boundary value problems for fractional order …

BOUNDARY VALUE PROBLEMS FOR FRACTIONAL ORDERDIFFERENTIAL EQUATIONS: EXISTENCE THEORY AND

NUMERICAL SIMULATIONS

by

HASIB KHAN

A Dissertation

Submitted To The University Of Malakand In Partial Fulfillment Of Requirements For The Degree Of

DOCTOR OF PHILOSOPHY

In Mathematics

Supervisor: Prof. Dr Rahmat Ali Khan

DEPARTMENT OF MATHEMATICS UNIVERSITY OF MALAKAND, CHAKDARA,

PAKISTAN

2015

i

AUTHORIZATION TO SUBMIT DISSERTATION

This dissertation of Mr. Hasib Khan submitted for the Degree of Doctor of Philosophy in Mathematics

and titled “BOUNDARY VALUE PROBLEMS FOR FRACTIONAL ORDER DIFFERENTIAL

EQUATIONS: EXISTENCE THEORY AND NUMERICAL SIMULATIONS” has been reviewed

in final form. Permission, as indicated by the signatures given below, is now granted to submit the copies

to University of Malakand for the final evaluation process.

Research Supervisor

Prof. Dr Rahmat Ali Khan ...............................................................................

Committee members

....................................................................................................................................................

....................................................................................................................................................

Chairman Department of Mathematics

Prof. Dr Gul Zaman .............................................................................

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Acknowledgement

First and foremost, I am grateful to Almighty Allah for all His blessings and mercies in every walks

of my life. Then, I do hereby greatly acknowledge hereby the Holy Prophet Muhammad (PBUH), whose

life and principles serve as a symbol of perpetual peace and knowledge for the whole mankind.

I am thankful to the prime minister on the reimbursement scheme which have supported me in the

fee of my PhD studies at the Mathematics department, University of Malakand, Chakdara, Dir (L).

I am also thankful to the higher education commission of Pakistan (HEC) for providing the scholars

a very good research and higher studies environment in the universities of Pakistan, which have enabled

me for the PhD degree in my homeland.

I am thankful to my respected supervisor Professor Dr Rahmat Ali Khan, Dean of Sciences, Uni-

versity of Malakand, Chakdara, for his guidance throughout my PhD, during course and research work.

His very kind support helped me getting along the track and out of the hurdles in my studies. Thanks

are also due to Professor Gul Zaman, Chairman of Mathematics Department, for providing such a nice

environment for the studies.

My sincere thanks to the Ex-Vice Chancellor and founder of Shaheed Benazir Bhutto University

Sheringal, Dir Upper, Professor Dr Jahandar Shah for his support during my PhD enrollment and the

provision of No Objection Certificate, in 2012. I pray for his long and peaceful life.

Thanks to Professor Dr Gul Hassan, Shaheed Benazir Bhutto University, Sheringal, Dir Upper, for

his help and support in my travel to Iran during his Chairmanship which has resolved a lot of difficulties

in my research work.

Thanks to Professor Dr Hossein Jafari, Department of Mathematical Sciences, University of South

Africa, for his support in different aspects of my studies. My special thanks to him for the hospitality

during my short stay at the Department of Mathematics, University of Mazandaran, Iran, and for the

financial support in the publications of my research as article processing charges. I found him a very

good person and a great mathematician. I am also thankful to Assistant Professor Dr Mohsen Alipour

and Dr Haleh Tajadodi for their help in different problems for the numerical programing. May Allah

bless all of them. Ameen!

Thanks also go to Professor Dr Dumitru Baleanu, Department of Mathematics Computer Science,

Cankaya University, Ankara, Turkey, for his help and support. He is indeed a kind human being and a

good professional. Thank you Sir.

I am also indebted to Professor Dr Xiao-Jun Yang, Department of Mathematics and Mechanics,

China University of Mining and Technology, for his help in the LF differential equations. He provided

me a lot of research materials and guided me for the work.

iii

Thanks are also extended to Professor Dr Mohammad Mehdi Rashidi, Tongji University, China, for

sharing of his research expertise and with whom I am going to interact for my research in which I will

work in the nearest future.

My humble thanks to my parents, whose unending support and prayers got me every thing of my

life. May Allah bless them with every happiness in their lives and hereafter!

My thanks to my brothers Suliman Khan (School Teacher) and Aziz Khan (PhD Scholar) for their

love and help in my life and a special thanks to my sisters for their support and prayers. The thanks are

extended to my wife for her support, help and prayers.

My love for my nephews Shah Fahad Khan, Shah Zaib Khan, Mashaal Khan, Amsaal Khan, Ahmad

Khan, Khalid Khan, Tanveer Ahmad Khan and Tauseef Ahmad Khan. They have given me good time in

playing Chess and kept me joyful when we were at home. May Allah succeed all of them. Ameen!

HASIB KHAN

iv

DEDICATION

I dedicate my PhD thesis to my dear parents who bolstered me long lasting and pray for their happiest

life.

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List of publications

This thesis consist of the following published papers:

1. Dumitru Baleanu, Hasib Khan, Hossein Jafari, Rahmat Ali Khan and Mohsen Alipour, “On existence

results for solutions of a coupled system of hybrid boundary valueproblems with hybrid conditions’, Ad-

vances in Difference Equations (2015) 2015:318

2 D. Baleanu, R. P. Agarwal, Hasib Khan, R. A. Khan and H. Jafari, “On the existence of solution

for fractional differential equations of order 3 < δ1 ≤ 4”, Advances in difference Equations 2015:362

(2015).

3. H. Jafari, D. Baleanu, Hasib Khan, R. A. Khan, and A. Khan, “Existence criterion for the solutions

of fractional order p–Laplacian boundary value problems” Boundary Value Problems 2015:164, (2015).

4. D. Baleanu, Hasib Khan, H. Jafari, and R. A. Khan, “On the Exact Solution of Wave Equations on

Cantor Sets,” Entropy 17, 6229–6237 (2015).

5. D. Baleanu, H. Jafari, Hasib Khan, and S. J. Johnston, “Results for solution of fractional coupled

hybrid boundary value problems,” Open Mathematics, 13, 601–608 (2015).

6. Hasib Khan, H. Jafari, R. A. Khan, H. Tajadodi, and S. J. Johnston, “Numerical Solutions of the

Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials,” The Scientific World Jour-

nal Vol. 2014, Article ID 257484, 7 pages (2014).

7. Hasib Khan, M. Alipour, R. A. Khan, H. Tajadodi, and A. Khan, “On Approximate Solution of

Fractional Order Logistic Equations by OMBPs,” Journal of Mathematics and Computer Science 14,

222–232 (2015).

8. Hasib Khan, M. Alipour, H. Jafari, R. A. Khan, “Approximate Analytical Solution of a Coupled

System of Fractional Partial Differential Equations by Bernstein Polynomials,” Int. J. Appl. Comput.

Math., (2015). DOI 10.1007/s40819-015-0052-8

9. Hasib Khan, R. A. Khan, and M. Alipour “On the existence of solution for fractional boundary value

problem,” Journal of Fractional Calculus 6(1), 83–93, (2015).

10. R. A. Khan, Hasib Khan, “Existence of solution for a three point boudary value problem of frac-

tional differential equation,” Journal of Fractional Calculus 5(1), 156–164, (2014).

11. Existence and uniqueness of results for coupled system of fractional q–difference equations with

boundary conditions, J. Comput. Complex. Appl., 1(2), 79–88 (2015).

Accepted papers are:

1. M. Alipour, R. A. Khan, Hasib Khan “Applications of local fractional series expansion method to

vi

fractal Vehicular traffic flow model, Sindh University Research Journal (Accepted)

2. D. Baleanu, H. Jafari, R. A. Khan and Hasib Khan “On approximations of local fractional Schrodinger

equations on Cantor sets, Sindh University Research Journal (Accepted)

3. M. Alipour, R. A. Khan and Hasib Khan “Computational method Based on Bernstein polynomials

for solving a fractional optimal control problem”, Punjab University Journal of Mathematics (Accepted)

vii

Abstract

The study of fractional calculus has been initiated in the Seventeenth century and has received much

attention in the last few decades. Because of the fractional order derivatives, scientists have developed

excellent approach for the description of memory and hereditary properties of different problems in

science and engineering. Therefore, we see the applications of fractional calculus in the fields such

as; signal processing, diffusion process, physics, fluid mechanics, bioscience, chemistry, economics,

polymer rheology and many others.

In this thesis, we are concerned with the existence and uniqueness of positive solutions for different

classes of boundary value problems for fractional differential equations (FDEs). We also study numerical

solutions of FDEs and for some classes exact analytical solutions of local FDEs.

Existence and uniqueness theory for positive solutions is developed for following classes of bound-

ary value problems (BVPs) for FDEs:

Class of two point BVPs for FDEs, class of three point BVPs for FDEs, Class of multi point BVPs for

FDEs, a general class of BVPs with p-Laplacian operator, BVPs for coupled systems of FDEs, BVPs for

coupled system of fractional order differential integral equations, BVPs for coupled system of fractional

order q–difference equations, and BVPs for coupled systems of hybrid FDEs.

For numerical solutions, Bernstein polynomials (BPs) are used and operational matrices (OM) for

fractional order integrations and differentiations are developed. Based on these OM, numerical schemes

for numerical solutions are developed for the following classes of of FDEs;

fractional partial differential equations, coupled systems of FDEs, optimal control problems. We also

use B-Spline functions and develop operational matrices of B-Spline functions for the numerical solution

of a coupled system of FDEs.

We also study exact solutions of some local FDEs, we use different mathematical methods for differ-

ent local fractional (LF) problems. In this work, we produce iterative techniques for the approximation

of solutions of different problems in LF calculus and the efficiency of the schemes are tested by many

examples.

viii

Contents

1 Preliminaries 6

1.1 Fractional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 q–Fractional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 Local fractional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.3 Local fractional Laplace Transform (LFLT) . . . . . . . . . . . . . . . . . . . . 9

1.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Bernstein polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Approximation of a function by Bernstein polynomials . . . . . . . . . . . . . . 11

1.2.3 Linear Cardinal B-Spline function . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.4 Function approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Fixed point theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Existence and uniqueness of solutions for FDEs 17

2.1 Two point BVPs for FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.1 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Three point BVPs for FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Multi point BVPs for FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Existence and uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 p–Laplacian BVP for FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35


3 Coupled systems of BVPs for fractional differential equations 43

3.1 Coupled systems of BVPs for FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44


3.2 Coupled systems of BVPs for fractional order q–difference equations . . . . . . . . . . . 53

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3.2.1 Existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.2 Uniqueness of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Coupled systems of hybrid FDEs 61

4.1 Coupled system of hybrid FDEs I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


4.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Coupled system of hybrid FDEs II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72


4.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Local fractional differential equations 79

5.1 Iterative technique for solutions of LF wave equations . . . . . . . . . . . . . . . . . . . 80

5.1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Iteration technique for solutions of LF Schrodinger equation . . . . . . . . . . . . . . . 83

5.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Iteration technique for the solution of fractal Vehicular Traffic flow model . . . . . . . . 87

5.3.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Numerical solutions of fractional differential equations 90

6.1 Operational Matrix for fractional order derivatives using BPs . . . . . . . . . . . . . . . 90

6.2 Numerical scheme for solutions of Fractional Telegraph equations . . . . . . . . . . . . 92

6.2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.3 Numerical scheme for solutions of Fractional partial differential equations (FPDEs) . . . 95

6.3.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7 Fractional integral and product matrices of BPs 100

7.1 Operational matrix for fractional order integration . . . . . . . . . . . . . . . . . . . . 100

7.2 Operational matrix for multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3 Fractional order Logistic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.3.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.4 Fractional order Brusselator system (FOBS) . . . . . . . . . . . . . . . . . . . . . . . . 107

7.4.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.5 Optimal controle problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

x

7.5.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8 B-Spline functions 115

8.1 Operational matrix for fractional integration . . . . . . . . . . . . . . . . . . . . . . . . 115

8.2 Operational matrix of multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8.3 Fractional order Brusselator system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.3.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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List of Figures

6.1 Graph for the approximation of u(x, t) when m = 10 and α = 1.5, 1.75, 2.0 and the exact

solution in6.2.1, by our numerical technique (6.2.6)– (6.2.7). . . . . . . . . . . . . . . . . . . 93

6.2 Plot for error in our approximation for u(x, t), by the proposed technique and our numerical

technique (6.2.6), (6.2.7) for α = 2 with m = 10 in 6.2.1. . . . . . . . . . . . . . . . . . . . 93

6.3 Graph for the approximate values of u(x, t) for α = 1.5, 1.75, 2.0, m = 10 and the exact

solution at t = 0.25. The graph for our approximation is the dashed for α = 2 and the green line

represents the exact solution in 6.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4 Graph for the approximation of u(x, t) when m = 10 and α = 1.5, 1.75, 2.0 and the exact

solution in example 6.2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.5 Plot for the error in our approximation for u(x, t), by the proposed technique and our numerical

technique (6.2.6)– (6.2.7) for α = 2 with m = 10 in example 6.2.2. . . . . . . . . . . . . . . . 95

6.6 Graph for the approximate values of u(x, t) for α = 1.5, 1.75, 2.0, m = 10 and the exact

solution at t = 0.25. The graph for our approximation is the dashed for α = 2 and the green line

represents the exact solution in example 6.2.2. . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.7 Error in our approximation for u(x, t), for m = 5 and α, β = 1. . . . . . . . . . . . . . . . . 97

6.8 Error in our approximation for v(x, t), for m = 5 and α, β = 1. . . . . . . . . . . . . . . . . 97

6.9 Plot of the approximation solutions u(x, t), for m = 3, α = 1 and β = 0.8, 0.9, 1. . . . . . . . . 97

6.10 Plot of the approximation solutions v(x, t), for m = 3, α = 1 and β = 0.8, 0.9, 1. . . . . . . . . 98

6.11 Error in our approximation for u(x, t), for m = 3 and α, β = 1. . . . . . . . . . . . . . . . . 98

6.12 Error in our approximation for v(x, t), for m = 3 and α, β = 1. . . . . . . . . . . . . . . . . 99

6.13 Plot of the approximation solutions u(x, t), for m = 3, α = 1 and β = 0.8, 0.9, 1. . . . . . . . . 99

6.14 Plot of the approximation solutions v(x, t), for m = 3, α = 1 and β = 0.8, 0.9, 1. . . . . . . . . 99

7.1 Approximation of FOLE in example 7.3.1, where the black line represents the exact solution,

Red line is for α = 1, Green line is for α = .99, Doted line is for α = .98, and x ∈ [0, 2] . . . . . 104

7.2 Error in our approximation for the FOLE in example 7.3.1 for α = 1 when x ∈ [0, 1]. . . . . . . 105

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7.3 Approximation of FOLE in example 2 via BPs operational matrices. Here Black line=Exact

solution, Green line is for α = 1, Doted line is for α = .99, and the Red line is for α = .95, and

x ∈ [0, 2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.4 Error in our approximation of the FOLE in example 7.3.2 for α = 1 when x ∈ [0, 1]. . . . . . . 105

7.5 In this figure, we present comparison of our approximation with the exact solution in example

7.3.3, for different values of α. Here the Dashed line represents exact solution at α = 1, where

our approximations are: Green line is for α = 1, Red line is for α = .99, Gray line is for

α = .98, Black line is for α = .95, and x ∈ [0, 2]. . . . . . . . . . . . . . . . . . . . . . . . 106

7.6 Error in our approximate solution of the FOLE in example 7.3.3, when α = 1, x ∈ [0, 1]. . . . . 106

7.7 The exact solution: (black line) and approximation solutions when α = 1, β = 1 and m = 12

(dotted), m = 8(dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.8 The exact solution: (black line) and approximation solutions when m = 12 and α = .98, β = 1

(dotted), α = .95, β = .99(dashed), α = 0.9, β = .98(Long-dashed). . . . . . . . . . . . . . 109

7.9 The exact solution: (black line) and approximation solutions when α = 1, β = 1 and m = 6

(dotted), m = 4(dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.10 Fig.1. The exact solution: (black line) and approximation solutions when m = 6 and α =

1, β = .98 (dotted), α = .98, β = .95(dashed). . . . . . . . . . . . . . . . . . . . . . . . 110

7.11 Plot of x(t) for m = 7 and α = .8, .9, 1 in example 7.5.1. . . . . . . . . . . . . . . . . . . . 112

7.12 Plot of u(t) for m = 7 and α = .8, .9, 1 in example 7.5.1. . . . . . . . . . . . . . . . . . . . 113

7.13 Plot of x(t) for m = 8 and α = .8, .9, 1 in example 7.5.2. . . . . . . . . . . . . . . . . . . . 114

7.14 Plot of u(t) for m = 8 and α = .8, .9, 1 in example 7.5.2. . . . . . . . . . . . . . . . . . . . 114

8.1 The exact solution (black line) and approximation solutions when α = 1, β = 1 and J = 5

(dotted) and J = 4 (dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.2 The exact solution (black line) and approximation solutions when J = 5 and α = 0.98, β = 1

(dotted) and α = 0.95, β = 0.98 (dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.3 The exact solution: (black line) and approximation solutions when α = 1, β = 1 and J = 5

(dotted), J = 4(dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.4 The exact solution (black line) and approximation solutions when J = 5 and α = 0.98, β = 0.99

(dotted) and α = 0.97, β = 0.98 (dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xiii

Introduction

Differential equations have been considered by a large number of scientists in many different scientific

fields. The scientists used different techniques for the exploration of different important features of

differential equations, including; existence of positive solutions, numerical solutions and exact solutions.

For these types of studies different mathematical tools have been used.

The study of differential equations has been initiated in the seventeenth century by Isaac Newton

(1642–1727) and Gottfried Leibniz (1646–1716). The concept of the differential operator was then

generalized by Leibniz and Marquis de L’Hospital. Leibniz introduced the notion for nth derivative of

a function y(x) for n ∈ N by Dnydxn . L’Hospital raised a question to Leibniz ”what will be the result if

n = 12?” Leibniz replied that ”this is an apparent paradox from which, one day, useful consequences will

be drawn”. This leads to the concept of fractional derivatives and S. F. de Lacroix (1819) introduced the

fractional derivative and then a large number of scientists have shown their contributions in the fractional

calculus. Some of the useful and important contributions in fractional calculus are due to I. Podlubny [1],

R. P. Agarwal [45], D. Baleanu [76, 88], R. A. Khan [19, 22, 24, 33, 58, 73], H. Jafari [80, 83, 97, 98], X.

J. Yang [122, 123], K. Maleknejad [82, 100], A. Akgul and A. Kilicman [72], K. B. Oldhalm and J.

Spainer [9], M. A. E. Herzallah [63], B. Ahmad [13, 14, 41, 42, 49], M. Rehman [21, 22, 59], R. A. C.

Ferreira [40], W. Yang [34], G. A. Anastassiou [56], M. Benchohra [26], J. Sabatier [10], A. A. Kilbas

et al. [2], R. Hilfer [6], A. A. Kilbas [7], B. C. Dhage [66, 70, 71, 104] and G. Wang [103].

There are many types of fractional order derivatives, including; the Riemann, the Marchaud, the

RiemannLiouville, the Weyl, the Caputo, the Hadamard, the Granwald-Letnikov, the Liouville, the

Riesz-Feller. In this dissertation, we are related to the Riemann-Liouville and the Caputo fractional

derivatives.

Scientists of different fields are exploring different features of fractional calculus. Among the differ-

ent features, existence of solutions for FDEs, fractional order integro-differential equations, FDEs with

p-Laplacian operator, hybrid FDEs, coupled systems of FDEs, coupled system of hybrid FDEs, q–FDEs,

and many others have been considered by a large number of scientists using different mathematical meth-

ods. For the existence of solutions, scientists are utilizing different mathematical methods, including;

1

Schauder fixed point theorem, Banach’s Contraction Principle, Arzela Ascoli theorem, Dhage’s theorem,

monotone iterative technique, etc., [17–29, 31, 32, 63].

The numerical solutions of FDEs is another important aspect of the fractional calculus. In this

area, scientists are using different mathematical techniques, including; wavelets, operational matrices for

fractional order integration of different polynomials, operational matrices for fractional order derivative

of different polynomials, operational matrices for fractional order integration of B–Spline functions

[73, 74, 76, 77, 81, 82, 129].

The fractal curves [105], are everywhere continuous but nowhere differentiable and therefore, the

classical calculus cannot be used to interpret the motions in Cantor time-space [107]. Local FC [110–

113], is considered as one of the useful ways to handle the fractals and other functions that are con-

tinuous but non-differentiable. Mandelbrot [105, 117], described that fractal geometry is a workable

geometric middle ground between the excessive geometric order of Euclid and the geometric case of

general mathematics, and extensively illustrated wide range of applications of fractals in many scientific

fields like in, physics, engineering, mathematics, geophysics. Christianto and Rahul [118], gave the

derivation of Proca equations on Cantor sets. Hao et al. [119], investigated the Helmholtz and Diffusion

equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora [120], studied

diffusion problems in fractal media in Cantor sets. Zhang et al. studied LF wave equations under fixed

entropy. Many techniques are utilized for handling the LF problems in both ordinary and partial deriva-

tives. For instance, The Local fractional Laplace Transform [110], the LFLVIM [113] and many others.

These methods are widely used in different scientific fields [112–116].

Organization of the thesis: In Chapter 2, we give details about the preliminary results, including

the basic definitions and results from fractional calculus and provide some related theorems, includ-

ing; Banach’s contraction principle, Arzela Ascoli theorem, Dhage’s theorem, Krasnoselskii’s theorem,

Leray–Schauder Alternative. The introduction of the q–Fractional calculus, basic results about the Bern-

stein polynomials are also provided in this chapter.

In Chapter 3, we study sufficient conditions for existence and uniqueness of solutions (EUS) to some

classes of boundary value problems for FDEs. The first class of problems of our interest consist of FDEs

of order α ∈ (3, 4] with two point boundary conditions of the type:

Dαx(t) = −H(t, x(t)),

Dα1x(1) = 0 = I3−αx(0) = I4−αx(0), x(1) =Γ(α− α1)

Γ(α)Iα−α1H(t, x(t))(1),

(0.0.1)

where t ∈ [0, 1], α1 ∈ (1, 2], the fractional derivatives Dα, Dα1 are in the sense of Riemann–Liouville

of ordersα, α1, H : C([0, 1]×R → R) and Iα−α1H(t, x(t))(1) = 1Γ(α−α1)

∫ 10 (1−s)

α−α1−1H(s, x(s))ds.

2

We study existence of positive solutions (EPS). The second class of problems of our interest is the class

of three-point BVPs of the type:

Dαx(t) + f(t, x(t), x′(t)) = 0, 1 < α ≤ 2, (0.0.2)

x(0) = 0, Dpx(1) = δDpx(η), 0 < p < 1,

where 0 < µ < p < 1, 0 < η ≤ 1, Dα is Caputo’s fractional derivative of order α ∈ (1, 2]. We develop

upper and lower solutions (EUS) method for the existence of solutions. The third class of problems is

the class of multi-point BVPs:

Dαx(t) = f(t, x(t),Dpx(t)) 2 < α < 3, 0 < p < 1, (0.0.3)

x(0) = 0, Dpx(1) =m−2∑i=1

ζiDpx(ηi), x′′(1) = 0,

where Dα, Dp are Caputo’s fractional derivative of order q, p respectively. The fourth class of problems

is the higher order p-laplacian class:

Dγ(ϕp(Dβx(t))) + a(t)f(x(t)) = 0, 3 < β, γ ≤ 4, t ∈ [0, 1],

x(0) = x′′′(0), ηDαx(t)|t=1 = x′(0), ξx′′(1)− x′′(0) = 0, 0 < α < 1,

ϕp(Dβx(t))|t=0 = 0 = (ϕp(Dβx(t)))′|t=0, (0.0.4)

(ϕp(Dβx(t)))′′|t=1 =1

2(ϕp(Dβx(t)))′′|t=0,

(ϕp(Dβx(t)))′′′|t=0 = 0,

where Dβ , Dγ and Dα are Caputo’s fractional derivative of fractional order β, γ, α, and β, γ ∈ (3, 4],

α ∈ (0, 1), ϕp(s) = |s|p−2s, p > 1, ϕ−1p = ϕq, 1

p + 1q = 1.

In Chapter 4, we study two classes of coupled systems for EUS. The first class is the coupled systems

of BVPs for fractional order integro-differential equations (FDIEs) of orders α, β ∈ (3, 4] of the form

Dαx(t) = −G1(t, Iαx(t), Iβy(t)),

Dβy(t) = −G2(t, Iαx(t), Iβy(t)),(0.0.5)

subject to the boundary conditions

Dδx(1) = 0 = I3−αx(0) = I4−αx(0), x(1) =Γ(α− δ)

Γ(α)Iα−δG1(t, Iαx(t), Iβy(t))(1),

Dνv(1) = 0 = I3−βv(0) = I4−βv(0), v(1) =Γ(β − ν)

Γ(β)Iβ−νG2(t, Iαx(t), Iβy(t))(1),

where t ∈ [0, 1], δ, ν ∈ (1, 2]. The functions G1, G2 : [0, 1] × R × R → R, satisfy the Caratheodory

conditions. The fractional derivatives Dα, Dβ, Dδ, Dν are in Riemann–Liouville sense and Iαx(t) =

3

∫ t0

(t−s)α−1x(s)Γ(α) ds, Iβy(t) =

∫ t0

(t−s)β−1y(s)Γ(β) ds. The second class we study the EUS for a coupled sys-

tems of fractional order q–difference equations for α, β ∈ (n− 1, n], k ≤ n− 1, of the type:

Dαq x(t) = −f(t,Dk

q y(t),Dp1q y(t)),

Dβq y(t) = −g(t,Dk

qx(t),Dp2q x(t)),

(0.0.6)

subject to the following initial and boundary conditions

x(1) = 0, Dqx(1) = 0, Diqx(0) = 0, i = 2, 3, ..., n− 1,

y(1) = 0, Dqy(1) = 0, Djqy(0) = 0, j = 2, 3, ..., n− 1,

for t ∈ [0, 1] and n ≥ 3. Dγq , Dβ

q , Dp1q , Dp2

q , are Caputo’s fractional q–difference operators of order

γ, β, p1, p2 where p1, p2 ∈ (0, 1) and Dkq , Di

q, Djq are q–difference operators of order k ≤ n−1, i =

2, 3, . . . , n− 1, j = 2, 3, . . . , n− 1, respectively and f, g ∈ C([0, 1]×R2, R).

In Chapter 5, we study two systems of problems for coupled hybrid systems of FDEs. The first

coupled hybrid system of problems is given below:

Dα(x(t)

H(t, x(t), y(t))) = −K1(t, x(t), y(t)), α ∈ (2, 3],

Dβ(y(t)

G(t, x(t), y(t))) = −K2(t, x(t), y(t)), β ∈ (2, 3],

x(t)

H(t, x(t), y(t))|t=1 = 0, Dµ(

x(t)

H(t, x(t), y(t)))|t=δ1 = 0, x(2)(0) = 0,

y(t)

G(t, x(t), y(t))|t=1 = 0, Dν(

y(t)

G(t, x(t), y(t)))|t=δ2 = 0, y(2)(0) = 0,

(0.0.7)

where δ1, δ2 ∈ (0, 1), α, β ∈ (2, 3], µ, ν ∈ (0, 1), t ∈ [0, 1] and Dα, Dβ , Dµ, Dν are Caputo’s frac-

tional derivative of orders α, β, µ, ν, respectively, K1,K2 ∈ C([0, 1] × R2,R), H,G ∈ C([0, 1] ×

R2,R− {0}). x, y ∈ C([0, 1],R). The second coupled system of problems is:

Dα

(x

H(t, x, y)

)= −T1(t, x, y), α ∈ (n− 1, n],

Dβ

(y

G(t, x, y)

)= −T2(t, x, y), β ∈ (n− 1, n], (0.0.8)(

x

H(t, x, y)

)(i)

|t=0 = 0 =

(x

H(t, x, y)

)(1)

|t=1,x

H(t, x, y)|z=δ1 = δ1Iα−1

1 T1(t, x, y),(y

G(t, x, y)

)(j)

|z=0 = 0 =

(y

G(t, x, y)

)(1)

|t=1,y

G(t, x, y)|t=δ2 = δ2Iβ−1

1 T2(t, x, y),

i, j = 2, 3, . . . , n− 1,

t ∈ [0, 1]. where Dα, Dβ are Caputo’s fractional derivative of orders α, β respectively, T1, T2 ∈

C([0, 1]× R2,R), H,G ∈ C([0, 1]× R2,R− {0}) and x, y ∈ C([0, 1],R).

4

In Chapter 5, we study three problems from LF calculus and obtain their exact analytical solutions.

We use some mathematical methods to solve the LF wave equation, LF Schrodinger’s equation and

Vehicular traffic flow model in Cantore sets.

In Chapter 6, we develop numerical schemes for solutions of the fractional Telegraph equation (FTE)

and fractional partial differential equations with the help of OMBPs and also provide lustrative examples.

In Chapter 7, we develop operational matrices for fractional order integration of Bernstein polyno-

mials and use them for the numerical solutions of fractional order Logistic equations, fractional order

Brusselator systems and optimal control problems. We develop numerical techniques and provide illus-

trative examples.

In Chapter 8, we use B–Spline functions for the numerical solutions of the fractional order Brus-

selator systems. We develop a numerical technique by the using operational matrix of fractional order

integration of B–Spline functions and provide illustrative examples.

5

Chapter 1

Preliminaries

For the study of existence and uniqueness of solutions (EUS) to boundary value problems for FDEs,

numerical solutions and exact solutions of local FDEs, we use different tools from calculus, functional

and numerical analysis. In the following, we provide some basic definitions and known results, which

are useful in our study.

1.1 Fractional calculus

In this section, we recall some basic definitions and fundamental results from fractional calculus which

we use in this thesis.

Definition 1.1.1. [63] If f(t) ∈ L1(a, b), the set of all integrable functions, and α > 0, then the

Riemann-Liouville fractional integral of order α is defined by

Iαf(t) =1

Γ(α)

∫ t

a(t− s)α−1f(s)ds (1.1.1)

provided the integral converges.

Definition 1.1.2. [63] For α > 0, the Caputo fractional derivative of order α is defined by

Dαf(t) =1

Γ(n− α)

∫ t

0(t− s)n−α−1Dnf(s)ds, (1.1.2)

where n ∈ N is such that n− 1 < α < n and D = dds .

From this definition, we have:

Dαt f(t) =

1

Γ(n−α)

∫ t0

Dnf(s)(t−s)1+α−nds, n− 1 < α < n, n ∈ IN,

⌈n

dtnf(t), α = n.

(1.1.3)

Lemma 1.1.3. [7] For α, β > 0, the following relations hold:

Dαtβ =Γ(1 + β)

Γ(1 + β − α)tβ−α−1, β > n and Dαtk = 0, k = 0, 1, ..., n− 1.

6

Lemma 1.1.4. [56] Assume that a, b ≥ 0 and H ∈ L1[p, q], then Ia0+I

b0+H(t) = Ia+b

0+H(t) =

Ib0+I

a0+H(t) and cDb

0+Ib0+H(t) = H(t), for all t ∈ [p, q].

The following properties of fractional derivatives and integrals are useful for our study and are

available in [1, 8, 9]:

(i) DαxC = 0, (C is a constant),

(ii) Dαxx

β =

0, β ∈ IN0, β < ⌈α⌉ ,

Γ(1+β)Γ(1+β−α)x

β−α, β ∈ IN0, β ≥ ⌈α⌉ or β /∈ IN0, β > ⌊α⌋ ,(1.1.4)

(iii) IαxDα

xf(x) = f(x)−n−1∑k=0

f (k)(0+)xk

k!, n− 1 < α ≤ n, (1.1.5)

where Iαx is the fractional Riemann-Liouville integral defined by

Iαx f(x) =

1

Γ(α)

∫ x

0

f(t)

(x− t)1−αdt, α > 0. (1.1.6)

Lemma 1.1.5. [7] Fort β ≥ α > 0 and H(t) ∈ L1[a, b], the following holds

DαIβa+H(t) = Iβ−α

a+ H(t)

on the interval [a, b].

Lemma 1.1.6. [7] For H(t) ∈ C(0, 1), and n−1 < α ≤ n, solution of homogenous FDEDα0+H(t) = 0

is given by

H(t) = k1 + k2t+ k3t2 + ...+ knt

n−1, ki ∈ R, i = 1, 2, 3, ..., n. (1.1.7)

In some of our results, we deal with coupled systems of fractional order q–difference equations. For

this purpose, we need some basic definitions and results from q–Fractional calculus.

1.1.1 q–Fractional calculus

The q-Gamma function for real number α, is given by

Γq(α) =(1− q)(α−1)

(1− q)α−1,

where q ∈ R and satisfy Γq(α+ 1) = [α]qΓq(α),

(α− β)(γ) = αγ∞∏n=0

α− βqn

α− βqγ+n,

where n ∈ N and γ ∈ R.

For a function T , the q-derivative is defined by

(DqT )(x) =T (x)− T (qx)

(1− q)y, (DqT )(0) = lim

x→0(DqT )(x), x = 0.

7

For higher order q-derivative, we have

Dnq T (x) = Dq(D

n−1q T (x)).

The q-integral of a function defined in [0,1], is given by

IqT (x) =

∫ x

0T (s)dqs = x(1− q)

∞∑n=0

T (xqn)qn, x ∈ [0, 1].

For a ∈ [0, b], the q-integral from a to b is define by∫ b

af(x)dqx =

∫ b

0f(x)dqx−

∫ a

0f(x)dqx.

The fractional q-derivative in Caputo’s sense is defined by

Dαq,af(x) = I⌈α⌉−α

q,a D⌈α⌉q f(x),

where ⌈α⌉ denotes the smallest integer greater or equal to α.

Lemma 1.1.7. [40] Let α > 0, such that α ∈ R+ \ N. Then

Iαq Dα

q x(t) = x(t)−m−1∑k=0

tkDkqx(0)

Γq(k + 1),

where m is the smallest integer greater than or equal to α.

In some of our work, we study exact analytical solutions of some problems in fractal space based

on the techniques of local fractional(LF) calculus. We provide below some of the basic definitions and

results from LF calculus.

1.1.2 Local fractional calculus

A function f(x) is said to be a LF continuous function if f(x) satisfies

|f(x)− f(x0)| < aγ , (1.1.8)

where γ ∈ (0, 1], |x− x0| < b, for a, b > 0 and a, b ∈ R.

