
α = 1.5 α = 1.75
t x uHarr uADM uV IM uHarr uADM uV IM
0.2 0.25 0.26284409 0.26284061 0.26269693 0.26267320 0.26266989 0.26248505
0.50 0.55136710 0.55136246 0.55078773 0.55068355 0.55067959 0.55059402
0.75 0.86557129 0.86556550 0.86427239 0.86403415 0.86402909 0.86383654
0.4 0.25 0.27697847 0.27697113 0.27642739 0.27616359 0.27567202 0.27567202
0.50 0.60792985 0.60788455 0.60570958 0.60463521 0.60268808 0.60268808
0.75 0.99272600 0.99274024 0.98784655 0.98542677 0.98104818 0.98104818
0.6 0.25 0.29310668 0.29309481 0.29198616 0.29110091 0.28979084 0.28979084
0.50 0.66272310 0.67237923 0.66794464 0.66437353 0.65916338 0.65916339
0.75 1.14235387 1.13785532 1.12787544 1.11793023 1.10811762 1.10811764
Table 6.10: The Haar wavelet solutions and solutions obtained in [105], using ADM and VIM.
Example 6.2.6. Consider the fractional diffusion equation
∂u(x, t)
∂t= a(x)
∂1.8u(x, t)
∂x1.8+ f(x, t), (6.2.32)
with initial conditions u(x, 0) = x2(1 − x), and Dirichlet boundary Condition u(0, t) = 0, u(1, t) = 0.
In particular, for a(x) = Γ(1.2)x1.8, f(x, t) = 3x2(2x− 1)e−t, the problem has been studied for numerical
solutions in [66] by a method based upon Chebyshev approximations. It can be easily verified that the
exact solution is u(x, t) = x2(1 − x)e−t. The exact and numerical solutions by the Haar wavelets are
plotted in Figure 6.13(a). Figure 6.13(b) show the maximum absolute error between exact and numerical
solutions.
Example 6.2.7. Finally, we consider the fractional convection–diffusion equation
∂γu(x, t)
∂tγ= −a(x)∂
αu(x, t)
∂xα+ b(x)
∂βu(x, t)
∂xβ+ f(t, x), (6.2.33)
122
where 0 < γ ≤ 2, 1 < α ≤ 2, 0 < β ≤ 1 and, with boundary conditions u(x, 0) = u(x, 1) = 0, u(0, t) =
u(1, t) = 0. We solve this problem for
a(x) =Γ(2 + β)Γ(5− α− β)xβ , b(x) = Γ(2β − α+ 2)Γ(5− 2α)xα,
f(x, t) =(2π)x2β+1 − x4−α)t1−γE2,2−γ(−(2πt)2) +Γ(2β + 2)(Γ(5− α− β)
− Γ(5− 2α))x2β+1 + Γ(5− α)(Γ(2β − α+ 2)− Γ(β + 2))x4−αsin(2πt).
The exact solution of the problem is u(x, t) = (x2β+1 − x4−α) sin(2πt). Numerical solutions are obtained
for different values of α, β and γ. The results are shown in Figure 6.14. The absolute error computed
between exact and numerical solutions by the Haar wavelets, for J = 7 and different values of α, β, γ is
shown in figure Figure 6.15.
123
00.2
0.40.6
0.81
0
0.5
1−0.06
−0.04
−0.02
0
0.02
0.04
0.06
t
x
u(x,
t)
HaarExact
(a) Exact and numerical solutions for J = 5, α = 2.0, β = 0.35,
γ = 0.50.
00.2
0.40.6
0.81
0
0.5
1−0.1
−0.05
0
0.05
0.1
tx
u(x,
t)
HaarExact
(b) Error for J = 5, α = 2.0, β = 0.35, γ = 0.50.
00.2
0.40.6
0.81
0
0.5
1−0.2
−0.1
0
0.1
0.2
t
x
u(x,
t)
HaarExact
(c) For J = 5, α = 2.0, β = 0.75, γ = 0.50.
Figure 6.14: Exact and numerical solutions for different values of J , α, β and γ.
124
00.2
0.40.6
0.810
0.5
10
2
4
6
8
x 10−4
xt
Err
or
(a) For J = 5, α = 2.0, β = 0.75, γ = 0.50.
00.2
0.40.6
0.810
0.20.4
0.60.8
10
0.2
0.4
0.6
0.8
1
x 10−3
x
t
Err
or
(b) For J = 5, α = 0.75, β = 0.50, γ = 0.75.
00.2
0.40.6
0.810
0.20.4
0.60.8
1
0
2
4
6
8
x 10−3
x
t
Err
or
(c) Error for J = 7, α = 0.75, β = 0.5, γ = 0.75.
Figure 6.15: The absolute error between exact and numerical solutions for different values of α, β and γ.
Chapter 7
Numerical solutions to fractional
differential equations by the Legendre
wavelets
Approximation by orthogonal families of functions have been playing a vital role in the development of
physical sciences engineering and technology. These functions have received considerable attention in
the evaluation of new numerical techniques to solve various problems of dynamic system and optimal
controls. The most commonly used classes of orthogonal functions include the sets of piecewise constant
basis functions such as block-pulse functions and the Walsh functions; the sets of orthogonal polynomials,
for example the Laguerre, the Chebshev, and the widely used sine-cosine functions in the Fourier series.
The application of the Legendre wavelets for solving differential and integral equations is thoroughly
considered in [10, 67, 117, 147] and the references therein. The main characteristics of this technique is
that it converts the underlying problems into equivalent algebraic systems. The objective of the present
chapter is to introduce a Legendre wavelet operational matrix of fractional order integration and use it to
solve different types of fractional differential equations. The derivation of the Legendre wavelet operational
matrix of integration is similar to the derivation of the Haar wavelet operational matrix of integration [80]
and CAS operational matrix of integration [120, 121]. The Legendre wavelet method efficiently works
for initial value and boundary value problems for fractional order differential equations. The illustrative
examples are provided to demonstrate the applicability of the numerical scheme based on the Legendre
wavelet operational matrix of integration. Throughout in this chapter, we consider the derivative in the
Caputo sense.
7.0.3 The Legendre wavelets
The Legendre polynomials denoted by Pm(t), m ∈ N, are defined on [−1, 1] and can be determined by
recurrence formulae:
P0 = 1, P1 = t, Pm+1(t) =2m+ 1
m+ 1tPm(t)−
m
m+ 1Pm−1(t), m = 1, 2, 3, . . . .
125
126
0 0.2 0.4 0.6 0.8 1−3
−2
−1
0
1
2
3
4
5
6
t
L1,0
L1,1
L1,2
L2,0
L2,1
L2,2
Figure 7.1: The Legendre wavelets for M = 3, k = 2.
The Legendre wavelets are defined on interval [0, 1) as [10]
Ln,m(t) =
2k2√2m+ 1Pm(2
kt− n), n−12k
≤ t < n+12k,
0, elsewhere,
where k = 2, 3, . . . , n = 2n − 1, n = 1, 2, 3, . . . , 2k−1 and, for some fixed positive integer M , m =
0, 1, 2, . . . ,M − 1 is the order of the Legendre polynomial.
7.0.4 Function approximations by the Legendre wavelets
Any function g(t) ∈ L2[0, 1) can be expanded into Legendre wavelet series as [10]:
g(t) =
∞∑n=1
∞∑m=0
cn,mLn,m(t), (7.0.1)
where cn,m = ⟨f(t), Ln,m(t)⟩. When working with the infinite series representation (7.0.1) in practice, we
can only deal with finite number of terms.
g(t) =
2k−1∑n=1
M−1∑m=0
cn,mLn,m(t), (7.0.2)
where Cm and Lm are m× 1 (m = 2k−1M) matrices, given by
Cm =[c1,0, c1,1, . . . , c1,M−1, c2,0, c2,1, . . . , c2,M−1, . . . , c2k−1,0, c2k−1,1, . . . , c2k−1,m−1]T ,
Lm(t) =[L1,0(t), L1,1(t), . . . , L1,M−1(t), L2,0(t), L2,1(t), . . . ,
L2,M−1(t), . . . , L2k−1,0(t), L2k−1,1(t), . . . , L2k−1,m−1(t)]T .
The convergence of the Legendre wavelet series (7.0.2) is established in [82]. A function of two variables
k(s, t) ∈ L2([0, 1)× [0, 1)) can be expanded into Legendre wavelet series as
k(s, t) =
2k−1∑n1=1
M−1∑m1=0
2k−1∑n2=1
M−1∑m2=0
⟨Ln1,m1(s), ⟨k(s, t), Ln2,m2(t)⟩⟩Ln1,m1(s)Ln2,m2(t). (7.0.3)
127
In order to simplify notations, we define i =M(n−1)+m+1, where n = 1, 2, 3, . . . , 2k−1, (k = 2, 3, . . . , n =
2n− 1, m = 0, 1, 2, . . . ,M − 1). Thus, we can write (7.0.2) as
g(t) ≈m∑i=1
ciLi(t) = CTmLm(t), (7.0.4)
where, Cm = [c1, c2, · · · , cm]T , Lm(t) = [L1(t), L2(t), · · · , Lm(t)]T . Similarly, (7.0.3) can be represented as
k(s, t) =
m∑i=1
m∑j=1
kijLi(s)Lj(t) = LTm(s)Km×mLm(t), (7.0.5)
where Km×m = [kij ]m×m and kij = ⟨Li(s), ⟨k(s, t), Lj(t)⟩⟩. The number of needed terms increases fast
with the desired accuracy. Therefore, these type of calculations are performed with computers.
7.1 An operational matrices of fractional order integration
The Legendre function vector Lm(t) can be represented in terms of block-pulse functions as
Lm(t) = Lm×mBm(t), (7.1.1)
We recall the fractional integral of block-pulse function vector
(IαBm)(t) = Fαm×m Bm(t), (7.1.2)
where Fαm×m is the block-pulse operational matrix of the fractional order integration [73].
On the other hand, since (IαLm)(t) ∈ L2[0, 1], there fore we have following Legendre wavelet series
(IαLm)(t) = Rαm×mLm(t), (7.1.3)
where Rαm×m is the Legendre wavelet operational matrix of fractional order integration. Substituting
(7.1.1), (7.1.2) into (7.1.3), we have
Rαm×m = Lm×mFαm×mL
−1m×m.
In particular, let α = 0.75, M = 3, k = 2, the operational matrix of fractional order integration Rαm×m
is given by
R0.756×6 =
0.3741 0.5006 0.1738 −0.0468 −0.0082 0.0066
0 0.3741 0 0.1738 0 −0.0082
−0.1873 0.0392 0.0721 −0.0196 0.0609 0.0046
0 −0.1873 0 0.0721 0 0.0609
0.1660 0.2773 −0.0004 −0.0323 0.0404 0.0057
0 0.1660 0 −0.0004 0 0.0404
.
Remark 7.1.1. The Legendre wavelet operational matrix can be obtained by computing inner products
directly. But this approach leads to higher computational complexity. Based on above arguments, we
have developed an efficient Matlab program, given in Appendix A, which can be utilized to compute the
Legendre operational matrix of fractional order integration easily.
128
0 0.2 0.4 0.6 0.8 1 1
1.5
2
0.2
0.4
0.6
0.8
1
1.2
α
t
y
(a)
0 0.2 0.4 0.6 0.8 1
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
t
y
α=1.0
α=1.0 (Exact)
α=1.2
α=1.4
α=1.6
α=1.6
α=2.0
α=2.0 (Exact)
(b)
Figure 7.2: Numerical and exact solutions of problem (7.2.1), (7.2.2) for 1 ≤ α ≤ 2.
7.2 Numerical solutions of fractional differential equations
The general formulism of the Legendre wavelet method is much similar to the Haar wavelet method for
solutions of different types of fractional differential equations discussed in previous chapter. Since there
is no general formulism for the comparison of the Legendre wavelet method to other existing methods
that can be applied to solve fractional differential equations. Therefore we will compare the method with
the Haar wavelet method and some other methods by considering some specific examples for which exact
solutions are available.
Example 7.2.1. Consider the fractional differential equation
cDαy(t) + y(t) = 0, 0 < α ≤ 2, (7.2.1)
satisfying
y(0) = 1, y′(0) = 0. (7.2.2)
This problem have been studied by P. Kumar and O.P. Agrawal [76], K. Diethelm, N.J. Ford and A.D.
Freed [38]. The second initial condition is taken into account only if 1 < α ≤ 2. Using the Laplace
transform, the exact solution of the problem (7.2.1), (7.2.2) is given by
y(t) = Ea(−tα), (7.2.3)
where Ea(t) is the Mittag–Leffler function of order α. The corresponding integral representation for equa-
tions (7.2.1), (7.2.2) is
y(t) = −Iαy(t) + y(0) + y′(0)t = −Iαy(t) + 1. (7.2.4)
Now, approximating the solution y(t) in term of the Legendre wavelets as
y(t) = CTmLm(t). (7.2.5)
Applying the fractional integral operator Iα on both sides of (7.2.5), we have
Iαy(t) = CTmIαLm(t),
= CTmRα
m×mLm(t).(7.2.6)
129
Inserting (7.2.5) and (7.2.6) into equation (7.2.4) we have following system of algebraic equations.
CTm
(Im×m + Rα
m×m)Lm(t) = [1, 0, 0, . . . , 0]T . (7.2.7)
Solving the linear system (7.2.7), for the vector Cm, and substituting it in (7.2.5) we get approximate
solution of the initial value problem (7.2.1), (7.2.2). First, we consider the case when α is an integer i.e.,
α = 1,or α = 2. For the case when α = 1, we observe that the exact solution is y(t) = e−t and for
α = 2 we have y = cos(t). The numerical and exact solutions plotted in the Figure 7.2 indicate that, as
α approaches to 1 or 2, the numerical solutions converge to the exact solution. The absolute error for
k = 3, 4, . . . , 8, α = 1.5 is shown in Table 7.1. It can be observed that the error decreases with increasing
k.
t k = 3 k = 4 k = 5 k = 6 k = 7 k = 8
0.1 5.094(−4) 1.051(−4) 2.698(−5) 6.731(−6) 1.682(−6) 4.207(−7)
0.2 1.915(−4) 4.990(−5) 1.244(−5) 3.109(−6) 7.776(−7) 1.944(−7)
0.3 5.720(−5) 1.455(−5) 3.649(−6) 9.124(−7) 2.282(−7) 5.705(−8)
0.4 4.650(−5) 1.178(−5) 2.948(−6) 7.368(−7) 1.842(−7) 4.605(−8)
0.5 1.314(−4) 3.282(−5) 8.205(−5) 2.051(−6) 5.128(−7) 1.282(−7)
0.6 1.987(−4) 4.974(−5) 1.244(−5) 3.111(−6) 7.777(−7) 1.944(−7)
0.7 2.525(−4) 6.324(−5) 1.581(−5) 3.954(−6) 9.886(−7) 2.471(−7)
0.8 2.942(−4) 7.365(−5) 1.842(−5) 4.605(−6) 1.151(−6) 2.878(−7)
0.9 3.235(−4) 8.126(−5) 2.032(−5) 5.081(−6) 1.270(−6) 3.176(−7)
Table 7.1: Absolute error for M = 3 and different values of k.
Example 7.2.2. Consider the problem [46],
cDαy(t) + ωα−βcDβy(t) = 0, y(0) = A, y′(0) = B, (7.2.8)
where α ∈ (1, 2], β ∈ (0, 1]. β = 0 the equations (7.2.8) describes fractional oscillator and has been studied
in details by B.N. Narahari Achar et.al. in [107].
The corresponding integral equation for fractional order differential equation (7.2.8) is
y(t) = −ωα−βIα−βy(t) + (1− ωα−β)A+Bt. (7.2.9)
We approximate y as
y = CTmLm(t). (7.2.10)
Applying Iα on both sides of (7.2.10), we have
IαcDβy(t) = Iα−βy(t)
= CTmIα−βLm(t)
= CTmRα−β
m×mLm(t).
(7.2.11)
130
Substituting (7.2.10) and (7.2.11) into equation (7.2.9) we have system of algebraic equations.
CTm = FTm
(Im×m + ωα−βCT
mRα−βm×m
)−1, (7.2.12)
where f(t;α, β) := (1− ωα−β)A+Bt = FTmLm(t). Thus the numerical solution of the problem (7.2.8) is
y(t) = FTm(Im×m + ωα−βCT
mRα−βm×m
)−1Lm(t). (7.2.13)
For the classical case α = 2, β = 0 with the constants A = 0 and B = 1, the exact solution is
y(t) = 1ω sin(ωt). Also, for α = 2, β = 1 (A = 0, B = 1), the exact solution is y(t) = − 1
ω (eωt − 1).
Numerical results for different values of α and β are shown in Figure 7.3. From Figure 7.3, we see that,
as α approaches to 1 or 2, the numerical solution converges to that of differential equations of order 1 or
2 respectively.
Example 7.2.3. Consider following linear non–homogenous boundary value problem with constant coef-
ficients
acDαy(t) + bcDβy(t) + cy(t) = g(t), y(0) = A, y′(0) = B, (7.2.14)
where α = 2, β ∈ (1, 2), a = 0 and b, c ∈ R. For α = 2, β = 3/2, equation (7.2.14) reduces to the
Bargely–Torvik equation studied in [17] for modeling of a rigid plate immersed in a Newtonian fluid.