The LF derivative of a function f(x) ∈ Cγ(a, b) of order γ is defined by

Dγf(x)

dxγ=

∆γ(f(x)− f(x0))

(x− x0)γ, (1.1.9)

where

∆γ(f(x)− f(x0)) = Γ(1 + γ)(f(x)− f(x0)), (1.1.10)

8

andDγ

dxγ

(xmγ

Γ(mγ + 1)

)=

x(m−1)γ

Γ((m− 1)γ + 1), m ∈ N. (1.1.11)

The local fractional integral of a function g(t) on [a, b] is defined by

I(γ)[a,b]f(x) =

1

Γ(γ + 1)

∫ b

ag(x)(dx)γ =

1

Γ(γ + 1)lim

∆x→0

i=M−1∑i=0

g(xi)(∆xi)γ , (1.1.12)

where [a, b] is divided into M − 1 sub-intervals (xi, xi+1) and ∆xi = xi+1 − xi, with a = x0, b = xM

and

I(γ)0,x

(xmγ

Γ(mγ + 1)

)=

x(m+1)γ

Γ((1 +m)γ + 1). (1.1.13)

The following results describing Mittage-Leffler function and its variants in fractal space are useful in

our study

Eγ(xγ) =

∞∑j=0

xjγ

Γ(jγ + 1),

sinhγ(xγ) =

∞∑j=0

x(2jγ+1)

Γ((2jγ + 1)γ + 1),

coshγ(xγ) =

∞∑j=0

x2jγ

Γ(2jγγ + 1),

Dγ

dxγsinhγ(x

γ) = coshγ(xγ),

Dγ

dxγcoshγ(x

γ) = − sinhγ(xγ).

1.1.3 Local fractional Laplace Transform (LFLT)

In this subsection, we provide definition and some basic results related to the Yang-Laplace transforms

(LFLT). For a function f(t) satisfying the following inequality:

1

Γ(γ + 1)

∫ ∞

0|f(x)|(dx)γ < m <∞, (1.1.14)

the LFLT is defined by

Lγ{f(x)} =1

Γ(γ + 1)

∫ ∞

0Eγ(−sγxγ)f(x)(dx)γ , (1.1.15)

where γ ∈ (0, 1] and sγ = βγ + iγαγ . Here, iγ is the fractal imaginary unit and Re(sγ) = βγ > 0. We

use the following properties of the LFLT in our study:

Lγ{ah(x) + bg(x)} = aLγ{h(x)}+ bLγ{g(x)}, (1.1.16)

Lγ{h(t, x)}(mγ) = smγLγ{h(t, x)} − s(m−1)γLγh(0, x)− s(m−2)γh(0, x)(γ)

− s(m−3)γh(0, x)(2γ) . . .− h(0, x)((m−1)γ), (1.1.17)

9

Lγ{xmγ} =Γ(mγ + 1)

s(m+1)γ, (1.1.18)

coshγ(xγ) =

∞∑i=0

t2jγ

Γ(2jγ + 1). (1.1.19)

1.2 Numerical Analysis

In this section, we introduce two types of polynomial approximations, namely, Bernstein polynomials

and Linear Cardinal B-Spline functions, for numerical solutions of fractional order differential equa-

tions.

1.2.1 Bernstein polynomials

The Bernstein polynomials (BPs) [77] of degree m on the interval [0, 1] are defined by

Hi,m =

(m

i

)xi(1− x)m−i, 0 ≤ i ≤ m. (1.2.1)

These polynomials satisfy the following recursive relation:

Hi,m(x) = (1− x)Hi,m−1(x) + xHi−1,m−1(x), i = 0, 1 . . . ,m. (1.2.2)

Using the binomial expansion of (1− x)m−i, the BPs can be written in terms of a linear combination of

the basis functions as follows

Hi,m(x) =

(m

i

)xi(1− x)m−i =

(m

i

)xi

(m−i∑k=0

(−1)k(m− i

k

)xk

)

=

m−i∑k=0

(−1)k(m

i

)(m− i

k

)xi+k, i = 0, 1, . . . ,m. (1.2.3)

Let us denote ϕ(x) = [H0,m(x),H1,m(x), . . . ,Hm,m(x)], then we have ϕ(x) = ATm(x), where

Tm(x) = [1, x, x2, . . . , xm]T and A is (m+ 1)× (m+ 1) matrix defined by

A =

(−1)0(m0

)(−1)1

(m0

)(m−01−0

). . . (−1)m−0

(m0

)(m−0m−0

). . .

......

...

0 (−1)i(mi

). . . (−1)m−i

(mi

)(m−im−i

)...

. . . . . ....

0 . . . 0 (−1)m(mm

)

,

since det(A) = |A| = Πmi=0

(mi

)= 0, it means that A is an invertible matrix.

10

1.2.2 Approximation of a function by Bernstein polynomials

Any squared integrable function F(x) on the interval (0, 1) be approximated in terms of the basis func-

tions {H0,m(x),H1,m(x), . . . ,Hm,m(x)} of BPs (cf. [82]) as follows

F(x) ≈m∑i=0

ciHi,m(x) = CTϕ(x), (1.2.4)

where CT = [c0, c1, . . . , cm], χT (x) = [H0,m(x),H1,m(x), . . . ,Hm,m(x)]. The vector CT can be

obtained from (1.2.4) by using the following relation

CT ⟨ϕ(x), ϕ(x)⟩ = ⟨F(x), ϕ(x)⟩, (1.2.5)

where

⟨F(x), ϕ(x)⟩ =∫ 1

0F(x)ϕ(x)Tdx = [⟨F(x),H0,m⟩, ⟨F(x),H1,m(x)⟩, . . . , ⟨F(x),Hm,m(x)⟩], (1.2.6)

and ⟨ϕ(x), ϕ(x)⟩ = Q is known by dual matrix of ϕ(x) and is given by the following relation

Q = ⟨ϕ(x), ϕ(x)⟩ =∫ 1

0ϕ(x)ϕ(x)Tdx. (1.2.7)

The matrix Q is an invertible symmetric matrix of order (m+ 1)× (m+ 1) whose entries are given by

the relation

qi+1,j+1 =

∫ 1

0Hi,m(x)Hj,m(x)dx

=

(m

i

)(m

j

)∫ 1

0(1− x)2m−(i+j)xi+jdx

=

(mi

)(mj

)(2m+ 1)

(2mi+j

) i, j = 0, 1, . . . ,m.

From (1.2.5) and (1.2.7), it follows that

CT =

(∫ 1

0f(x)ϕ(x)Tdx

)Q−1. (1.2.8)

Similarly, we can approximate a function F(x, t) ∈ L2([0, 1]× [0, 1]) by BPs as follow

F(x, t) ≈m∑j=0

m∑i=0

CijBi,m(x)Hj,m(t)

=m∑j=0

m∑i=0

Hi,m(x)CijHj,m(t) = ϕT (x)C�ϕ(t),

(1.2.9)

where C� is a square matrix of (m+ 1)× (m+ 1), given by

11

C� =

c00 c01 . . . c0m

c1,0 c11 . . . c1m...

. . . . . ....

cm0 cm1 . . . cmm

,

where C� can be obtained from (1.2.9) by the relation

C� = Q−1

(∫ 1

0

∫ 1

0F(x, t)ϕ(x)ϕT (t)dxdt

)Q−1. (1.2.10)

1.2.3 Linear Cardinal B-Spline function

For a given scaling function in L2(R) the scaling function can be utilized to expand it. We know that the

scaling functions are defined R, therefore they can not be inside of the domain of the investigated issue.

To bypass this aspect, in this desertations we consider the B-spline scaling functions on [0, 1] [99, 129]:

Nm(x) =1

(m− 1)!

m∑k=0

(m

k

)(−1)k(x− k)m−1

+ , (1.2.11)

denotes the cardinal B-spline function of orderm ≥ 2 (degree m-1) for the knot sequence {. . . ,−1, 0, 1, . . .}

and sup[Nm(x)] = [0,m]. Also let N1(x) = χ[0,1](x).

Boor et al. [99] define explicit expression of N2(x) in the following form:

N2(x) =

2∑k=0

(2

k

)(−1)k(x− k)+ =

x, x ∈ [0, 1),

2− x, x ∈ [1, 2),

0, otherwise.

Suppose Nj,k(x) = N2(2jx − k), j, k ∈ Z and Bj,k = sup[Nj,k] = close {x : Nj,k = 0}. By

inspection we have that

Bj,k = [2−jk, 2−j(2 + k)], j, k ∈ Z. (1.2.12)

The set of indices is:

Cj = {k : Bj,k ∩ (0, 1) = 0} , j ∈ Z. (1.2.13)

In view of (1.2.13), we have min {Cj} = −1 and max {Cj} = 2j − 1, j ∈ Z. The support of Nj,k(x)

can be outside of [0, 1],therefore we have to define Nj,k(x) on [0, 1]. Thus, we conclude that

ϕj,k = Nj,k(x)χ[0,1](x), j ∈ Z, k ∈ Cj . (1.2.14)

12

As a result we have

ϕj,k(x) =2∑

i=0

(2

i

)(−1)i(2jx− (k + i))+ =

2jx− k, k

2j≤ x < k+1

2j,

2− (2jx− k), k+12j

≤ x < k+22j,

0, otherwise.

1.2.4 Function approximations

A function f(x) ∈ [0, 1] can be approximated by linear B-spline scaling functions for a fixed j = J , as

fJ(x) ≈2j−1∑k=−1

ckϕJ,k(x) = CTϕJ(x), (1.2.15)

where C and ϕJ(x) are (2j + 1) column vectors given by

C = [c−1, c0, . . . , c2j−1]T , (1.2.16)

ϕJ(x) = [ϕJ,−1(x), ϕJ,0(x), . . . , ϕJ,2j−1(x)]T . (1.2.17)

Assume that eJ(x) is the error of the approximation of f(x) by linear B-spline scaling functions then

|eJ(x)| = ∥f(x)− fJ(x)∥ = O(2−2J) (1.2.18)

For further details, See [100]. Then, ck can be obtained by

ck =

∫ 1

0f(x)ϕJ,k(x)dx, k = −1, . . . , 2j − 1, (1.2.19)

where ϕJ,k are the vector of dual basis of ϕJ . By using the linear combinations of ϕJ,k, the ϕJ,k are

obtained as:

ϕJ,k = P−1ϕJ , (1.2.20)

where

P =

∫ 1

0ϕJ(x)ϕ

TJ (x)dt =

1

2J−2

112

124

124

16

124

. . . . . . . . .

124

16

124

124

112

,

where P is a symmetric (2J + 1)× (2J + 1) matrix. Replacing (1.2.20) in (1.2.19) we get

cTk =

(∫ 1

0f(x)ϕTJ (x)dx

)P−1. (1.2.21)

13

In most of the results in this thesis, we investigate sufficient conditions for existence and uniqueness of

solutions to boundary value problems for fractional order differential equations. We use the results from

fixed point theory.

1.3 Fixed point theory

In this section, we provide some definitions and results from fixed point theory which we use in this

thesis.

Theorem 1.3.1. [31](The Arzela Ascoli Theorem) Let F be an equicontinuous, uniformly bounded fam-

ily of real valued functions f on an interval I (finite or infinite).Then F contains a uniformly convergent

sequence of function in f , converging to a function f∈C(I) where C(I) denotes the space of all con-

tinuous bounded functions on I . Thus, any sequence in F contains a uniformly bounded Convergent

subsequence on I and consequently F have a compact closure in C(I).

Definition 1.3.2. Let X be a metric space, and f : X → X , f is said to be a contraction if there exist

some constant 0 < k < 1, such that

d(f(x), f(y)) ≤ kd(x, y), ∀x, y ∈ X.

The infimum of such k’s is called the contraction coefficient.

Theorem 1.3.3. (Banach fixed point theorem) LetX be a complete metric space, and f be a contraction

on X . Then, there exists a unique fixed point of f , that is, f(x∗) = x∗.

Proof. For the existence of x∗, such that f(x∗) = x∗, assume some x0 ∈ X and define a recursive

sequence as xn+1 = f(xn), then the seqence {xn}∞n=1, is Cauchy. For this, assume that k be the

constructive coefficient of f , then:

d(x2, x1) = d(f(x1), f(x0)) ≤ kd(x1, x0),

d(x3, x2) = d(f(x2), f(x1)) ≤ kd(x2, x1) ≤ k2d(x1, x0),

......

d(xn+1, xn) ≤ knd(x1, x0),

Thus, for any p, q > p+ 1, we have:

d(xq, xp) ≤ d(xq, xq−1) + d(xq−1, xq−2) + . . .+ d(xp+1, xp) (1.3.1)

≤ (kq−1 + kq−2 + . . .+ xp)d(x1, x0) = kp(1− kq−1)

1− kd(x1, x0), (1.3.2)

as q → ∞, we have Kq−1 → 0. And we get d(xq, xp) ≤ kp

1−kd(x1, x0), this implies that the sequence is

a Cauchy sequence. Since, the metric spaceX is complete therefore, the sequence {xn}∞n will converge

14

to a point say z∗ in X . Thus, for any ϵ > 0, there is N sucn that d(z∗, xn) < ϵ2 ∀n > N . Hence,

d(f(z∗), z∗) ≤ d(f(z∗), (xn+1)) + d(z∗, xn+1) = (f(z∗), f(xn)) + d(z∗, xn+1) (1.3.3)

≤ kd(z∗, xn) + d(z∗, xn+1) <ϵ

2+ϵ

2= ϵ.

And this is valid for any ϵ > 0, that is, it must be d(f(z∗), z∗) = 0, this implies that f(z∗) = z∗. Thus,

z∗ is a fixed point of f .

Now to prove the uniqueness of the fixed point, assume that z is another fixed point of the f , then

we can have d(f(z), f(z∗)) = d(z, z∗), which is a contradiction, since d(f(z), f(z∗)) < kd(z, z∗) for

k < 1.

Lemma 1.3.4. [62] (Leray–Schauder Alternative) Let F : E → E be a completely continuous operator

(i.e., a map that is restricted to any bounded set in B is compact ). Let Y = {x ∈ E : x = µFx for some

0 < µ < 1}. Then either the set Y is unbounded or F has at least one fixed point.

Theorem 1.3.5. [31](Shauder-Tychnoff Fixed Point Theorem) Let X be a locally convex, topological

vector space. Let B be a compact, convex subset of X and T a continuous map of B into itself. Then Thas a fixed point x ∈ B.

Theorem 1.3.6. (Krasnoselskii Theorem) [32, 68, 69] Let E be a Banach space and B ⊂ E be a cone.

Assume that Q1, Q2 are open sets contained in E such that 0 ∈ Q1 and Q1 ⊂ Q2. Assume, further, that

F : B ∩ (Q2 \ Q1) → B is a completely continuous operator. If either

• (A1) ∥Fv∥ ≤ ∥v∥ for v ∈ B ∩ ∂Q1 and ∥Fv∥ ≥ ∥v∥ for v ∈ B ∩ ∂Q2, or

• (A2) ∥Fv∥ ≥ ∥v∥ for v ∈ B ∩ ∂Q1 and ∥Fv∥ ≤ ∥v∥ for v ∈ B ∩ ∂Q2,

then F has at least one fixed point in B ∩ (Q2 \ Q1).

Theorem 1.3.7. [63, 104] ( Dhage’s Theorem) Let Xa(0) and Xa(0) be open and closed balls in a

Banach algebra X centered at the origin 0 of radius a, for some real number a, and let A, B : Xa(0) →X be two operators satisfying the following:

• (1) A is Lipschitz with Lipschitz constant λ;

• (2) B is continuous and compact;

• (3) λM < 1, where M = ∥B(Xa(0))∥ = sup{∥B(x)∥ : x ∈ Xa(0)}.

Then, either

• (a) The equation A(x)B(x) = x has a solution in Xa(0), or

• (b) there is an element x ∈ X such that ∥x∥ = a satisfying µAxBx = x, for some 0 < µ < 1.

15

Definition 1.3.8. [63] Let X be a Banach space. A mapping F : X → X is called D– Lipschitzian if

there exist a continuous and nondecreasing function ψ : R→ R such that ∥Fx−Fy∥ ≤ ψ∥x− y∥ for

all x, y ∈ X , where ψ(0) = 0. If ψ is not necessarily nondecreasing and satisfies ψ(a) < a, for a > 0,

the mapping F is called a nonlinear contraction with a contraction function ψ.

Definition 1.3.9. [126] Let B be a solid cone in a real Banach space E , T : B0 → B0 be an operator and

0 < α < 1. Then T is called θ-concave operator if T (ku) ≥ kαT (x) for any 0 < k < 1 and x ∈ B0.

Lemma 1.3.10. [126] Assume that B is a normal solid cone in a real Banach space X , 0 < α < 1 and

T : P 0 → B0 is a α–concave Increasing operator. Then T has only one fixed point in B0.

16

Chapter 2

Existence and uniqueness of solutions forFDEs

Fractional calculus (FC) has recently attracted the attentions of researchers working in various scientific

fields. It is due to the applications of FC in mathematical modeling of various phenomena in many

scientific fields such as chemistry, physics, biophysics, biology, control theory, aerodynamics, physics,

blood flow problems, the fluid-dynamics, polymer rheology, traffic model, variation in thermodynamics,

economics, etc. [1, 2]. Fractional differential equations (FDEs) are capable of describing memory and

hereditary properties of certain important materials and processes. This area of research contributed in

the rapid popularity and development of the subject in the last few years. Theoretical aspects of the

FDEs such as existence, uniqueness, multiplicity, asymptotic behavior and periodicity of solutions have

recently been studied by many researchers, we refer to [11–13, 15, 16, 30] and the references therein.

In this chapter, we discuss existence and uniqueness of solutions to some important classes of

boundary value problems for FDEs. Most of the results of this chapter are published in international

journals [130, 132, 137, 138].

In section 2.1, we study existence of solutions to a class of two point BVPs for higher order FDEs

of order α ∈ (3, 4]. In section 2.2, we study a a general class of three point BVPs for FDEs where the

nonlinearity explicitly depends on the derivative and develop conditions for the EUS. In section 2.3, we

study a class of multi point BVPs for FDEs and develop sufficient conditions for the EUS. In section

2.4, we generalize the results to more general class p–Laplacian for FDEs and develop conditions for

the EUS.

17

2.1 Two point BVPs for FDEs

In this section, we study existence of positive solutions to a class of two point boundary value problems

for FDEs of order α ∈ (3, 4] of the form

Dαx(t) = −H(t, x(t)), t ∈ J = [0, 1]

Dα1x(1) = 0 = I3−αx(0) = I4−αx(0), x(1) =Γ(α− α1)

Γ(α)Iα−α1H(t, x(t))(1),

(2.1.1)

where α1 ∈ (1, 2], the fractional derivatives Dα, Dα1 are in Riemann–Liouville sense of orders α, α1,

respectively. The nonlinear function is assumed to be continuous, that is, H : C([0, 1]×R → R) and the

operator in the boundary conditions is given by Iα−α1H(t, x(t))(1) = 1Γ(α−α1)

∫ 10 (1−s)

α−α1−1H(s, x(s))ds.

Lemma 2.1.1. For s, t ∈ [0, 1], solutions of (2.1.1) are solutions of the following integral equation

x(t) =

∫ 1

0G(t, s)H(s, x(s))ds, (2.1.2)

where G(t, s) is Green’s function and is given by

G(t, s) =

−(t−s)α−1

Γν1+ tα−1(1−s)α−α1−1

Γ(α) + tα−2(1−s)α−1

Γ(α) , t ≥ s,

tα−1(1−s)α−α1−1

Γ(α) + tα−2(1−s)α−1

Γ(α) , t ≤ s.(2.1.3)

Proof. Applying the operator Iα0 on (2.1.1) and using lemma 1.1.6, we get the problem (2.1.1) in the

following equivalent integral form

x(t) = −IαH(t, x(t)) + c1tα−1 + c2t

α−2 + c3tα−3 + c4t

α−4. (2.1.4)

The boundary conditions I3−αx(0) = I4−αx(0) = 0 yield c3 = c4 = 0 and the boundary conditions

Dα1x(1) = 0 and x(1) = Γ(α−α1)Γ(α) Iα−α1H(t, x(t))(1) yield

c1 =

∫ 1

0

(1− s)α−α1−1H(s, x(s))ds

Γ(α)

c2 =

∫ 10 (1− s)α−1H(s, x(s))ds

Γ(α).

(2.1.5)

Hence, it follows that

x(t) =−∫ t

0

(t− s)α−1

Γ(α)H(s, x(s))ds+ tα−1

∫ 1

0

(1− s)α−α1−1

Γ(α)H(s, x(s))ds

+ tα−2

∫ 1

0

(1− s)α−1

Γ(α)H(s, x(s))ds =

∫ 1

0K(t, s)H(s, x(s))ds,

(2.1.6)

where G(t, s) is the Green’s functions which is given by (2.1.3).

Lemma 2.1.2. The Green’s function G(t, s) given by (2.1.3) satisfies the following properties

(A1) G(t, s) is continuous and G(t, s) ≥ 0 for each t, s ∈ J ;

(A2) maxt∈J G(t, s) = G(1, s) for each s ∈ J ;

(A3) mint∈[ 13,1]G(t, s) ≥ λ0G(1, s) for some λ0 ∈ (0, 1).

18

Proof. The continuity of the Green’s function G(t, s) is obvious from the definition in (2.1.3). For

t, s ∈ J such that t ≥ s, since s ≤ st implies that −(1 − s) ≤ −(1 − s

t ) and α − α1 − 1 < α − 1

implies that (1− s)α−α1−1 > (1− s)α−1. Hence, it follows that

G(t, s) =−(t− s)α−1

Γ(α)+tα−1(1− s)α−α1−1

Γ(α)+tα−2(1− s)α−1

Γ(α)

= −(1− s

t )α−1

Γ(α)tα−1 +

(1− s)α−α1−1

Γ(α)tα−1 +

(1− s)α−1

Γ(α)tα−2

≥ −(1− s)α−1

Γ(α)tα−1 +

(1− s)α−α1−1

Γ(α)tα−1 +

(1− s)α−1

Γ(α)tα−2

= ((1− s)α−α1−1 − (1− s)α−1)tα−1

Γ(α)+tα−2(1− s)α−1

Γ(α)≥ 0.

(2.1.7)

From (2.1.3), for t ≤ s it is obvious that G(t, s) ≥ 0. For (A2), we consider s, t ∈ J , such that t ≥ s.

For α ∈ (3, 4], α1 ∈ (1, 2], we have α−α1 − 1 ≤ α− 2, this implies that (1− s)α−α1−1 ≥ (1− s)α−2

and

∂

∂tG(t, s) =

−(α− 1)(t− s)α−2

Γ(α)+

(α− 1)tα−2(1− s)α−α1−1

Γ(α)+

(α− 2)tα−3(1− s)α−1

Γ(α)

= (α− 1)

[−(t− s)α−2 + tα−2(1− s)α−α1−1

Γ(α)

]+

(α− 2)tα−3(1− s)α−1

Γ(α)

= (α− 1)

[−(1− st )

α−2 + (1− s)α−α1−1

Γ(α)

]tα−2 +

(α− 2)(1− s)α−1

Γ(α)tα−3

≥ (α− 1)

[−(1− s)α−2 + (1− s)α−α1−1

Γ(α)

]tα−2 +

(α− 2)(1− s)α−1

Γ(α)tα−3 ≥ 0.

(2.1.8)

For the proof of (A3), utilizing (A1), (A2) in the following calculations:

mint∈[ 1

3,1]G(t, s) =

G(13 , s), s ∈ (0, 13 ],

G(13 , s), s ∈ [13 , 1],

=

−( 1

3−s)α−1+( 1

3)α−1(1−s)α−α1−1+( 1

3)α−2(1−s)α−1

Γ(α) , t ≥ s,

( 13)α−1(1−s)α−α1−1+( 1

3)α−2(1−s)α−1

Γ(α) , t ≤ s.(2.1.9)

For t ≥ s, we have

−(13 − s)α−1 + (13)α−1(1− s)α−α1−1 + (13)

α−2(1− s)α−1

−(1− s)α−1 + (1− s)α−α1−1 + (1− s)α−1

=−(13 − s)α−1 + (13)

α−2(1− s)α−α1−1[13 + (1− s)α

](1− s)α−α1−1

≥−(13 − 1

3)α−1 + (13)

α−2(1− 13)

α−α1−1[13 + (1− 1

3)α1]

(1− 13)

α−α1−1

= (1

3)α−2

[1

3+ (

2

3)α1

].

(2.1.10)

19

For t ≤ s, we deduce that

(13)α−1(1− s)α−α1−1 + (13)

α−2(1− s)α−1

(1− s)α−α1−1 + (1− s)α−1

=(13)

α−2(1− s)α−α1−1[13 + (1− s)α1

](1− s)α−α1−1 [1 + (1− s)α1 ]

≥(13)

α−2[13 + (1− 1

3)α1][

1 + (1− 13)

α1] =

(13)α−2

[13 + (23)

α1][

1 + (23)α1] .

(2.1.11)

Choose

λ0 = min{(1

3)α−2

[13+ (

2

3)α1],(13)

α−2[13 + (23)

α1][

1 + (23)α1] }

. (2.1.12)

Therefore, from (2.1.10), (2.1.11) and (2.1.12), we have λ0 ∈ (0, 1) such that

mint∈[ 1

3, 1]G(t, s) ≥ λ0max

t∈JG(t, s) = λ0G(1, s). (2.1.13)

2.1.1 Existence of solutions

In this section, we study existence of positive solutions to the BVPs (2.1.1) by using Krasnosel’ski1

fixed point theorem.

Consider the Banach space E = {x(t) : x(t) ∈ C(J )}, endowed with the norm ∥x(t)∥ =

maxt∈J |x(t)|. Define an operator T : E → E by:

T x(t) =−∫ t

0

(t− s)α−1


∫ 1

0

(1− s)α−α1−1

Γ(α)H(s, x(s))ds

+ tα−2

∫ 1

0

(1− s)α−1

Γ(α)H(s, x(s))ds =

∫ 1

0K(t, s)H(s, x(s))ds.

(2.1.14)

In view of Lemma (2.1.1), solutions of the BVPs (2.1.1) are fixed points of T . Assume that the following

conditions hold:

(A1) There exist a real number k1 > 0 such that H(t, s) ≤ ξk1, when ever 0 ≤ x ≤ k1;

(A2) There exist a real number k2 > 0 such that H(t, s) ≥ νk2, when ever λ0k2 ≤ x ≤ k2, where λ0 is

constant defined by (2.1.12).

Use the notations ξ =[∫ 1

0 G(1, s)ds]−1

and ν =[∫ 1

13G(23 , s)ds

]−1.

Theorem 2.1.3. Assume that H(t, s) ≥ 0 and is continuous, then under the conditions (A1) and (A2),

the BVP (2.1.1) has at least one positive solution.

Proof. The continuity of the the operator T follows from the continuity of the Green’s function G(t, s)

and and that of H(t, x(t)) ∈ C(J × R → R). Define S = {x(t) ∈ E : ∥x∥ ≤ ∆} where ∆ =

20

maxt∈J H(t, s) + 1, then for any x(t) ∈ S, we have

|T x(t)| = | −∫ t

0

(t− s)α−1


∫ 1

0

(1− s)α−α1−1

Γ(α)H(s, x(s))ds

+ tα−2

∫ 1

0

(1− s)α−1

Γ(α)H(s, x(s))ds|

≤ ∆

[(t− s)α

Γ(1 + α)|0t + tα−1 (1− s)α−α1

(α− α1)Γ(α)|01 + tα−2 (1− s)α

Γ(1 + α)|01]

= ∆

[tα

Γ(1 + α)+

tα−1

(α− α1)Γ(α)+

tα−2

Γ(1 + α)

]≤ ∆

[2

Γ(1 + α)+

1

(α− α1)Γ(α)

]<∞,

(2.1.15)

which implies that the operator T is bounded. Next, for x(t) ∈ S, t1, t2 ∈ J such that t2 > t1, we

have

|T x(t2)− T x(t1)|

= | −∫ t2

0

(t2 − s)α−1

Γ(ν1)H(s, x(s))ds+

∫ t1

0

(t1 − s)α−1

Γ(ν1)H(s, x(s))ds

+ (tα−12 − tα−1

1 )

∫ 1

0

(1− s)α−α1−1

Γ(α)H(s, x(s))ds

+ (tα−22 − tα−2

1 )

∫ 1

0

(1− s)α−1

Γ(α)H(s, x(s))ds|

≤ ∆

[tα2 − tα1Γ(1 + α)

+tα−12 − tα−1

1

(α− α1)Γ(α)+tα−22 − tα−2

1

Γ(1 + α)

].

(2.1.16)

From the definition of the norm and (2.1.16), we have: ∥T x(t2)− T x(t1)∥ → 0 as t1 → t2, which im-

plies that T is equicontinuous and by Arzela–Ascoli theorem, the operator T is completely continuous.

Consider a cone B = {x(t) ∈ E : x(t) ≥ 0 and mint∈[ 13,1] x(t) ≥ λ0∥x∥} in E , then for any x ∈ B,

we have that

mint∈[ 1

3,1](T x)(t) ≥ λ0

∫ 1

0G(1, s)H(s, x(s))ds = λ0max

t∈J

∫ 1

0G(t, s)H(s, x(s))ds = λ0∥T x∥,

(2.1.17)

this implies that T x(t) ∈ B. Let Q∞ = {x ∈ B : ∥x∥ > k1}, then for any x ∈ ∂Q1 implies that

∥x∥ = k1, so (A1) is satisfied for all x ∈ ∂Q1. For x ∈ B ∩ ∂Q1, we obtain

∥T x∥ = maxt∈J

∫ 1

0G(t, s)H(s, x(s))ds ≤ ξk1

∫ 1

0G(t, s)ds = k1, (2.1.18)

by (2.1.18), we get that ∥T x∥ ≤ ∥x∥ for x ∈ B ∩ ∂Q1. Assume Q2 = {x ∈ B : ∥x∥ < k2}, for

x ∈ ∂Q2, we have ∥x∥ = k2, this implies that the condition (C2) is satisfied for x ∈ B ∩ ∂Q2, further,

we have that

(T x)(23) =

∫ 1

0K(

2

3, s)H(s, x(s))ds ≥

∫ 1

13

G(2

3, s)H(s, x(s))ds ≥ νk2

∫ 1

13

G(2

3, s)ds = k2.

(2.1.19)

21

Thus, (2.1.19) implies ∥T x∥ ≥ ∥x∥ for x ∈ B ∩ ∂Q2. Therefore, by the help of lemma 1.3.6, the

operator T has a fixed point say x0 such that k1 ≤ ∥x0∥ ≤ k2.

Example 2.1.4. Consider the following BVP

Dαx(t) =

(∫ 1

13

G(2

3, s)ds

)−1(∫ 1

0G(1, s)ds

)−1 1 + 2maxx∈(0,1] |x(t)|2

, (2.1.20)

with the boundary conditions defined in (2.1.1).

For x ∈ Q2, we have

H(t, x(t)) ≥

(∫ 1

13

G(2

3, s)ds

)−1(∫ 1

0G(1, s)ds

)−1

(1 + 2x(t)

2) ≥ νx(t),

where ν =(∫ 1

13G(23 , s)ds

)−1and for x ∈ B ∩ ∂Q1, we obtain

H(t, x(t)) =

(∫ 1

13

G(2

3, s)ds

)−1(∫ 1

0G(1, s)ds

)−1

(1 + 2maxx∈(0,1] |x(t)|

2) ≤ ξk1,

for k1 =(∫ 1

13G(23 , s)ds

)−1(1+3maxt∈(0,1] |x(t)|

2 ). Therefore, the assumptions (A1) and (A2) are satis-

fied and hence by Theorem 2.1.3, the problem (2.1.20) has a solution.

Now in the following sections, we generalize the results to a class of three point boundary value

problems with a more general nonlinearity in the sense that the nonlinearity explicitly depends of the

first derivative.

2.2 Three point BVPs for FDEs

This section is devoted to the study of existence and uniqueness of solutions for three point BVPs for

FDEs of the type:

Dαx(t) + f(t, x(t), x′(t)) = 0, 1 < α ≤ 2,

x(0) = 0, Dpx(1) = δDpx(η), 0 < p < 1,(2.2.1)

where 0 < µ < p < 1, 0 < η ≤ 1, and Dα is Caputo’s fractional derivative of order α. Our results

are based on some classical results from fixed point theory. We impose some growth and continuity

conditions on f(t, x, x′). Let us use the following notations:

∆ =1− δη1−p

Γ(2− p), G1(t, s) =

−1

Γ(α)(t− s)α−1,

G2(t, s) =−δt

∆Γ(α− p)(η − s)α−p−1, G3(t, s) =

t

∆Γ(α− p)(1− s)α−p−1.

22

Lemma 2.2.1. For h(t) ∈ C[0, 1], 0 < δ < p < 1, and 0 < η ≤ 1, the BVP for fractional differential

equation

Dαx(t) + h(t) = 0, 1 < α ≤ 2,

x(0) = 0, Dpx(1) = δDpx(η), 0 < p < 1,(2.2.2)

has a solution of the form

x(t) =

∫ 1

0G(t, s)h(s)ds, (2.2.3)

where

G(t, s) =

G1(t, s) +G2(t, s) +G3(t, s) 0 ≤ s ≤ t ≤ η ≤ 1 ,

G2(t, s) +G3(t, s) 0 ≤ t ≤ s ≤ η ≤ 1 ,

G3(t, s) 0 ≤ t ≤ η ≤ s ≤ 1,

G1(t, s) +G2(t, s) +G3(t, s) 0 ≤ s ≤ η ≤ t ≤ 1 ,

G1(t, s) +G3(t, s) 0 ≤ η ≤ s ≤ t ≤ 1 ,

G3(t, s) 0 ≤ η ≤ t ≤ s ≤ 1 .

Proof. Applying the operator Iα on (2.2.2) and using lemma 1.1.6, we obtain

x(t) = −Iαh(t) + c1 + c2t. (2.2.4)

The boundary condition x(0) = 0 implies c1 = 0, thus we have

x(t) = −Iαh(t) + c2t, which implies Dpx(t) = −Iα−ph(t) + c2t1−p

Γ(2− p).

The boundary condition Dpx(1) = δDpx(η) yields c2 = 1∆(−δIα−ph(η) + Iα−ph(1)). Hence, (2.2.4)

takes the form

x(t) = −Iαh(t) +t

∆(−δIα−ph(η) + Iα−ph(1)), (2.2.5)

which can be rewritten as

x(t) =−1

Γ(α)

∫ t

0(t− s)α−1h(s)ds+

t

∆Γ(α− p)(−δ

∫ η

0(η − s)α−p−1h(s)ds

+

∫ 1

0(1− s)α−p−1h(s)ds) =

∫ 1

0G(t, s)h(s)ds.