We approximate cDαy(t) ascDαy(t) = CT
mLm(t). (7.2.15)
Then
cDβy(t) = Iα−β(cDαy)(t)
= CTmIα−βLm(t)
= CTmRα−β
m×mLm(t).
(7.2.16)
Applying Iα on both sides of (7.2.16) and using Lemma 2.2.24, we have
y(t) = CTmRα−β
m×mLm(t) +A+Bt. (7.2.17)
Substituting equation (7.2.15), (7.2.16) and (7.2.17) into equation (7.2.14), we have following system of
algebraic equations.
aCTmLm(t) + bCT
mRα−βm×mLm(t) + cCT
mRα−βm×mLm(t) = FTmLm(t), (7.2.18)
where the function f(t) := g(t)− c(A+Bt) = FTmLm(t). We choose α = 2, β = 32 , a = 12, b = 8
17 , c =59 ,
g(t) = 72t + 6417
√πt32 + 5
9 t3 and A = B = 0, one can verify that the exact solution of the boundary value
problem (7.2.14) is y = t3. We compare the numerical solution with exact solution for different values of
k and the results are given in the Table 7.2. For the comparison of the Legendre wavelet method with
some other numerical methods, we choose α = 2, β = 1/2, a = b = c = 1 and g(t) = 8. The numerical
solutions of this problem are discussed in Ref. [12] and [104] by some other numerical methods namely,
the Adomian decomposition method, variational iteration method, fractional finite difference method,
131
0
0.2
0.4
0.6
0.8
1 00.2
0.40.6
0.81
−0.1
0
0.1
0.2
0.3
0.4
βt
y
(a) α = 1.3, 0 ≤ β ≤ 1
0
0.5
1 00.2
0.40.6
0.81
−0.1
0
0.1
0.2
0.3
βt
y
(b) α = 1.6, 0 ≤ β ≤ 1
0
0.2
0.4
0.6
0.8
1 0 0.2 0.4 0.6 0.8 1
−0.1
−0.05
0
0.05
0.1
βt
y
(c) α = 2, 0 ≤ β ≤ 1
0
0.5
1 1
1.5
2
−0.1
−0.05
0
0.05
0.1
αt
y
(d) 1 ≤ α ≤ 2, β = 0
0
0.5
1 11.2
1.41.6
1.82
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
αt
y
(e) 1 ≤ α ≤ 2, β = 1.5
0
0.5
1 1 1.2 1.4 1.6 1.8 2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
αt
y
(f) 1 ≤ α ≤ 2, β = 1
Figure 7.3: Solutions y(t) for Example 7.2.2 for ω = 11, y0 = 0, y1 = 1 and different values of α, β.
132
t k = 4 k = 5 k = 6 k = 7 k = 8, k = 9
0.1 1.293(−4) 3.229(−5) 8.070(−6) 2.016(−6) 5.041(−7) 1.260(−7)
0.2 2.574(−4) 6.433(−5) 1.607(−5) 4.017(−6) 1.004(−6) 2.510(−7)
0.3 3.849(−4) 9.617(−5) 2.403(−5) 6.006(−6) 1.501(−6) 3.753(−7)
0.4 5.117(−4) 1.278(−4) 3.195(−5) 7.984(−6) 1.996(−6) 4.988(−7)
0.5 6.378(−4) 1.593(−4) 3.982(−5) 9.952(−6) 2.487(−6) 6.218(−7)
0.6 7.632(−4) 1.907(−4) 4.765(−5) 1.191(−5) 2.976(−6) 7.441(−7)
0.7 8.879(−4) 2.218(−4) 5.544(−5) 1.385(−5) 3.463(−6) 8.656(−7)
0.8 1.012(−3) 2.528(−4) 6.317(−5) 1.579(−5) 3.946(−6) 9.865(−7)
0.9 1.135(−3) 2.836(−4) 7.087(−5) 1.771(−5) 4.427(−6) 1.107(−6)
Table 7.2: Absolute error for M = 3 and different values of k.
fractional differential transform method. The numerical results in table 7.3 indicate that, at least for this
particular problem the Legendre wavelet method is in good agreement with above mentioned numerical
methods. It is worth mentioning that some of the numerical methods, such as Adomian decomposition
method require stronger differentiability assumptions on y(t). On the other hand the Haar wavelet and
the Legendre wavelet methods require y to be an element of L2(R). How ever the operational matrices
approach has some draw backs too. For example, it becomes cumbersome to deal the nonlinear problems
by this approach, since the method reduces nonlinear boundary value problems to nonlinear system of
algebraic equations. To solve these nonlinear algebraic systems, we have to look for some iterative methods,
such as Newton’s method, which requires some appropriate initial guess. Also, since we need to solve
large algebraic systems which may cause grater computational complexity. However for well–posed linear
problems the Haar and the Legendre wavelet methods work effectively. Also the Haar and the Legendre
wavelet methods are much simpler then other methods.
Example 7.2.4. In example (7.2.5), we take α = 2, 0 ≤ β ≤ 1, a = b = c = 1, A = 0, B = 0 and
g(t;α, β) = 6(
t−α
Γ(4−α) −t−β
Γ(4−β)
)t3. The exact solution in this case is y(t) = t3. The numerical results
obtained for the Haar wavelet and the Legendre wavelets are shown in Table 7.4. The Legendre wavelet
method seems to giving almost the same results as the Haar wavelet method. However the calculations
with the Legendre wavelets are much complicated as compared with the Haar wavelets. This causes
relatively greater computational complexity and large storage requirements.
Example 7.2.5. Consider the following boundary value problem for nonlinear fractional order differential
equation
cDαy(t) + a(y(t))n = g(t), y(0) = A, y(1) = B, (7.2.19)
where < α ≤ 2, A,B ∈ R, n ∈ N and g(t) is a given function. Using properties of fractional derivatives
and integrals the differential equation (7.2.19) can be reduced into following integral equation
y(t) = −aIα(y(t))n + atIα(y(1))n + f(t), (7.2.20)
133
t yFDM [12] yADM [104] yFDTM [12] yVIM [104] yLWM yexact
0.0 0.000000 0.000000 0.000000 0.000000 0.000000 00000000
0.1 0.039473 0.039874 0.039750 0.039874 0.039750 0.039750
0.2 0.157703 0.158512 0.157036 0.158512 0.157035 0.157036
0.3 0.352402 0.353625 0.347370 0.353625 0.347370 0.347370
0.4 0.620435 0.622083 0.604695 0.622083 0.604695 0.604695
0.5 0.957963 0.960047 0.921768 0.960047 0.921767 0.921768
0.6 1.360551 1.363093 1.290457 1.363093 1.290456 1.290457
0.7 1.823267 1.826257 1.702008 1.826257 1.702007 1.702008
0.8 2.340749 2.344224 2.147287 2.344224 2.147286 2.147287
0.9 2.907324 2.911278 2.617001 2.911278 2.617000 2.617001
1.0 3.517013 3.521462 3.101906 3.521462 3.101905 3.101906
Table 7.3: Numerical results with comparison to Ref. [104] and [12].
(k = 10)
Haar Legendre
β m = 8 m = 32 m = 128 m = 512 m = 6 m = 24 m = 96 m = 384
0.25 0.0076 4.80(−4) 3.00(−5) 1.87(−6) 0.0135 8.54(−4) 5.34(−5) 3.33(−6)
0.50 0.0071 4.47(−4) 2.80(−5) 1.75(−6) 0.0127 7.96(−4) 4.97(−5) 3.11(−6)
0.75 0.0066 4.16(−4) 2.60(−5) 1.62(−6) 0.0118 7.40(−4) 4.63(−5) 2.89(−6)
1.00 0.0062 3.90(−4) 2.44(−5) 1.52(−6) 0.0111 6.94(−4) 4.34(−5) 2.71(−6)
Table 7.4: Maximum absolute error for the Haar wavelet and the Legendre wavelets.
where,
f(t) = Iαg(t) + (y1 − y0)t− tIαg(1) + y0. (7.2.21)
We approximate y(t) as
y(t) = CTmLm(t) = CT
mLm×mBm(t), (7.2.22)
Let CTmLm×m = [s1, s2, . . . , sm], then using orthogonality of block-pulse function, we obtain
(y(t))n = [sn1 , sn2 , . . . , s
nm]Bm(t) = STmBm(t). (7.2.23)
where Sm = [sn1 , sn2 , . . . , s
nm]T . Now
Iα(y(t))n = STmIαBm(t) = STmFαm×mBm(t). (7.2.24)
134
Let µ(t) ∈ L2[0, 1], then
µ(t)Iα(y(1))n =
∫ 1
0
µ(t)(1− s)α−1
Γ(α)(y(s))nds
=
∫ 1
0Kα(s, t)(y(s))
nds
= STm
∫ 1
0Bm(s)L
Tm(s)K
αm×mLm(t)ds
= STm(∫ 1
0Bm(s)BT
m(s)ds)LTm×mKα
m×mLm(t)
=1
mSTmL
Tm×mKα
m×mLm×mBm(t).
(7.2.25)
We take µ(t) = at and substitute (7.2.22), (7.2.24) and (7.2.25) into equation (7.2.20), then
CTmLm(t) = −aSTmFαm×mBm(t) +
1
mSTmLm×mKα
m×mLm×mBm(t) + FTmLm(t), (7.2.26)
where the function f(t) is approximated by f(t) = FTmLm(t). For α = 1.5, a = e−2π, A = 0, B = 1, n = 2,
and g(t) = t2(e2πt5 + 10532 ), the exact solution is y(t) = t
72 . We compare the numerical solution with exact
solution for M = 3 and different values of k. The results are given below in the Table 7.5.
t α = 1.1 α = 1.3 α = 1.5 α = 1.7 α = 1.9, α = 2
0.1 2.9111(−4) 1.9756(−4) 9.6996(−5) 4.7793(−5) 2.8511(−5) 2.4492(−5)
0.2 5.4223(−3) 2.2168(−3) 9.3927(−4) 4.0536(−4) 1.7477(−4) 1.1436(−4)
0.3 6.0275(−3) 3.0160(−3) 1.5087(−3) 7.5162(−4) 3.7264(−4) 2.6165(−4)
0.4 1.3892(−3) 6.8755(−4) 3.3989(−4) 1.6667(−4) 7.8889(−5) 5.3659(−5)
0.5 8.4144(−3) 4.5061(−3) 2.4163(−3) 1.2916(−3) 6.8293(−4) 4.9443(−4)
0.6 1.2710(−3) 6.5427(−4) 3.1023(−4) 1.1972(−4) 1.3952(−5) 1.6752(−5)
0.7 5.7669(−3) 3.0298(−3) 1.4799(−3) 5.9606(−4) 9.2122(−5) 7.0255(−5)
0.8 5.1134(−3) 2.3439(−3) 6.3407(−4) 4.5289(−4) 1.1664(−3) 1.4317(−3)
0.9 3.4295(−3) 4.0232(−3) 4.6701(−3) 5.3913(−3) 6.1995(−3) 6.6398(−3)
Table 7.5: The absolute error for M = 3, k = 3 and different values of α.
Chapter 8
Summery and Conclusions
In the first sections of Chapter 2, we introduced some spacial functions that were required for the devel-
opment of our results. In the second section some fundamental concepts and definitions from fractional
calculus including some basic results about the Riemann–Liouville and the Caputo fractional operators
are provided. In the remaining two sections of the Chapter 2 some fixed point theorems are stated and
some basic properties of the Haar wavelets are reviewed.
With the help of classical tools from functional analysis, operator theory and fixed point theory,
the theory on existence and uniqueness of solutions to boundary value problems for nonlinear fractional
differential equations, with the Riemann–Liouville and the Caputo fractional derivatives, is developed. In
Chapter 3, we established sufficient conditions for existence and uniqueness results for different classes
of nonlinear boundary value problems involving fractional derivatives, subject to two–point three–point,
multi–point and integral boundary conditions.
In Chapter 4, several existence results for positive and multiple positive solutions to two different
types boundary value problems for fractional differential equations are obtained. For the boundary value
problem 4.1, the existence of at least one, at least two and at least three positive solutions is guaranteed in
a specially constructed cone in the Banach space B. It is observed that the Green functions Gα(t, s) and
Gα,β(t; η, s) for the boundary value problem (4.2.1), (4.2.2), satisfy some interesting and useful properties
and they are related to each other. This helps us to construct a cone in the Banach space B. Then we
established existence and uniqueness results for positive solutions in this cone. The existence of positive
solution is assured only if λ1 ≤ cαρ1 and λ2 ≤ cβρ1.
In chapter Chapter 5, we derived some existence results for positive and multiple positive solutions
for two different types of boundary value problems concerning systems of nonlinear fractional differential
equations. The existence and multiplicity results for the boundary value problem (5.1.1), (5.1.2) are
established by assuming that the nonlinear functions f and g satisfy superlinear–sublinear type conditions.
For some of the established results, we found the explicit intervals for the parameter λ for which the
boundary value problem (5.1.1), (5.1.2) has positive or multiple positive solutions. In section 5.2, we have
established some existence theorems for positive and multiple positive solutions for a higher order system
of nonlinear fractional differential equations with three–point boundary conditions. Explicate intervals for
the parameters λ, µ are derived, for which the system possesses positive solutions. Validity of proposed
135
136
results is shown by two examples.
Owing to the challenging difficulties of having an exact analytic solutions for fractional boundary value
problems, we have used the Haar and the Legendre wavelets to obtain numerical solutions. We obtained
interesting results demonstrating that these mathematical tools are useful to develop efficient numerical
schemes to treat the wider class of problems.
In Chapter 6, a numerical scheme, based on the Haar wavelets, is proposed to solve the boundary
value problems for fractional differential equations. A new operational matrix Qα,η,nm×m is obtained and is
used along with some existing Haar wavelet operational matrices of integration to solve different types of
boundary value problems for fractional differential equations. The method is successfully applied to linear
problems with constant and variable coefficients and also to some nonlinear problems. Also we noticed
that the matrix Qα,η,nm×m is very useful to deal with three–point (or multi–point) boundary value problems
as well. The wavelet based method reduces the problem to a system of algebraic equations. The numerical
results obtained are compared with exact solutions by tabulating their absolute error and by comparing
their respective graphs. It is worth mentioning that results obtained, agree well with exact solutions even
for small number of collocation points. The method seems to be more efficient and convenient for solving
linear boundary value problems for fractional order differential equations. It takes care of the boundary
conditions automatically. However much is needed to be done for nonlinear problems. The method, when
applied to nonlinear problems, reduces the problem to system of nonlinear algebraic equations leading to
extra difficulties.
Despite of effectiveness, efficiency, and simplicity of the wavelet method, there are some issues, related
to this approach, needed to be addressed. While solving the linear systems for wavelet coefficients ci, we
have to invert some matrix which of course involves the Haar matrix. The Haar wavelet basis leads to a
sparse matrix. Thus for large values of m, the matrix appears to be nearly singular. In this situation, the
calculation of the wavelet coefficients becomes impossible with required accuracy.
In Chapter 6, we have used the Haar basis to represent the solutions of boundary value problems.
However, it is possible to have a wavelet representation of solution for boundary value problems on an
infinitely many different wavelet bases. In Chapter 7 we used the Legendre polynomials as a basis for the
wavelet representation of solutions of fractional differential equations. An operational matrix of fractional
order integration is obtained. The procedure for obtaining solution remains the same as in the case of
the Haar wavelets. The results obtained by the numerical scheme based on the Legendre wavelets are
compared with exact solutions and also with solutions obtained by some other numerical methods. It has
been observed that, in some situations, the method gives good results as compared with other methods.
The method is convenient for solving linear initial value problems as well as boundary value problems,
science the boundary conditions are taken care of automatically.
Appendix A
Matlab and Mathematica programs
In this chapter, we present some of the most important Matlab and Mathematica codes that have been
developed for the numerical analysis of the fractional differential equations.
A.1 Computations of some operational matrices by Matlab
The following Matlab code generates the Haar matrix of order m by m. Here we point out that F. Chanal
have developed a Matlab code for computing Haar matrices, available at
http://www.mathworks.com/matlabcentral/fx_files/4619/1/haarmtx.m. The computation by the code
developed here are little bit faster then that of F. Chanal.
function H=Hmtx(J)
format long J=3; m=2.^J;
for l=1:m;
x=(2*l-1)/(2*m);
G(1,l)=1;
end
for l=1:m;
for j=0:J-1
for k=0:2.^j-1;
i=2.^j+k+1;
a=(k)/2^j; b=(2*k+1)/(2*2^j); c=(k+1)/2^j;
x=(2*l-1)/(2*m); cc=min(x,c);
if x >= a & x <= b;
G(i,l) = 1;
elseif x >= b & x <= c;
G(i,l) =-1;
else
G(i,l) =0;
end
137
138
end
end
end
H=G
The following programm is developed to compute the block–pulse operational matrix of fractional order
integration. The computations of most of the other operational matrices of fractional order integration
are based on this matrix.