Now, we study existence and uniqueness of solutions of the BVP (2.2.1)

23

2.2.1 Existence and uniqueness

Consider the space E = {x(t) ∈ C[0, 1] : x′(t) ∈ C[0, 1]} with the norm defined by ∥x∥ = maxt∈[0,1] |x(t)|+

maxt∈[0,1] |x′(t)|, E is a Banach space [23]. Define an operator T : E → E by:

T x(t) = −∫ t

0

(t− s)α−1

Γ(α)f(s, x(s), x′(s))ds

+t

∆Γ(α− p)(−δ

∫ η

0(η − s)α−p−1f(s, x(s), x′(s))ds+

∫ 1

0(1− s)α−p−1f(s, x(s), x′(s))ds)

(2.2.6)

In view of Lemma 2.2.1, solutions of the problem (2.2.1) are fixed points of the operator T . By conti-

nuity of f : [0, 1] × R × R → R and G on [0, 1] × [0, 1], the operator T is continuous and in view of

(2.2.6), we have

(T u)′(t) = −∫ t

0

(t− s)α−2

Γ(α− 1)f(s, x(s), x′(s))ds

+1

∆Γ(α− p)(−δ

∫ η

0(η − s)α−p−1f(s, x(s), x′(s))ds+

∫ 1

0(1− s)α−p−1f(s, x(s), x′(s))ds).

(2.2.7)

Assume that the following growth conditions hold:

(A1) There exists a nonnegative function m(t) ∈ L1([0, 1]) such that

|f(t, x(t), x′(t))| ≤ m(t) + a1|x(t)|λ1 + a2|x′(t)|λ2 ,

where a1, a2 ∈ R are nonnegative constants and 0 < λ1, λ1 < 1.

(A2) There exists a nonnegative function m(t) ∈ L1([0, 1]) such that

|f(t, x(t), x′(t)))| ≤ m(t) + a1|x(t)|λ1 + a2|x′(t)|λ2 ,

where a1, a2 ∈ R are nonnegative constants and λ1, λ1 > 1.

(A3) There exist a constant k > 0 such that

|f(t, x(t), x′(t)))− f(t, y(t), y′(t)))| ≤ k(|x(t)− y(t)|+ |x′(t)− y′(t)|),

for each t ∈ [0, 1] and all x, y, x′, y′ ∈ R .

24

For convenience in calculations, the following notations are introduced:

h1(t) =

∫ 1

0G(t, s)m(s)ds+

1

Γ(α− 1)

∫ t

0(t− s)α−2m(s)ds+

1

∆(

δ

Γ(α− p)

∫ η

0(η − s)α−p−1m(s)ds+

1

Γ(α− p)

∫ 1

0(1− s)α−p−1m(s)ds),

h2(t) =tα

Γ(1 + α)+

(t+ 1)(δηα−p + 1)

∆Γ(α− p+ 1)+

1

Γ(α)tα−1,

ϖ1(t) =1

Γα+ 1tα +

t

∆Γ(α− p+ 1)(δηα−p−1 + 1), ϖ2(t) =

tα−1

Γ(α)+

δηα−p + 1

∆Γ(α− p+ 1),

(2.2.8)

and ϖ = ϖ1 +ϖ2. Choose ρ ≥ max{3h1, (3k1h2)1

1−λ1 , (3k2h2)1

1−λ2 }, where k1 = maxt∈[0,1] a1(t),

k2 = maxt∈[0,1] a2(t). Consider a closed bounded subset B = {x(t) ∈ E : ∥x∥ ≤ ρ, t ∈ [0, 1]} of E .

Theorem 2.2.2. Under the assumption (A1) the BVP (2.2.1), has at least one solution.

Proof. For x ∈ B using the assumption (A1), we have:

|T x(t)| ≤ |∫ 1

0|G(t, s)|m(s)ds+ (a1(t)|x|λ1 + a2(t)|x′|λ2)

∫ 1

0|G(t, s)|ds

≤∫ 1

0|G(t, s)|m(s)ds+ (a1(t)|x|λ1 + a2(t)|x′|λ2)(

∫ t

0(|G1(t, s)) +G2(t, s)

+G3(t, s)|)ds+∫ η

t|G2(t, s) +G3(t, s)|ds+

∫ 1

η|G3(t, s)|ds)

=

∫ 1

0G(t, s)m(s)ds+ (a1(t)|x|λ1 + a2(t)|x′|λ2)(

tα

Γ(1 + α)+

δtηα−p

∆Γ(α− p+ 1)

+t

∆Γ(α− p+ 1))

≤∫ 1

0G(t, s)m(s)ds+ (k1|x|λ1 + k2|x′|λ2)(

tα

Γ(1 + α)

+δtηα−p

∆Γ(α− p+ 1)+

t

∆Γ(α− p+ 1)).

(2.2.9)

In view of (2.2.7), we have

|T ′x(t)| ≤ 1

Γ(α− 1)(

∫ t

0(t− s)α−2m(s)ds+ (k1|x|λ1 + k2|x′|λ2)

∫ t

0(t− s)α−2ds)

+1

∆(

δ

Γ(α− p)(

∫ η

0(η − s)α−p−1m(s)ds+ (k1|x|λ1 + k2|x′|λ2)

∫ η

0(η − s)α−p−1ds)

+1

Γ(α− p)(

∫ 1

0(1− s)α−p−1m(s)ds+ (k1|x|λ1 + k2|x′|λ2)

∫ 1

0(1− s)α−p−1ds))

=( 1

Γ(α− 1)

∫ t

0(t− s)α−2 +

1

∆(

δ

Γ(α− p)

∫ η

0(η − s)α−p−1

+1

Γ(α− p)

∫ 1

0(1− s)α−p−1)

)m(s)ds+ (

1

Γ(α)tα−1 +

1

∆

δηα−p + 1

Γ(α− p+ 1))(k1|x|λ1

+ k2|x′|λ2).

(2.2.10)

25

From the definition of the norm and (2.2.9), (2.2.10), we have:

∥T x(t)∥ ≤∫ 1

0G(t, s)m(s)ds+ (k1|x|λ1 + k2|x′|λ2)(

tα

Γ(1 + α)+

δtηα−p

∆Γ(α− p+ 1)

+t

∆Γ(α− p+ 1)) + (

1

Γ(α− 1)

∫ t

0(t− s)α−2 +

1

∆(

δ

Γ(α− p)

∫ η

0(η − s)α−p−1

+1

Γ(α− p)

∫ 1

0(1− s)α−p−1))m(s)ds+ (

1

Γ(α)tα−1 +

1

∆

δηα−p + 1

Γ(α− p+ 1))(k1|x|λ1

+ k2|x′|λ2)

≤ (

∫ 1

0G(t, s) +

1

Γ(α− 1)

∫ t

0(t− s)α−2 +

1

∆(

δ

Γ(α− p)

∫ η

0(η − s)α−p−1

+1

Γ(α− p)

∫ 1

0(1− s)α−p−1))m(s)ds+ (

tα

Γ(1 + α)+

(t+ 1)(δηα−p + 1)

∆Γ(α− p+ 1)

+1

Γ(α)tα−1)(k1|ρ|λ1 + k2|ρ|λ2).

(2.2.11)

Using (2.2.8) and (2.2.11), we have

∥T x(t)∥ ≤ h1 + (k1|ρ|λ1 + k2|ρ|λ2)h2 ≤ρ

3+ρ

3+ρ

3= ρ,

for h1 = supt∈[0,1] |h1(t)|, this implies that T B ⊂ B.

Now, we show that T is completely continuous operator. Let M = max{|f(t, x(t), x′(t))| : t ∈[0, 1], x ∈ B}, then

|T x(t)− T x(τ)| = |∫ 1

0G(t, s)f(s, x(s), x′(s))ds−

∫ 1

0G(τ, s)f(s, x(s), x′(s))ds|

≤M |∫ 1

0(G(t, s)−G(τ, s))ds|

=M |∫ t

0(t− s)α−1ds−

∫ τ

0(τ − s)α−1ds+

t− τ

∆Γ(α− p)(−δ

∫ η

0(η − s)α−p−1ds

+

∫ 1

0(1− s)α−p−1ds)| ≤M(

1

Γ(1 + α)(tα − τα) +

t− τ

∆Γ(α− p+ 1)(δηα−p + 1)),

(2.2.12)

and

|T ′x(t)− T ′x(τ)| = | − Iα−1(f(t, x(t), x′(t))− f(τ, x(τ), x′(τ))|

≤ M

Γ(α− 1)(

∫ t

0(t− s)α−2ds−

∫ τ

0(τ − s)α−2ds) =

M

Γ(α)(tα−1 − τα−1).

(2.2.13)

From the definition of the norm, (2.2.12) and (2.2.13), we have:

∥T x(t)− T x(τ)∥ ≤M(1

Γ(1 + α)(tα − τα) +

t− τ

∆Γ(α− p+ 1)(δηα−p + 1))

+M

Γ(α)(tα−1 − τα−1).

(2.2.14)

Since the functions tα−1, τα−1, tα, τα, are uniformly continuous on the interval [0,1], therefore, from

(2.2.14), it follows that T is equicontinuous and by Arzela-Ascoli theorem T is completely continuous.

Hence, by Schauder fixed point theorem, T has a fixed point x ∈ B.

26

Lemma 2.2.3. Under the assumption (A2), the BVP (2.2.1) has a solution.

Proof. The proof is similar to that of Theorem 2.2.2, so we exclude the proof.

Theorem 2.2.4. Assume that (A3) hold. If kϖ < 1, then the problem (2.2.1), has a unique solution.

Proof. In view of supposition (A3), we have the following estimates

|T x(t)− T y(t)| = 1

Γ(α)

∫ t

0(t− s)α−1|f(s, x(s), x′(s))− f(s, y(s), y′(s))|ds

+t

∆Γ(α− p)(δ

∫ η

0(η − s)α−p−1|f(s, x(s), x′(s))− f(s, y(s), y′(s))|ds

+

∫ 1

0(1− s)α−p−1|f(s, x(s), x′(s))− f(s, y(s), y′(s))|ds)

≤ 1

Γ(α+ 1)tα(k{|x− y|+ |x′ − y′|})) + t

∆Γ(α− p+ 1)(δηα−p−1(k(|x− y|

+ |x′ − y′|)) + (k{|x− y|+ |x′ − y′|}))

≤ k{|x− y|+ |x′ − y′|}( 1

Γ(α+ 1)tα +

t

∆Γ(α− p+ 1)(δηα−p−1 + 1))

= kϖ1∥x− y∥.

(2.2.15)

By the help of (2.2.7) we have the following estimates

|T ′x(t)− T ′y(t)| = | 1

Γ(α− 1)

∫ t

0(t− s)α−2|f(s, x(s), x′(s))− f(s, y(s), y′(s))|ds

+1

∆Γ(α− p)(−δ

∫ η

0(η − s)α−p−1|f(s, x(s), x′(s))− f(s, y(s), y′(s))|ds

+

∫ 1

0(1− s)α−p−1|f(s, x(s), x′(s))− f(s, y(s), y′(s))|)ds|

≤k{|x− y|+ |x′ − y′|}( tα−1

Γ(α)+

δηα−p + 1

∆Γ(α− p+ 1)) = kϖ2∥x− y∥.

(2.2.16)

Using the definition of the norm, (2.2.15) and (2.2.16), we have:

∥T x(t)− T y(t)∥ ≤ kϖ1∥x− y∥+ kϖ2∥x− y∥

= k(ϖ1 +ϖ2)∥x− y∥ = kϖ∥x− y∥.(2.2.17)

Thus, by contraction mapping principle, the BVP (2.2.1) has a unique solution.

Example 2.2.5. Consider the following FDE as a special case of the problem (2.2.1):

D32t x(t) =

(5+7|x(t)|+5|x′(t)|) cos(t)35 ,

x(0) = 0, D13x(1) = 1

10D13u(13).

(2.2.18)

For the unique solution of problem (2.2.18), we apply theorem (2.2.4) with f(t, x(t), x′(t))

= (5+7|x(t)|+5|x′(t)|) cos(t)35 , t ∈ [0, 1], α = 3

2 , p = 13 , δ = 1

10 , η = 13 . For x, x′, y, y′ ∈ R, we have

|f(t, x(t), x′(t))− f(t, y(t), y′(t))| ≤ 15{|x− y|+ |x′(t)− y′(t)|} which is condition (A4). By simple

calculations, we get ϖ = 2.4389 and kϖ = .4878 < 1. Thus by theorem (2.2.4) the problem (2.2.18)

has a unique solution.

27

In next section, we study further generalization of the problem (2.2.1).

2.3 Multi point BVPs for FDEs

In this section, we are concerned with a class of multi point boundary value problems (BVP) for FDE of

the type:

Dαx(t) = f(t, x(t),Dpx(t)), 2 < α < 3, 0 < p < 1,

x(0) = 0, Dpx(1) =

m−2∑i=1

ζiDpx(ηi), x′′(1) = 0, 0 < ηi, ζi ≤ 1, i = 0, 1, . . . ,m(2.3.1)

where Dα, Dp are Caputo’s fractional derivative of order α, p respectively.

Lemma 2.3.1. The BVP for fractional differential equation (2.3.1) is equivalent to the integral equation

x(t) =

∫ 1

0G(t, s)f(t, x(t),Dpx(t))dt, (2.3.2)

where

G(t, s) =

(t−s)α−1

Γ(α) + t∆(− 1

Γ(α−p)(1− s)α−p−1 + 1Γ(3−p)Γ(α−2)(1− s)α−3

+∑m−2

i=1ζi

Γ(α−p)(ηi − s)α−p−1 −∑m−2

i=1 ζi(ηi)2−p

Γ(3−p)Γ(α−p) (1− s)α−p−1)

− t2

2Γ(2−p)(1− s)α−3, s ≤ t,

t∆(− 1

Γ(α−p)(1− s)α−p−1 + 1Γ(3−p)Γ(α−2)(1− s)α−3

+∑m−2

i=1ζi(ηi−s)α−p−1

Γ(α−p) −∑m−2

i=1 ζi(η)2−p

Γ(3−p)Γ(α−p) (1− s)α−p−1)

− t2

2Γ(2−p)(1− s)α−3, t ≤ s.

(2.3.3)

Proof. Applying fractional integral of order α on the fractional differential equation (2.3.1), we have:

x(t) = Iαf(t, x(t),Dpx(t)) + c0 + c1t+ c2t2. (2.3.4)

Using the boundary conditions in (2.3.1), we obtain c0 = 0, c1 = 1∆(−Iα−pf(1) + 1

Γ(3−p)Iα−2f(1) +∑m−2

i=1 ζiIα−pf(ηi) −∑m−2

i=1ζi(ηi)

2−p

Γ(3−p) Iα−2f(1)), c2 = −Iα−2f(1)2 , where ∆ =

1−∑m−2

i=1 ζi(ηi)1−p

Γ(2−p) .

Hence (2.3.4) takes the form

x(t) =1

Γ(α)

∫ t

0(t− s)α−1f(s, x(s),Dpx(s))ds

+t

∆(

−1

Γ(α− p)

∫ 1

0(1− s)α−p−1f(s, x(s),Dpx(s))ds

+1

Γ(3− p)

1

Γ(α− 2)

∫ 1

0(1− s)α−3f(s, x(s),Dpx(s))ds

(2.3.5)

28

+1

Γ(α− p)

m−2∑i=1

ζi

∫ ηi

0(ηi − s)α−p−1f(s, x(s),Dpx(s))ds

− 1

Γ(3− p)Γ(α− 2)

m−2∑i=1

ζi(ηi)2−p

∫ 1

0(1− s)α−3f(s, x(s),Dpx(s))ds)

− t2

2Γ(α− 2)

∫ 1

0(1− s)α−3f(s, x(s),Dpx(s))ds.

For 0 < t < ηi, we have the following estimates

x(t) =

∫ t

0[

1

Γ(α)(t− s)α−1 +

t

∆(

−1

Γ(α− p)(1− s)α−p−1 +

(1− s)α−3

Γ(3− p)Γ(α− 2)

+

∑m−2i=1 ζi(ηi − s)α−p−1

Γ(α− p)− 1

Γ(3− p)Γ(α− 2)

m−2∑i=1

ζi(ηi)2−p(1− s)α−3)

− t2(1− s)α−3

2Γ(α− 2)]f(s, x(s),Dpx(s))ds+

∫ η1

t{ t∆(

−1

Γ(α− p)(1− s)α−p−1

+1

Γ(3− p)Γ(α− 2)(1− s)α−3 +

∑m−2i=1 ζi(ηi − s)α−p−1

Γ(α− p)

+1

Γ(3− p)Γ(α− 2)(1− s)α−3 +

∑m−2i=1 ζi(ηi − s)α−p−1

Γ(α− p)

−∑m−2

i=1 ζi(ηi)2−p

Γ(3− p)Γ(α− 2)(1− s)α−3)− t2(1− s)α−3

2Γ(α− 2)}f(s, x(s),Dpx(s))ds

+m−2∑i=2

∫ ηi

ηi−1

{ t∆(

−1

Γ(α− p)(1− s)α−p−1 +

1

Γ(3− p)Γ(α− 2)(1− s)α−3

+1

Γ(α− p)

m−2∑i=1

ζi(ηi − s)α−p−1 −∑m−2

i=1 ζi(ηi)2−p(1− s)α−3

Γ(3− p)Γ(α− p))

− t2(1− s)α−3

2Γ(α− 2)}f(s, x(s),Dpx(s))ds+

∫ 1

ηm−2

{ t∆(

−1

Γ(α− 2)(1− s)α−p−1

+(1− s)α−3

Γ(3− p)Γ(α− 2)−∑m−2

i=1 ζi(ηi)2−p(1− s)α−3)

Γ(3− p)Γ(α− 2)

− t2

2Γ(α− 2)(1− s)α−3}f(s, x(s),Dpx(s))ds =

∫ 1

0G(t, s)f(s, x(s),Dpx(s))ds.

(2.3.6)

For ηi−1 < t < ηi, η0 = 0, ηm−1 = 1 where i = 1, 2, ...,m− 1, we have

x(t) =

i−1∑k=1

∫ ηk

ηk−1

[1

Γ(α)(t− s)α−1 +

t

∆(

−1

Γ(α− p)(1− s)α−p−1

+(1− s)α−3

Γ(3− p)Γ(α− 2)+

∑m−2i=1 ζi(ηi − s)α−p−1

Γ(α− p)−∑m−2

i=1 ζi(ηi)2−p(1− s)α−3

Γ(3− p)Γ(α− 2))

− t2

2Γ(α− 2)(1− s)α−3]f(s, x(s),Dpx(s))ds+

∫ t

ηi−1

{∫ t

0[

1

Γ(α)(t− s)α−1

+t

∆(−(1− s)α−p−1

Γ(α− p)+

(1− s)α−3

Γ(3− p)Γ(α− 2)+

∑m−2i=1 ζi(ηi − s)α−p−1

Γ(α− p)

29

−∑m−2

i=1 ζi(ηi)2−p(1− s)α−3

Γ(3− p)Γ(α− 2))− t2

2Γ(α− 2)(1− s)α−3]}f(s, x(s),Dpx(s))ds

+

∫ ηi

t{ t∆(−(1− s)α−p−1

Γ(α− p)+

(1− s)α−3

Γ(3− p)Γ(α− 2)+

∑m−2i=1 ζi(ηi − s)α−p−1

Γ(α− p)

−∑m−2

i=1 ζi(ηi)2−p(1− s)α−3

Γ(3− p)Γ(α− 2))− t2

(1− s)α−3

2Γ(α− 2)}f(s, x(s),Dpx(s))ds

+

m−1∑k=i+1

∫ ηk

ηk−1

{ t∆(−(1− s)α−p−1

Γ(α− p)+

(1− s)α−3

Γ(3− p)Γ(α− 2)+

∑m−2i=1 ζi(ηi − s)α−p−1

Γ(α− p)

−∑m−2

i=1 ζi(ηi)2−p(1− s)α−3

Γ(3− p)Γ(α− 2))− t2

2Γ(α− 2)(1− s)α−3}f(s, x(s),Dpx(s))ds

=

∫ 1

0G(t, s)f(s, x(s),Dpx(s))ds.

2.3.1 Existence and uniqueness of solutions

Let I = [0, 1], and C(I) be the space of continuous functions defined on I . The space E = {x(t) ∈

C(I,R) : Dpx(t)∈C(I,R)} with the norm ∥x∥ = maxt∈I |x(t)|+maxt∈I |Dpx(t)|, is a Banach space

[23]. We assume the spermium value:

G∗ = supt∈I

∫ 1

0|G(t, s)|ds.

Lemma 2.3.2. Assume that the following hold:

(A1) f∈C([0, 1]×R×R,R),

(A2) there exist a constant k > 0 such that for each t ∈ I = [0, 1] and for all x, y ∈ R

|f(t, x,Dpx)− f(t, y,Dpy)|≤k{|x− y|+ |Dpx−Dpy|}, (2.3.7)

and

max{2kG∗, 2k{ 1

Γ(α− p+ 1)+

1

∆Γ(2− p)(

1

Γ(α− p+ 1)+

1

Γ(3− p)Γ(α− 1)

+

∑m−2i=1 ζi(ηi)

α−p

Γ(α− p+ 1)+


2−p

Γ(3− p)Γ(α− 1)) +

1

Γ(3− p)Γ(α− 1)}} = r < 1,

then, the BVP (2.3.1) has a unique solution.

Proof. Define an operator T : E → E , by:

T x(t) =∫ 1

0G(t, s)f(s, x(s),Dpx(s))ds. (2.3.8)

30

We show that, T is a contraction mapping. For this, we have:

|T x− T y| =∫ 1

0|G(t, s)||f(s, x,Dpx)− f(s, y,Dpy)|ds

≤ supt∈I

G(t, s)

∫ 1

0|f(s, x,Dpx)− f(s, y,Dpy)|ds

≤ G∗k{|x− y|+ |Dpx−Dpy|}

≤G∗k(maxt∈I

|x− y|+maxt∈I

|Dpx−Dpy|)=G∗k∥x− y∥.

(2.3.9)

From (2.3.5), we have

DpT x(t) =∫ t

0

(t− s)α−p−1

Γ(α− p)f(s, x(s),Dpx(s))ds

+t1−p

∆Γ(2− p)(

∫ 1

0

−(1− s)α−p−1

Γ(α− p)f(s, x(s),Dpx(s))ds

+

∫ 1

0

(1− s)α−3

Γ(3− p)Γ(α− 2)f(s, x(s),Dpx(s))ds

+

∑m−2i=1 ζi

Γ(α− p)

∫ ηi


−∑m−2

i=1 ζi(ηi)2−p

Γ(3− p)Γ(α− 2)

∫ 1


− t2−p

∫ 1

0

(1− s)α−3

Γ(3− p)Γ(α− 2)f(s, x(s),Dpx(s))ds,

(2.3.10)

which implies that

|DpT x−DpT y|

≤ 1

Γ(α− p)

∫ t

0(t− s)α−p−1|f(s, x(s),Dpx(s))− f(s, y(s),Dpy(s))|ds

+t1−p

∆Γ(2− p)(

1

Γ(α− p)

∫ 1

0(1− s)α−p−1|f(s, x(s),Dpx(s))− f(s, y(s),Dpy(s))|ds

+1

Γ(3− p)Γ(α− 2)

∫ 1

0(1− s)α−3|f(s, x(s),Dpx(s))− f(s, y(s),Dpy(s))|ds

+

∑m−2i=1 ζi

Γ(α− p)

∫ ηi

0(ηi − s)α−p−1|f(s, x(s),Dpx(s))− f(s, y(s),Dpy(s))|ds

+


2−p

Γ(3− p)Γ(α− 2)

∫ 1

0(1− s)α−3|f(s, x(s),Dpx(s))− f(s, y(s),Dpy(s))|ds)

+t2−p

Γ(3− p)

∫ 1

0(1− s)α−3|f(s, x(s),Dpx(s))− f(s, y(s),Dpy(s))|ds

≤ tα−p

Γ(α− p+ 1)k{|x− y|+ |Dpx(s)−Dpy(s)|}+ t1−p

∆Γ(2− p)(

1

Γ(α− p+ 1){|x− y|

+ |Dpx(s)−Dpy(s)|}ds+ 1

Γ(3− p)Γ(α− 1){|x− y|+ |Dpx(s)−Dpy(s)|}

(2.3.11)

31

+

∑m−2i=1 (ηi)

α−p

Γ(α− p+ 1){|x− y|+ |Dpx(s)−Dpy(s)|}+


2−p

Γ(3− p)Γ(α− 1){|x− y|

+ |Dpx(s)−Dpy(s)|}) + t2−p

Γ(3− p)

1

Γ(α− 1){|x− y|+ |Dpx(s)−Dpy(s)|}

≤ k{ 1

Γ(α− p+ 1)+

1

∆Γ(2− p)(

1

Γ(α− p+ 1)+

1

Γ(3− p)Γ(α− 1)

+


α−p

Γ(α− p+ 1)+


2−p

Γ(3− p)Γ(α− 1)) +

1

Γ(3− p)Γ(α− 1)}∥x− y∥.

From (2.3.9) and (2.3.11), we have the following estimates

∥T x− T y∥ = maxt∈I

|T x− T y|+maxt∈I

|DpT x−DpT y|

≤ {kG∗∥x− y∥+ k{ 1

Γ(α− p+ 1)+

1

∆Γ(2− p)(

1

Γ(α− p+ 1)

+1

Γ(3− p)Γ(α− 1)+


α−p

Γ(α− p+ 1)+


2−p

Γ(3− p)Γ(α− 1))

(2.3.12)

+1

Γ(3− p)Γ(α− 1)}∥x− y∥

≤ max{2kG∗, 2k{ 1

Γ(α− p+ 1)+

1

∆Γ(2− p)(

1

Γ(α− p+ 1)

+1

Γ(3− p)Γ(α− 1)+


α−p

Γ(α− p+ 1)+


2−p

Γ(3− p)Γ(α− 1))

+1

Γ(3− p)Γ(α− 1)}}∥x− y∥ = r∥x− y∥.

Thus, T is contraction mapping and hence by Theorem 1.3.5 the BVP (2.3.1), has a unique fixed point

x.

Theorem 2.3.3. Assume that the following hold

(A1). f∈C(I × R× R);(A2). There exist h = h(t)∈C(I,R) and a continuous non-decreasing function ϕ : [0,∞] → (0,∞)

such that

|f(t, x,Dpx)|≤hϕ|(Dpx)| ∀ t ∈ I, x, y ∈ R; (2.3.13)

(A3). There exists constant r > 0, such that

max{2G∗h∗ϕ|Dpx(t)|, 2h∗ϕ|Dpx(t)|{ 1

Γ(α− p+ 1)+

1

∆Γ(2− p)(

1

Γ(α− p+ 1)

+1

Γ(3− p)Γ(α− 1)+


α−p

Γ(α− p+ 1)+


2−p

Γ(3− p)Γ(α− 1)) +

1

Γ(3− p)Γ(α− 1)}}

= r < 1,

where h∗ = sup{h(t), t ∈ I}, then (2.3.1), has a solution x(t) such that |x| < r for t ∈ [0, 1].

Proof. Step 1: Let {xn(t)} be a convergent sequence such that {xn(t) → x(t)} as n → ∞, then it is

easy to show that T (t, xn,Dpxn) → T (t, x,Dpx). Hence, it follows that

|T (t, xn,Dpxn)− T (t, x,Dpx)| → 0 or T (t, xn,Dpxn) → T (t, x,Dpx) as n→ ∞.

32

Step 2: let B = {x(t) ∈ E : ∥x∥ ≤ r} be a bounded subset of E . We show that T (B) is also bounded.

For x(t) ∈ B, we have

|T x| ≤ |∫ 10 G(t, s)f(s, x(s),D

px(s))ds| ≤ G∗ ∫ 10 |f(s, x(s),Dpx(s))|ds ≤ G∗h∗ϕ|Dpx(t)|.

(2.3.14)

From equation (2.3.10), we have

|DpT (x)| = | 1

Γ(α− p)

∫ t

0(t− s)α−p−1f(s, x(s),Dpx(s))ds

+t1−p

∆Γ(2− p)(

−1

Γ(α− p)

∫ 1


+1

Γ(3− p)Γ(α− 2)

∫ 1


+1

Γ(α− p)

m−2∑i=1

ζi

∫ ηi


−∑m−2

i=1 ζi(ηi)2−p

Γ(3− p)Γ(α− 2)

∫ 1


− t2−p

Γ(3− p)Γ(α− 2)

∫ 1

0(1− s)α−3f(s, x(s),Dpx(s))ds|

≤h∗ϕ|Dpx(t)|{ 1

Γ(α− p+ 1)tα−pds+

t1−p

∆Γ(2− p)(

1

Γ(α− p+ 1)

+1

Γ(3− p)Γ(α− 1)+

∑m−2i=1 (ηi)

α−p

Γ(α− p+ 1)+


2−p

Γ(3− p)Γ(α− p+ 1))

+t2−p

Γ(3− p)

1

Γ(α− 1)}


Γ(α− p+ 1)+

1

∆Γ(2− p)(

1

Γ(α− 1)+

1

Γ(3− p)Γ(α− 1)

+


α−p

Γ(α− p+ 1)+


2−p

Γ(3− p)Γ(α− 1)) +

1

Γ(3− p)Γ(α− 1)}.

(2.3.15)

Using the definition of the norm and (2.3.10) and (2.3.15), we obtain

∥T (x)∥ = maxt∈I

|T (x)|+maxt∈I

|DpT (x)|

≤G∗h∗ϕ|Dpx(t)|+ h∗ϕ|Dpx(t)|{ 1

Γ(α− p+ 1)

+1

∆Γ(2− p)(

1

Γ(α− p+ 1)+

1

Γ(3− p)Γ(α− 1)+


α−p

Γ(α− p+ 1)

+


2−p

Γ(3− p)Γ(α− 1)) +

1

Γ(3− p)Γ(α− 1)}

≤max{2G∗h∗ϕ|Dpx(t)|, 2h∗ϕ|Dpx(t)|{ 1

Γ(α− p+ 1)

+1

∆Γ(2− p)(

1

Γ(α− p+ 1)+

1

Γ(3− p)Γ(α− 1)+


α−p

Γ(α− p+ 1)

+


2−p

Γ(3− p)Γ(α− 1)) +

1

Γ(3− p)Γ(α− 1)}} = r < 1,

33

which implies that T x ∈ B ∀ x ∈ B. Hence, T (B)⊆B.

Step 3: For equicontinuity, choose t1, t2 ∈ I such that t1 < t2 and x ∈ B, we have

|T x(t2)− T x(t1)| ≤∫ 1

0|G2(t, s)−G1(t, s)||f(s, x(s),Dpx(s))|ds

≤ h∗ϕ|Dpx(s)|∫ 1

0|G(t2, s)−G(t1, s)|ds.

This implies that

|T x(t2)− T x(t1)| → 0 as t1 → t2 . (2.3.16)

By (2.3.10) and the conditions (A1)− (A3), we have:

|DpT (x)(t2)−DpT (x)(t1)|

≤ | 1

Γ(α− p)(

∫ t2

0(t2 − s)α−p−1f(s)ds−

∫ t1

0(t1 − s)α−p−1f(s, x(s),Dpx(s))ds)

+t1−p2 − t1−p

1

∆Γ(2− p)(

−1

Γ(α− p)

∫ 1


+1

Γ(3− p)Γ(α− 2)

∫ 1


+1

Γ(α− p)

m−2∑i=1

ζi

∫ ηi


−∑m−2

i=1 ζi(ηi)2−p

Γ(3− p)Γ(α− 2)

∫ 1


− t2−p2 − t2−p

1

Γ(3− p)Γ(α− 2)

∫ 1

0(1− s)α−3f(s, x(s),Dpx(s))ds|


Γ(α− p+ 1)(tα−p

2 − tα−p1 ) +

t1−p2 − t1−p

1

∆Γ(2− p)(

1

Γ(α− p+ 1)

+1

Γ(3− p)Γ(α− 1)+

∑m−2i=1 (ηi)

α−p

Γ(α− p+ 1)+


2−p

Γ(3− p)Γ(α− 1))

+t2−p2 − t2−p

1

Γ(3− p)

1

Γ(α− 1)}.

This implies that

|DpT (x)(t2)−DpT (x)(t1)| → 0 as t1 → t2, (2.3.17)

Thus, by (2.3.16) and (2.3.17), we have

∥T x(t2)− T x(t1)∥ = maxt∈I

|T x(t2)− T x(t1)|+maxt∈I

|DpT (x)(t2)−DpT (x)(t1)|

≤ h∗ϕ|Dpx(s)|∫ 1

0|G(t2, s)−G(t1, s)|ds

+ h∗ϕ|Dpx(t)|{ tα−p2 − tα−p

1

Γ(α− p+ 1)+t1−p2 − t1−p

1

∆Γ(2− p)(

1

Γ(α− p+ 1)

+1

Γ(3− p)Γ(α− 1)+

∑m−2i=1 (ηi)

α−p

Γ(α− p+ 1)+


2−p

Γ(3− p)Γ(α− 1))

+(t2−p

2 − t2−p1 )

Γ(3− p)Γ(α− 1)}.

34

Therefore, ∥T x(t2)−T x(t1)∥ → 0 as t1 → t2. By Arzela Ascoli theorem, the operator T is completely

continuous and hence by Shauder fixed point theorem, T has a fixed point x ∈ B, which is a solution of

the BVP (2.3.1).

Example 2.3.4. Consider a special case of the FDE (2.3.1):

D5/2x(t) = f(t, x(t),D0.5x(t))

x(0) = 0, D0.5x(1) =

m−2∑i=1

ζiD0.5x(ηi) = 1/5, x′′(1) = 0.(2.3.18)

where, ζi = 1/2, ηi = 1/2, m = 3 and f(t, x(t),D0.5x(t)) = 140et(1+3|x(t)|+5|D0.5x(t)|) . It follows that

|f(t, x,Dpx)− f(t, y,Dpy)| ≤ 1

40et(

3|x| − 3|up|+ 5|y| − 5|Dpy|(1 + 3|x|+ 5|up|)(1 + 3|y|+ 5|Dpy|)

)

≤ 1

40(5|x−Dpx|+ 5|y −Dpy|) = 1

8(|x−Dpx|+ |y −Dpy|).

(2.3.19)

From (2.3.19), we have k = 18 , and by simple calculations, we obtain G∗ = 2.55, 2kG∗ = 0.64 < 1, and

2k{ 1

Γ(α− p+ 1)+

1

∆Γ(2− p)(

1

Γ(α− p+ 1)+

1

Γ(3− p)Γ(α− 1)

+1

Γ(α− p+ 1)

m−2∑i=1

ζi(ηi)α−p +

1

Γ(3− p)Γ(α− 1)

m−2∑i=1

ζi(ηi)2−p)

+1

Γ(3− p)Γ(α− 1)} < 0.4752 < 1.

(2.3.20)

Therefore, the conditions (A1) and (A2) are satisfied and hence by Lemma (2.3.2), the BVP (2.3.18),

has a unique solution.

Now we study more general class of BVP for FDEs

2.4 p–Laplacian BVP for FDEs

In this section, we study existence and uniqueness of solutions of the following p-Laplacian BVP for

FDEs

Dγ(ϕp(Dβx(t))) + a(t)f(x(t)) = 0, 3 < β, γ ≤ 4, t ∈ [0, 1],

x(0) = x′′′(0), ηDαx(t)|t=1 = x′(0), ξx′′(1)− x′′(0) = 0, 0 < α < 1,

ϕp(Dβx(t))|t=0 = 0 = (ϕp(Dβx(t)))′|t=0, (2.4.1)

(ϕp(Dβx(t)))′′|t=1 =1

2(ϕp(Dβx(t)))′′|t=0,

(ϕp(Dβx(t)))′′′|t=0 = 0,

where Dβ , Dγ and Dα are Caputo’s fractional derivative of fractional order β, γ, α, β, γ ∈ (3, 4],

α ∈ (0, 1) and ϕp(s) = |s|p−2s, p > 1. The p–Laplacian operator satisfies ϕ−1p = ϕq, 1

p + 1q = 1.

35

Lemma 2.4.1. Let f(t) ∈ C[0, 1], the BVP

Dβx(t) = f(t), 3 < β ≤ 4, (2.4.2)

x(0) = x′′′(0), ηDαx(1) = x′(0), ξx′′(1)− x′′(0) = 0, 0 < α < 1, (2.4.3)


x(t) =

∫ 1

0G(t, s)f(t)ds, (2.4.4)

where

G(t, s) =

tη1− η

Γ(2−α)

[(1−s)β−α−1

Γ(β−α) + ξ(1−ξ)Γ(3−α)

(1−s)β−3

Γ(β−2)

]+ t2ξ

2(1−ξ)(1−s)β−3

Γ(β−2) + 1Γ(β)(t− s)β−1, 0 < s ≤ t < 1,

tη1− η

Γ(2−α)

[(1−s)β−α−1

Γ(β−α) + ξ(1−ξ)Γ(3−α)

(1−s)β−3

Γ(β−2)

]+ t2ξ

2(1−ξ)(1−s)β−3

Γ(β−2) , 0 < t ≤ s < 1.

(2.4.5)

Proof. Applying the operator Iα on the differential equation in (2.4.2), we obtain

x(t) = Iβf(t) + c1 + c2t+ c3t2 + c4t

3. (2.4.6)

The boundary conditions x(0) = 0 = x′′′(0) yield to c1 = 0 = c4, and by using the boundary condition

ξx′′(1) − x′′(0) = 0, we have c3 = ξ2(1−ξ)I

β−2h(1). Using ηDαx(1) = x′(0), we obtain c2 =

η1− η

Γ(2−α)

[Iβ−αf(1) + ξ

(1−ξ)Γ(3−α)Iβ−2f(1)

]. Substituting the values of c1, c2, c3, c4, in (2.4.6), we

get

x(t) = Iβf(t) +tη

1− ηΓ(2−α)

[Iβ−αh(1) +

ξ

(1− ξ)Γ(3− α)Iβ−2f(1)

](2.4.7)

+t2ξ

2(1− ξ)Iβ−2f(1).

The integral form is given as

x(t) =

∫ t

0

(t− s)β−1f(s)ds

Γ(β)+

t2ξ

2(1− ξ)

∫ 1

0

(1− s)β−3h(s)ds

Γ(β − 2)

+tη

1− ηΓ(2−α)

[

∫ 1

0

(1− s)β−α−1f(s)ds

Γ(β − α)(2.4.8)

+ξ

(1− ξ)Γ(3− α)

∫ 1

0

(1− s)β−3f(s)ds

Γ(β − 2)] =

∫ 1

0G(t, s)f(s)ds.

Lemma 2.4.2. For 3 < β, γ ≤ 4, assume that a, x : (0, 1) → [0,+∞) and f : [0,+∞) → [0,+∞) are

continuous functions, the BVP with p–Laplacian operator

Dγ(ϕp(Dβx(t))) + a(t)f(x(t)) = 0, 3 < β, γ ≤ 4, 0 < α < 1 t ∈ [0, 1],

x(0) = x′′′(0), ηDαy(t)|t=1 = x′(0), ξx′′(1)− x′′(0) = 0,

ϕp(Dβx(t))|t=0 = 0 = (ϕp(Dβx(t)))′|t=0, (2.4.9)

(ϕp(Dβx(t)))′′|t=1 =1

2(ϕp(Dβx(t)))′′|t=0,

(ϕp(Dβx(t)))′′′|t=0 = 0,

36


x(t) =

∫ 1

0G(t, s)ϕq(

∫ 1

0H(s, z)a(z)f(x(z))dz)ds, (2.4.10)

where

H(t, s) =

− (t−x)γ−1

Γγ + t2

Γ(γ−2)(1− s)γ−3, 0 < x ≤ t < 1,

t2

Γ(γ−2)(1− s)γ−3, 0 < t ≤ x < 1,(2.4.11)

and G(t, s) is given by (2.4.5).

Proof. Applying integral Iγ on the differential equation in (2.4.9) and using lemma 9, we obtain

ϕp(Dβx(t)) = −Iγa(t)f(x(t)) + c1 + c2t+ c3t2 + c4t

3. (2.4.12)

The boundary conditions ϕp(Dβx(0)) = (ϕp(Dβx(0)))′ = (ϕp(Dαx(0)))′′′ = 0 lead to c1 = c2 =

c4 = 0. From (2.4.12) and c1 = c2 = c4 = 0, we deduce

(ϕp(Dβx(t)))′′ = −Iγ−2a(t)f(x(t)) + 2c3. (2.4.13)

The boundary condition (ϕp(Dβx(1)))′′ = 12(ϕp(D

βx(0)))′′, yields c3 = 1Γ(γ−2)

∫ 10 (1−s)

γ−3a(z)f(x(z))dz.

Consequently, (2.4.12) takes the form

ϕp(Dβx(t)) = − 1

Γ(γ)

∫ 1

0(t− s)γ−1a(s)f(x(s))ds

+t2

Γ(γ − 2)

∫ 1

0(1− s)γ−3a(s)f(x(s))ds (2.4.14)

=

∫ 1

0H(t, s)a(s)f(x(s))ds.

The BVP (2.4.9) reduces to the following problem

Dβx(t) = ϕq(

∫ 1

0H(t, s)a(s)f(x(s))ds) (2.4.15)

x(0) = x′′′(0), ηDαx(1) = x′(0), ξx′′(1)− x′′(0) = 0, 0 < α < 1,

which in view of Lemma 2.4.1, yields the required result, as under:

x(t) =

∫ 1

0G(t, s)ϕq(

∫ 1

0H(s, z)a(z)f(x(z))dz)ds. (2.4.16)

Lemma 2.4.3. Let 3 < γ ≤ 4. The Green’s function H(t, s) is a continuous function and satisfies

(A ) H(t, s) ≥ 0, H(t, s) ≤ H(1, s), for t, s ∈ (0, 1];

(B) H(t, s) ≥ tγ−1H(1, s) for t, s ∈ (0, 1].

37

Proof. The continuity of H(t, s) is ensured by its definition in (2.4.11). For (A), consider the case when

0 < s ≤ t ≤ 1, we have

H(t, s) = −(t− s)γ−1

Γ(γ)+

t2

Γ(γ − 2)(1− s)γ−3

= −tγ−1(1− s

t )γ−1

Γ(γ)+

t2

Γ(γ − 2)(1− s)γ−3 (2.4.17)

≥ − tγ−1(1− s)γ−1

Γ(γ)+

tγ−1

Γ(γ − 2)(1− s)γ−3

≥ tγ−1(1− s)γ−1

Γ(γ)Γ(γ − 2)[Γ(γ)− Γ(γ − 2)] > 0.

In case, when 0 < t ≤ s ≤ 1, the H(t, s) > 0 is obvious. Now, for H(t, s) ≤ H(1, s), for t, s ∈ (0, 1],

it follow that

∂

∂tH(t, s) =

− (t−s)γ−2

Γ(γ−1) + 2 tΓ(γ−2)(1− s)γ−3, 0 < s ≤ t ≤ 1,

2tΓ(γ−2)(1− s)γ−3, 0 < t ≤ s ≤ 1.

For x, t ∈ (0, 1], such that 0 < s ≤ t ≤ 1. From (2.4.18), we deduce

∂

∂tH(t, s) = −(t− s)γ−2

Γ(γ − 1)+

2t

Γ(γ − 2)(1− s)γ−3

= −tγ−2(1− s

t )γ−2

Γ(γ − 1)+

2t

Γ(γ − 2)(1− s)γ−3 (2.4.18)

≥ − tγ−2(1− s)γ−2

Γ(γ − 1)+

2tγ−2

Γ(γ − 2)(1− s)γ−3

≥ tγ−2(1− s)γ−2

Γ(γ − 1)Γ(γ − 2)[2Γ(γ − 1)− Γ(γ − 2)] ≥ 0,

from (2.4.18), we have ∂∂tH(t, s) ≥ 0. In case, when 0 < t ≤ s ≤ 1, (2.4.18), implies that ∂

∂tH(t, s) ≥ 0

is obvious which implies that H(t, s) is an increasing function w.r.t t. Hence H(t, s) ≤ H(1, s).

For part (B), consider the case, when 0 < x ≤ t ≤ 1, we proceed:

H(t, s)

H(1, s)=

−(t−x)γ−1

Γ(γ) + t2

Γ(γ−2)(1− s)γ−3

−(1−s)γ−1

Γ(γ) + 1Γ(γ−2)(1− s)γ−3

=

−tγ−1(1−xt)γ−1

Γ(γ) + t2

Γ(γ−2)(1− s)γ−3

−(1−s)γ−1

Γ(γ) + 1Γ(γ−2)(1− s)γ−3

(2.4.19)

≥−tγ−1(1−s)γ−1

Γ(γ) + tγ−1

Γ(γ−2)(1− s)γ−3

−(1−s)γ−1

Γ(γ) + 1Γ(γ−2)(1− s)γ−3

= tγ−1

−(1−s)γ−1

Γ(γ) + 1Γ(γ−2)(1− s)γ−3

−(1−s)γ−1

Γ(γ) + 1Γ(γ−2)(1− s)γ−3

= tγ−1.

38

The proof of the result for the case of 0 < t ≤ s ≤ 1 is similar, therefore, we omit it.

Assume that the following conditions are satisfied:

(A1) 0 <∫ 10 H(1, s)a(s)ds < +∞;

(A2) There exist 0 < δ < 1 and ρ > 0 such that f(t) ≤ δLϕp(t), for 0 ≤ t ≤ ρ,

where 0 < L ≤ (ϕp(ϖ)δ∫ 10 H(1, s)a(s)ds)−1;

(A3) There exist b > 0, such that f(t) ≤Mϕp(t), for x > b,

0 < M < (ϕp(ϖ2α−1)

∫ 1

0H(1, s)a(s)ds)−1;

(A4) f(t) is non-decreasing with respect to t;

(A5) There exist 0 ≤ β < 1 such that f(kt) ≥ (ϕp(k))βf(t), for any 0 ≤ k < 1, and 0 < t < +∞.


Theorem 2.4.4. Under the assumption (A1) and (A2), the BVP with p-Laplacian operator (2.4.1) has at

least one positive solution.

Proof. Consider a closed convex set K1 = {x(t) ∈ C[0, 1] : 0 ≤ x(t) ≤ ρ on [0, 1]} [128]. Define an

operator T : K1 → C[0, 1] by

T x(t) =∫ 1

0G(t, s)ϕq(

∫ 1

0H(s, z)a(z)f(x(z))dz)ds. (2.4.20)

By (2.4.2), x(t) is a solution of the BVP with p-Laplacian operator (2.4.1) if and only if x(t) is a fixed

point of T defined by (2.4.20). The compactness of the operator T can be easily shown. Consider,∫ 1

0G(t, s)ds =

∫ t0 (t− s)β−1ds

Γ(β)+

tη

1− ηΓ(2−α)

[

∫ 10 (1− s)β−α−1ds

Γ(β − α)

+ξ∫ 10 (1− s)β−3ds

(1− ξ)Γ(3− α)Γ(β − 2)] +

t2ξ

2(1− ξ)

∫ 10 (1− s)β−3ds

Γ(β − 2)(2.4.21)

=(t− s)β|0tΓ(1 + β)

+tη

1− ηΓ(2−α)

[(1− s)β−α|01Γ(β − α+ 1)

+ξ

(1− ξ)Γ(3− α)

(1− s)β−2|01Γ(β − 1)

] +t2ξ

2(1− ξ)

(1− s)β−2|01Γ(β − 1)

=tβ

Γ(1 + β)+

tη

1− ηΓ(2−α)

[1

Γ(β − α+ 1)

+ξ

(1− ξ)Γ(3− α)

1

Γ(β − 1)] +

t2ξ

2(1− ξ)

1

Γ(β − 1)

39

≤ 1

Γ(1 + β)+

η

1− ηΓ(2−α)

[1

Γ(β − α+ 1)

+ξ

(1− ξ)Γ(3− α)

1

Γ(β − 1)] +

ξ

2(1− ξ)

1

Γ(β − 1)= ϖ,

where ϖ = 1Γ(1+β) +

η1− η

Γ(2−α)

[1

Γ(β−α+1) +ξ

(1−ξ)Γ(3−α)1

Γ(β−1)

]+ ξ

2(1−ξ)1

Γ(β−1) . For any x ∈ K1,

using (A2) and (2.4.3), we obtain

T (x(t)) =

∫ 1

0G(t, s)ϕq(

∫ 1

0H(s, z)a(z)f(x(z)dz))ds

≤∫ 1

0G(t, s)ϕq(

∫ 1

0H(1, z)a(z)Lδϕp(ρ)dz)ds

≤ ϖϕq(

∫ 1

0H(1, z)a(z)Lδϕp(ρ)dz) (2.4.22)

= ϖϕq(Lδ

∫ 1

0H(1, z)a(z)dz)ρ ≤ ρ,

this implies that T (K1) ⊆ K1. Therefore, by Schauder’s fixed point theorem, T has a fixed point

x(t) ∈ K1.

Theorem 2.4.5. Under the assumptions (A1), (A3) the BVP with p-Laplacian operator (2.4.1) has at

least one positive solution.

Proof. Let b > 0 be as given in (A3). Define F∗ = max0≤x≤b f(x), then F∗ ≥ f(x) for 0 ≤ x ≤ b. In

view of (J3), we have

ϖ12α−1ϕq(M)ϕq(

∫ 1

0H(1, s)a(x)dx) < 1. (2.4.23)

Choose b∗ > b, large enough such that

ϖ12α−1(ϕq(F∗) + ϕq(M)b∗)ϕq(

∫ 1

0H(1, z)a(z))dz) < b∗. (2.4.24)

Define K2 = {x(t) ∈ C[0, 1] : 0 ≤ x(t) ≤ b∗ on [0, 1]}, S1 = {t ∈ [0, 1] : 0 ≤ x(t) ≤ b},

S2 = {t ∈ [0, 1] : b < x(t) ≤ b∗}, then S1 ∪ S2 = [0, 1] and S1 ∩ S2 = φ. For t ∈ K2, (A3) implies

f(x(t)) ≤Mϕp(x(t)) ≤Mϕp(b∗). Further, for t ∈ [0, 1], we have:

0 ≤ T (x(t)) =

∫ 1

0G(t, s)ϕq(

∫ 1

0H(s, z)a(z)f(x(z))dz)ds (2.4.25)

≤ ϖϕq(

∫S1

H(1, z)a(z)f(x(z))dz +

∫S2

H(1, z)a(z)f(x(z))dz)

≤ ϖϕq(F∗∫S1

H(1, z)a(z))dz +Mϕp(b∗)

∫S2

H(1, z)a(z))dz)

≤ ϖϕq(F∗ +Mϕp(b∗))ϕq(

∫ 1

0H(1, z)a(z))dz).

From (2.4.24) and using the inequality (a+ b)r ≤ 2r(ar + br) for any a, b, r > 0, (2.4.25) gets the form

0 ≤ T y(t) ≤ ϖ12α−1(ϕq(F∗) + ϕq(M)b∗)ϕq(

∫ 1

0H(1, z)a(z))dz) ≤ b∗, (2.4.26)

40

thus T (K2) ⊆ K2. Hence by Schauder’s fixed point theorem T has a fixed point x ∈ K2, which is the

positive solution of FDE with BVP and p–Laplacian operator (2.4.1).

Theorem 2.4.6. Assume that (A1), (A4), (A5) hold. Then the BVP with p-Laplacian operator (2.4.1)

has a unique positive solution.

Proof. Consider the set B = {x(t) ∈ C[0, 1] : y(t) ≥ 0 on [0, 1]}, where B is a normal solid cone in

C[0, 1] with B0 = {x(t) ∈ C[0, 1] : x(t) > 0 on [0, 1]}. Let T : B0 → C[0, 1] be defined by (2.4.20),

we prove that T is θ–concave and increasing operator. For x1, x2 ∈ B0 with x1 ≥ x2 the assumption

(A4) implies that the operator T is an increasing operator, i.e., T (x1(t)) ≥ T (x2(t)) on t ∈ [0, 1]. By

the help of f(kx) ≥ ϕp(kθ)f(x), we proceed

T (ky(t)) ≥∫ 1

0G(t, s)ϕq(

∫ 1

0H(t, x)ϕp(k

θ)a(z)f(x(z))dz)ds (2.4.27)

= kθ∫ 1

0G(t, s)ϕq(

∫ 1

0H(t, x)a(z)f(x(z))dz)ds = kθT (y(t)),

which implies that T is θ–concave operator and by the help of Lemma 2.4.4, the operator T has a unique

fixed point in B0, which is the unique positive solution of the FDE with BVP and p-Laplacian operator

(2.4.1).

Example 2.4.7. Consider the following BVP as a special case of the FDE with BVP and p–Laplacian

operator (2.4.1):

D3.5(ϕp(D3.5x(t))) + tx(t) = 0,

x(0) = 0 = x′′′(0),1

2D0.5x(t)|t=1 − x′(0) = 0,

1

2x′′(1)− x′′(0) = 0,

ϕp(D3.5x(t))|t=0 = 0 = (ϕp(D3.5x(t)))′|t=0, (2.4.28)

(ϕp(D3.5x(t)))′′|t=1 =1

2(ϕp(D3.5x(t)))′′, (ϕp(D3.5x(t)))′′|t=0 = 0,

where, α = β = 3.5, α = 0.5, ξ = γ = 12 , a(t) = t, f(x(t)) = x(t). By simple computation, we

obtain 0 < L ≤ 1.9092, choose L = 1.5, δ = 1, p = 2, then the conditions (A1), (A2) are satisfied.

Hence, by Theorem 2.4.4, the BVP with p-Laplacian operator (2.4.28) has at least one positive solution.

Example 2.4.8. Consider the following BVP as a special case of the FDE with BVP and p–Laplacian

operator (2.4.1):

D3.5(ϕp(D3.5x(t))) + t 3√x(t) = 0,

x(0) = 0 = x′′′(0), 0.1D0.5x(t)|t=1 − x′(0) = 0, 0.1x′′(1)− x′′(0) = 0,

ϕp(D3.5x(t))|t=0 = 0 = (ϕp(D3.5x(t)))′|t=0, (2.4.29)

(ϕp(D3.5x(t)))′′|t=1 =1

2(ϕp(D3.5x(t)))′′, (ϕp(D3.5x(t)))′′|t=0 = 0,

where, α = β = 3.5, ξ = η = 0.1, a(t) = t, f(x(t)) = 3√x(t). By simple computations, we get

M < 4.4792, choosing M = 4.00, b = 1 and p = q = 2, we see that (2.4.29) satisfy (A1) and (A3).

41

Hence by Theorem 2.4.5, the FDE with BVP and p-Laplacian operator (2.4.29) has at least one positive

solution.

Example 2.4.9. Consider the following BVP as a special case of the p–Laplacian BVP for FDE (2.4.1)

D3.5(ϕp(D3.5x(t))) + tx(t) = 0,

x(0) = 0 = x′′′(0),1

3D0.5x(t)|t=1 − x′(0) = 0,

1

3x′′(1)− x′′(0) = 0, (2.4.30)

ϕp(D3.5x(t))|t=0 = 0 = (ϕp(D3.5x(t)))′|t=0,

(ϕp(D3.5x(t)))′′|t=1 =1

2(ϕp(D3.5x(t)))′′, (ϕp(D3.5x(t)))′′|t=0 = 0.

Here, β = γ = 3.5, α = 0.5, ξ = γ = 13 , a(t) = t, f(x(t)) = x(t). It is clear that (2.4.30) satisfy

conditions (A1) and (A4). Also considering θ = 12 , (A5) is satisfied. Thus, by Theorem 2.4.6, BVP

(2.4.30) has a unique solution.

42

Chapter 3

Coupled systems of BVPs for fractionaldifferential equations

Existence of solutions for coupled systems of BVPs for FDEs have attracted the attention of researcher

quite recently. For example, Rehman and Khan [59], studied EPS for a coupled system of FDEs in the

Caputo’s sense. Li and Liu [60], discussed EUS for coupled system of FDEs by utilizing by the help of

some classical results. Ntouyas and Obaid [61], discussed a coupled system FDEs with integral BVP in

Caputo’s sense. Xu and Liu [101], developed iterative techniques for the existence and approximation

solutions for a coupled systems of two point BVPs for FDEs. Wei et al. [102], used monotone itera-

tive techniques for Initial value problems for FDEs, where the fractional differential operators were in

Riemann–Liouville Sense. Wang et al. [103], discussed the existence of solutions for FDEs by using

Monotone iterative method.

In this chapter, we study two types of coupled systems for existence and uniqueness of solutions.

In section 3.1, we study existence and uniqueness (EU) of maximal and minimal solutions of a general

class of coupled systems of BVPs and also provide error estimates. In section 3.2, we study existence

and uniqueness of solutions to coupled systems of BVPs of fractional order q–difference equations. The

results of this section are published [140]. The basic concept of the fractional q-difference calculus

are available in the research works of Al- Salam [44] and Agarwal [45]. Recently, q–difference FDEs

have studied by many scientists, for example, Ferreira [40], studied non-trivial solutions for fractional

q-difference boundary value problems. Ahmad et al. [42], studied nonlinear fractional q-difference

equations with nonlocal and sub-strip type boundary conditions. Ahmad and Ntouyas [41], considered

fractional q-difference hybrid equations and inclusions with Dirichlet boundary conditions. Yang [34],

studied positive solutions for BVPs involving nonlinear fractional q-difference equations. Zhao et al.

[35], studied q–difference equations with nonlocal q–integral boundary conditions. The detail study

43

about q–difference equations can be found in [36, 38, 39, 45–51, 53–55, 67].

3.1 Coupled systems of BVPs for FDEs

In this section, we develop monotone iterative technique for existence and uniqueness of minimal and

maximal solutions of the following coupled systems of BVPs

Dαx(t) = −G1(t, Iαx(t), Iβy(t)), α, β ∈ (3, 4]

Dβy(t) = −G2(t, Iαx(t), Iβy(t)),(3.1.1)

with conditions

Dδx(1) = 0 = I3−αx(0) = I4−αx(0), x(1) =Γ(α− δ)

Γ(α)Iα−δG1(t, Iαx(t), Iβy(t))(1),

Dνy(1) = 0 = I3−βy(0) = I4−βv(0), y(1) =Γ(β − ν)

Γ(β)Iβ−νG2(t, Iαx(t), Iβy(t))(1),

where t ∈ [0, 1], δ, ν ∈ (1, 2]. The functions G1, G2 : [0, 1] × R × R → R, satisfy the Caratheodory

conditions. The fractional derivatives Dα, Dβ, Dδ, Dν are in Riemann–Liouville sense and Iαx(t) =∫ t0


∫ t0

(t−s)β−1y(s)Γ(β) ds.

Let E = C[0, 1], with the norm ∥x∥ = maxt∈[0,1] |x(t)|, E ×E with the norm ∥(x, y)∥ = ∥x∥+∥y∥,

then obviously, E , E × E are Banach spaces. Let B be a cone in E and define partially order in E

as; for u, v ∈ E , v ≥ u if and only if v − u ∈ B. Further, define partially order in E × E as, for

(u, v), (x, y) ∈ E ×E , (u, v) ≥ (x, y) if and only if u ≥ x, v ≥ y. Then E , E ×E are partially ordered

Banach spaces. The following conditions are assumed:

(A1) G1, G2 : [0, 1]× R2 → R satisfy Caratheodory conditions;

(A2) (x0, y0) ∈ E × E , (x∗, y∗) ∈ E × E are lower and upper solutions respectively of the problem

(6.3.1), with x0 ≤ x∗, y0 ≤ y∗;

(A3) For any x, y, x1, y1 ∈ S with x ≥ x1, y ≥ y1, there exist nonnegative constants L1, L2, M1, M2

such that

0 ≤ G1(t, x(t), y(t))− G1(t, x1(t), y1(t)) ≤ L1(x− x1) + L2(y − y1),

0 ≤ G2(t, x(t), y(t))− G2(t, x1(t), y1(t)) ≤ N1(x− x1) +N2(y − y1).

Lemma 3.1.1. For s, t ∈ [0, 1], solutions of the coupled system (3.1.1) are solutions of the following

coupled systems of integral equations

x(t) =

∫ 1

0G1(t, s)G1(s, Iαx(s), Iβy(s))ds, (3.1.2)

y(t) =

∫ 1


44

where Iαx(t) =∫ t0


∫ t0

(t−s)β−1y(s)Γ(β) ds and Gi(t, s)(i = 1, 2) are Green’s

functions given by

G1(t, s) =

−(t−s)α−1

Γα + tα−1(1−s)α−δ−1

Γ(α) + tα−2(1−s)α−1

Γ(α) , t ≥ s,

tα−1(1−s)α−δ−1

Γ(α) + tα−2(1−s)α−1

Γ(α) , t ≤ s,(3.1.4)

and

G2(t, s) =

−(t−s)β−1

Γβ + tβ−1(1−s)β−ν−1

Γ(β) + tβ−2(1−s)β−1

Γ(β) , t ≥ s,

tα−1(1−s)β−ν−1

Γ(β) + tβ−2(1−s)β−1

Γ(β) , t ≤ s.(3.1.5)

Proof. Applying the operator Iα0 on (3.1.1), we get

x(t) = −IαG1(t, Iαx(t), Iβy(t)) + c1tα−1 + c2t

α−2 + c3tα−3 + c4t

α−4,

y(t) = −IβG1(t, Iαx(t), Iβy(t)) + d1tβ−1 + d2t

β−2 + d3tβ−3 + d4t

β−4.(3.1.6)

Initial conditions I3−αx(0) = I4−αx(0) = 0 = I3−βy(0) = I4−βy(0) in (3.1.1) implies that c3 =

c4 = d3 = d4 = 0. Applying conditions Dδx(1) = 0 = Dνy(1) on (3.1.6), we get

c1 =

∫ 1

0

(1− s)α−δ−1G1(s, Iαx(s), Iβy(s))ds

Γ(α), (3.1.7)

d1 =

∫ 1

0

(1− s)β−ν−1G2(s, Iαx(s), Iβy(s))ds

Γ(β). (3.1.8)

From conditions x(1) = Γ(α−δ)Γ(α) Iα−δG1(t, Iαx(t), Iβy(t))(t = 1), y(1) = Γ(β−ν)

Γ(β) Iβ−νG2 (t, Iαx(t),

Iβy(t))(t = 1), we deduce that

c2 =

∫ 10 (1− s)α−1G2(s, Iαx(s), Iβy(s))ds

Γ(α), (3.1.9)

d2 =

∫ 10 (1− s)β−1G2(s, Iαx(s), Iβy(s))ds

Γ(β). (3.1.10)

By substituting the values of c1, c2, c3, c4, d1, d2, d3, d4 in the coupled system (3.1.6), we have

x(t) = −∫ t

0

(t− s)α−1

Γ(α)G1(s, Iαx(s), Iβy(s))ds+ tα−1

∫ 1

0

(1− s)α−δ−1

Γ(α)G1(s, Iαx(s), Iβy(s))ds

+ tα−2

∫ 1

0

(1− s)α−1

Γ(α)G1(s, Iαx(s), Iβy(s))ds =

∫ 1

0G1(t, s)G1(s, Iαx(s), Iβy(s))ds,

(3.1.11)

and

y(t) =

∫ 1


where Gi(i = 1, 2) are the Green’s functions defined by (3.1.4), (3.1.5) respectively.

45

Lemma 3.1.2. For t, s ∈ [0, 1], α, β ∈ (3, 4], ν, δ ∈ (1, 2]. The Green’s functions defined by (3.1.4), (3.1.5)

satisfy the following:

(A1) Gi(t, s) ≥ 0, for i = 1, 2;

(A2)∫ 10 G1(t, s) ≤ 2

(α−δ)Γ(α) and∫ 10 G2(t, s) ≤ 2

(β−ν)Γ(β) .

Proof. ConsiderG1(t, s), for t, s ∈ [0, 1] such that t ≥ s. Since s ≤ st implies that −(1−s) ≤ −(1− s

t )

and α− δ − 1 < α− 1 implies that (1− s)α−δ−1 > (1− s)α−1, and

G1(t, s) =−(t− s)α−1

Γα+tα−1(1− s)α−δ−1

Γ(α)+tα−2(1− s)α−1

Γ(α)

=−tα−1(1− s

t )α−1

Γα+tα−1(1− s)α−δ−1

Γ(α)+tα−2(1− s)α−1

Γ(α)

(3.1.13)

≥ −tα−1(1− s)α−1

Γα+tα−1(1− s)α−δ−1

Γ(α)+tα−2(1− s)α−1

Γ(α)

=tα−1

Γ(α)((1− s)α−δ−1 − (1− s)α−1) +

tα−2(1− s)α−1

Γ(α)≥ 0.

From (3.1.4), for t ≤ s it is obvious that G1(t, s) ≥ 0. Similarly, we can show that G2(t, s) ≥ 0. This

completes the proof of (A1). For (A2), we consider s, t ∈ [0, 1], such that t ≥ s, we have:∫ 1

0G1(t, s)ds =

∫ 1

0(−(t− s)α−1

Γα+tα−1(1− s)α−δ−1

Γ(α)+tα−2(1− s)α−1

Γ(α))ds

=−(t− s)α

αΓα|01 +

tα−1(1− s)α−δ

(α− δ)Γ(α)|01 +

tα−2(1− s)α

αΓ(α)|01

= − tα

αΓ(α)+

tα−1

(α− δ)Γ(α)+

tα−2

αΓ(α)=

tα−2

α(α− δ)Γ(α)((t− t2)α+ α+ t2δ − δ)

≤ tα−2

α(α− δ)Γ(α)(tα+ α+ t2δ − δ) ≤ 1

α(α− δ)Γ(α)(α+ α+ δ − δ) =

2

(α− δ)Γ(α).

(3.1.14)

For s ≥ t, we have∫ 1

0G1(t, s)ds =

∫ 1

0(tα−1(1− s)α−δ−1

Γ(α)+tα−2(1− s)α−1

Γ(α))ds =

tα−1(1− s)α−δ

(α− δ)Γ(α)|01 +

tα−2(1− s)α

αΓ(α)|01

=tα−1

(α− δ)Γ(α)+

tα−2

αΓ(α)=

tα−2

α(α− δ)Γ(α)(tα+ α− δ) ≤ 1

α(α− δ)Γ(α)(2α− δ).

(3.1.15)

Since 1α(α−δ)Γ(α)(2α − δ) < 2

(α−δ)Γ(α) . Therefore, we have∫ 10 G1(t, s)ds ≤ 2

(α−δ)Γ(α) . Similarly, we

can show that∫ 10 G2(t, s)ds ≤ 2

(β−ν)Γ(β) .

46


In view of Lemma (3.1.1), the coupled systems (3.1.6) is equivalent to the following system of integral

equations:

x(t) =

∫ 1

0G1(t, s)G1(s, Iα

∫ 1

0G1(t, s)

G1(s, Iαx(s), Iβy(s))ds, Iβ

∫ 1

0G2(t, s)G2(s, Iαx(s), Iβy(s))ds)ds,

y(t) =

∫ 1

0G2(t, s)G1(s, Iα

∫ 1

0G1(t, s)


∫ 1

0G2(t, s)G2(s, Iαx(s), Iβy(s))ds)ds.

(3.1.16)

Define an operator T : B × B → E × E by

T (x(t), y(t)) = (T1(x(t), y(t)), T2(x(t), y(t))), (3.1.17)

where,

T1(x(t), y(t)) =∫ 1

0G1(t, s)G1(s, Iα

∫ 1

0G1(t, s)


∫ 1


(3.1.18)

T2(x(t), y(t)) =∫ 1

0G2(t, s)G2(s, Iα

∫ 1

0G1(t, s)


∫ 1

0G2(t, s)G2(s, Iαx(s), Iβy(s))ds)ds.

(3.1.19)

It is clear that the fixed points of the operator T are solutions of the system (3.1.1).

Now, we define the lower and upper solutions for the operator equation

T (x, y) = (x, y). (3.1.20)

We say a lower solution to the pair (x0, y0) of the operator equation (3.1.20), if (I − T )(x0, y0) ≤ 0,

and (x∗, y∗) be the upper solution of the operator equation (3.1.20) if (I − T )(x∗, y∗) ≥ 0.

The following notations are defined for the simplification in calculations:

M1 =2

(α− δ)Γ(α)(

L1

Γ(1 + α)

2

(α− δ)Γ(α)+

L2

Γ(1 + β)

2

(β − ν)Γ(β))(

N1

Γ(1 + α)+

N2

Γ(1 + β)),

(3.1.21)

M2 =2

(β − ν)Γ(β)(

L1

Γ(1 + α)

2

(α− δ)Γ(α)+

L2

Γ(1 + β)

2

(β − ν)Γ(β))(

N1

Γ(1 + α)+

N2

Γ(1 + β)),

(3.1.22)

and M = M1 +M2.