function F=Falpha(J,alpha) m=2^J;
for j=1:m;
for i=1:m;
if i>j;
xi(i,j)=0;
elseif i<=j&i>=j;
xi(i,j)=1;
else
k=j-i;
xi(i,j)=(k+1).^(alpha+1)-2.*(k).^(alpha+1)+(k-1).^(alpha+1);
end
end
end
F=xi; F=(1./(m.^(alpha)*gamma(alpha+2)))*xi
Next, we present a Matlab code for the computation of the operational matrix of fractional order integration
Pαm×m.
function P=PFalpha(J,alpha)
m=2^J;
for j=1:m;
for i=1:m;
if i>j;
xi(i,j)=0;
elseif i<=j&i>=j;
xi(i,j)=1;
else
k=j-i;
xi(i,j)=(k+1).^(alpha+1)-2.*(k).^(alpha+1)+(k-1).^(alpha+1);
end
end
end
R=xi;
139
F=(1./(m.^(alpha)*gamma(alpha+2)))*xi;
P=H*F*inv(H);
The following code is developed to compute an other important operational matrix, namely Qαm×m.
functionQ=Qmtx(J,alpha)
m=2.^J;
eta=1;
psi=@(s)s;
psi2=@(s)((eta-s).^(alpha-1))./gamma(alpha); U(1)=quad8(psi,0,1);
for j=1:J;
for k=1:2.^(j-1);
i=2.^(j-1)+k;
a=(k-1)./2.^(j-1); b=(2*k-1)./(2.^j); c=k./2^(j-1);
h0=@(s)2.^((j-1)./2)*psi;
U(i)=quad8(h0,a,b)-quad8(h0,b,c);
end
end
V(1)=quad8(psi2,0,eta);
for j=1:J;
for k=1:2.^(j-1);
i=2.^(j-1)+k;
a=(k-1).*eta./2.^(j-1); b=min(eta,(2*k-1).*eta./(2.^j));
c=min(eta,k.*eta./2^(j-1));
h0=@(s)2.^((j-1)./2)*((eta-s).^(alpha-1))./gamma(alpha);
V(i)=quad8(h0,a,b)-quad8(h0,b,c);
end
end for i=1:m;
for j=1:m;
Qe(j,i)=[U(i)*V(j)];
end
end
Q=Qe;
The Legendre matrix can be computed by using following code.
function L=Lmtx(k)
M=3; N=2.^(k-1); m=M*N; for i=1:m;
t=(2*i-1)./(2*m);
for n=1:N;
a=(2*n-2)./2.^k;
b=(2*n)./2.^k;
140
if t>=a & t<b;
FF(n,i)=2.^(k./2).*(1./2).^(1./2)+0*n;
FF(n+N,i)=2.^(k./2).*(3./2).^(1./2)*(2.^k*t-2*n+1);
FF(n+2*N,i)=2.^(k./2).*(5./2).^(1./2)*((5./2).*(2.^k*t-2*n+1).^2-1./2);
elseif t>b;
FF(n,i)=0;
end
end
end L=FF;
The following programme is developed to compute the Legendre wavelet operational matrix of fractional
order integration.
function P=PFalpha(J,alpha)
m=2^J; for j=1:m;
for i=1:m;
if i>j;
xi(i,j)=0;
elseif i<=j&i>=j;
xi(i,j)=1;
else
k=j-i;
xi(i,j)=(k+1).^(alpha+1)-2.*(k).^(alpha+1)+(k-1).^(alpha+1);
end
end
end R=xi; F=(1./(m.^(alpha)*gamma(alpha+2)))*xi;
P=haarmtx(J)*F*inv(haarmtx(J));
In the following we give an other important code in the context of boundary value problems for fractional
order differential equations.
function KL=KLmtx(k,alpha)
syms t;
M=3; N=2.^(k-1); m=M*N; psi=t.^(alpha-1);
ptt=((1-t).^(alpha-1))./gamma(alpha);
for n=1:N;
a=(2*n-2)./2.^k;
b=(2*n)./2.^k;
L0=2.^(k./2).*(1./2).^(1./2)*psi;
L1=2.^(k./2).*(3./2).^(1./2)*(2.^k*t-2*n+1)*psi;;
L2=2.^(k./2).*(5./2).^(1./2)*((5./2).*(2.^k*t-2*n+1).^2-1./2)*psi;
U(n)=eval(int(L0,t,a,b));
141
U(n+N)=eval(int(L1,t,a,b));
U(n+2*N)=eval(int(L2,t,a,b));
end
for n=1:N;
aa=(2*n-2)./2.^k;
bb=(2*n)./2.^k;
L0=(2.^(k./2).*(1./2).^(1./2)+0.*n+0.*t)*ptt;
L1=2.^(k./2).*(3./2).^(1./2)*(2.^k*t-2*n+1)*ptt;
L2=2.^(k./2).*(5./2)...
.^(1./2)*((5./2).*(2.^k*t-2*n+1).^2-1./2)*ptt;
V(n)=eval(int(L0,t,aa,bb));
V(n+N)=eval(int(L1,t,aa,bb));
V(n+2*N)=eval(int(L2,t,aa,bb));
end
for i=1:m;
for j=1:m;
Ke(j,i)=[U(i)*V(j)];
end
end KL=Ke;
142
A.2 Computations by Mathematica
143
Appendix B
Useful results from Analysis
For convenience, in this chapter we review some useful result from analysis.
Definition B.0.1. A function f [a, b] → R is said to be absolutely continuous on [a, b], if for each ε > 0,
there is a δ > 0, such that given any collection (ak, bk) : 1 ≤ k ≤ n of pairwise disjoint open subintervals
of [a, b], we haven∑k=1
(bk − ak) < δ impliesn∑k=1
(f(bk)− f(ak)) < ε.
Absolutely continuous functions on [a, b] are uniformly continuous but converse is not true. The space
of absolutely continuous functions is denoted by AC[a, b].
We define the space
ACm[a, b] =f : [a, b] → R and Dm−1f(x) ∈ AC[a, b]
.
Theorem B.0.2. [68] Let f : [a, b] → R and m ∈ N. Then f ∈ ACm[a, b] if and only if f can be
represented as
f(x) =
m−1∑k=0
ck(x− a)k + Ima φ,
where k = 0, 1, . . . ,m− 1, φ ∈ L(a, b).
Theorem B.0.3. (Taylor’s Theorem) Let f ∈ ACm[a, b], m ∈ N. The for any x ∈ [a, b]
f(x) =
m−1∑j=0
(x− a)j
j!Djf(a) + Ima Dmf(x), (B.0.1)
where Dmf(x) ∈ L[a, b].
Definition B.0.4. [56] Let Ω ⊂ Rn. A set M ⊂ C(Ω) is said to be equicontinuous if and only if, for
every ε > 0 there exist a δ > 0 such that |f(x), f(y)| < ε with ∥x− y∥ < δ, for all x, y ∈ Ω and all f ∈ M.
Theorem B.0.5. [56] (The Arzela-Ascoli Theorem) Let Ω be a bounded subset of Rn, and let M be a
subset of C(Ω). Then M is relatively compact if and only if it is bounded and equicontinuous.
144
145
The space obtained by generalizing Rn or Cn with an infinite sequence f = (f1, f2, f3, . . . ) as its
element is called the sequence space and is denoted by ℓp. For 0 ≤ p ≤ ∞, the sequence space ℓp with the
norm defined by
∥f∥p =
∞∑n=1
|fn|p 1
p
(1 ≤ p <∞), ∥f∥∞ = sup ∥fn∥,
is a Banach space.
Now we define an other important space which is a continues analog of sequence space.
Definition B.0.6. Let f be measureable and set
∥f∥p =∫
|f |pdµ 1
p
(1 ≤ p <∞), ∥f∥∞ = ess sup |f |,
where µ is the Lebesgue measure defined on a σ-algebra of subsets of a set X in Rn. The space Lp(X) is
the space of all measurable functions with the ∥f∥p <∞.
The space Lp, (p ≥ 1) is a Banach space under the norm ∥.∥p. In particular, the space L2 is the Hilbert
space supplied with the inner product ⟨f, g⟩ =∫fgdµ.
We use the notation Lp[0, 1] for the space of measurable functions such that∫ 10 |f(s)|ds <∞, where the
integral is understood in the sense of Lebesgue and the norm on Lp[0, 1] is defined as ∥f∥p = (∫ 10 |f(s)|pds)
1p
where 1 < p <∞.
Theorem B.0.7. (Fubini’s Theorem) Suppose that f : R2 → R is measureable and one of the integrals∫ ∫|f(t, x)|dtdx,
∫dt∫|f(t, x)|dx,
∫dx∫|f(t, x)|dt is finite. Then the functions g(t) =
∫f(t, x)dx and
h(x) =∫f(t, x)dt are measureable and∫
dt
∫f(t, x)dx =
∫dx
∫f(t, x)dt =
∫ ∫f(x, y)dxdy.