47

Theorem 3.1.3. Assume that the conditions (A1) and (A2) hold and M < 1, then the system (3.1.1) has

a minimal solution (x∗, y∗) and maximal solution (x∗∗, y∗∗). Assume the initial iterations x0, x∗0 and

form the following iterative sequence for each n ∈ Z,

xn(t) =

∫ 1

0G1(t, s)G1(s, Iα

∫ 1

0G1(t, s)

G1(s, Iαxn−1(s), Iβyn−1(s))ds, Iβ

∫ 1

0G2(t, s)G2(s, Iαxn−1(s), Iβyn−1(s))ds),

(3.1.23)

x∗n(t) =

∫ 1

0G1(t, s)G1(s, Iα

∫ 1

0G1(t, s)

G1(s, Iαx∗n−1(s), Iβy∗n−1(s))ds, Iβ

∫ 1

0G2(t, s)G2(s, Iαx∗n−1(s), Iβy∗n−1(s))ds),

(3.1.24)

yn(t) =

∫ 1

0G2(t, s)G2(s, Iα

∫ 1

0G1(t, s)

G1(s, Iαn−1(s), Iβyn−1(s))ds, Iβ

∫ 1

0G2(t, s)G2(s, Iαxn−1(s), Iβyn−1(s))ds),

(3.1.25)

y∗n(t) =

∫ 1

0G2(t, s)G2(s, Iα

∫ 1

0G1(t, s)

G1(s, Iαx∗n−1(s), Iβy∗n−1(s))ds, Iβ

∫ 1

0G2(t, s)G2(s, Iαx∗n−1(s), Iβy∗n−1(s))ds),

(3.1.26)

and

x0 ≤ x1 ≤ . . . ≤ xn ≤ . . . ≤ x∗n ≤ . . . ≤ x∗1 ≤ x∗0, (3.1.27)

y0 ≤ y1 ≤ . . . ≤ yn ≤ . . . ≤ y∗n ≤ . . . ≤ y∗1 ≤ y∗0, (3.1.28)

where

x∗ = limn→∞

xn(t), y∗ = limn→∞

yn(t), y∗ =

∫ 1

0G2(t, s)G2(s, Iαx∗(s), Iβy∗(s))ds, (3.1.29)

x∗∗ = limn→∞

x∗n(t), y∗∗ = lim

n→∞y∗n(t), y

∗∗ =

∫ 1

0G2(t, s)G2(s, Iαx∗∗(s), Iβy∗∗(s))ds, (3.1.30)

and errors estimations for the minimal, maximal solutions are given by

∥x∗(t)− xn(t)∥+ ∥y∗(t)− yn(t)∥ ≤ Mn

1−M{∥x1(t)− x0(t)∥+ ∥y1(t)− y0(t)∥},

∥x∗(t)− x∗∗n (t)∥+ ∥y∗(t)− y∗∗n (t)∥ ≤ Mn

1−M{∥x∗0(t)− x∗1(t)∥+ ∥y∗0(t)− y∗1(t)∥}.

Proof. Assume (x0, y0) ∈ E × E as the lower solution of (3.1.1), then we have:

x0(t) ≤∫ 1

0G1(t, s)G1(s, Iαx(s), Iβy(s))ds, y0(t) ≤

∫ 1


(3.1.31)

this leads us to

Iαx0(t) ≤ Iα

∫ 1


Iβy0(t) ≤ Iβ

∫ 1

0G2(t, s)G2(s, Iαx(s), Iβy(s))ds.

(3.1.32)

48

Thus

x0(t) ≤∫ 1

0G1(t, s)G1(s, Iαx(s), Iβy(s))ds

≤∫ 1

0G1(t, s)G1(s, Iα

∫ 1

0G1(t, s)


∫ 1


= T1(x, y),

(3.1.33)

and

y0(t) ≤∫ 1

0G2(t, s)G1(s, Iαx(s), Iβy(s))ds

≤∫ 1

0G2(t, s)G1(s, Iα

∫ 1

0G1(t, s)


∫ 1


= T2(x, y).

(3.1.34)

Therefore, we have x0 lower solution of T1. From (A3),we have that for any x, y ∈ B such that

x0 ≤ x, y0 ≤ y, we get T1(x0, y0) ≤ T1(x, y), i.e., T1 is an increasing operator. Also from (3.1.23), for

any n ∈ Z+, we have

xn(t) = T1(xn−1(t), yn−1(t), yn(t) = T2(xn−1(t), yn−1(t)). (3.1.35)

By the help of assumption A2, we have:

x0 ≤ x1 ≤ . . . ≤ xn, y0 ≤ y1 ≤ . . . ≤ yn. (3.1.36)

For any x, y, z, p ∈ B such that x ≤ y, z ≤ p, it is obvious Iαx ≤ Iαy, Iβz ≤ Iβp and using C3 we

get G1(t, Iαx, Iβz) ≤ G1(t, Iαy, Iβp), G2(t, Iαx, Iβz) ≤ G2(t, Iαy, Iβp),∫ 1

0G1(t, s)G1(s, Iαx(s), Iβz(s))ds ≤

∫ 1

0G1(t, s)G1(s, Iαy(s), Iβp(s))ds,

Iα

∫ 1

0G1(t, s)G1(s, Iαx(s), Iβz(s))ds ≤ Iα

∫ 1


(3.1.37)

and ∫ 1

0G2(t, s)G2(s, Iαx(s), Iβz(s))ds ≤

∫ 1


Iβ

∫ 1

0G2(t, s)G2(s, Iαx(s), Iβz(s))ds ≤ Iβ

∫ 1

0G2(t, s)G2(s, Iαy(s), Iβp(s))ds.

(3.1.38)

49

It follows from (A3), (A2), that

0 ≤ G1(t, Iα

∫ 1

0K1(s, ξ)G1(ξ, Iαy, Iβp)dξ, Iβ

∫ 1

0K2(s, ξ)G2(ξ, Iαy, Iβp)dξ)

− G1(t, Iα

∫ 1

0K1(s, ξ)G1(ξ, Iαx, Iβz)dξ, Iβ

∫ 1

0K2(s, ξ)G2(ξ, Iαx, Iβz)dξ)

≤ L1

Γ(α)

∫ t

0(t− s)α−1

∫ 1

0K1(s, ξ)(N1(Iαy − Iαx) +N2(Iβp− Iβz))dξds

+L2

Γ(β)

∫ t

0(t− s)β−1

∫ 1

0K1(s, ξ)(N1(Iαy − Iαx) +N2(Iβp− Iβz))dξds.

(3.1.39)

Consequently, we have

∥G1(t, Iα

∫ 1

0K1(s, ξ)G1(ξ, Iαy, Iβp)dξ, Iβ

∫ 1

0K2(s, ξ)G2(ξ, Iαy, Iβp)dξ)

− G1(t, Iα

∫ 1

0K1(s, ξ)G1(ξ, Iαx, Iβz)dξ, Iβ

∫ 1

0K2(s, ξ)G2(ξ, Iαx, Iβz)dξ)∥

≤ L1

Γ(1 + α)

2

(α− δ)Γ(α)(

N1

Γ(1 + α)∥y − x∥+ N2

Γ(1 + β)∥p− z∥)

+L2

Γ(1 + β)

2

(β − ν)Γ(β)(

N1

Γ(1 + α)∥y − x∥+ N2

Γ(1 + β)∥p− z∥).

≤ (L1

Γ(1 + α)

2

(α− δ)Γ(α)+

L2

Γ(1 + β)

2

(β − ν)Γ(β))(

N1

Γ(1 + α)

+N2

Γ(1 + β))(∥y − x∥+ ∥p− z∥).

(3.1.40)

Using (3.1.21), (3.1.40), we have

∥T1(y(t), p(t))− T1(x(t), y(t))∥ ≤ M1(∥y − x∥+ ∥p− z∥). (3.1.41)

Similarly, we can have

∥T2(y(t), p(t))− T2(x(t), y(t))∥ ≤ M2(∥y − x∥+ ∥p− z∥). (3.1.42)

By the help of (3.1.41), (3.1.42), we deduce

∥T (y(t), p(t))− T (x(t), y(t))∥ ≤ M(∥y − x∥+ ∥p− z∥). (3.1.43)

Using (3.1.23), (3.1.25) and (3.1.43), we can have that

∥x2 − x1∥+ ∥y2 − v1∥ = ∥T (x1, y1)− T (x0, x0)∥ ≤ M(∥x1 − x0∥+ ∥y1 − y0∥). (3.1.44)

50

Similarly, we can get

∥x3 − x2∥+ ∥y3 − y2∥ = ∥T (x2, y2)− T (x1, y1)∥ ≤ M(∥x2 − x1∥+ ∥y2 − y1∥)

≤ M2(∥x1 − x0∥+ ∥y1 − y0∥),

∥x4 − x3∥+ ∥y4 − y3∥ ≤ M3(∥x1 − x0∥+ ∥y1 − y0∥),

∥x5 − x4∥+ ∥y5 − y4∥ ≤ M4(∥x1 − x0∥+ ∥y1 − y0∥),

∥x6 − x5∥+ ∥y6 − y5∥ ≤ M5(∥x1 − x0∥+ ∥y1 − y0∥),...

......

∥xn+1 − xn∥+ ∥yn+1 − yn∥ ≤ Mn(∥x1 − x0∥+ ∥y1 − y0∥).

(3.1.45)

For n, p ∈ Z+, we have that

∥xn+p − xn∥+ ∥yn+p − yn∥

≤ ∥xn+p − xn+p−1∥+ ∥yn+p − vn+p−1∥+ ∥xn+p−1 − xn+p−2∥

+ ∥yn+p−1 − yn+p−2∥+ . . .+ ∥xn+1 − x0n∥+ ∥yn+1 − yn∥

≤ Mn+p−1(∥x1 − x0∥+ ∥y1 − y0∥) +Mn+p−2(∥x1 − x0∥+ ∥y1 − y0∥) + . . .

+Mn(∥x1 − x0∥+ ∥y1 − y0∥)

= Mn 1−Mp

1−M(∥x1 − x0∥+ ∥y1 − y0∥).

(3.1.46)

For M ∈ (0, 1), we have that ∥xn+p − xn∥ + ∥yn+p − yn∥ → 0 as n → +∞. This implies that

∥xn+p − xn∥ → 0, ∥yn+p − yn∥ → 0 as n → +∞. Thus an = {xn}, bn = {yn} are Cauchy

sequences in B. Let x∗(t) = limn→∞ xn(t), y∗(t) = limn→∞ xn(t) thus T1(x∗(t), y∗(t)) = x∗ and

T2(x∗(t), y∗(t)) = y∗. From (3.1.46) we have, p → +∞, implies that ∥x∗ − xn∥ + ∥y∗ − yn∥ ≤Mn

1−M(∥x1 − x0∥+ ∥y1 − y0∥). Since the operators T1, T2 are increasing operators, therefore, we have

the sequence x0 ≤ x1 ≤ . . . ≤ xn ≤ . . . ≤ x∗n ≤ . . . ≤ x∗2 ≤ x∗1 ≤ x∗0.

Thus, the system (3.1.1) has a pair of solution (x∗(t), y∗(t)) = T (x∗(t), y∗(t)). Similarly, we can

show for the upper solution of the system (3.1.1), i.e., x∗∗ = limn→+∞ x∗n, y∗∗(t) = limn→+∞ x∗n. That

is, p→ +∞ implies ∥x∗n(t)−x∗∗(t)∥+ ∥y∗n(t)− y∗∗(t)∥ ≤ Mn

1−M(∥x1 −x0∥+ ∥y1− y0∥). Therefore,

(x∗, y∗) is a minimal solution and (x∗∗, y∗∗) is the maximal solution of the CSFDIEs (3.1.1).

Lemma 3.1.4. Under the assumptions C1 − C3 and M < 1. The minimal and maximal solutions of the

system (3.1.1) are unique.

Proof. Let us assume that (x∗, y∗) , (x∗∗, y∗∗) be the minimal, maximal solutions respectively for the

operator equation (3.1.16). Let there exist another pair of minimal solution(µ0, λ0) ∈ B × B. Then

µ0 ≤ T1(µ0, λ0) = µ1 and λ0 ≤ T2(µ0, λ0) = λ1. Consequently, we are having the sequence µ0 ≤µ1 ≤ . . . ≤ µn and λ0 ≤ λ1 ≤ . . . ≤ λn such that limn→∞ µn = µ∗ and limn→∞ λn = λ∗. Where

∥µ∗−xn∥+∥λ∗−yn∥ ≤ Mn

1−M(∥µ0−x0∥+∥λ0−y0∥). As n→ +∞ then ∥µ∗−x∗∥+∥λ∗−y∗∥ → 0, i.e.,

∥µ∗ − x∗∥ → 0, ∥λ∗ − y∗∥ → 0 consequently, we have µ∗ → x∗ and λ∗ → y∗. Therefore the minimal

51

solution (x∗, y∗) of the operator equation (3.1.16) is unique. Similarly, we can show the uniqueness of

the maximal solution (x∗∗, y∗∗).

Example 3.1.5. Consider the following system of coupled FDIEs as a special case of the coupled system

(3.1.1), for the EU of minimal and maximal solutions:

D3.5x(t) = −sin2(t)I3.5x(t) + cos2(t)I3.5y(t)

50 + t,

D3.5y(t) = −sin2(t)I3.5x(t) + cos2(t)I3.5y(t)

70 + t,

D1.3x(1) = 0 = I−0.5x(0) = I2.2x(0), x(1) =Γ(2.2)

Γ(3.5)I2.2G1(t, I3.5x(t), I3.5y(t))(1),

D1.3y(1) = 0 = I−0.5y(0) = I2.2y(0), y(1) =Γ(2.2)

Γ(3.5)I2.2G2(t, I3.5x(t), I3.5y(t))(1).

(3.1.47)

From (3.1.47), we have that β = α = 3.5, δ = ν = 1.3. G1(t, Iαx(t), Iβy(t)) = sin2(t)I3.5x(t)+cos2(t)I3.5y(t)50+t ,

G2(t, Iαx(t), Iβy(t)) = sin2(t)I3.5x(t)+cos2(t)I3.5y(t)70+t . For any (u, v), (x, y) ∈ E × E , such that (u, v) ≥

(x, y), we have

0 ≤ G1(t, Iαu(t), Iβv(t))− G1(t, Iαx(t), Iβy(t))

≤ 1

50(Iαu(t)− Iαx(t)) +

1

50(Iβv(t)− Iβy(t)),

0 ≤ G2(t, Iαu(t), Iβv(t))− G2(t, Iαx(t), Iβy(t))

≤ 1

70(Iαu(t)− Iαx(t)) +

1

70(Iβv(t)− Iβy(t)).

(3.1.48)

From (3.1.48), we have L1 = L2 = 150 , M1 = M2 = 1

70 . From these values, one can easily calculate

M = 1.26414 × 10−6 < 1. Therefore, by 3.1.4, there exist unique minimal and maximal solutions of

the problem (3.1.47).

Example 3.1.6. Consider the following system of FDIEs for error estimations:

D3.5x(t) = −sin2(t)I3.5x(t) + cos2(t)I3.5y(t)

50 + t,

D3.5y(t) = −sin2(t)I3.5x(t) + cos2(t)I3.5y(t)

70 + t,

D1.3x(1) = 0 = I−0.5x(0) = I2.2x(0), x(1) =Γ(2.2)

Γ(3.5)I2.2G1(t, I3.5x(t), I3.5y(t))(1),

D1.3y(1) = 0 = I−0.5y(0) = I2.2y(0), y(1) =Γ(2.2)

Γ(3.5)I2.2G2(t, I3.5x(t), I3.5y(t))(1).

(3.1.49)

From (3.1.49), we have that β = α = 3.5, δ = ν = 1.3, M = 1.26414× 10−6. We consider the lower

solution (−0.5,−0.5) and the upper solution (0.5, 0.5). For n = 1, we deduce that

∥x∗(t)− xn(t)∥+ ∥y∗(t)− yn(t)∥ ≤ Mn

1−M{∥x1(t)− x0(t)∥+ ∥y1(t)− y0(t)∥}

≤ 2.52828× 10−6,

∥x∗(t)− x∗∗n (t)∥+ ∥y∗(t)− y∗∗n (t)∥ ≤ Mn

1−M{∥x∗0(t)− x∗1(t)∥+ ∥y∗0(t)− y∗1(t)∥}

≤ 2.52828× 10−6.

52

The errors can further be decreased by increasing the iteration values.

3.2 Coupled systems of BVPs for fractional order q–difference equations

We study existence and uniqueness of solutions for a coupled systems of fractional order q-difference

equations of the form:

Dαq x(t) = −f(t,Dk

q y(t),Dp1q y(t)), γ ∈ (n− 1, n], k ≤ n− 1,

Dβq y(t) = −g(t,Dk

qx(t),Dp2q x(t)), β ∈ (n− 1, n], k ≤ n− 1,

(3.2.1)

with initial and boundary conditions

x(1) = 0, Dqx(1) = 0, Diqx(0) = 0, i = 2, 3, ..., n− 1,

y(1) = 0, Dqy(1) = 0, Djqy(0) = 0, j = 2, 3, ..., n− 1,

for t ∈ [0, 1] and n ≥ 3. Dγq , Dβ

q , Dp1q , Dp2

q , are Caputo’s fractional q–difference operators of order

α, β, p1, p2 where p1, p2 ∈ (0, 1) and Dkq , Di

q, Djq are q–difference operators of order k ≤ n−1, i =

2, 3, . . . , n− 1, j = 2, 3, . . . , n− 1, respectively; and f, g ∈ C([0, 1]× R2,R).

Lemma 3.2.1. For a given h(t) ∈ C([0, 1],R), the BVP

Dαq x(t) = −h(t), n− 1 < α ≤ n, n ≥ 3, t ∈ [0, 1] (3.2.2)

x(1) = 0, Dqx(1) = 0, Diqx(0) = 0, i = 2, 3, ..., n− 1, (3.2.3)

has a solution x(t) of the form

x(t) =−1

Γq(α)

∫ t

0(t− qs)α−1h(s)dqs+

1

Γq(α)

∫ 1

0(1− qs)α−1h(s)dqs

− 1

Γq(α− 1)

∫ 1

0(1− qs)α−2h(s)dqs+

t

Γq(α− 1)

∫ 1

0(1− qs)α−2h(s)dqs.

(3.2.4)

Proof. Applying fractional q integral Iαq on (3.2.2) and using lemma 1.1.7, we obtain

x(t) = −Iαq h(t) + a0 + a1t+ a2t

2 + ...+ an−1tn−1. (3.2.5)

Using the given conditions (3.2.3), we get

a0 = Iαq h(t)|t=1 − Iα−1

q h(t)|t=1, a1 = Iα−1q h(t)|t=1, a2 = a3 = ... = an−1 = 0.

Thus, (3.2.5) takes the form

x(t) = −Iαq h(t) + Iα

q h(t)|t=1 − Iα−1q h(t)|t=1 + tIα−1

q h(t)|t=1. (3.2.6)

53

The q-fractional integral form of (3.2.6) is

x(t) =−1

Γq(α)

∫ t


1

Γq(α)

∫ 1


− 1

Γq(α− 1)

∫ 1


t

Γq(α− 1)

∫ 1

0(1− qs)α−2h(s)dqs.

(3.2.7)

Conversely, applying fractional q–differential operator Dαq on (3.2.7) and using the basic property

Dαq Iα

q h(t)(t) = h(t), we have

Dαq x(t) = Dα

q

(− 1

Γq(α)

∫ t


1

Γq(α)

∫ 1


− 1

Γq(α− 1)

∫ 1


t

Γq(α− 1)

∫ 1

0(1− qs)α−2h(s)dqs)

= −Dαq Iα

q h(t) = −h(t),

and the boundary conditions (3.2.3) are also true, which can be easily verified from (3.2.7).

Consider E = {x(t) : Dpqx(t) ∈ C and x(t) ∈ C(n)}, with the norm, ∥x∥ = maxk≤n−1

maxt∈[0,1] |Dkqx(t)|+maxt∈[0,1] |D

pqx(t)|, where k ∈ N, p ∈ (0, 1); the space E × E equipped with the

norm, ∥(x, y)∥ = ∥x∥+ ∥y∥. Obviously, E , E ×E are Banach spaces. Let us define T : E ×E → E ×E

by

T (x, y)(t) = (T1(y), T2(x))(t),

where,

T1(y(t)) =−1

Γq(α)

∫ t

0(t− qs)α−1f(s,Dk

q y(s),Dp1q y(s))dqs

+1

Γq(α)

∫ 1

0(1− qs)α−1f(s,Dk


− 1

Γq(α− 1)

∫ 1



+t

Γq(α− 1)

∫ 1


q y(s),Dp1q y(s))dqs,

(3.2.8)

and

T2(x(t)) =−1

Γq(β)

∫ t

0(t− qs)β−1g(s,Dk

qx(s),Dp2q x(s))dqs

+1

Γq(β)

∫ 1

0(1− qs)β−1g(s,Dk

qx(s),Dp2q x(s))dqs

− 1

Γq(β − 1)

∫ 1


qx(s),Dp2q x(s))dqs

+t

Γq(β − 1)

∫ 1


qx(s),Dp2q x(s))dqs.

(3.2.9)

54

We define the following terms for our convenience:

ϖ1 =1

Γq(α− k + 1)+

1

Γq(α− p1)+

1

Γ(2− p1)Γq(α), (3.2.10)

ϖ2 =1

Γq(β − k + 1)+

1

Γq(β − p2)+

1

Γ(2− p2)Γq(β), (3.2.11)

and

M = min{1− (k1ϖ1 + λ1ϖ2), 1− (λ2ϖ2 +ϖ1k2)}. (3.2.12)

3.2.1 Existence of solutions

Theorem 3.2.2. For f, g ∈ C( [0, 1]×R2,R), let there exist real constantsm1, m2, k1, k2, λ1, λ2 such

that for all u1, v1 ∈ R, we have

|f(t,Dkq y(t),Dp1

q y(t))| ≤ m1 + k1|Dkq y(t)|+ k2|Dp1

q y(t)|,

|g(t,Dkqx(t),Dp2

q x(t))| ≤ m2 + λ1|Dkqx(t)|+ λ2|Dp2

q x(t)|.

In addition,

k1ϖ1 + λ1ϖ2 < 1, λ2ϖ2 +ϖ1k2 < 1,

where ϖ1, ϖ2 are defined by (3.2.10), (3.2.11), respectively. Then, the system (3.2.1) has a positive

solution.

Proof. It is enough to show that the operator T has a fixed point. We divide the proof in three steps.

Step 1: We prove that the operator T maps a bounded set into a bounded. For this, consider Sρ =

{(x, y)(t) : ∥(x, y)∥ ≤ ρ} ⊆ E × E . Then for any (x, y)(t) ∈ Sρ and t ∈ [0, 1], there exist constants

M1,M2 such that

|f(t,Dkq y(t),Dp1

q y(t))| ≤ M1, |g(t,Dkqx(t),Dp2

q x(t))| ≤ M2,

then, by the virtue of (3.2.3), we get

|DkqT1(y(t))| = | − 1

Γq(α− k)

∫ t

0(t− qs)α−k−1f(s,Dk

q y(s),Dp1q y(s))dqs|

≤ M1

Γq(α− k + 1),

(3.2.13)

and

|Dp1q T1(y(t))| = | − 1

Γq(α− p1)

∫ t

0(t− qs)α−p1−1f(s,Dk


+t1−p1

Γ(2− p1)Γq(α− 1)

∫ 1



≤ M1{1

Γq(α− p1)+

1

Γ(2− p1)Γq(α)}.

(3.2.14)

55

Using (3.2.13), (3.2.14) and definition of the norm, we deduce that

∥T1(y)∥ ≤ M1{1

Γq(α− k + 1)+

1

Γq(α− p1)+

1

Γ(2− p1)Γq(α)}, (3.2.15)

∥T2(x)∥ ≤M2{1

Γq(β − k + 1)+

1

Γq(β − p2)+

1

Γ(2− p2)Γq(β)}. (3.2.16)

From (3.2.15), (3.2.16), we have:

∥T (x, y)∥ ≤ {M1{1

Γq(α− k + 1)+

1

Γq(α− p1)+

1

Γ(2− p1)Γq(α)}

+M2{1

Γq(β − k + 1)+

1

Γq(β − p2)+

1

Γ(2− p2)Γq(β)}}.

(3.2.17)

Thus ∥T (x, y)∥ ≤ ϖ1M1 +ϖ2M2 and hence T is uniformly bounded.

Step 2: In this step, we show that the operator T is an equicontinuous operator. For this, we assume

(x, y)(t) ∈ Sρ and t1, t2 ∈ [0, 1] with t1 < t2. Then, from (3.2.8), we have the following estimates:

|DkqT1(y(t2))−Dk

qT1(y(t1))|

= | − 1

Γq(α− k)

∫ t2

0(t2 − qs)α−k−1f(s,Dk


+1

Γq(α− k)

∫ t1



≤ M1tα−k2 − tα−k

1

Γq(α− k + 1),

(3.2.18)

and

|Dp1q T1(y(t2)) − Dp1

q T1(y(t1))|

= | − 1

Γq(α− p1)

∫ t2

0(t2 − qs)α−p1−1f(s,Dk


+1

Γq(α− p1)

∫ t1

0(t1 − qs)α−p1−1f(s,Dk

q y(s),Dp1q y(s))dqs (3.2.19)

+t1−p12 − t1−p1

1

Γ(2− p1)Γq(α− 1)

∫ 1



≤ M1tα−k2 − tα−k

1

Γq(α− k + 1)+

t1−p12 − t1−p1

1

Γ(2− p1)Γq(α).

From (3.2.18), (3.2.19), we get

∥T1(y(t2))− T1(y(t1))∥ = | − 1

Γq(α− k)

∫ t2



+1

Γq(α− k)

∫ t1



≤ M1{tα−k2 − tα−k

1

Γq(α− k + 1)+

tα−k2 − tα−k

1

Γq(α− k + 1)+

t1−p12 − t1−p1

1

Γ(2− p1)Γq(α)},

(3.2.20)

∥T2(x(t2))− T2(x(t1))∥ ≤ M2{tβ−k2 − tβ−k

1

Γq(β − k + 1)+

tβ−k2 − tβ−k

1

Γq(β − k + 1)+

t1−p22 − t1−p2

1

Γ(2− p1)Γq(β)}. (3.2.21)

56

From (3.2.20)–(3.2.21), we deduce that

∥T (x, y)(t2)− T (x, y)(t1)∥

≤ M1{tα−k2 − tα−k

1

Γq(α− k + 1)+

tα−k2 − tα−k

1

Γq(α− k + 1)+

t1−p12 − t1−p1

1

Γ(2− p1)Γq(α)}

+M2{tβ−k2 − tβ−k

1

Γq(β − k + 1)+

tβ−k2 − tβ−k

1

Γq(β − k + 1)+

t1−p22 − t1−p2

1

Γ(2− p1)Γq(β)},

(3.2.22)

which tends to zero independent of (x, y) as t1 → t2. So by Arzela Ascoli theorem, we have that T is

completely continuous operator.

Step 3: In this step, we show that the set:

S∗ = {(x, y)(t) ∈ E × E : µT (x, y)(t) = (x, y)(t), 0 < µ < 1},

is a bounded set. For this, we take (x, y) ∈ S∗. Then (x, y) = µT(x, y), implies that

x(t) = µT1(y(t)), y(t) = µT2(x(t)),

|Dkqx(t)| ≤

1

Γq(α− k + 1)(m1 + k1|Dk

q y(t)|+ k2|Dp1q y(t)|), (3.2.23)

and

|Dp1q x| ≤ { 1

Γq(α− p1)+

1

Γ(2− p1)Γq(α)}(m1 + k1|Dk

q y(t)|+ k2|Dp1q y(t)|). (3.2.24)

Therfore, by (3.2.10), (3.2.23), (3.2.24), we have

∥x∥ ≤ { 1

Γq(α− k + 1)+

1

Γq(α− p1)+

1

Γ(2− p1)Γq(α)}(m1 + k1|Dk

q y(t)|+ k2|Dp1q y(t)|)

= ϖ1(m1 + k1|Dkq y(t)|+ k2|Dp1

q y(t)|).(3.2.25)

Similarly, from (3.2.9) we have

∥y∥ ≤ ϖ2(m2 + λ1|Dkqx(t)|+ λ2|Dp2

q x(t)|). (3.2.26)

From (3.2.25), (3.2.26), it follows that

∥(x, y)∥ ≤ ϖ1m1 +ϖ2m2 + (k1ϖ1 + λ1ϖ2)∥x∥+ (λ2ϖ2 +ϖ1k2)∥y∥. (3.2.27)

Consequently, we get

∥(x, y)∥ ≤ ϖ1m1 +ϖ2m2

M, (3.2.28)

where M is given by (3.2.12). It shows that S∗ is a bounded set. Thus, Leray-Schauder alternative

affirms that the operator T , has a fixed point. Hence, the problem (3.2.1) has at least one positive

solution.

57

3.2.2 Uniqueness of solutions

In the following result, we discuss the uniqueness of solution for the problem (3.2.1) with the help of

Banach’s contraction principle.

Theorem 3.2.3. Let f, g ∈ C([0, 1],R2,R) and there exist real constants µ1, µ2, ν1, ν2 such that for all

t ∈ [0, 1] and x1, x2, y1, y2 ∈ R,

|f(t,Dkq y2,Dp1

q y2)−f(t,Dkq y1,Dp1

q y1)| ≤ µ1|Dkq y2 −Dk

q y1|+ µ2|Dp1q y2 −Dp1

q y1|,

and

|g(t,Dkqx2,Dp2

q x2)−g(t,Dkqx1,Dp2

q x1)| ≤ ν1|Dkqx2 −Dk

qx1|+ ν2|Dp2q x2 −Dp2

q x1|.

in addition, provided that

ϖ1(µ1 + µ2) +ϖ2(ν1 + ν2) < 1,

where ϖ1, ϖ2 are defined by (3.2.10)–(3.2.11). Then, the problem (3.2.1) has a unique solution.

Proof. We define the terms, supt∈[0,1] f(t, 0, 0) = N1 <∞, supt∈[0,1] g(t, 0, 0) = N2 <∞ and the set

Sν = {(x, y)(t) ∈ E × E : ∥(x, y)∥ ≤ ν}, where

ν >N1ϖ1 +N2ϖ2

1−ϖ1(µ1 + µ2)−ϖ2(ν1 + ν2).

Let (x, y) ∈ Sν . Then

|DkqT1(y(t))| = | − 1

Γq(α− k)

∫ t



≤ 1

Γq(α− k)

∫ t

0(t− qs)α−k−1{(|f(s,Dk

q y(s),Dp1q y(s))− f(s, 0, 0)|)

+ |f(s, 0, 0)|}dqs

≤ 1

Γq(α− k + 1){(|f(s,Dk

q y(s),Dp1q y(s))− f(s, 0, 0)|) + |f(s, 0, 0)|}

≤ 1

Γq(α− k + 1){µ1|Dk

q y|+ µ2|Dp1q y|+N1}

≤ { 1

Γq(α− k + 1)(µ1 + µ2)ν +N1},

(3.2.29)

and

|Dp1q T1(y(t))| = | − 1

Γq(α− p1)

∫ t

0(t− qs)α−p1−1f(s,Dk


+t1−p1

Γ(2− p1)Γq(α− 1)

∫ 1



≤ 1

Γq(α− p1)

∫ t

0(t− qs)α−p1−1{|f(s,Dk

q y(s),Dp1q y(s))− f(s, 0, 0)|

(3.2.30)

58

+ |f(s, 0, 0)|}dqs+ t1−p1

∫ 1

0

(1− qs)α−2

Γ(2− p1)Γq(α− 1){|f(s,Dk

q y(s),Dp1q y(s))

− f(s, 0, 0)|+ |f(s, 0, 0)|}dqs|

≤ (1

Γq(α− p1 + 1)+

1

Γ(2− p1)Γq(α)){µ1|Dk

q y|+ µ2|Dp1q y|+N1},

taking the norm, we get

∥T1(y)∥ ≤ ϖ1{(µ1 + µ2)ν +N1}, (3.2.31)

similarly,

∥T2(x)∥ ≤ ϖ2{(ν1 + ν2)ν +N2}. (3.2.32)

Consequently, we have

∥T (x, y)∥ ≤ ν. (3.2.33)

Therefore, T Sν ⊂ Sν . Now, we show that T is contraction. For this, we have

|DkqT1(y2(t))−Dk

qT1(y1(t))|

= | − 1

Γq(α− k)

∫ t


q y2(s),Dp1q y2(s))− f(s,Dk

q y1(s),Dp1q y1(s))dqs|

≤ 1

Γq(α− k + 1){µ1|Dk

q y2(s)−Dkq y1(s)|+ µ2|Dp1

q y2(s)−Dp1q y1(s)|},

(3.2.34)

and

|Dp1q T1(y2(t))−Dp1

q T1(y1(t))|

= | 1

Γq(α− p1)

∫ t

0(t− qs)α−p1−1|f(s,Dk


q y1(s),Dp1q y1(s))|dqs

+t1−p1

Γ(2− p1)Γq(α− 1)

∫ 1

0(1− qs)α−2|f(s,Dk


q y1(s),Dp1q y(s))|dqs

≤ (1

Γq(α− p1 + 1)+

1

Γ(2− p1)Γq(α)){µ1|Dk

q y2(s)−Dkq y1(s)|+ µ2|Dp1

q y2(s)−Dp1q y1(s)|}.

(3.2.35)

It follows from (3.2.34)–(3.2.35), we get

∥T1(y2)− T1(y1)∥

≤ { 1

Γq(α− k + 1)+

1

Γq(α− p1 + 1)+

1

Γ(2− p1)Γq(α)}{µ1|Dk

q y2(s)−Dkq y1(s)|

+ µ2|Dp1q y2(s)−Dp1

q y1(s)|}

= ϖ1{µ1|Dkq y2(s)−Dk

q y1(s)|+ µ2|Dp1q y2(s)−Dp1

q y1(s)|}

≤ ϖ1(µ1 + µ2){|Dkq y2(s)−Dk

q y1(s)|+ |Dp1q y2(s)−Dp1

q y1(s)|}.

(3.2.36)

Similarly, we can obtain

∥T2(x2)− T2(x1)∥ ≤ ϖ2(ν1 + ν2){|Dkqx2(s)−Dk

qx1(s)|+ |Dp1q x2(s)−Dp1

q x1(s)|}. (3.2.37)

59

From (3.2.36) − (3.2.37), we deduce that

∥T (x2, y2)− T (x1, y1)∥ ≤ (ϖ1(µ1 + µ2) +ϖ2(ν1 + ν2)){∥x2 − x1∥+ ∥y2 − y1∥}. (3.2.38)

whereϖ1(µ1+µ2)+ϖ2(ν1+ν2) < 1. Thus, T is contraction and Banach’s fixed point theorem affirms

the fixed point of T . So the problem (3.2.1) has a unique positive solution.

3.2.3 Examples

In this section, we are concerned with two illustrative examples for the applications of our work, i.e.,

Theorems 3.2.2 and 3.2.3. In these examples, we consider n = 3.

Example 3.2.4. We investigate the existence of positive solution for the following problem of q-difference

equations

D2.5.5 x(t) = − t+D2

0.5y +D0.50.5y

70,

D2.5.5 y(t) = − t+D2

0.5x+D0.50.5x

60,

x(1) = 0, D0.5x(1) = 0, D(2)0.5x(0) = 0,

y(1) = 0, D0.5y(1) = 0, D(2)0.5y(0) = 0.

(3.2.39)

β = γ = 2.5, p1 = p2 = 0.5 = q, k = 2, k1 = k2 = 170 , λ1 = λ2 = 1

60 . From these values and

(3.2.11)–(3.2.12), we find thatϖ1 = ϖ2 = 27.2018 and k1ϖ1+λ1ϖ2 = λ2ϖ2+ϖ1k2 = 0.84196 < 1.

Therefore, by Theorem 3.2.2, the coupled system (3.2.39), has a solution.

Example 3.2.5. Consider the problem:

D2.5.5 x(t) = −

(t+

D20.5y

100+

D0.50.5y

110

),

D2.5.5 y(t) = −

(t+

D20.5x

150+

D0.50.5x

160

),

x(1) = 0, D0.5x(1) = 0, D20.5x(0) = 0,

y(1) = 0, D0.5y(1) = 0, D20.5y(0) = 0.

(3.2.40)


|f(t,D20.5y2,D0.5

0.5y2)−f(t,D20.5y1,D0.5

0.5y1)| ≤1

100|D2

0.5y2−D20.5y1|+

1

110|D0.5

0.5y2−D0.50.5y1|, (3.2.41)

|g(t,D20.5y2,D0.5

0.5y2)−g(t,D20.5y1,D0.5

0.5y1)| ≤1

150|D2

0.5y2−D20.5y1|+

1

160|D0.5

0.5y2−D0.50.5y1|, (3.2.42)

and (3.2.41)–(3.2.42) implies that µ1 = 1100 , µ2 = 1

110 , ν1 = 1150 , ν2 = 1

160 . Using these values, we

find that ϖ1(µ1 + µ2) + ϖ2(ν1 + ν2) = 0.870663 < 1. Therefore, by Theorem 3.2.3, the coupled

system (3.2.40), has a unique solution.

60

Chapter 4

Coupled systems of hybrid FDEs

Recently, the theory on EUS and multiplicity of nonlinear BVPs for FDEs have attracted much attentions

and is fast growing area of research, we refer to [57, 58] and the references therein for some of valuable

and recent achievements in the study of existence theory for BVPs corresponding to FDEs. Existence

of positive solution (EPS) for coupled system of hybrid FDEs were considered by many scientists, for

example, Hedrih [65] and Dhage [66,71] explored some aspects of Hybrid differential equations, Ahmad

et al. [62], studied a coupled system of hybrid FDEs by classical approach, they considered the fractional

derivative in Caputo’s sense. Herzallah and Baleanu [63], studied existence of solutions for HFDEs in

Caputo’s sense. Ahmad and Ntouyas [64], discussed the existence of solutions for hybrid Hadamard

FDEs with initial value problems by a fixed point theorem due to Dhage.

In this chapter, we consider the EUS for two types of coupled systems of HFDEs. The results of this

chapter are published in international journal [131, 134]. In section 4.1, we consider the EUS for the

coupled system I. The results for EUS are obtained in subsection 4.1.1 and are applied to two coupled

systems of HFDEs in subsection 4.1.2. In section 4.2, the coupled system of HFDEs II is considered

for the EUS. The results are obtained in the subsection 4.2.1 and applied to two problems in subsection

4.2.2.

4.1 Coupled system of hybrid FDEs I

In this section, we study EUS for the following coupled systems of HFDEs

Dα(x(t)

H(t, x(t), y(t))) = −K1(t, x(t), y(t)), α ∈ (2, 3],

Dβ(y(t)

G(t, x(t), y(t))) = −K2(t, x(t), y(t)), β ∈ (2, 3], (4.1.1)

61

x(t)

H(t, x(t), y(t))|t=1 = 0, Dµ(

x(t)

H(t, x(t), y(t)))|t=δ1 = 0, x(2)(0) = 0,

y(t)

G(t, x(t), y(t))|t=1 = 0, Dν(

y(t)

G(t, x(t), y(t)))|t=δ2 = 0, y(2)(0) = 0,

where, t ∈ [0, 1]. δ1, δ2 ∈ (0, 1), α, β ∈ (2, 3], and Dα, Dβ , Dµ, Dν are Caputo’s fractional derivative

of orders α, β, µ, ν respectively, K1,K2 ∈ C([0, 1]×R2,R), H,G ∈ C([0, 1]×R2,R−{0}) and x, y ∈

C([0, 1],R). Here we remark that H(t, x, y) can not be of O(x) and G(t, x(t), y(t)) can not be of O(t).

Because if H(t, x, y) is of O(x) and G(t, x(t), y(t)) is of O(t), then the system (4.1.1) will reduce to a

coupled system of algebraic equations and will lose its hybrid nature. It is investigated that dynamics of

multi deformable bodies such as beams, plates, membranes etc coupled by standard light fractional order

discrete continuous layers is described by coupled fractional order hybrid system dynamics [65]. These

are generally perturbed FDEs and are mainly classified as perturbed differential equations of the first and

second type. The Perturbation itself is of two types, namely, linear and quadratics whose details can be

studied in [65, 66, 71]. Here, our system 4.1.1 is not an artificial one but in fact a quadratic perturbation

of the second type. These types of systems are generally tackled with the use of hybrid fixed point

theory. The significance of the hybrid system of differential equations lies in the fact that they include

several dynamical systems as special cases. For example, if we choose H(t, x, y) = 1, G(t, x, y) = 1,

as constant functions, then our system 4.1.1 reduce to a class of coupled system of boundary value

problems for FDEs of the type:

Dαx(t) = −K1(t, x(t), y(t)), α ∈ (2, 3],

Dβy(t) = −K2(t, x(t), y(t)), β ∈ (2, 3],

x(t)|t=1 = 0, Dµx(t)|t=δ1 = 0, x(2)(0) = 0,

y(t)|t=1 = 0, Dνy(t)|t=δ2 = 0, z(2)(0) = 0.

Lemma 4.1.1. For x ∈ AC3[0, 1] and any h(t) ∈ C[0, 1], the solution of the hybrid FDE:

Dα(x(t)

H(t, x, z)) = −h(t), α ∈ (2, 3], (4.1.2)

x(t)

H(t, x, y)|t=1 = 0, Dµ(

x(t)

H(t, x, y))|t=δ1 = 0, x(2)(0) = 0, µ ∈ (0, 1),

is given by

x(t) = H(t, x, y)[−∫ t

0

(t− s)α−1

Γ(α)h(s)ds+

∫ 1

0

(1− s)α−1

Γ(α)h(s)ds

+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1h(s)ds]. (4.1.3)

Proof. Applying the operator Iα (4.1.2), we obtain

x(t)

H(t, x, y)= −Iαh(t) + c1 + c2t+ c3t

2, (4.1.4)

62

by the use of initial condition x(2)(0) = 0, we get c3 = 0, and hence, (4.1.4) takes the form

x(t)

H(t, x, y)= −Iαh(t) + c1 + c2t. (4.1.5)

Applying Caputo’s fractional derivative of order µ, on (4.1.5), we get

Dµ(x(t)

H(t, x, y)) = −Iα−µh(t) + c2

t1−µ

Γ(2− µ). (4.1.6)

The boundary condition Dµ( x(t)H(t,x,α))|t=δ1 = 0, yields c2 = Iα−µ

∆1Z(δ1), where ∆1 =

δ1−µ1

Γ(2−µ) and the

boundary condition xH(t,x,y) |t=1 = 0 yields c1 = IαZ(1) − Iα−µ

∆1Z(δ1). Substituting the values of

c1, c2, in (4.1.6), we get:

x(t)

H(t, x, y)= −Iαh(t) + IαZ(1) + (t− 1)

Iα−µ

∆1Z(δ1). (4.1.7)

The equivalent integral form of (4.1.7) is:

x(t) = H(t, x, y)[−∫ t

0

(t− s)α−1

Γ(α)h(s)ds+

∫ 1

0

(1− s)α−1

Γ(α)h(s)ds

+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1h(s)ds

]. (4.1.8)

Thus, the proof is completed.


In this section, we are concerned with the EUS for the coupled system of HFDEs (4.1.1) by using

some classical results. For this, consider the Banach space E = {x(t) : x(t) ∈ C[0, 1]} with norm

∥x∥ = maxt∈[0,1] |x(t)| and (E × E , ∥(., .)∥) with norm ∥(x, y)∥ = ∥x∥ + ∥y∥. We define an operator

T : E × E → E × E by

T (x, y)(t) = (T1(x, y)(t), T2(x, y)(t)), (4.1.9)

where

T1(x, y)(t) = H(t, x, y)[−∫ t

0

(t− s)α−1

Γ(α)K1(s, x(s), y(s))ds

+

∫ 1

0

(1− s)α−1

Γ(α)K1(s, x(s), y(s))ds (4.1.10)

+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1K1(s, x(s), y(s))ds

],

and

T2(x, y)(t) = G(t, x, y)[−∫ t

0

(t− s)β−1

Γ(β)K2(s, x(s), y(s))ds

+

∫ 1

0

(1− s)β−1

Γ(β)K2(s, x(s), y(s))ds (4.1.11)

+(t− 1)

∆2Γ(β − ν)

∫ δ2

0(δ2 − s)β−ν−1K2(s, x(s), y(s))ds

],

63

where ∆1 =δ1−µ1

Γ(2−µ) , ∆2 =δ1−ν2

Γ(2−ν) . In view of Lemma 4.1.1, the solutions of the coupled system of

HFDEs (4.1.1), is a fixed point of the operator T . Define the following terms:

N1 =2

Γ(1 + α)+

2δα−µ1

∆1Γ(α− µ+ 1), (4.1.12)

N2 =2

Γ(1 + β)+

2δβ−ν2

∆2Γ(β − ν + 1). (4.1.13)

Theorem 4.1.2. Assume the following:

• (A1) There exists real constants µi > 0 (for i = 1, 2, 3, 4) such that |H(t, x1, y1)−H(t, x2, y2)| ≤µ1|x1−x2|+µ2|y1−y2|, |G(t, x1, y1)−G(t, x2, y2)| ≤ µ3|x1−x2|+µ4|y1−y2|, for (t, xj , yj) ∈([0, 1]× R× R), j = 1, 2.

• (A2) There exist functions Mi(t) ∈ C[0, 1] (for i = 1, 2) such that |K1(t, x, y)| ≤ M1(t), |K2(t, x, y)| ≤M2(t) for (t, x, y) ∈ ([0, 1]× R× R).

• (A3) γ(N1∥M1∥+N2∥M2∥) < 1, for γ = µ1 + µ2 + µ3 + µ4.

• (A4)y

1−ρ

(H0N1∥M1∥1−λN1∥M1∥ + G0N2∥M2∥

1−λN2∥M2∥

)< r for H0 = maxt∈[0,1] |H(t, 0, 0)| and G0 = maxt∈[0,1] |G(t, 0, 0)|

and ρ = max{ λN1∥M1∥1−λN1∥M1∥ ,

λN2∥M2∥1−λN2∥M2∥}, where λ ∈ (0, 1).

then, the coupled system of HFDEs (4.1.1), has a solution.

Proof. We give the proof in the following four steps. The operators T1, T2 given in (4.1.10), (4.1.11)

are equivalent to:

T1(x(t), y(t)) = f1(x(t), y(t))g1(x(t), y(t)) = (x(t), y(t)) t ∈ [0, 1], (4.1.14)

T2(x(t), y(t)) = f2(x(t), y(t))g2(x(t), y(t)) = (x(t), y(t)) t ∈ [0, 1]. (4.1.15)

Thus, the operator T defined in (4.1.9) can be expressed in the form:

T (x, y)(t) = (T1(x, y)(t), T2(x, y)(t)) = A(x(t), y(t))B(x(t), y(t)), (4.1.16)

where F1 = (f1, f2), F2 = (g1, g2) and fi, gi : Br(0) → E for i = 1, 2 are defined by

f1(x(t), y(t)) = H(t, x, y), f2(x(t), y(t)) = G(t, x, y), (4.1.17)

g1(x(t), y(t)) = −∫ t

0

(t− s)α−1


+

∫ 1

0

(1− s)α−1

Γ(α)K1(s, x(s), y(s))ds (4.1.18)

+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1K1(s, x(s), y(s))ds,

64

g2(x(t), y(t)) = −∫ t

0

(t− s)β−1


+

∫ 1

0

(1− s)β−1

Γ(β)K2(s, x(s), y(s))ds (4.1.19)

+(t− 1)

∆2Γ(β − ν)

∫ δ2

0(δ2 − s)β−ν−1K2(s, x(s), y(s))ds,

Step 1. The operators f1, f2 are Lipschitz on E . For this, let (x1, z1), (x2, z2) ∈ E × E and t ∈ [0, 1];

then by (A1) we get

|f1(x1, y1)(t)− f1(x2, y2)(t)| = |H(x1, y1)−H(x2, y2)|

≤ µ1|x1(t)− y1(t)|+ µ2|x2(t)− y2(t)| (4.1.20)

≤ (µ1 + µ2)∥(x1, y1)− (x2, y2)∥.

Similarly, we obtain

|f2(x1, y1)(t)− f2(x2, y2)(t)| = |G(x1, y1)− G(x2, y2)|

≤ µ3|x1(t)− y1(t)|+ µ2|x4(t)− y2(t)| (4.1.21)

≤ (µ3 + µ4)∥(x1, y1)− (x2, y2)∥.

From (4.1.20) and (4.1.21), we obtain

|A(x1, y1)(t) − A(x2, y2)(t)| = |(f1, f2)(x1, y1)− (f1, f2)(x2, y2)|

≤ (µ1 + µ2 + µ3 + µ4)∥(x1, y1)− (x2, y2)∥. (4.1.22)

Thus, (4.1.22) implies that F1 is Lipschitz on E × E with Lipschitz constant µ1 + µ2 + µ3 + µ4.

Step II In this step, we show that the operator F2 is continuous. For this, let us assume that {xn}, {yn}be convergent sequences such that xn → x, yn → y as n→ +∞. From the continuity of K1, it follows

that K1(t, xn(t), yn(t)) → K1(t, x(t), y(t)), K2(t, xn(t), yn(t)) → K2(t, x(t), y(t)). By Lebesgue

dominated theorem, we have

limn→∞

g1(xn, yn) = limn→∞

(−∫ t

0

(t− s)α−1

Γ(α)K1(s, xn(s), yn(s))ds

+

∫ 1

0

(1− s)α−1

Γ(α)K1(s, xn(s), yn(s))ds

+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1K1(s, xn(s), yn(s))ds) (4.1.23)

= −∫ t

0

(t− s)α−1

Γ(α)limn→∞

K1(s, xn(s), yn(s))ds

+

∫ 1

0

(1− s)α−1

Γ(α)limn→∞


+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1 lim

n→∞K1(s, xn(s), yn(s))ds

65

= −∫ t

0

(t− s)α−1


+

∫ 1

0

(1− s)α−1


+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1K1(s, x(s), y(s))ds

= g1(x, y).


limn→0

g2(xn, yn) = limn→∞

(−∫ t

0

(t− s)β−1

Γ(β)K2(s, xn(s), yn(s))ds

+

∫ 1

0

(1− s)β−1

Γ(β)K2(s, xn(s), yn(s))ds

+(t− 1)

∆2Γ(β − ν)

∫ δ2

0(δ2 − s)β−ν−1K2(s, xn(s), yn(s))ds)),

= −∫ t

0

(t− s)β−1

Γ(β)limn→∞


+

∫ 1

0

(1− s)β−1

Γ(β)limn→∞


+(t− 1)

∆2Γ(β − ν)

∫ δ2

0(δ2 − s)β−ν−1 lim

n→∞K2(s, xn(s), yn(s))ds))

= −∫ t

0

(t− s)β−1


+

∫ 1

0

(1− s)β−1


+(t− 1)

∆2Γ(β − ν)

∫ δ2

0(δ2 − s)β−ν−1K2(s, x(s), y(s))ds

= g2(x, y).

By the help of (4.1.23), (4.1.24) we proved that the operator F2 = (g1, g2) is continuous for all t ∈ [0, 1].

Step III In this step, we show that the operator F2 = (g1, g2) is compact on Br(0).

|g1(x, y)(t)| = | −∫ t

0

(t− s)α−1


+

∫ 1

0

(1− s)α−1


+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1K1(s, x(s), y(s))ds| (4.1.24)

≤∫ t

0

(t− s)α−1

Γ(α)|M1(s)|ds+

∫ 1

0

(1− s)α−1

Γ(α)|M1(s)|ds

+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1|M1(s)|ds

≤( 1

Γ(1 + α)+

1

Γ(1 + α)+

2δα−µ1

∆1Γ(α− µ+ 1)

)∥M1∥

= N1∥M1∥.

66

Taking supremum over t ∈ [0, 1], we have

∥g1(x, y)∥ ≤ N1∥M1∥ ∀ (x, y) ∈ Br(0)× Br(0). (4.1.25)


|g2(x(t), y(t))| = | −∫ t

0

(t− s)β−1


+

∫ 1

0

(1− s)β−1


+(t− 1)

∆2Γ(β − ν)

∫ δ2

0(δ2 − s)β−ν−1K2(s, x(s), y(s))ds|

≤∫ t

0

(t− s)β−1

Γ(β)|M2(s)|ds+

∫ 1

0

(1− s)β−1

Γ(β)|M2(s)|ds (4.1.26)

+(t− 1)

∆2Γ(β − ν)

∫ δ2

0(δ2 − s)β−ν−1|M2(s)|ds

≤( 1

Γ(1 + β)+

1

Γ(1 + β)+

2δβ−ν2

∆1Γ(β − ν + 1)

)∥M2∥

= N2∥M2∥.

Taking the supremum of (4.1.26), we get

∥g2(x, y)∥ ≤ N2∥M2∥ ∀ (x, y) ∈ Br(0)× Br(0).

From (4.1.25) and (4.1.26),it follows that

∥F(x, y)∥ ≤ N1∥M1∥+N2∥M2∥ ∀ (x, y) ∈ Br(0)× Br(0), (4.1.27)

which implies that F2(Br(0)× Br(0)) is a uniformly bounded set in E × E .Now, we prove that the operator F2 is an equicontinuous. For this, let us assume 0 ≤ t1 ≤ t2 ≤ 1, then

we have

|g1(x, y)(t2) − g1(x, y)(t1)| = |∫ t2

0

(t2 − s)α−1


−∫ t1

0

(t1 − s)α−1


+(t2 − t1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1K1(s, x(s), y(s))ds|) (4.1.28)

≤∫ t2

0

(t2 − s)α−1

Γ(α)M1(s)ds−

∫ t1

0

(t1 − s)α−1

Γ(α)M1(s)ds

+(t2 − t1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1M1(s)ds)

≤( tα2 − tα1Γ(1 + α)

+(t2 − t1)δ

α−ν1

∆1Γ(α− µ+ 1)

)∥M1∥.

67

Similarly, we have

|g2(x, y)(t1) − g2(x, y)(t2)| = |∫ t2

0

(t2 − s)β−1


−∫ t1

0

(t1 − s)β−1


+(t2 − t1)

∆2Γ(β − ν)

∫ δ2

0(δ2 − s)β−ν−1K2(s, x(s), y(s))ds|

≤∫ t2

0

(t2 − s)β−1

Γ(β)M2(s)ds−

∫ t1

0

(t1 − s)β−1

Γ(β)M2(s)ds (4.1.29)

+(t2 − t1)

∆2Γ(β − ν)

∫ δ2

0(δ2 − s)β−ν−1M2(s)ds

≤( tβ2 − tβ1Γ(1 + β)

+(t2 − t1)δ

β−ν2

∆2Γ(β − ν + 1)

)∥M2∥.


|F(x, y)(t1)−F(x, y)(t2)| ≤[tα2 − tα1Γ(1 + α)

+(t2 − t1)δ

α−ν1

∆1Γ(α− µ+ 1)

]∥M1∥ (4.1.30)

+

[tβ2 − tβ1Γ(1 + β)

+(t2 − t1)δ

β−ν2

∆2Γ(β − ν + 1)

]∥M2∥

which implies that F(x, y)(t1) − F(x, y)(t2) → 0 as t1 → t2. This shows that the operator B is an

equicontinuous and by Arzela Ascoli theorem, the operator B is a compact.

Step IV We have γF2,max < 1, where F2,max = ∥F2(Br(0) × Br(0))∥ = sup{∥F2(x, y)∥ : x, y ∈Br(0) × Br(0)}, and gi,max = ∥gi(Br(0))∥ = sup{∥gi(x) : x ∈ Br(0)∥} for i = 1, 2. Using (4.1.27)

and hypothesis (A3), we have

F2,max ≤ γ(N1∥M1∥+N2∥M2∥) < 1, ∀ (x, y) ∈ Br(0)× Br(0)). (4.1.31)

Now, we prove that the operator T maps bounded subset of E×E into a bounded set. For this, assume that

(x, y) ∈ Br(0)×Br(0) and t ∈ [0, 1], λ ∈ (0, 1) such that ∥(x, y)∥ = r and (x, y) = λA(x, y)F(x, y),

using (A2), (A4), we proceed

|x(t)| = λ|H(t, x, y)||(−∫ t

0

(t− s)α−1


+

∫ 1

0

(1− s)α−1


+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1K1(s, x(s), y(s))ds)| (4.1.32)

≤ λ|H(t, x, y)−H(t, 0, 0) +H(t, 0, 0)||(−∫ t

0

(t− s)α−1

Γ(α)M1(s)ds

+

∫ 1

0

(1− s)α−1

Γ(α)M1(s)ds+ (t− 1)

∫ δ1

0

(δ1 − s)α−µ−1M1(s)ds)

∆1Γ(α− µ)|

≤ λ(µ1|x(t)|+ µ2|y(t)|+H0)(2

Γ(1 + α)+

2δα−µ1

∆1Γ(α− µ+ 1))∥M1∥

= λ1(µ1|x(t)|+ µ2|y(t)|+H0)N1∥M1∥.

68

Taking supremum over t ∈ [0, 1] and using (A4), we obtain

∥x∥ ≤ λ1(µ1∥x∥+ µ2∥y∥+H0)N1∥M1∥. (4.1.33)

Which implies that

∥x∥ ≤ λ(µ2∥y∥+H0)

1− λµ1N1∥M1∥N1∥M1∥. (4.1.34)

Similarly, we have

∥y∥ ≤ λ(µ1∥x∥+ G0)

1− λµ2N2∥M2∥N2∥M2∥. (4.1.35)

By the help of (4.1.34) and (4.1.35), we get

∥(x, y)∥ = ∥x∥+ ∥y∥ ≤ λ(µ2∥y∥+H0)

1− λµ1N1∥M1∥N1∥M1∥

+ λ(µ1∥x∥+ G0)

1− λµ2N2∥M2∥N2∥M2∥ (4.1.36)

= λµ2∥y∥

1− λN1∥M1∥N1∥M1∥+ λ

H0

1− λµ1N1∥M1∥N1∥M1∥

+ λµ1∥x∥

1− λµ2N2∥M2∥N2∥M2∥+ λ

G0

1− λµ2N2∥M2∥N2∥M2∥.

From (4.1.36), we obtain

∥(x, y)∥ ≤ 1

1− ρ

(λH0N1∥M1∥

1− λµ1N1∥M1∥+

λG0N2∥M2∥1− µ2λN2∥M2∥

)< r, (4.1.37)

which contradicts our supposition ∥(x, y)∥ = r. Thus, the operator (x, y) = F1(x, y)F2(x, y) has a

solution in Br(0) × Br(0) which is a solution of the system (4.1.1). This completes the proof of the

theorem.

Theorem 4.1.3. Assume that (A1) and (A2) hold. If µ(N1+N2)(∥M1∥+∥M2∥) < 1, then the coupled

system of HFDEs (4.1.1), has a unique solution.

Proof. Define H0 = supt∈[0,1] |H(t, 0, 0)|, G0 = supt∈[0,1] |G(t, 0, 0)| and µ = µ1 + µ2 + µ3 + µ4.

Choose ρ ≥ µ(H0+G0)(N1+N2)(∥M1∥+∥M2∥)1−µ(N1+N2)(∥M1∥+∥M2∥) where N1, N2 are defined in (4.1.12), (4.1.13) respectively.

Define a set Sρ = {(x, y)(t) ∈ E × E : ∥(x, y)∥ ≤ ρ}. Then, for (x, y) ∈ Sρ, we have

|T1(x(t), y(t))| = |H(t, x, y)||(−∫ t

0

(t− s)α−1


+

∫ 1

0

(1− s)α−1


+(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1K1(s, x(s), y(s))ds)|

≤ |H(t, x, y)−H(t, 0, 0) +H(t, 0, 0)|(∫ t

0

(t− s)α−1

Γ(α)∥M1(s)∥ds (4.1.38)

+

∫ 1

0

(1− s)α−1

Γ(α)∥M1(s)∥ds+

(t− 1)

∆1Γ(α− µ)

∫ δ1

0(δ1 − s)α−µ−1∥M1(s)∥ds)

69

≤ (µ1|x(t)|+ µ2|y(t)|+H0)(2

Γ(1 + α)+

2δα−µ1

∆1Γ(α− µ+ 1))∥M1∥

= (|µ1x(t)|+ µ2|y(t)|+H0)N1∥M1∥.

Taking supremum over t ∈ [0, 1], we obtain

∥T1(x, y)∥ ≤ (µ1 + µ2)(∥x∥+ ∥y∥+H0)N1∥M1∥ (4.1.39)

≤ (µ1 + µ2)(β +H0)N1∥M1∥.

Similarly, we have

∥T2(x, y)∥ ≤ (µ3 + µ4)(∥x∥+ ∥y∥+ G0)N2∥M2∥ (4.1.40)

≤ (µ3 + µ4)(β + G0)N2∥M2∥.

By the help of (4.1.39), (4.1.40), we get

∥T (x, y)∥ ≤ (µ1 + µ2)(β +H0)N1∥M1∥

+ (µ3 + µ4)(β + G0)N2∥M2∥ (4.1.41)

≤ (µ1 + µ2 + µ3 + µ4)(β +H0 + G0)(N1

+ N2)(∥M1∥+ ∥M2∥) ≤ β.

Therefore, (4.1.40) implies that the operator T is a bounded operator. Now, for (x1, y1)(t),

(x2, y2)(t) ∈ E × E and any t ∈ [0, 1], we have

|T1(x2, y2)(t)− T1(x1, y1)(t)|

≤ |H(t, x2, y2)−H(t, x1, y1)|{∫ t

0

(t− s)α−1

Γ(α)|M1(s)|ds (4.1.42)

+

∫ 1

0

(1− s)α−1

Γ(α)|M1(s)|ds+ |t− 1|

∫ δ1

0

(δ1 − s)α−µ−1|M1(s)|ds∆1Γ(α− µ)

}

≤ (µ1|x2 − x1|+ µ2|y2 − y1|)( 2

Γ(1 + α)+

2δα−µ1

∆1Γ(α− µ+ 1)

)|M(t)|

≤ (µ1 + µ2)N1|M1(t)|(|x2 − x1|+ y2 − y1|).

Taking the supremum over t ∈ [0, 1], we obtain

∥T1(x2, y2) − T1(x1, y1)∥ (4.1.43)

≤ (µ1 + µ2)N1∥M1∥(∥x2 − x1∥+ ∥y2 − y1∥).


∥T2(x2, y2) − T2(x1, y1)∥ (4.1.44)

≤ (µ3 + µ4)N2∥M2∥(∥x2 − x1∥+ ∥y2 − y1∥).

70


∥T (x2, y2) − T (x1, y1)∥ (4.1.45)

≤ µ(N1 +N2)(∥M1∥+ ∥M2∥)(∥x2 − x1∥+ ∥y2 − y1∥),

which implies that T is a contraction and by the help of Banach’s contraction principle the operator Thas a unique fixed point which is the unique solution of the coupled system of HFDEs (4.1.1).

4.1.2 Examples

Example 4.1.4. Consider the following coupled system of HFDEs as a special case of the problem

(4.1.1):

D2.5( x(t)

100 + (|x(t)|+|y(t)|) sin(t)50

)=

1 + x(t) sin(t)

1 + |x(t)|+ |y(t)|, (4.1.46)

D2.5( y(t)

100 + (|x(t)|+|y(t)|) cos(t)50

)=

1 + x(t) cos(t)

1 + |x(t)|+ |y(t)|,

with initial and boundary conditions as defined in (4.1.1), for δ1 = δ2 = µ = ν = 0.5 and t ∈ [0, 1].

From (4.1.46), we have H(t, x, y) = 100 + (|x(t)|+|y(t)|) sin(t)50 , G(t, x, y) = 100 + (|x(t)|+|y(t)|) cos(t)

50 .

|H(t, x2, y1) − H(t, x1, y1)| ≤ 150(∥x2 − x1∥ + ∥y2 − y1∥), |G(t, x2, y1) − G(t, x1, y1)| ≤ 1

50(∥x2 −x1∥+∥y2−y1∥) which implies µi = 1

50 for i = 1, 2, 3, 4.By using these values, we deduce ρ ≥ 67.7022,

we choose ρ = 70. Thus, by the Theorem 4.1.2, the coupled system of HFDEs (4.1.46) has a solution

in B70(0)× B70(0).

Example 4.1.5. Consider the following coupled system of HFDEs as a special case of the problem

(4.1.1):

D2.5( x(t)

10 + (|x(t)|+|y(t)|) sin(t)50

)=

1 + x(t) sin(t)

2 + |x(t)|+ |y(t)|, (4.1.47)

D2.5( y(t)

10 + (|x(t)|+|y(t)|) cos(t)50

)=

1 + x(t) cos(t)

2 + |x(t)|+ |y(t)|,

with initial and boundary conditions as defined in (4.1.1), for δ1 = δ2 = µ = ν = 0.5. From

system (4.1.47), we have H(t, x, y) = 10 + (|x(t)|+|y(t)|) sin(t)50 , G(t, x, y) = 10 + (|x(t)|+|y(t)|) cos(t)

50 .

|H(t, x2, y1)−H(t, x1, z1)| ≤ 150(∥x2−x1∥+∥y2−y1∥), |G(t, x2, y1)−G(t, x1, z1)| ≤ 1

50(∥x2−x1∥+∥y2 − y1∥) which implies µi = 1

50 for i = 1, 2, 3, 4. By using these values, we deduce ρ ≥ 6.77022, we

choose ρ = 7. µ(N1 +N2)(∥M1∥+ ∥M2∥) ≤ 0.252917 < 1. Therefore, the system (4.1.47) satisfies

the assumptions of Theorem 4.1.3. Thus, the system (4.1.47), has a unique solution in B7(0)× B7(0).

71

4.2 Coupled system of hybrid FDEs II

In this section, we are interested in the unique solution of the coupled system of HFDEs of the type:

Dα

(x

H(t, x, y)

)= −K1(t, x, y), α ∈ (n− 1, n],

Dβ

(y

G(t, x, y)

)= −K2(t, x, y), β ∈ (n− 1, n],(

x

H(t, x, y)

)(i)

|t=0 = 0 =

(x

H(t, x, y)

)(1)

|t=1,x

H(t, x, y)|t=δ1 = δ1Iα−1

1 T1(t, x, y),(y

G(t, x, y)

)(j)

|t=0 = 0 =

(x

G(t, x, y)

)(1)

|t=1,y

G(t, x, y)|t=δ2 = δ2Iβ−1

1 T2(t, x, y),

i, j = 2, 3, . . . , n− 1,

(4.2.1)

where Dα, Dβ are Caputo’s fractional derivative of orders α, β respectively, K1,K2 ∈ C([0, 1]×R2,R),

H,G ∈ C([0, 1]× R2,R− {0}) and x, y ∈ C([0, 1],R).

Lemma 4.2.1. Let h(t) be a continuous function then the mild solution of

Dα

(x

H(t, x, y)

)= −h(t), α ∈ (n− 1, n], (4.2.2)

with (x

H(t, x, y)

)(i)

|t=0 = 0 =

(x(t)

H(t, x, y)

)(1)

|t=1,x

H(t, x, y)|z=δ1 = δ1Iα−1h(1), (4.2.3)

for i = 2, . . . , n− 1, is given by

x(t) = H(t, x, y)[−∫ t

0

(t− s)α−1

Γ(α)h(s)ds+

∫ δ

0

(δ − s)α−1

Γ(α)h(s)ds

+ t

∫ 1

0

(1− s)α−2

Γ(α− 1)h(s)ds].

(4.2.4)

Proof. Applying the operator Iα0 on (4.2.2) and using Lemma 1.1.6, we obtain

x(t)

H(t, x, y)= −Iαh(t) + c1 + c2t+ c3t

2 + . . .+ cntn−1. (4.2.5)

Initial conditions(

xH(t,x,y) |z=0

)(i)= 0, for i = 2, . . . , n − 1 implies c3 = . . . = cn = 0, and (4.2.5)

gets the formx(t)

H(t, x, y)= −Iαh(t) + c1 + c2t. (4.2.6)

Applying the boundary condition(

x(t)H(t,x,y) |t=1

)(1)= 0, on (4.2.6), we have c2 = Iα−1h(1) and (4.2.6)

gets the formx(t)

H(t, x, y)= −Iαh(t) + c1 + tIα−1h(1). (4.2.7)

72

By the help of condition xH(t,x,y) |t=δ1 = δ1Iα−1h(1), and (4.2.7), we get c0 = Iαh(δ1). Putting the

values of c0, c1, . . . , cn in (4.2.7), we get

x(t)

H(t, x, y)= −Iαh(t) + Iαh(δ) + tIα−1h(1), (4.2.8)

and the integral form of the solution x(t), is given by

x(t) = H(t, x, y)[−∫ t

0

(t− s)α−1

Γ(α)h(s)ds+

∫ δ

0

(δ − s)α−1

Γ(α)h(s)ds

+ t

∫ 1

0

(1− s)α−2

Γ(α− 1)h(s)ds].