Definition B.0.8. A function f : [0, 1]× Rn → R satisfies the Carathéodory conditions if
(i) f(t, x) is Lebesgue measureable on [0, 1] for each x ∈ Rn .
(ii) f(t, x) is continuous on Rn for almost every t ∈ [0, 1].
(iii) For each r > 0, there exists an ψr ∈ Lp([0, 1]) such that |x| ≤ r implies that |f(t, x)| ≤ ψr(t) for
a.e., t ∈ [0, 1].
Definition B.0.9. A sequence of elements φkk∈Z in a Hilbert space H is a Riesz basis if for every
f ∈ H there exists a unique sequence αkk∈Z ∈ ℓ2(Z) such that the sequence φkk∈Z is complete and
A∥f∥22 ≤∑k∈Z
|αk|2 ≤ B∥f∥22
with 0 < A ≤ B <∞ constants independent of f ∈ H.
References
[1] O. Abdulaziz, I. Hashim and S. Momani, Solving systems of fractional differential equations by
homotopy–perturbation method, Phy. Lett. A, 372 (2008) 451–459.
[2] R.P. Agarwal, V. Lakshmikantham and J. J. Nieto, On the concept of solution for fractional differential
equations with uncertainty, Nonlin. Anal: TMA, 72 (2010) 2859–2862
[3] R. P. Agarwal, M. Benchohra and S. Hamani, A Survey on existence results for boundary value
problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 109 (2010)
973–1033.
[4] B. Ahmad and J.J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-
order nonlinear fractional differential equations, Abs. Appl. Anal., 2009, Article ID 494720, 9 pages,
doi:10.1155/2009/494720.
[5] B. Ahmad and J.J. Nieto, Existence results for nonlinear boundary value problems of fractional
integrodifferential equations with integral boundary conditions, Boun. Val. Prob., (2009), Article ID
708576, 11 pages, doi:10.1155/2009/708576.
[6] W.M. Ahmad and R. El-Khazalib, Fractional–order dynamical models of love, Cha. Solit. Fract., 33
(2007) 1367–1375.
[7] B. Ahmad and S. Sivasundaram, On four–point nonlocal boundary value problems of nonlinear
integro–differential equations of fractional order, Appl. Math. Comput., 217 (2010) 480–487.
[8] B. Ahmad and J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential
equations with three–point boundary conditions, Comp. Math. Appl., 58 ( 2009) 1838–1843.
[9] Q.M. Al–Mdallal, M.I. Syam and M.N. Anwar, A collocation–shooting method for solving fractional
boundary value problems, Commun. Nonlin. Sci. Num. Simulat., 15 (2010) 3814–3822.
[10] S. Ali Yousefi, Legendre wavelets method for solving differential equations of Lane–Emden type, Appl.
Math. Comp., 181 (2006) 417–1422.
[11] T.M. Atanackovic and B. Stankovic, On a system of differential equations with fractional derivatives
arising in rod theory, J. Phys. A, 37 (2004) 1241–1250.
146
147
[12] A. Arikoglu and I. Ozkol, Solution of fractional differential equations by using differential transform
method, Chaos, Solit. Fract., 34 (2007) 1473–1481.
[13] A. Babakhani and V. Daftardar–Gejji, On calculus of local fractional derivatives , J. Math. Anal.
Appl., 270 (2002) 66–79.
[14] A. Babakhani, Positive solutions for system of nonlinear fractional differential equations in two di-
mensions with delay, 2010, Article ID 536317, 16 pages doi:10.1155/2010/536317.
[15] E. Babolian and A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the
second kind using Haar wavelets, J. Comput. Appl. Math., 225 (2009) 87–95.
[16] E. Babolian and F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev
wavelet operational matrix of integration, Appl. Math. Comput., 188 (2007) 417–426.
[17] R.L. Bagley and P.J. Torvik, On the appearance of the fractional derivative in the behavior of real
materials, ASME J. Appl. Mech., 51 (1984) 294–298.
[18] C. Bai and J. Fang, The existence of positive solution for singular coupled system of nonlinear
fractional equations, Appl. Math. Comput., 150 (2004) 611–621.
[19] Z. Bai and Y. Zhang The existence of solutions for a fractional multi–point boundary value problem,
Comput. Math. Appl., 60 (2010) 2364–2372.
[20] Z. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlin. Anal., 72
(2010) 916–924.
[21] M. Benchohra, S. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with
fractional order, Surv. Math. Appl., 3 (2008) 1–12.
[22] A.V. Bitsadze and A.A. Samarski, Some elementary generalizations of linear elliptic boundary value
problems, Dokl. Akad. Nauk SSSR 185 (1969) 739–740 (in Russian).
[23] R. F. Camargo, A. O. Chiacchio, and E. C. Oliveira, Differentiation to fractional orders and the
fractional telegraph equation, J. Math. Phy., 49 (2008) Article ID 033505.
[24] J.R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl.
Math., 21 (1963) 155–160.
[25] A. Carpinteri and F. Mainardi (Ed.), Fractals and Fractional Calculus in Continuum Mechanics
(CISM International Centre for Mechanical Sciences), Springer, 1997.
[26] R. C. Cascaval, E. C. Eckstein, C. L. Frota, and J. A. Goldstein, Fractional telegraph equations, J.
Math. Anal. Appl., 276 (2002) 145–159.
[27] C. Chen and C. Hsiao, Haar wavelet method for solving lumped and distributed–parameter systems,
IEE P.-Contr. Theor. Ap., 144 (1997) 87–94.
148
[28] Y. Chen, Y. Yan and Kewei Zhang, On the local fractional derivative, J. Math. Anal. Appl., 362
(2010) 17–33.
[29] Y.S. Choi and K.Y. Chan, A parabolic equation with nonlocal boundary conditions arising from
electrochemistry, Nonlin. Anal.: TMA, 18 (1992) 317–331.
[30] J. Chen, F. Liu, and V. Anh, Analytical solution for the time–fractional telegraph equation by the
method of separating variables, J. Math. Anal. Appl.,, 338 (2008) 1364–1377.
[31] Y. Chen, Y. Wua, Y. Cuib, Z. Wanga and D. Jin, Wavelet method for a class of fractional convection–
diffusion equation with variable coefficients, Journal of Computational Science 1 (2010) 146–149.
[32] R.E. Ewing and T. Lin, A class of parameter estimation techniques for fluid flow in porous media,
Adv. Water Resour., 14 (1991) 89–97.
[33] V. Daftardar-Gejji and S. Bhalekar, Solving fractional boundary value problems with Dirichlet bound-
ary conditions using a new iterative method, Comput. Math. Appl., 59 (2010) 1801–1809.
[34] M. Denche and A. Memou, Boundary value problem with integral conditions for a linear third–order
equation, J. Appl. Math., 11 (2003) 533–567.
[35] S. Das, K. Vishal, P.K. Gupta and A. Yildirim, An approximate analytical solution of time–fractional
telegraph equation, Appl. Math. Comput., 217 (2011) 7405–7411.
[36] K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extrapola-
tion, Numer. Algo., 16 (1997) 231–253.
[37] K. Diethelm and N.J. Ford, Numerical solution of the Bagley–Torvik equation, BIT, 42 (2002) 490–
507.
[38] K. Diethelm, N.J. Ford and A.D. Freed, A predictor–corrector approach for the numerical solution of
fractional differential equation, Nonlin. Dyn., 29 (2002) 3–22.
[39] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics Series
Springer 2010.
[40] Z. Ding, A. Xiao and M. Li, Weighted finite difference methods for a class of space fractional partial
differential equations with variable coefficients, Journal of Computational and Applied Mathematics
233 (2010) 1905–1914.
[41] F.B.M. Duarte and J.A. Tenreiro Machado, Chaotic phenomena and fractional–order dynamics in the
trajectory control of redundant manipulators, Nonlin. Dyn., 29 (2002), 342–362.
[42] V. Daftardar-Gejji and H. Jafari, Solving a multi–order fractional differential equation using Adomian
decomposition method, Appl. Math. Comput., 189 ( 2007) 541–548.
[43] I. Daubechies, Ten Lectures on Wavelets, SIAM 1992.
149
[44] M. El-Shahed and J.J. Nieto, Nontrivial solutions for a nonlinear multi–point boundary value problem
of fractional order, Comput. Math. Appl., 59 (2010) 3438–3443.
[45] P.W. Eloe and B. Ahmad, Positive solutions of a nonlinear nth order boundary value problems with
nonlocal conditions, Appl. Math. Let., 18 (2005) 521–527.
[46] V.S. Erturk, S. Momani amd Z. Odibat, Application of generalized differential transform method to
multi-order fractional differential equations, Commun. Nonlin. Sci. Num. Simul., 13 (2008) 1642-1654.
[47] L. Formaggia and F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modelling of the circulatory
system: a preliminary analysis, Comput. Visual. Sci., 2 (1999) 75–83.
[48] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag New Yark, Inc, 2003.
[49] R. Gorenflo, A.A. Kilbas, S.V. Rogozin, On the generalized Mittag–Leffler type function, Integr.
Transf. Spec. Funct., 7 (1998) 215–224.
[50] D.J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Vol 5 Notes and Reports
in Mathematics in Science and Engineering, Academic press, Boston, Mass, USA 1988.
[51] C.P. Gupta, S.K. Ntouyas and P.C. Tsamatos, Solvability of m–point boundary value problem for
second order ordinary differential equations, J. Math. Anal. Appl., 189 (1995) 575–584.
[52] I. Hashim, O. Abdulaziz and S. Momani, Homotopy analysis method for fractional IVPs, Commun.
Nonlin. Sci. Num. Simul., 14 (2009) 674–684.
[53] J. W. Hanneken , D. M. Vaught, and B. N. Narahari Achar, Enumeration of the real zeros of the
Mittag-Leffler function Eα(z), 1 < α < 2; J. Sabatier et al. (eds.), Advances in Fractional Calculus:
Theoretical Developments and Applications in Physics and Engineering, (2007) 15–26.
[54] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company,
2000.
[55] F. Huang, Analytical Solution for the Time-Fractional Telegraph Equation, J. Appl. Math., (2009),
Article ID 890158, doi:10.1155/2009/890158.
[56] V. Hutson, J. Sydney Pym and Michael J. Cloud, Applications of functional analysis and operator
theory, Elsevier Science 2005.
[57] R. Hilfer, Threefold introduction to fractional derivatives, Anomalous Transport: Foundations and
Applications, R. Klages et al. (eds.), Wiley-VCH, Weinheim, 2008, page 17.
[58] V.A. Il’in and E.I. Moiseev, Nonlocal boundary value problem of the second kind for a Sturm–Liouville
operator, Diff. Equat., 23 (1987) 979–987.
[59] V.A. Il’in and E.I. Moiseev, Nonlocal boundary value problem of the first kind for a Sturm–Liouville
operator in its differential and finite difference aspects, Diff. Equat., 23 (1987) 803–810.
150
[60] H. Jafari and V. Daftardar-Gejji, Positive solutions of nonlinear fractional boundary value problems
using Adomian decomposition method, App. Math. Comp., 180 (2006) 700–706.
[61] W. Jiang, Y. Lin, Approximate solution of the fractional advection–dispersion equation, Computer
Physics Communications 181 (2010) 557–561.
[62] M.T. Kajani, M. Ghasemi and E. Babolian, Comparison between the homotopy perturbation method
and the sine–cosine wavelet method for solving linear integro-differential equations, Comp. Math.
Appl., 54 (2007) 1162–1168.
[63] R. A. Khan and M. Rafique, Existence and multiplicity results for some three–point boundary value
problems, Nonlin. Anal.: TMA, 66(2007) 1686–1697.
[64] R.A. Khan, The generalized method of quasilinearization and nonlinear boundary value problems
with integral boundary conditions, Electron. J. Qual. Theory. Differ. Equ., 10 (2003) 1–15.
[65] R. A. Khan, Mujeeb ur Rehman, Existence of multiple positive solutions for a general system of
fractional differential equations, Commun. Appl. Nonlin. Anal. 18 (2011) 25–35.
[66] M.M. Khader, On the numerical solutions for the fractional diffusion equation, Commun Nonlinear
Sci Numer Simulat, 16 (2011) 2535–2542.
[67] F. Khellat and S.A. Yousefi, The linear Legendre mother wavelets operational matrix of integration
and its application, J. Frank. Inst., 343 (2006) 181–190.
[68] A. A. Kilbas, Megumi Saigo and R. K. Saxena, Generalized mittag–leffler function and generalized
fractional calculus operators, Integ. Tran. Spec. Func., 15 (2004) 31–49.
[69] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential
Equations, vol. 204 (North-Holland Mathematics Studies), Elsevier, 2006.
[70] V. Kiryakova, The special functions of fractional calculus as generalized fractional calculus operators
of some basic functions, Comput. Math. Appl., 59 (2010) 1128–1141.
[71] V. Kiryakova, All the special functions are fractional differintegrals of elementary functions, J. Phys.
A: Math. Gen., 30 (1997) 50–85.
[72] V. Kiryakova, The multi-index Mittag-Leffler functions as an important class of special functions of
fractional calculus, Comput. Math. Appl., 59 (2010) 1885–1895.
[73] A. Kilicman and Z.A.A. Al Zhour, Kronecker operational matrices for fractional calculus and some
applications, Appl. Math. Comp., 187 (2007) 250–265.
[74] N. Kosmatov, A singular boundary value problem for nonlinear differential equations of fractional
order, J. Appl. Math. Comput., 29 (2009) 125–135.
151
[75] M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, Netherland,
1964.
[76] P. Kumar and O.P. Agrawal, An approximate method for numerical solution of fractional differential
equations, Signal Proc., 86 (2006) 2602–2610.
[77] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems,
Cambridge Scientific Publishers, Cambridge, 2009.
[78] Ü. Lepik, Solving fractional integral equations by the Haar wavelet method, Appl. Math. Comp., 214
(2009) 468–478.
[79] Ü. Lepik, Solving PDEs with the aid of two–dimensional Haar wavelets Comput. Math. Appl., 61 (
2011) 1873–1879.
[80] Y. Li and W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications
in solving the fractional order differential equations, Appl. Math. Comp., 216 (2010) 2276–2285.
[81] C.F. Li, X. N. Luo and Y. Zhou, Existence of positive solutions of the boundary value problem for
nonlinear fractional differential equations, Comp. Math. Appl., 59 (2010) 363–1375.
[82] N. Liu and En-Bing Lin, Legendre Wavelet method for numerical solutions of partial differential
equations, Siam J. MAath. Anal., 29 (1998) 1040–1065.
[83] R. Leggett and L. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach
spaces, Indiana Univ. Math. J., 28 (1979) 673–688.
[84] C. Lubich, Fractional linear multistep methods for Abel–Volterra integral equations of the second
kind, Math. Comp., 45 (1985) 463–469.
[85] Mujeeb ur Rehman and R.A. Khan, Positive solutions to nonlinear higher-order nonlocal boundary
value problems for fractional differential equations, Abs. Appl. Anal., Volume 2010, Article ID 501230,
15 pages, doi:10.1155/2010/501230
[86] Mujeeb ur Rehman and R.A. Khan, Existence and uniqueness of solutions for multi–point boundary
value problem for fractional differential equations, Appl. Math. Lett., 23 (2010) 1038–1044.
[87] Mujeeb ur Rehman, R.A. Khan and N.A. Asif, Three point boundary value problems for nonlinear
fractional differential equations, Acta Math. Sci., 31 (2011).
[88] Mujeeb ur Rehman and R.A. Khan, Positive Solutions to Coupled System of Fractional Differential
Equations, Int. J. Nonlin. Sci., 10 (2010) 96–104.
[89] R.A. Khan, Mujeeb ur Rehman and J. Hendersom, Existence and uniqueness of solutions for nonlinear
fractional differential equations with integral boundary conditions, Fract. Diff. Cal., 1 (2011) 29–43.
152
[90] Mujeeb ur Rehman and R.A. Khan, Existence and uniqueness of solutions for fractional order differ-
ential equations with nonlocal boundary conditions, Int. J. Math. Anal., (Accapted).
[91] Mujeeb ur Rehman and R.A. Khan, The Legendre wavelet method for solving fractional differential
equations, Commun. Nonlin. Sci. Numer. Simul., 16 (2011) 4163–4173.
[92] Mujeeb ur Rehman and R.A. Khan, Numerical solutions of boundary value problems for fractional
differential equations by Haar wavelets, (submitted)
[93] Mujeeb ur Rehman, R.A. Khan, and Paul W. Eloe, Positive solutions to three-point boundary value
problem for higher order fractional differential system, Dyn. Syst. Appl., (Accepted)
[94] Mujeeb ur Rehman and R.A. Khan, A numerical method for solving boundary value problems for frac-
tional differential equations, Appl. Math. Model. (Accepted) (2011), doi: 10.1016/j.apm.2011.07.045
[95] Mujeeb ur Rehman, R.A. Khan, A note on a boundary value problem of a coupled system of differential
equations of fractional order, Comp. Math. Appl., 61 (2011) 2630–2637.
[96] Mujeeb ur Rehman, Numerical solutions to a class of partial fractional differential equations, (Sub-
mitted)
[97] R. Ma, Existence theorems for a second order m-point boundary value problems, J. Math. Anal.
Appl., 211 (1997) 545–555.