(4.2.9)


In this section, our interest is in two lemmas for the EMS and unique solution of the coupled system

(4.2.1). For these two lemmas, we divide this section in two subsections. Consider the Banach space

E = {x(t) : x(t) ∈ C[0, 1]} with norm ∥x∥ = max{|x(t)| for t ∈ [0, 1]} and (E × E , ∥(., .)∥) with

norm ∥(x, y)∥ = ∥x∥+ ∥y∥. We define an operator T : E × E → E × E by

T (x, y)(t) = (T1(x, y)(t), T2(x, y)(t)), (4.2.10)

where

T1(x, y)(t) = H(t, x, y)[−∫ s

0

(t− s)α−1

Γ(α)K1(s, x, y)ds+

∫ δ1

0

(δ1 − s)α−1

Γ(α)K1(s, x, y)ds

+ t

∫ 1

0

(1− s)α−2

Γ(α− 1)T1(s, x, y)ds],

(4.2.11)

and

T2(x, y)(t) = G(t, x, y)[−∫ z

0

(z − y)β−1

Γ(β)K2(s, x, y)ds+

∫ δ2

0

(δ2 − y)β−1

Γ(β)K2(s, x, y)ds

+ t

∫ 1

0

(1− y)β−2

Γ(β − 1)K2(s, x, y)ds],

(4.2.12)

We define the following terms for simplifying our calculations

N1 =1

Γ(1 + α)+

δα1Γ(1 + α)

+1

Γ(α), (4.2.13)

N2 =1

Γ(1 + β)+

δβ2Γ(1 + β)

+1

Γ(β). (4.2.14)

Assume the following two conditions:

(A1) The functions H(t, x, y),G(t, x, y) are continuous and bounded such that |H(t, x, y)| ≤ µ1,

73

|G(t, x, y)| ≤ µ2 for all (t, x, y) ∈ ([0, 1]× R× R) and µ1, µ2 ∈ R;

(A2) There exist real constants λj , ηj , ξj > 0 for j = 1, 2 such that |K1(t, x, y)| ≤ γ1 + η1|x| + ξ1|y|,

|K2(t, x, y)| ≤ γ2 + η2|x| + ξ2|y| for all x, y ∈ R and t ∈ [0, 1] also µ1N1(γ1 + η1|x| + ξ1|y|) +

µ2N2(γ2 + η2|x|+ ξ2|y|) < +∞.

Theorem 4.2.2. Assume that (A1), (A2) are satisfied and there exist real constants M1,M2 ∈ R, such

that

|K1(t, x, y)| ≤ M1, |K2(t, x, y)| ≤ M2,

for all t ∈ [0, 1], then the system (4.2.1), has a solution.

Proof. The proof is given in the following three steps.

Step 1: In this step, we prove that the operator T maps bounded subset of E × E into a bounded. For

this, we consider (x, y) ∈ E × E for t ∈ [0, 1], then (4.2.11) deduce the following:

|T1(x, y)(t)| = |H(t, x, y)|[| −∫ t

0

(t− s)α−1

Γ(α)K1(s, x, y)ds

+

∫ δ1

0

(δ1 − s)α−1

Γ(α)K1(s, x, y)ds+ z

∫ 1

0

(1− y)α−2

Γ(α− 1)K1(s, x, y)ds|]

≤ µ1

[1

Γ(1 + α)+

δα1Γ(1 + α)

+1

Γ(α)

]M1 = µ1N1M1,

(4.2.15)

similarly, (4.2.12) implies that:

|T2(x, y)(t)| ≤ µ2

[1

Γ(1 + β)+

δβ2Γ(1 + β)

+1

Γ(β)

]M2 = µ2N2M2. (4.2.16)

Using the definition of the norm and (4.2.15), (4.2.16), we have:

∥T (x, y)(t)∥ ≤ µ1N1M1 + µ2N2M2 <∞, (4.2.17)

thus, T is uniformly bounded.

Step 2: Now, to show that T is an equicontinuous operator. For this, let (x, y)(t) ∈ E × E and t1, t2 ∈[0, 1], such that t1 < t2. Then, we have

|T1(x, y)(t2)− T1(x, y)(t1)| = µ1[|∫ t2

0

(t2 − s)α−1

Γ(α)K1(s, x, y)ds

−∫ t1

0

(t1 − y)α−1

Γ(α)K1(s, x, y)ds+ (t2 − t1)

∫ 1

0

(1− y)α−2

Γ(α− 1)K1(s, x, y)ds|]

≤ µ1

[tα2 − tα1Γ(1 + α)

+t2 − t1Γ(α)

]M1,

(4.2.18)

similarly, we can get

|T2(x,y)(t2)− T2(x, y)(t1)| ≤ µ2

[tβ2 − tβ1Γ(1 + β)

+t2 − t1Γ(β)

]M2, (4.2.19)

74

by the help of (4.2.18) and (4.2.19), we have

|T (x, y)(t2)− T (x, y)(t1)| ≤ µ1

[tα2 − tα1Γ(1 + α)

+t2 − t1Γ(α)

]M1

+ µ2

[tβ2 − tβ1Γ(1 + β)

+t2 − t1Γ(β)

]M2,

(4.2.20)

∥T (x, y)(t2)− T (x, y)(t1)∥ → 0 as t2 → t1. By step 1, step 2 and Arzela Ascoli theorem, we have Tis completely continuous operator.

Step 3: Here, we show that the set

S∗ = {(x, y)(t) ∈ E × E : µT (x, y)(t) = (x, y)(t)},

where 0 < µ < 1, is bounded. From the set S∗, we have µT1(x, y)(t) = x(t) and µT2(x, y)(t) = x(t),

consequently,

|x(t)| ≤ µ|T1(x, y)(t)|

≤ |H(t, x, y)|[| −∫ t

0

(t− s)α−1

Γ(α)K1(s, x, y)ds

+

∫ δ1

0

(δ1 − y)α−1

Γ(α)K1(s, x, y)ds+ t

∫ 1

0

(1− s)α−2

Γ(α− 1)K1(s, x, y)ds|]

≤ µ1

[1

Γ(1 + α)+

δα1Γ(1 + α)

+1

Γ(α)

](γ1 + η1|x|+ ξ1|y|)

= µ1N1(γ1 + η1|x|+ ξ1|y|),

(4.2.21)

and

|y(t)| ≤ µ|T2(x, y)(t)|

≤ |G(t, x, y)|[| −∫ t

0

(t− s)β−1

Γ(β)K2(s, x, y)ds+

∫ δ2

0

(δ2 − s)β−1

Γ(β)K2(s, x, y)ds

+ t

∫ 1

0

(1− y)β−2

Γ(β − 1)K2(s, x, y)ds|]

≤ µ2

(1

Γ(1 + β)+

δβ2Γ(1 + β)

+1

Γ(β)

)(γ2 + η2|x|+ ξ2|y|)

≤ µ2N2(γ2 + η2|x|+ ξ2|y|).

(4.2.22)

From (4.2.21) and (4.2.22), we have

∥x∥ ≤ µ1N1(γ1 + η1|x|+ ξ1|y|), (4.2.23)

∥y∥ ≤ µ2N2(γ2 + η2|x|+ ξ2|y|), (4.2.24)

consequently,

∥(x, y)∥ ≤ µ1N1(γ1 + η1|x|+ ξ1|y|) + µ2N2(γ2 + η2|x|+ ξ2|y|) < +∞, (4.2.25)

75

this implies that S∗ is bounded. Thus, by the help of Step 1, 2, 3 we have that T is completely continuous

and by the help of lemma 1.3.4, T has fixed the point (x, y)(t) and is the solution of the coupled system

of HFDEs (4.2.1).

Theorem 4.2.3. Assume that K1,K2 ∈ C([0, 1]×R2,R) and there exist positive real constants γ1, γ2, γ3, γ4,

such that for all t ∈ [0, 1] and x1, x2, y1, y2 ∈ R,

|K1(t, x, y)−K1(t, x1, y1)| ≤ γ1|x− y1|+ γ2|y − y1|,

and

|K2(t, x, y)−K2(t, x1, y1)| ≤ γ3|x− y1|+ γ4|y − y1|,

with the condition that

µ1N1(γ1 + γ2) + µ2N2(γ3 + γ4) < 1,

where N1,N2 are defined by (4.2.13), (4.2.14). Then the problem (4.2.1), has a unique solution.

Proof. Assume that supt∈[0,1]K1(t, 0, 0) = ζ1 < ∞, supt∈[0,1]K2(t, 0, 0) = ζ2 < ∞ and the set

Sν = {(x, y)(t) ∈ E × E : ∥(x, y)∥ ≤ ν}, such that

ν ≥ µ1N1ζ1 + µ2N2ζ21− µ1N1(γ1 + γ2)− µ2N2(γ3 + γ4)

. (4.2.26)

From (4.2.11), we have:

|T1(x, y)(t)| ≤ |H(t, x, y)|[| −∫ z

0

(z − y)α−1

Γ(α)K1(s, x, y)ds

+

∫ δ1

0

(δ1 − s)α−1

Γ(α)K1(s, x, y)ds+ t

∫ 1

0

(1− y)α−2

Γ(α− 1)K1(s, x, y)ds|]

≤ µ1[

∫ t

0

(t− s)α−1

Γ(α)(|K1(s, x, y)−K1(s, 0, 0)|+ |K1(s, 0, 0)|)dy

+

∫ δ1

0

(δ1 − s)α−1

Γ(α)(|K1(s, x, y)−K1(s, 0, 0)|+ |K1(s, 0, 0)|)ds

+ t

∫ 1

0

(1− s)α−2

Γ(α− 1)(|K1(s, x, y)−K1(s, 0, 0)|+ |K1(s, 0, 0)|)ds]

≤ µ1

(1

Γ(1 + α)+

δα1Γ(1 + α)

+1

Γ(α)

)(γ1|x|+ γ2|y|+ ζ1)

≤ µ1N1((γ1 + γ2)ν + ζ1).

(4.2.27)

Similarly, (4.2.12)

|T2(x, y)(t)| ≤ µ2N2((γ3 + γ4)ν + ζ2). (4.2.28)

From (4.2.26), (4.2.27) and (4.2.28), we have

∥T (x, y)∥ ≤ ν. (4.2.29)

76

Let, (x1, y1)(t), (x2, y2)(t) ∈ E × E and for any t ∈ [0, 1], we have

|T1(x2, y2)(t)− T1(x1, y1)(t)|

≤ |H(t, x, y)|[∫ z

0

(z − y)α−1

Γ(α)(|K1(s, x2, y2)−K1(s, x1, y1)|)ds

+

∫ δ1

0

(δ1 − s)α−1

Γ(α)(|K1(s, x2, y2)−K1(s, x1, y1(y))|)dy

+ z

∫ 1

0

(1− s)α−2

Γ(α− 1)(|K1(s, x2, y2)−K1(s, x1, y1(y))|)ds]

≤ µ1

(1

Γ(1 + α)+

δα1Γ(1 + α)

+1

Γ(α)

)(γ1|x2 − x1|+ γ2|y2 − y1|)

≤ µ1N1(γ1 + γ2)(|x2 − x1|+ |y2 − y1|).

(4.2.30)

Similarly, from (4.2.12), we have:

|T2(x2, y2)(t)− T2(x1, y1)(t)| ≤ µ2N2(γ3 + γ4)(|x2 − x1|+ |y2 − y1|), (4.2.31)

Therefore, from (4.2.30), (4.2.32), we have:

∥T (x2, y2)− T (x1, y1)∥ ≤ (µ1N1(γ1 + γ2) + µ2N2(γ3 + γ4))(|x2 − x1|+ |y2 − y1|). (4.2.32)

Therefore, the function T is a contraction mapping and consequently, T fixes a point which is the unique

solution of (4.2.1).

4.2.2 Examples

Example 4.2.4. Consider the following coupled system of HFDEs as a special case of (4.2.1):

D3.5

(x

1 + | sinx|+ | cos y|

)= − t+ | sinx|+ | cos y|

40(1 + 2t),

D3.5

(y

1 + | sinx|+ | cos y|

)= − t+ | sinx|+ | cos y|

40(1 + 3t),(

x

H(t, x, y)

)(i)

|t=0 = 0 =

(x

H(t, x, y)

)(1)

|t=1,x

H(t, x, y)|t=0.5 = 0.5I2.5K1(t, x, y)|t=1,(

y

G(t, x, y)

)(i)

|t=0 = 0 =

(y

G(t, x, y)

)(1)

|t=1,y

G(t, x, y)|t=0.5 = 0.5I2.5K2(t, x, y)|t=1.

(4.2.33)

In this system, β = α = 3.5, δ1 = δ2 = 0.5, H = G = 1 + | sinx| + | cos y| ≤ 3 = µ1 = µ2,

H,G ∈ R− {0}. |K1| ≤ 140 + |x|

40 + |y|40 and |K2| ≤ 1

40 + |x|40 + |y|

40 , λi = ηi = ξi =140 for i = 1, 2 and

µ1N1(t1 + η1|x| + ξ1|y|) + µ2N2(t2 + η2|x| + ξ2|y|) < +∞ for any x, y ∈ C([0, 1],R). Therefore,

conditions (A1), (A2), are satisfied and by Theorem 4.2.2, the problem (4.2.33) has a solution.

77

Example 4.2.5. In this example, Theorem 4.2.3, is utilized for the unique solution of the following

coupled system of HFDEs:

D3.5

(x

1 + | sinx|+ | cos y|

)= − t+ | sinx|+ | cos y|

35(1 + 2t),

D3.5

(y

1 + | sinx|+ | cos y|

)= − t+ | sinx|+ | cos y|

35(1 + 3t),(

x

H(t, x, y)

)(i)

|t=0 = 0 =

(x

H(t, x, y)

)(1)

|t=1,x

H(t, x, y)|t=0.5 = 0.5I2.5K1(t, x, y)|t=1,(

y

G(t, x, y)

)(i)

|t=0 = 0 =

(y

G(t, x, y)

)(1)

|t=1,y

G(t, x, y)|t=0.5 = 0.5I2.5K2(t, x, y)|t=1.

(4.2.34)

In this coupled system of HFDEs, we have α = β = 3.5, δ1 = δ2 = 0.5, K1(t, 0, 0) ≤ 135 ,K2(t, 0, 0) ≤

135 and

|K1(t, x, y)−K1(t, x1, y1)| ≤1

35(|x− y1|+ |y − y1|),

|K2(t, x, y)−K2(t, x1, y1)| ≤1

35(|x− y1|+ |y − y1|),

(4.2.35)

from (4.2.35), we have γ1 = γ2 = γ3 = γ4 = 135 , N1 = N2 = 0.394472, ζ1 = ζ2 = 1, ultimately, we

get ν ≥ 0.0782001, and µ1N1(γ1 + γ2) + µ2N2(γ3 + γ4) = 0.135248 < 0.2. Therefore, by Theorem

4.2.3, the uniqueness of solution of the coupled system of HFDEs (4.2.34), in S0.1(0), is varified.

78

Chapter 5

Local fractional differential equations

Local fractional derivative was defined by Kolwankar and Gangal [108] and got the attention of scientists

in the fields like physics and engineering based on the fractals. A recent definition of derivatives on

fractals have been introduced by Golmankhanreh and Baleanu [109]. Where the fractal curves [105],

are everywhere continuous but nowhere differentiable and therefore, the classical calculus cannot be

used to interpret the motions in Cantor time-space [107]. LFC [110–113], started to be considered as

one of the useful way to handle the fractals and other functions that are continuous but non-differentiable.

Mandelbrot [105, 117], described that fractal geometry is a workable geometric middle ground between

the excessive geometric order of Euclid and the geometric chaos of general mathematics and extensively

illustrated wide range of applications fractals in many scientific fields like in, physics, engineering,

mathematics, geophysics. Christianto and Rahul [118], gave the derivation of Proca equations on Cantor

sets. Hao et al. [119], investigated the Helmholtz and Diffusion equations in Cantorian and Cantor-Type

Cylindrical Coordinates. Carpinteri and Sapora [120], studied diffusion problems in fractal media in

Cantor sets. Zhang et al. studied LF wave equations under fixed entropy. Many techniques are utilized

for handling the LF problems in both ordinary and partial derivatives. For instance, The Local fractional

Laplace Transform [110], the local fractional variational iteration method (LFLVIM) [113] and many

others. These methods are widely used in different scientific fields [106, 112–116, 120–122, 124]. This

area of research is much popular in the community of scientists and we are continuously observing

recent developments in it.

In this chapter, we are concerned with the analytical solutions of three problems in LF calculus. The

problems are; LF wave equations (LFWE), LF Schrodinger equation (LFSE) and a Fractal Vehicular

Traffic flow model (FVTFM). The results of this chapter are [134, 141, 142]. This chapter has three

sections. In section 5.1, an iterative technique for the solutions of the wave equations is developed by

the help of local fractional variational iteration method (LFLVIM) and the scheme is applied on two ex-

79

amples of LFWEs. In section 5.2, solutions of LFSE are obtained by the help of Yang Laplace transform

(LFLT) and Series expansion method (SEM). The iterative technique is applied on two examples. In sec-

tion 5.3, an iterative technique for solutions of FVTFM is developed which is applied for the solutions

of two illustrative examples in subsection 5.3.1.

5.1 Iterative technique for solutions of LF wave equations

In this section, we develop an iterative scheme for analytic solutions of the following LFWEs

vmγt (t, x)− p(x)vnγx (t, x) = 0, (5.1.1)

where m, n are orders of LF partial derivatives with respect to t and x respectively.

Applying the LF variation iteration method for the correction LF operator for (5.1.1), we have

vm+1(t, x) = vm(t, x) + I(γ)0,t

(X (x)γ

Γ(γ + 1)

)(vmγ

m,t(t, x)− p(x)vnγm,x(t, x)), (5.1.2)

where X (x)γ

Γ(γ+1) , is the Lagrange multiplier and (5.1.2) leads to

vm+1(t, x) = vm(t, x) + I(γ)0,t

(X (x− t)γ

Γ(γ + 1)

)(vmγ

m,t(t, x)− p(x)vnγm,x(t, x)). (5.1.3)

Applying LFLT on (5.1.3), we have

Lγ{vm+1(t, x)} = Lγ{vm(t, x)}+ Lγ{X (x)γ

Γ(γ + 1)}Lγ{(vmγ

m,t(t, x)− p(x)vnγm,x(t, x)}

= Lγ{vm(t, x)}+ Lγ{X (x)γ

Γ(γ + 1)}{smγLγ{vm(t, x)} (5.1.4)

− s(m−1)γvm(0, x)− s(m−2)γvm(0, x)(γ)

− s(m−3)γvm(0, x)(2γ) . . .− vm(0, x)((m−1)γ) − p(x)Lγ{vnγm,x(t, x)}}.

Taking the γ order local fractional variation of (5.1.4) w.r.t T , and assuming that the term p(x)vnγx (t, x)

be invariant, we have

δ(γ)Lγ{vm+1(t, x)} = δ(γ)Lγ{vm(t, x)}+ Lγ{X (x)γ

Γ(γ + 1)}δ(γ){smγLγ{vm(t, x)}

− s(m−1)γvm(0, x)− s(m−2)γvm(0, x)(γ) (5.1.5)

− s(m−3)γvm(0, x)(2γ) . . .− vm(0, x)((m−1)γ)}.

From (5.1.5), we obtain the Lagrange-Multiplier as follows

X (x)γ

Γ(γ + 1)=

−1

smγ, (5.1.6)

80

and by the help of (5.1.4) and (5.1.6), we have the following relation

Lγ{vm+1(t, x)} = Lγ{vm(t, x)} − 1

smγ{smγLγ{vm(t, x)}

− s(m−1)γvm(0, x)− s(m−2)γvm(0, x)(γ) (5.1.7)

− s(m−3)γvm(0, x)(2γ) . . .− vm(0, x)((m−1)γ) − p(x)vnγm,x(s, x)}.

Consequently, we have solution of (5.1.1) as given below:

v(t, x) = limm→∞

L−1γ {Lγ{vm(s, x)}}. (5.1.8)

5.1.1 Applications

As an application of the iterative technique (5.1.8), we consider the following examples.

Example 5.1.1. Consider the following wave equation on Cantor sets:

∂2γ

∂t2γv(t, x)− C

∂2γ

∂x2γv(t, x) = 0, (5.1.9)

subject to the following initial-boundary conditions

∂γ

∂tγv(0, x) = 0, v(0, x) = Eγ(x

γ). (5.1.10)

Using (5.1.7), we obtain

Lγ{vm+1(t, x)} = Lγ{vm(t, x)} − 1

s2γ{s2γLγ{vm(t, x)} (5.1.11)

− s(γ)vm(0, x)− vm(0, x)(γ) − C∂2γ

∂x2γvm(s, x)}.

Using the initial condition (5.1.10), we get

{v0(s, x)} = Lγ{v0(0, x)} =Eγ(x

γ)

sγ. (5.1.12)

From (5.1.10) to (5.1.12), we have the first iteration

Lγ{v1(t, x)} = Lγ{v0(t, x)} −1

s2γ{s2γLγ{v0(t, x)}

− s(γ)v0(0, x)− v0(0, x)(γ) − C

∂2γ

∂x2γv0(s, x)}

=1

s2γ

[sγEγ(x

γ) + CEγ(x

γ)

sγ

](5.1.13)

=Eγ(x

γ)

sγ+ C

Eγ(xγ)

s3γ= v1(s, x).

81

For the second iteration, we utilize (5.1.10) and (5.1.13) and obtain

Lγ{v2(t, x)} = Lγ{v1(t, x)} −1


− s(γ)v1(0, x)− v1(0, x)(γ) − C

∂2γ

∂x2γv1(s, x)} (5.1.14)

=1

s2γ

[sγEγ(x

γ) + C

(Eγ(x

γ)

sγ+ C

Eγ(xγ)

s3γ

)]=

Eγ(xγ)

sγ+ C

Eγ(xγ)

s3γ+ C2Eγ(x

γ)

s5γ= v2(s, x).

For the third iteration, using (5.1.10), (5.1.12) and (5.1.14), we have

Lγ{v3(t, x)} = Lγ{v2(t, x)} −1


− s(γ)v2(0, x)− v2(0, x)(γ) − C

∂2γ

∂x2γv2(s, x)} (5.1.15)

=1

s2γ

[sγEγ(x

γ) + C

(Eγ(x

γ)

sγ+ C

Eγ(xγ)

s3γ+ C2Eγ(x

γ)

s5γ

)].

=Eγ(x

γ)

sγ+ C

Eγ(xγ)

s3γ+ C2Eγ(x

γ)

s5γ+ C3Eγ(x

γ)

s7γ= v3(s, x).

Continuing this process upto nth iteration, we obtain

Lγ{vn(t, x)} =Eγ(x

γ)

sγ+ C

Eγ(xγ)

s3γ+ C2Eγ(x

γ)

s5γ+ C3Eγ(x

γ)

s7γ+ . . . (5.1.16)

+ Cn Eγ(xγ)

s(2n+1)γ.

Applying L−1γ on (5.1.16), we get

vn(t, x) = Eγ(xγ) + C

t2γEγ(xγ)

Γ(2γ + 1)+ C2 t4γEγ

Γ(4γ + 1)+ C3Eγ(x

γ)

s7γ+ . . .+ Cn Eγ(x

γ)

s(2n+1)γ

= Eγ(xγ) lim

n→∞

n∑j=0

Cj t2jγ

Γ(2jγ + 1)= Eγ(x

γ) coshγ(Ctγ). (5.1.17)

Example 5.1.2. Consider the following wave equation on Cantor sets:

∂2γv(t, x)

∂t2γ− xγ

Γ(γ + 1)

∂2γv(t, x)

∂x2γ= 0, (5.1.18)

the initial-boundary condition are as under

∂γv(0, x)

∂tγ= 0, v(0, x) =

x2γ

Γ(2γ + 1). (5.1.19)

From (5.1.19), we get

v(s, x) = Lγ{v0(0, x)} =x2γ

sγΓ(2γ + 1). (5.1.20)

82

From (5.1.7), (5.1.18) and (5.1.20), we get the first iteration

Lγ{v1(t, x)} = Lγ{v0(t, x)} −1

s2γ{s2γLγ{v0(t, x)} (5.1.21)

− s(γ)v0(0, x)− v0(0, x)(γ) − xγ

Γ(γ + 1)

∂2γ

∂x2γv0(s, x)}

=x2γ

sγΓ(2γ + 1)− xγ

s3γΓ(γ + 1)= v1(s, x).

For the second iteration, using (5.1.19) and (5.1.21), we obtain

Lγ{v2(t, x)} = Lγ{v1(t, x)} −1


− s(γ)v1(0, x)− v1(0, x)(γ) − xγ

Γ(γ + 1)

∂2γ

∂x2γv1(s, x)} (5.1.22)

=x2γ


s3γΓ(γ + 1)= v2(s, x).

For the third iteration, using (5.1.19) and (5.1.22), to get

Lγ{v2(t, x)} = Lγ{v1(t, x)} −1


− sγv1(0, x)− v1(0, x)(γ) − xγ

Γ(γ + 1)

∂2γ

∂x2γv1(s, x)} (5.1.23)

=x2γ


s3γΓ(γ + 1)= v2(s, x).

Continuing this process upto nth iteration, we get

Lγ{vn+1(t, x)} =x2γ


s3γΓ(γ + 1). (5.1.24)

Applying L−1γ on (5.1.24), we deduce the following result as the solution of (5.1.18), (5.1.19), by the

help of proposed method LFLVIM and our scheme in (5.1.8):

v(t, x) =x2γ

Γ(2γ + 1)− t2γxγ

Γ(2γ + 1)Γ(γ + 1). (5.1.25)

5.2 Iteration technique for solutions of LF Schrodinger equation

In this section, we develop an iterative technique for the analytical solutions of LFSE by the help of two

methods, namely; LFSEM and LFLT. For this, consider the following LFDE:

v(nγ)t (t, x)− λγv(t, x) = 0, (5.2.1)

where v(nγ)t (t, x) = ∂nγ

∂tnγ v(t, x) and λγ is a linear operator. The function v(t, x), can be considered as

multi-term separated function of the variables t and x as under:

v(t, x) =∞∑j=0

Tj(t)Xj(x), (5.2.2)

83

where Tj(t) and Xj(x) are LF continuous function. Expressing Tj(t) = tjγ

Γ(1+jγ) , then (5.2.2) takes the

form

v(t, x) =∞∑j=0

tjγ

Γ(1 + jγ)Xj(x). (5.2.3)

Taking LFLT of (5.2.4), we obtain

v(t, s) =∞∑j=0

tjγ

Γ(1 + jγ)Xj(s). (5.2.4)

By the help of (5.2.5), we produce

Lγ{v(nγ)t (t, x)} =∞∑j=0

tjγ

Γ(1 + jγ)Xj+n(s), (5.2.5)

Lγ{λγv(t, x)} =∞∑j=0

tjγ

Γ(1 + jγ)(λγXj)(s). (5.2.6)

By the use of (5.2.5), (5.2.6), in (5.2.1) yields

∞∑j=0

tjγ

Γ(1 + jγ)Xj+n(s) =

∞∑j=0

tjγ

Γ(1 + jγ)λγXj(s). (5.2.7)

From (5.2.7), we have the following recursion relation

Xj+n(s) = (λγXj)(s). (5.2.8)


Xj+1(s) = (λγXj)(s), (5.2.9)

and

Xj+2(s) = (λγXj)(s). (5.2.10)

Consequently, we get the solution of (5.2.1), as under:

v(x, s) =

∞∑j=0

tjγ

Γ(jγ + 1)Xj(s). (5.2.11)

Where the convergent relation is

limj→∞

(tjγ

Γ(1 + jγ))Xj(s) = 0. (5.2.12)

And the solution of (5.2.1) is obtained by the use of (5.2.9), as below

v(t, x) =∞∑j=0

tjγ

Γ(1 + jγ)L−1γ {Xj(s)}. (5.2.13)

84

5.2.1 Applications

In this subsection we utilize the iteration scheme (5.2.9) for the solution of LF Schrodinger equations.

Example 5.2.1. Consider the LFSE

∂γ

∂tγv(t, x) +

hγ2iγm

∂2γ

∂x2γv(t, x) =

1

iγhγv(t, x), (5.2.14)

where γ ∈ (0, 1], with the initial condition

v0(0, x) =x2γ

Γ(1 + 2γ)= X0(x). (5.2.15)

In this example, the operator λγ is given by

λγ =1

iγhγ− hγ

2iγm

∂2γ

∂x2γ. (5.2.16)

Applying the LFLT on (5.2.15), we deduce

X0(s) =x2γ

sγΓ(1 + 2γ). (5.2.17)

Applying the operator λγ on (5.2.17) and using the iteration formula (5.2.9), we get:

X1(s) =

(1

iγhγ− hγ

2iγm

∂2γ

∂x2γ

)x2γ

sγΓ(1 + 2γ)=

1

iγhγ

x2γ

sγΓ(1 + 2γ)− hγ

2iγmsγ. (5.2.18)

For the second iteration, applying the operator λγ on (5.2.18) and the recursion relation (5.2.9), we

proceed

X2(s) =

(1

iγhγ− hγ

2iγm

∂2γ

∂x2γ

)(

1

iγhγ

x2γ

sγΓ(1 + 2γ)− hγ

2iγmsγ) (5.2.19)

=1

i2γh2γ

x2γ

sγΓ(1 + 2γ)− 2

2i2γmsγ.

For the third iteration, applying the operator λγ on (5.2.19) and using the recursion relation (5.2.9), we

deduce

X3(s) =

(1

iγhγ− hγ

2iγm

∂2γ

∂x2γ

)(

1

i2γh2γ

x2γ

sγΓ(1 + 2γ)− 2

2i2γmsγ) (5.2.20)

=1

i3γh3γ

x2γ

sγΓ(1 + 2γ)− 3

2i3γhγmsγ.

Continuing this process, we get the following approximations:

X4(s) =1

i4γh4γ

x2γ

sγΓ(1 + 2γ)− 4

2i4γh2γmsγ,

X5(s) =1

i5γh5γ

x2γ

sγΓ(1 + 2γ)− 5

2i5γh3γmsγ,

X6(s) =1

i6γh6γ

x2γ

sγΓ(1 + 2γ)− 6

2i6γh4γmsγ, (5.2.21)

......

...

Xj(s) =1

ijγhjγ

x2γ

sγΓ(1 + 2γ)− j

2ijγhj−2γ msγ

, (5.2.22)

85

therefore, the solution of the LFSE (5.2.14), (5.2.15) is approximated by the help of (5.2.13) and (5.2.22),

as below

v(t, x) =

∞∑j=0

tjγ

Γ(jγ + 1)L−1γ { 1

ijγhjγ

x2γ

sγΓ(1 + 2γ)− j

2ijγhj−2γ msγ

}

=∞∑j=0

tjγ

Γ(jγ + 1){ 1

ijγhjγ

x2γ

Γ(1 + 2γ)− j

2ijγhj−2γ m

}.

Example 5.2.2.

∂γ

∂tγv(t, x) + i

∂2γ

∂x2γv(t, x) = 0, (5.2.23)

with initial condition

v(0, x) = Eγ(xγ) = X0(x). (5.2.24)

Applying LFLT on (5.2.24), we have

X0(s) =Eγ(x

γ)

sγ. (5.2.25)

From (5.2.23), we have the operator λγ = −i ∂2γ

∂x2γ . Applying the operator λγ on (5.2.25), we deduce

X1(s) = −i ∂2γ

∂x2γEγ(x

γ)

sγ= −iEγ(x

γ)

sγ. (5.2.26)

For the second approximation, we apply the operator λγ on (5.2.26) and using (5.2.9), we have

X2(s) = −i ∂2γ

∂x2γ

((−1)i

Eγ(xγ)

sγ

)= (−1)2i2

Eγ(xγ)

sγ. (5.2.27)

Continuing this process, we have

X3(s) = −i ∂2γ

∂x2γ

((−1)2i2

Eγ(xγ)

sγ

)= (−1)3i3

Eγ(xγ)

sγ, (5.2.28)

X4(s) = −i ∂2γ

∂x2γ

((−1)3i3

Eγ(xγ)

sγ

)= (−1)4i4

Eγ(xγ)

sγ, (5.2.29)

X5(s) = −i ∂2γ

∂x2γ

((−1)4i4

Eγ(xγ)

sγ

)= (−1)5i5

Eγ(xγ)

sγ, (5.2.30)

......

...

Xj(s) = (−1)jijEγ(x

γ)

sγ. (5.2.31)

From (5.2.13) and (5.2.31), we have the approximation of solution of the probelem (5.2.23), (5.2.24) as

under

v(t, x) =

∞∑j=0

tjγ

Γ(1 + jγ)L−1γ {(−1)jij

Eγ(xγ)

sγ}

= Eγ(xγ)Eγ(−it)γ = Eγ(x− it)γ . (5.2.32)

86

5.3 Iteration technique for the solution of fractal Vehicular Traffic flowmodel

In this section, we develop iterative scheme for analytic solution of FVTFM which is also known by

Lighthill-Richards model [123]. The model equation is given by

∂nγ

∂tnγv(t, x) + ρ

∂mγ

∂xmγv(t, x) = 0, (5.3.1)

where γ ∈ (0, 1), with the conditions

v(0, x) = f(x), v(t, 0) = g(t). (5.3.2)

For this, consider the following equation

vnγt (t, x) = ρλγv(t, x), (5.3.3)

where vnγt (t, x) = Dnγ

dtnγ v(t, x) and λγ is a linear operator. Expressing v(t, x) as a multi-term separated

functions of independent variables Ti and Xi as below:

v(t, x) =∞∑j=0

Tj(t)Xj(x), (5.3.4)

where Tj(t) and Xj(x) are local fractional continuous functions. Expressing the function Tj(t) =

tjγ

Γ(1+jγ) , then (5.3.4) takes the form:

v(t, x) =

∞∑j=0

cjtjγ

Γ(1 + jγ)Xj(x), (5.3.5)

assuming cj = 1, in (5.3.5), we have

v(t, x) =

∞∑j=0

tjγ

Γ(1 + jγ)Xj(x). (5.3.6)

From (5.3.6), we have the following equalities:

vnγt (t, x) =

∞∑j=0

tjγ

Γ(1 + jγ)Xj+n(x), (5.3.7)

λγv(t, x) =

∞∑j=0

tjγ

Γ(1 + jγ)λγXj(x). (5.3.8)

Using (5.3.7) and (5.3.8) in (5.3.3), we obtain

∞∑j=0

tjγ

Γ(1 + jγ)Xj+n(x) =

∞∑j=0

tjγ

Γ(1 + jγ)λγXj(x). (5.3.9)

87

From (5.3.9),we have the following recursion

Xj+n(s) = λγXj(s). (5.3.10)

By the recursion formula (5.3.10), we have

Xj+1(s) = λγXj(s), (5.3.11)

and

Xj+2(s) = λγXj(s). (5.3.12)

Finally, we get the solutions of (5.3.3) as below

v(x, s) =∞∑j=0

tjγ

Γ(1 + jγ)Xj(s). (5.3.13)

Where the convergent condition is

limj→∞

tjγ

Γ(1 + jγ)Xj(s) = 0. (5.3.14)

Therefore, the solution of (5.3.3), is obtained by the help of (5.3.11), as

v(t, x) =

∞∑j=0

tjγ

Γ(1 + jγ)L−1γ {Xj(s)}. (5.3.15)

5.3.1 Applications

This subsection is reserved for the application of the recursion scheme (5.3.11).