[98] R. Ma, D. O’Regan, Solvability of singular second order m-point boundary value problems, J. Math.
Anal. Appl., 301 (2005) 124–134.
[99] R. Ma, Multiplicity of positive solutions for second-order three-point boundary value problems, J.
Math. Anal. Appl., 268 (2002) 256–265.
[100] R. Ma, Existence and uniqueness of solutions to first-order three–point boundary value problems,
Appl. Math. Lett. 15 (2002) 211–216.
[101] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathe-
matical Models, Imperial College Press, 2010.
[102] K. Maleknejad, M. Shahrezaee and H. Khatami, Numerical solution of integral equations system of
the second kind by Block—Pulse functions Appl. Math. Comp., 166 (2005) 15–24.
[103] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential
equations, Wiley, New York, 1993.
[104] S. Momani and Z. Odibat, Numerical comparison of methods for solving linear differential equations
of fractional order, Chaos, Solit. Fract., 31 (2007) 1248–1255.
[105] S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising
in fluid mechanics, Physics Letters A 355 (2006) 271–279.
153
[106] S. Momani, Analytic and approximate solutions of the space and time–fractional telegraph equations,
Appl. Math. Comput., 170 (2005) 1126–1134.
[107] B.N. Narahari Achar, J.W Hanneken, T. Enck and T.Clarke, Dynamics of the fractional oscillator,
Physica A, 297 (2001) 361–367.
[108] J.J. Nieto, Maximum principles for fractional differential equations derived from Mittag–Leffler func-
tions, Appl. Math. Lett., 23 ( 2010) 1248–1251.
[109] Z. Odibat, S. Momani and V. Suat Erturk, Generalized differential transform method: Application
to differential equations of fractional order, Appl. Math. Comput., 197 (2008) 467–477.
[110] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[111] E. Orsingher and L. Beghin, Time–fractional telegraph equations and telegraph processes with brow-
nian time, Prob. Th. Relat. Fields, 128 (2004) 141–160.
[112] P. N. Paraskevopoulos, P. D. Sparis and S. G. Mouroutsos, The Fourier series operational matrix of
integration, Int. J. Syst. Sci., 16 (1985) 171 – 176.
[113] A. Pálfalvi, Efficient solution of a vibration equation involving fractional derivatives, Int. J. Non–Lin.
Mech., 45 ( 2010) 169–175.
[114] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[115] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differen-
tiation, Frac. Calc. Appl. Anal., 4 (2002) 367–386.
[116] M. Razzaghi and S. Yousefi, Sine–Cosine wavelets operational matrix of integration and its applica-
tions in the calculus of variations, Int. J. Syst. Sci., 33 (2002) 805 – 810.
[117] M. Razzaghi and S. Yousefi, The Legendre wavelets operational matrix of integration, Int. J. Syst.
Sci. 32 (2001) 495 – 502.
[118] K. Ram Pandey, P.O. Singh and K. Vipul Baranwal, An analytic algorithm for the space–time
fractional advection–dispersion equation, Computer Physics Communications 182 (2011) 1134–1144.
[119] J. Sabatier, O.P. Agrawal and J.A. Tenreiro Machado (Ed.), Advances in Fractional Calculus: The-
oretical Developments and Applications in Physics and Engineering, Springer, 2007.
[120] H. Saeedi, M. Mohseni Moghadam, N. Mollahasani and G.N. Chuev, A CAS wavelet method for
solving nonlinear Fredholm integro-differential equations of fractional order, Commun. Nonl. Sci.
Num. Simul., 16 (2011) 115–1163.
[121] H. Saeedi and M. Mohseni Moghadam, Numerical solution of nonlinear Volterra integro-differential
equations of arbitrary order by CAS wavelets, Appl. Math. Comput., 16 (2011) 1216–1226.
154
[122] S. Saha Ray and R.K. Bera, Analytical solution of the Bagley Torvik equation by Adomian decom-
position method, Appl. Math. Comp., 168 (2005) 398–410.
[123] H.A.H. Salem, On the fractional order m-point boundary value problem in reflexive Banach spaces
and weak topologies, J. Comput. Appl. Math., 224 (2009) 565–572.
[124] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and
Applications, Gordon and Breach Science Publishers, 1993.
[125] R. K. Saxena, and M. Saigo, Certain properties of fractional calculus operators associated with
generalized Mittag-Leffler function, Frac. Cal. App. Anal., 8 (2005) 141–54.
[126] A. Shi and S. Zhang, Upper and lower solutions method and a fractional differential equation bound-
ary value problem, Elect. J. Qualit. Th. Diff. Equat., 30 (2009) 1–13.
[127] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equa-
tions, Elect. J. Diff. Equat., 36 (2006) 1–12.
[128] Z. Shuqin, Existence of solution for boundary value problem of fractional order, Act. Math. Sci., 26
(2006) 220–228.
[129] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations,
Appl. Math. Lett., 22 (2009) 64–69.
[130] L. Su, W. Wang and Z. Yang, Finite difference approximations for the fractional advection–diffusion
equation, Physics Letters A 373 (2009) 4405–4408.
[131] P. Shi and M. Shillor, Design of contact patterns in one–dimensional thermoelasticity, Theoretical
Aspects of Industrial Design, SIAM, Philadelphia, PA (1992).
[132] L. Song, S. Xu, and J. Yang, Dynamical models of happiness with fractional order Commun. Nonlin.
Sci. Num. Simul., 15 (2010) 616–628.
[133] M. Tavassoli Kajani, A. Hadi Vencheh and M. Ghasemi, The Chebyshev wavelets operational matrix
of integration and product operation, Int. J. Comp. Math., 86 (2009) 1118–1125.
[134] J. Tenreiro Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun.
Nonlin. Sci. Numer. Simul., doi:10.1016/j.cnsns.2010.05.027.
[135] A comparative study of numerical integration based on Haar wavelets and hybrid functions, Comput.
Math. Appl., 59 (2010) 2026–2036.
[136] Z.H. Wang and X. Wang, General solution of the Bagley–Torvik equation with fractional–order
derivative, Commun. Nonlin. Sci. Num. Simul., 15 (2010) 1279–1285.
[137] H. Wang, On the existence of positive solutions for semilinear elliptic equations in the annulus, J.
Diff. Equat., 109 (1994) 1–7.
155
[138] K. Wang, H. Wang, A fast characteristic finite difference method for fractional advection–diffusion
equations, Advances in Water Resources, (2011) doi:10.1016/j.advwatres.2010.11.003.
[139] J. R. L. Webb, Positive solutions of some three–point boundary value problems via fixed point index
theory, Nonlinear Anal.: TMA, 47 (2001) 4319–4332.
[140] G. Wu and E.W.M. Lee, Fractional variational iteration method and its application, Phy. Lett. A,
374 (2010) 2506–2509.
[141] J. Wu, C.Chen and Chih-Fan Chen, A unified derivation of operational matrices for integration
in systems analysis, Proceedings. International Conference on Information Technology: Coding and
Computing, 2000.
[142] A. Yang and W. Ge, Positive solutions for boundary value problems of N-dimension nonlinear frac-
tional differential system, Bound. Val. Prob., 2008, Article ID 437453.
[143] A. Yu. Popov, On zeros of a certain family of Mittag-Leffler functions, J. Math. Sci., 144 (2007)
4228–4231.
[144] E. Zeidler, Nonlinear Functional Analysis an its Applications I: Fixed Point Theorems, Springer New
York 1986.
[145] S. Zhang, Existence results for positive solutions to boundary value problem for fractional differential
equation, Positivity, 13 (2009) 583–599.
[146] S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential
equation, Comp. Math. Appl., 59 (2010) 1300–1309.
[147] X. Zheng and Xiaofan Yang, Techniques for solving integral and differential equations by Legendre
wavelets, Int. J. Syst. Sci., 40 (2009) 1127–1137.
[148] X. Zhang, M. Feng and Weigao Ge, Existence result of second-order differential equations with
integral boundary conditions at resonance, J. Math. Anal. Appl., 333 ( 2007) 657–666.
[149] Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. Comput.,
215 (2009) 524–529.
[150] Y. Zhang, Z. Bai and T. Feng, Existence results for a coupled system of nonlinear fractional three–
point boundary value problems at resonance, Comp. Math. Appl., 61 (2011) 1032–1047.
[151] Y. Zhang, A finite difference method for fractional partial differential equations, Applied Mathematics
and Computation, 215 (2009) 524–529.
[152] S. Zhang, Nonnegative solution for singular nonlinear fractional differential equation with coefficient
that changes sign, Positivity, 12 (2008) 711–724.
[153] W. Zhong and W. Lin, Nonlocal and multiple-point boundary value problem for fractional differential
equations, Comp. Math. Appl., 59 (2010) 1345–1351.