Example 5.3.1. Consider the FVTFM:

∂γ

∂tγv(t, x) + ρ

∂γ

∂xγv(t, x) = 0, (5.3.16)


v(0, x) = Eγ(xγ) = X0(x). (5.3.17)

In this model, the operator λγ is

λγ = −ρ ∂γ

∂xγ. (5.3.18)

By the help of (5.3.11), we deduce

X1(x) = (−1)ρ∂γ

∂xγEγ(x

γ) = (−1)Eγ(xγ). (5.3.19)

Using (5.3.11) and (5.3.19), we deduce the second iteration

X2(x) = (−1)ρ∂γ

∂xγ(−1)Eγ(x

γ) = (−1)2Eγ(xγ). (5.3.20)

88

Continuing the process of approximating terms, we deduce

X3(x) = (−1)ρ∂γ

∂xγ(−1)2Eγ(x

γ) = (−1)3Eγ(xγ), (5.3.21)

X4(x) = (−1)ρ∂γ

∂xγ(−1)3Eγ(x

γ) = (−1)4Eγ(xγ), (5.3.22)

......

Xj(x) = (−1)jEγ(xγ). (5.3.23)

The solution of FVTFM (5.3.16), (5.3.17), is approximated by the help of (5.3.12) and (5.3.23), as:

v(t, x) =

∞∑j=0

tjγ

Γ(1 + jγ)(−1)jEγ(x

γ)

= Eγ(xγ)

∞∑j=0

t2jγ

Γ(1 + 2jγ)ρ2j −

∞∑j=0

ρ2j+1t(2j+1)γ

Γ(1 + (2j + 1)γ)

(5.3.24)

= Eγ(xγ)(cosγ(ρt

γ)− sinγ(ρtγ)).

Example 5.3.2. Consider the following FVTFM:∂γ

∂tγv(t, x) + ρ

∂γ

∂xγv(t, x) = 0, (5.3.25)


v(0, x) = sinhγ(xγ) = X0(x). (5.3.26)

From (5.3.25), (5.3.26) we have ρ = 1 and the operator λγ is

λγ = − ∂γ

∂xγ. (5.3.27)

And

X0(x) = sinhγ(xγ). (5.3.28)

For the first approximation of the model (5.3.25), (5.3.26), we use (5.3.11), (5.3.27) and (5.3.28) as

under

X1(x) = − ∂γ

∂xγ(sinhγ(x

γ)) = − coshγ(xγ). (5.3.29)

For the second approximation of the model (5.3.25), (5.3.26), using (5.3.11), (5.3.27) and (5.3.29), and

deduce

X2(x) = − ∂γ

∂xγ(− coshγ(x

γ)) = − sinhγ(xγ). (5.3.30)

Consequently, we get the following approximation

v(t, x) = limm→∞

(sinhγ(xγ)

m∑j=0

(−1)jt2jγ

Γ(1 + 2jγ)

− coshγ(xγ)

m∑j=0

(−1)jt(2j+1)γ

Γ(1 + (2j + 1)γ)) (5.3.31)

= sinhγ(xγ) cosγ(t

γ)− coshγ(xγ) sinγ(t

γ).

89

Chapter 6

Numerical solutions of fractionaldifferential equations

Numerical solutions of FDEs have studied by many researchers such as; Yousefi and Behroozifar [84],

developed operational matrices using Bernstein polynomials (BPs), Rostamy et al. [83], studied multi

term FDEs by the using operational matrices of BPs, Jafari et al. [81], studied Abel differential equation

of fractional order using Homotopy analysis method, Baleanu et al. [88], studied fractional quadratic

Riccati differential equations via operational matrices using Bernstein polynomials(OMBPs), Alipour

and Rosatamy [89], used OMBPs for solving Abel’s integral equation, Alipour and Rostamy [74], ob-

tained numerical solutions of nonlinear FDE via OMBPs.

In literature, some good and interesting results on the numerical solutions of telegraph equations

(TE) via different numerical tools are available, for example, Inc et al. [72], studied numerical solutions

of one dimensional Telegraph equation (TE) of second kind by reproducing kernel in Hilbert space,

Srivastava et al. [85], studied the solutions of TE by reduced differential transform method, Garg et

al. [86], studied space time FTE by generalized differential transform method.

In this chapter, we develop numerical scheme for solutions of FTE and a coupled system of farctional

partial differential equation (FPDE) by using OMBPs for fractional derivatives. Some of the results of

this chapter are published [136]. This chapter has three sections, in section 6.1, we develop OMBPs for

fractional derivatives. In section 6.2 use the OMBPs and develop s numerical technique for solutions of

FTEs. In section 6.3, a numerical technique for the solutions of a coupled system of FPDEs is developed

using OMBPs.

6.1 Operational Matrix for fractional order derivatives using BPs

90

The operational matrix of fractional order derivative is followed by (1.1.3) as

0Dαϕ(t) =1

Γ(n− α)

∫ t

0(t− x)n−α−1χ(n)

m (x)dx =1

Γ(n− α)tn−α−1 ∗ χ(n)

m (t), (6.1.1)

where the operator ∗ denotes the convolution product. Thus, we have

0Dαϕ(t) = A1

Γ(n− α)

(tn−α−1 ∗ T (n)

m (t))= ADαTm(x) = A [Dα1,Dαt, . . . ,Dαtm]T . (6.1.2)

where Dαti is as defined in (1.1.4). Therefore we can write,

DαTm(t) = HT (6.1.3)

where H is a square diagonal matrix of order (m+1)×(m+1) and T is a row matrix of order (m+1)×1.

The entries of these matrices are given by

Hij =

Γ(j+1)

Γ(j+1−α) i, j = ⌈α⌉ , . . . ,m and i = j,

0 otherwise,(6.1.4)

and

T ij =

0 i = 0, . . . , ⌈α− 1⌉ ,

ti−α i = ⌈α⌉ , . . . ,m.(6.1.5)

Now we approximating ti−α for i = ⌈α⌉ , . . . ,m using BPs as in (1.2.4), we have

ti−α ≈MTi ϕm(t) (6.1.6)

where Mi is a row matrix of order (m+ 1)× 1 and can be obtained by the relation

Mi = Q−1

(∫ 1

0ti−αϕm(t)dt

)= Q−1Mi (6.1.7)

where M i =[M i,0,M i,1, . . . ,M i,m

]T and

Mi,j =

∫ 1

0ti−αHj,m(t)dt =

m!Γ(j + i− α+ 1)

j!Γ(i+m− α+ 2)(6.1.8)

where i = ⌈α⌉ , . . . ,m and j = 0, 1, . . . ,m. Now we assume a square matrix V of order (m+1)×(m+

1) having zero vectors at first ⌈α⌉ columns andWi in its (⌈α⌉+1)th columns, for i = (⌈α⌉+1), . . . ,m.

Consequently we have

Dαϕm(t) ≈ Dαϕm(t) ≈ AHV Tϕm(t) (6.1.9)

where

Dα ≈ AHV T (6.1.10)

is the operational matrix of fractional derivative of order α using BPs.

91

6.2 Numerical scheme for solutions of Fractional Telegraph equations

In this section, we develope a numerical technique for solutions of fractional Telegraph equations (FTE)

via operational matrices of fractional order derivatives and operational matrix of multiplication based on

BPs. Consider the FT equation

∂αu(x, t)

∂tα− C2∂

2u(x, t)

∂x2+ a

∂u(x, t)

∂x+ bu(x, t) = f(x, t), α ∈ (1, 2] (6.2.1)

with initial conditions

u(x, 0) = f1(x), ut(x, 0) = f2(x),

and boundary conditions u(0, t) = g1(t), u(1, t) = g2(t), where x, t ∈ [0, 1], and a, b ∈ R. Using

(1.2.9), we have

u(x, t) ≈ ϕT (x)C�1ϕ(t), f(x, t) ≈ ϕT (x)C�2ϕ(t) (6.2.2)

and by (6.1.9) and (6.1.10), we have

∂2

∂x2u(x, t) ≈ (

∂2

∂x2ϕ(x))TC�1ϕ(t) ≈ (D2ϕ(x))

TC�1ϕ(t) = ϕ(x)TDT2 C�1ϕ(t),

∂

∂xu(x, t) ≈ (

∂

∂xϕ(x))TC�1ϕ(t) = (Dϕ(x))TC�1ϕ(t) = ϕ(x)TDTC�1ϕ(t)

∂α

∂tαu(x, t) ≈ ϕ(x)TC�1Dαϕ(t),

u(x, 0) = θ1(x) ≈ F T1 ϕ(x), ut(x, 0) = θ2(x) ≈ F T

2 ϕ(x).

(6.2.3)

Using (6.2.3) in (6.3.1), we have

ϕ(x)TC�1Dαϕ(t)− C2ϕ(x)TDT2 C�1ϕ(t) + aϕ(x)TDTC�1ϕ(t) + bϕ(x)TC�1ϕ(t)

≈ ϕ(x)TC�2ϕ(t)(6.2.4)


ϕ(x)TC�1ϕ(0) ≈ F T1 ϕ(x), ϕ(x)TC�1D1ϕ(0) ≈ F T

2 ϕ(x)

ϕ(0)TC�1ϕ(t) ≈ GT1 ϕ(t), ϕ(1)TC�1ϕ(t) ≈ GT

2 ϕ(t).(6.2.5)

We use Tau method for solutions of the system, so we get from (6.2.4) and (6.2.5), the following system∫ 1

0

∫ 1

0Hi(x)ϕ(x)

T (C�1Dα − C2DT2 C�1 + aDTC�1 + bC�1 − C�2)ϕ(t)Hj(t)dxdt = 0,

i = 0, 1, . . . ,m− 4

(6.2.6)

with

C�1ϕ(0)−F1 = 0, C�1D1ϕ(0)−F2 = 0, ϕ(0)TC�1 −GT1 = 0, ϕ(1)TC�1 −GT

2 = 0. (6.2.7)

Solving this system of algebraic equations for the unknown C�1 , C�2 and putting in (6.3.2), give us our

required approximations.

92

6.2.1 Applications

In this section, we provide some applications of the technique (6.2.6), (6.2.7) developed in the previous

section.

Example 6.2.1. Choose f1(x) = x, f2(x) = 0, f(x, t) = 1 − x − t2 and the constants C2 = 1, a =

−1, b = −1. For α = 2, the exact solution is given by u(x, t) = x+ t2. For the purpose of comparison,

we compare the exact solution with the solution obtained using the technique (6.2.6), (6.2.7). In Figure

6.1, the comparison of our scheme for α = 1.5, 1.75, 2 with the exact solution is given. In Figure 6.2, we

have given error in our scheme for by plotting the absolute value of the difference of our approximated

value and that of exact solution. We have considered the value of the approximating terms of BPs m =

10. In Figure 6.3, we have given the sketch of our approximation for different values of α = 1.5, 1.75, 2

and the exact solution for t = 0.25.

0.00.5

1.0

0.0

0.5

1.0

0

1

2

Figure 6.1: Graph for the approximation of u(x, t) when m = 10 and α = 1.5, 1.75, 2.0 and the exact solution

in6.2.1, by our numerical technique (6.2.6)– (6.2.7).

0.0

0.5

1.0 0.0

0.5

1.00

5.´10-8

1.´10-7

Figure 6.2: Plot for error in our approximation for u(x, t), by the proposed technique and our numerical technique

(6.2.6), (6.2.7) for α = 2 with m = 10 in 6.2.1.

93

0.2 0.4 0.6 0.8 1.0

1.0

1.5

2.0

Figure 6.3: Graph for the approximate values of u(x, t) for α = 1.5, 1.75, 2.0, m = 10 and the exact solution at

t = 0.25. The graph for our approximation is the dashed for α = 2 and the green line represents the exact solution

in 6.2.1.

Example 6.2.2. We assume a FTE with initial conditions θ1(x) = 0, θ2(x) = 0 and the constants

C2 = 2, a = 1, b = 1, with f(x, t) = 12t2x2 − 4t4 + 2xt4 + t4x2. For α = 2 the exact solution is

u(x, t) = t4x2, for α = 2. and comparison with the exact solution is given for the illustration of our

technique (6.2.6), (6.2.7). In Figure 6.4, the comparison of our scheme for α = 1.5, 1.75, 2 with the

exact solution is given. In Figure 6.5, we have given error in our scheme for by plotting the absolute

value of the difference of our approximated value and that of exact solution. We have considered the

value of the approximating terms of BPs m = 10. In Figure 6.6, we have given the sketch of our

approximation for different values of α = 1.5, 1.75, 2 and the exact solution for t = 0.25.

0.0

0.5

1.00.0

0.5

1.0

0.0

0.2

0.4

Figure 6.4: Graph for the approximation of u(x, t) when m = 10 and α = 1.5, 1.75, 2.0 and the exact solution

in example 6.2.2.

94

0.0 0.5 1.0

0.0

0.5

1.0

0

5.´10-6

0.00001

Figure 6.5: Plot for the error in our approximation for u(x, t), by the proposed technique and our numerical

technique (6.2.6)– (6.2.7) for α = 2 with m = 10 in example 6.2.2.

0.2 0.4 0.6 0.8 1.0

0.02

0.04

0.06

0.08

0.10

0.12

Figure 6.6: Graph for the approximate values of u(x, t) for α = 1.5, 1.75, 2.0, m = 10 and the exact solution at

t = 0.25. The graph for our approximation is the dashed for α = 2 and the green line represents the exact solution

in example 6.2.2.

6.3 Numerical scheme for solutions of Fractional partial differential equa-tions (FPDEs)

The numerical solutions of FPDEs is one of the important and hot research area which have got a good

attention of the scientists round the world [76–79, 81, 83, 84]. Motivated by the recent contributions of

the scientists [73,83,84], we develop a scheme for numerical solutions of the following coupled system

of FPDEs using OMBPs K1

∂αu(x,t)∂xα +K2

∂βv(x,t)∂tβ

= 0,

K3∂βv(x,t)

∂xβ +K4∂αu(x,t)

∂tα = 0,

(6.3.1)


u(0, t) = f(t), v(0, t) = g(t).

where x, t ∈ [0, 1], α, β ∈ (0, 1] and K1,K2,K3,K4, are constants. Using (1.2.9), we have

u(x, t) = ϕT (x)C1ϕ(t), v(x, t) = ϕT (x)C2ϕ(t). (6.3.2)

95

By the use of (6.1.10) and (6.3.2), we obtain

∂α

∂xαu(x, t) ≈ (

∂α

∂xαϕ(x))TC1ϕ(t) ≈ (Dαϕ(x))

TC1ϕ(t) = ϕ(x)TDTαC1ϕ(t),

∂β

∂xβv(x, t) ≈ (

∂β

∂xβϕ(x))TC2ϕ(t) ≈ (Dβϕ(x))

TC2ϕ(t) = ϕ(x)TDTβC2ϕ(t),

∂α

∂tαu(x, t) ≈ ϕ(x)TC1Dαϕ(t),

∂β

∂tβv(x, t) ≈ ϕ(x)TC2Dβϕ(t),

u(0, t) = f(t) ≈ F Tϕ(t), v(0, t) = g(t) ≈ GTϕ(t).

(6.3.3)

Using (6.3.3) in (6.3.1), we have

K1ϕ(x)TDT

αC1ϕ(t) +K2ϕ(x)TC2Dβϕ(t) = 0,

K3ϕ(x)TDT

βC2ϕ(t) +K4ϕ(x)TC1Dαϕ(t) = 0,

(6.3.4)


ϕ(0)TC1ϕ(t) = F Tϕ(t), ϕ(0)TC2ϕ(t) = GTϕ(t). (6.3.5)

By using Tau method [38], we obtain the following system of algebraic equations∫ 1

0

∫ 1

0Bi(x)ϕ(x)

T(K1DT

αC1 +K2C2Dβ

)ϕ(t)Bj(t)dxdt = 0,∫ 1

0

∫ 1

0Bi(x)ϕ(x)

T(K3DT

βC2 +K4C1Dα

)ϕ(t)Bj(t)dxdt = 0,(

ϕ(0)TC1 − F T)

Q = 0,(ϕ(0)TC2 −GT

)Q = 0,

(6.3.6)

where i = 0, 1, . . . ,m − 1 and j = 0, 1, 2, . . . ,m. Solving this system of algebraic equations for the

unknown C1, C2 and putting in (6.3.2), we get the desired result.

6.3.1 Applications

Example 6.3.1. Here we consider our coupled system with initial conditions f(t) = sin(t), g(t) =

cos(t) and the constants K1 = 1,K2 = 1,K3 = 1,K4 = −1. Since with α = β = 1 our system has

the exact solutions for u(x, t) = ex sin(t) and v(x, t) = ex cos(t). In Figures 6.7 and 6.8, we show the

errors of obtained results for m = 5 and α = β = 1. So, we can see the approximate solutions are good

agreement with analytical solutions.In [73], this example has been solved and errors in their approximate

solutions for x and v have been recorded by 10−3 with m = 5. While in our results which are based on

the OMBPs, are better as compared to [73]. In present work, the errors in the approximate solutions for

x, v are rounded by 10−4 for the same m. Also, figures 6.9 and 6.10 show the approximate solutions for

α = 1, β = 0.8, 0.9, 1 and m = 3. These figures show that for fixed α = 1, as β approaches close to 1,

the approximate solutions approach to the solutions for β = 1 as expected.

96

Figure 6.7: Error in our approximation for u(x, t), for m = 5 and α, β = 1.

Figure 6.8: Error in our approximation for v(x, t), for m = 5 and α, β = 1.

Figure 6.9: Plot of the approximation solutions u(x, t), for m = 3, α = 1 and β = 0.8, 0.9, 1.

97

Figure 6.10: Plot of the approximation solutions v(x, t), for m = 3, α = 1 and β = 0.8, 0.9, 1.

Example 6.3.2. Here we consider our coupled system with initial conditions f(t) = −t2, g(t) = 0 and

the constants K1 = 1,K2 = −1,K3 = 1,K4 = 1. Since with α = β = 1 our system has the exact

solutions for u(x, t) = x2 − t2 and v(x, t) = 2xt. The errors of approximate solutions with m = 3

and α = β = 1 are shown in figures 6.11 and 6.12. The results show the approximate solutions have

high accuracy. In [73], this example has been solved and errors in their approximate solutions for x and

V have been approximated by 10−3. While in our results which are based on OMBPs, are too better

as compared to [73]. In present work, the errors in the approximate solutions for x, V are rounded

by 10−15. Also, figures 6.13 and 6.14 show the approximate solutions for α = 1, β = 0.8, 0.9, 1 and

m = 3. Similar to previous example, These figures show that for α = 1, as β approaches close to 1, the

approximate solutions approach to the solutions for β = 1 as expected.

Figure 6.11: Error in our approximation for u(x, t), for m = 3 and α, β = 1.

98

Figure 6.12: Error in our approximation for v(x, t), for m = 3 and α, β = 1.

Figure 6.13: Plot of the approximation solutions u(x, t), for m = 3, α = 1 and β = 0.8, 0.9, 1.

Figure 6.14: Plot of the approximation solutions v(x, t), for m = 3, α = 1 and β = 0.8, 0.9, 1.

99

Chapter 7

Fractional integral and product matricesof BPs

In this chapter, we propose Bernstein polynomials (BPs) to achieve the numerical solutions of nonlinear

fractional-order chaotic system known by fractional order Brusselator system (FOBS).

By the help of OMBPs for fractional integration and multiplications, a numerical techniques is estab-

lished for the solution of fractional order Logistic equation (FOLE), fractional order Brusselator system

(FOBS) and for an optimal controle problem (OCP). This is joint work with Professor Dumitru Baleanu,

Professor Hossein Jafari, Assistant Professor Mohsen Alipour and Haleh Tajadodi [135, 139, 143]. In

section 7.1, we discuss OMBPs for fractional integration. In section 7.2, detail about OMBPs for mul-

tiplication is given. In section 7.3, a numerical technique for the solution of fractional order Logistic

equation is produced which is applied on two examples in subsection 7.3.1. In section 7.4, a numerical

technique for the solution of fractional order Brusselator system(FOBS) is developed which is applied

for the solution of two problems of FOBS in subsection 7.4.1. In section 7.5, a numerical technique for

the solution of a fractional order optimal controle problem (OCP) is obtained. The technique for OCP is

applied for the solution of two problems of fractional OCPs in subsection 7.5.1.

7.1 Operational matrix for fractional order integration

Lemma 7.1.1. For ϕ(t) = [B0,m(t), B1,m(t), . . . , Bm,m(t)]T , the OMBPs for fractional integral Iα for

α an arbitrary real valued number, is given by:

Iαt ϕ(t) ≈ Iαϕ(t), (7.1.1)

where Iα = ADET , is a square matrix of (m+ 1)× (m+ 1).

100

Proof. The operational matrices of fractional Integration of the vector ϕ(t) can be approximated as [83]:

0Iαt ϕ(t) ≃ Iαϕ(t), (7.1.2)

where Iα is the (m + 1) × (m + 1) Riemann-Liouville fractional operational matrix of integration for

BPs.

0Iαt ϕ(t) =

1

Γ(α)

∫ t

0(t− τ)α−1ϕm(τ)dτ =

1

Γ(α)tα−1 ∗ ϕ(t), (7.1.3)

where ∗ denotes the convolution product and by substituting ϕ(t) = ATm(t) and from Eq. (1.1.4) we

get:

1

Γ(α)tα−1 ∗ ϕ(t) =

1

Γ(α)tα−1 ∗ (ATm(t)) =

1

Γ(α)A(tα−1 ∗ Tm(t))

=A

Γ(α)

[tα−1 ∗ 1, tα−1 ∗ t, . . . , tα−1 ∗ t

]T= A [Iα1, Iαt, . . . , Iαtm]T

= A

[0!

Γ(1 + α)tα,

1!

Γ(α+ 2)tα+1, . . . ,

m!

Γ(α+m+ 1)tα+m

]T= ADTm,

where D and Tm are given by

D =

0!

Γ(1+α) 0 . . . 0

0 1!Γ(α+2) . . . 0

......

. . ....

0 0 . . . m!Γ(α+m+1)

and Tm =

tα

tα+1

...

tα+m

.

Now we approximate tk+α by m+ 1 terms of the Bernstein basis

tα+i ≃ ETi ϕm(t), (7.1.4)

where,

Ei = Q−1

(∫ 1

0tα+iϕ(t)dt

)(7.1.5)

= Q−1

[∫ 1

0tα+iB0,m(t)dt,

∫ 1

0tα+iB1,m(t)dt,. . . ,

∫ 1

0tα+iBm,m(t)dt

]T=Q−1Ei,

Ei = [Ei,0, Ei,1, . . . , Ei,m] and

Ei,j =

∫ 1

0tα+iBi,j(t)dt =

m!Γ(i+ j + α+ 1)

j!Γ(i+m+ α+ 2), i, j = 0, 1, . . . ,m, (7.1.6)

where E is an (m+1)× (m+1) matrix that has vector Q−1Ei for ith columns. Therefore, we can write

Iαϕ(t) = AD[ET0 ϕ(t), E

T1 ϕ(t), . . . , E

Tmϕ(t)]

T = ADETϕ(t). (7.1.7)

Finally, we obtain

0Iαt ϕ(t) ≃ Iαϕ(t) (7.1.8)

where

Iα = ADET , (7.1.9)

is operational matrix for fractional integration.

101

7.2 Operational matrix for multiplication

Lemma 7.2.1. Suppose y(t) ≈ cTϕ(t), x(t) ≈ DTϕ(t) and C(m+1)×(m+1) be the operational matrix of

product using BPs for vector c. Then the approximation of product x(t)y(t) can be expressed by

x(t)y(t) ≈ ϕ(t)Cd. (7.2.1)

Proof. The operational matrices for the product C is given by

CTϕ(t)ϕ(t)T ≃ ϕ(t)T C, (7.2.2)

where C is an (m+ 1)× (m+ 1) matrix. So we have

CTϕ(t)ϕ(t)T = CTϕ(t)(Tm(t)TAT

)= [CTϕ(t), t(CTϕm(t)), . . . , tm(CTϕm(t))]AT

=

[n∑

i=0

ciBi,m(t),

n∑i=0

citBi,m(t), . . . ,

n∑i=0

citmBi,m(t)

]. (7.2.3)

Now, we approximate all functions tkBi,n(t) in terms of {Bi,m}mi=0 for i, k = 0, 1, . . . ,m. By (1.2.4),

we have

tkBi,m(t) ≃ eTk,iϕm(t). (7.2.4)

where ek,i = [e0k,i, e1k,i, . . . , e

mk,i]

T , then we obtain the components of the vector of ek,i

ejk,i = Q−1

(∫ 1

0tkBi,m(t)ϕ(t)dt

)= Q−1

[∫ 1

0tkBi,m(t)B0,m(t)dt,

∫ 1

0tkBi,m(t)B1,m(t)dt,. . . , tkBi,m(t)Bm,m(t)dt

]T=

Q−1

2m+ k + 1

[ (m0

)(2m+ki+k

) , (m1

)(2m+ki+k+1

) , . . . , (mm

)(2m+ki+k+m

)]T i, k = 0, 1, . . . ,m. (7.2.5)

Finally, we obtain

n∑i=0

citkBi,m(t) =

n∑i=0

ci

n∑j=0

ejk,iBj,m(t)

=

n∑j=0

Bj,m(t)

(n∑

i=0

ciejk,i

)

= ϕm(t)T

[n∑

i=0

cie0k,i,

n∑i=0

cie1k,i, . . . ,

n∑i=0

ciemk,i

]T= ϕm(t)T [ek,0, ek,1, . . . , ek,m]C = ϕm(t)TVk+1C, (7.2.6)

where Vk+1(k = 0, 1 . . . ,m) is an (m+ 1)× (m+ 1) matrix that has vectors ek,i(i = 0, 1, . . . ,m) for

each column. If we choose an (m+ 1)× (m+ 1) matrix

C = [V1c, V1c, . . . , Vm+1c], then from (7.2.3) and (7.2.6), we write

CTϕ(t)ϕ(t)T ≃ ϕ(t)T CAT , (7.2.7)

where C = CAT is the operation matrix of product.

102

Corollary 7.2.1. Suppose y(t) ≈ cTϕ(t) and C(m+1)×(m+1) be the operational matrix of product using

BPs for vector c, then the approximation of function yk(t), (k ∈ N) is given by

yk(t) ≈ ϕ(t)Ck. (7.2.8)

where Ck ≈ Ck−1c.

7.3 Fractional order Logistic equation

In this section, we are interested to develop a numerical technique for solutions of fractional order

Logistic equations (FOLE) using operational matrices of fractional order integration and multiplication

based on BPs

Dαx(t) = βx(t)(1− x(t)), t > 0, β > 0, α ∈ (0, 1] (7.3.1)

with initial condition

x(0) = b (7.3.2)

We expand fractional derivative in (6.3.1) by the Bernstein basis ϕ as follows:

Dαx(t) = HTϕ(t) (7.3.3)

where

HT = [c0, c1, . . . , cm], ϕT = [Υ0,m,Υ1,m, . . . ,Υm,m], (7.3.4)

are unknowns. Using initial condition (7.3.2), (1.1.5) and (7.3.1), we have

x(t) = 0Iαt Dα

t x(t) + x(0) ≈ (HTIα +DT )ϕ(t) = GTϕ(t), (7.3.5)

where (HTIα+DT ) = GTα and Iα is the fractional operational matrix for integration of order α defined

by (7.1.9) and DT = [b, b, . . . , b]. By substituting the approximations (7.4.3), (7.4.5) in (7.3.1), we have

HTϕ(t) ≈ βGTϕ(t)(Id −GTϕ(t)) = β(GTϕ(t)−GTϕ(t)ϕT (t)G)

= β(GTϕ(t)− ϕT (t)GG) = β(GT −GT GT )ϕ(t)(7.3.6)

where Id is the identity matrix. Thus from (7.3.6), we obtain the following system

HT − β(GT −GT GT ) = 0 (7.3.7)

from this system we getH and hence we approximate the solution x(t) of FOLE (7.3.1) by using (7.3.5).

103

7.3.1 Applications

Here we consider three examples of FOLE for the numerical solutions in order to check the accuracy of

our scheme. We consider the solutions for different values of α and compare with the exact solutions in

these examples.

Example 7.3.1. We consider the following FOLE [96]:

Dαx(t) =1

2x(t)(1− x(t)), t > 0, 0 < α ≤ 1,

x(0) =1

2,

(7.3.8)

for the numerical solutions for different values of α by our prescribed scheme. The solution is plotted in

comparison with the exact solution

x(t) =Exp( t2)

(1 + Exp( t2)).

Our approximations are too close to the exact solution in interval [0, 1] and beyond this interval gradual

deviation is observed. We provide the solutions in [0, 2], by 7.1 and the error plot is given in interval

[0, 1], by 7.2.

0.5 1.0 1.5 2.0x

0.55

0.60

0.65

0.70

yHxL

Figure 7.1: Approximation of FOLE in example 7.3.1, where the black line represents the exact solution, Red

line is for α = 1, Green line is for α = .99, Doted line is for α = .98, and x ∈ [0, 2] .

Example 7.3.2. We consider a FOLE [96]:

Dαx(t) =1

4x(t)(1− x(t)), t > 0, 0 < α ≤ 1,

x(0) =1

3,

(7.3.9)

for the numerical solutions for different values of α by our prescribed scheme. The approximate solu-

tions are plotted in comparison with the exact solution

x(t) =Exp( t4)

(2 + Exp( t4))

104

0.2 0.4 0.6 0.8 1.0x

0.0002

0.0004

0.0006

0.0008

yHxL

Figure 7.2: Error in our approximation for the FOLE in example 7.3.1 for α = 1 when x ∈ [0, 1].

which shows too better results in interval [0, 1], and beyond this interval little deviation is observed. We

provide the solutions in interval [0, 2], by 7.3.3, and the error plot is given in interval [0, 1], by 7.4.

0.5 1.0 1.5 2.0x

0.35

0.40

0.45

yHxL

Figure 7.3: Approximation of FOLE in example 2 via BPs operational matrices. Here Black line=Exact solution,

Green line is for α = 1, Doted line is for α = .99, and the Red line is for α = .95, and x ∈ [0, 2].

0.2 0.4 0.6 0.8 1.0x

0.00005

0.00010

0.00015

yHxL

Figure 7.4: Error in our approximation of the FOLE in example 7.3.2 for α = 1 when x ∈ [0, 1].

105

Example 7.3.3. We consider a FOLE:

Dαx(t) =1

3x(t)(1− x(t)), t > 0, 0 < α ≤ 1,

x(0) =1

3,

(7.3.10)

for the numerical solutions for different values of α by our prescribed scheme. The solution is plotted in

comparison with the exact solution

x(t) =Exp( t3)

(2 + Exp( t3)).

The solutions obtained are tending close to the exact solution in interval [0, 1] when α approaches 1 and

beyond this interval deviation is observed. We provide the solutions in interval [0, 2], by 7.5 and the

error plot is given in the interval [0, 1], by 7.6.

0.5 1.0 1.5 2.0x

0.35

0.40

0.45

0.50

yHxL

Figure 7.5: In this figure, we present comparison of our approximation with the exact solution in example 7.3.3,

for different values of α. Here the Dashed line represents exact solution at α = 1, where our approximations are:

Green line is for α = 1, Red line is for α = .99, Gray line is for α = .98, Black line is for α = .95, and x ∈ [0, 2].

0.2 0.4 0.6 0.8 1.0x

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035

yHxL

Figure 7.6: Error in our approximate solution of the FOLE in example 7.3.3, when α = 1, x ∈ [0, 1].

106

7.4 Fractional order Brusselator system (FOBS)

Chaos theory is considered to be an important tool for viewing and understanding our universe and

different ways are utilized in order to reduce problems produced by the unusual behaviors of chaotic

system including chaos control [92,93]. In literature, several authors have considered the chaotic system

FOBS recently. Such as: Gafiychuk and Datsko [95], investigated stability of FOBS. Wang and Li [94],

proved that the solution of FOBS has a limit cycle using numerical method. Jafari et al. [81], used the

variational iteration method for the investigation of the approximate solutions of a FOBS.

Here we are interested in getting the numerical solutions of the nonlinear FOBS:

Dαt x(t) = a− (µ+ 1)x(t) + x2(t)y(t), (7.4.1)

Dβt y(t) = µx(t)− x2(t)y(t),


x(0) = c1, y(0) = c2 (7.4.2)

via operational matrices of fractional order integration and multiplication of BPs. Here, a > 0, µ > 0,

α, β ∈ (0, 1], and c1, c2 are constants and Dα,Dβ represent Caputo’s derivatives of order α, β respec-

tively. Expanding Dαy(t) in the basis of Bernstein polynomials, we have

Dαy(t) = CTϕ(t), (7.4.3)

where

CT = [c0, c1, . . . , cm], ϕT = [B0,m, B1,m, . . . , Bm,m], (7.4.4)

are unknowns. Using initial condition (7.4.2), (1.1.5) and (7.1.9), we approximate x(t) by

x(t) = 0Iαt Dα

t x(t) + x(0) ≈ (CT1 Iα +DT )ϕ(t) = GTϕ(t), (7.4.5)

where (CT1 Iα +DT ) = GT

α and Iα is the operational matrix for fractional order integration of order α

defined by (7.1.9) and

DT = [x0, x0, . . . , x0]. (7.4.6)

Similarly, we approximate y(t) from (7.4.1) by BPs as

y(t) = 0Iαt D

βt x(t) + y(0) ≈ (CT

2 Iβ + dT2 )ϕ(t) = HTϕ(t), (7.4.7)

where (CT2 Iβ +DT ) = HT

β and Iβ is the fractional operational matrix of integration of order β defined

by (7.1.9) and

dT2 = [y0, y0, . . . , y0]. (7.4.8)

107

Substituting (7.4.3)–(7.4.5) and (7.4.7) in (7.4.1), we get

CT1 ϕ(t) = ATϕ(t)− (µ+ 1)GTϕ(t) +HTϕ(t)Gϕ(t)ϕT (t)GT

1 ,

CT2 ϕ(t) = µGTϕ(t)−HTϕ(t)GTϕ(t)ϕT (t)G.

(7.4.9)

Now using matrix of multiplication (7.2.7) in (7.4.9), we have:

CT1 ϕ(t) = ATϕ(t)− (µ+ 1)GTϕ(t) + ϕT (t)HGG, (7.4.10)

CT2 ϕ(t) = µGTϕ(t)− ϕT (t)HGG.

From (7.4.10) we have the following system(CT1 −AT + (µ+ 1)GT −GT GT HT

)ϕ(t) = 0, (7.4.11)(

CT2 − µGT +GT GT HT

)ϕ(t) = 0. (7.4.12)

Using the independent property of BPs, we get

CT1 −AT + (µ+ 1)GT −GT GT HT = 0, (7.4.13)

CT2 − µGT +GT GT HT = 0, (7.4.14)

solving this system for the vectors C1, C2, we can approximate x(t) and y(t) from (7.4.5), (7.4.7)

respectively.

7.4.1 Applications

Below we use the presented approach in order to solve two examples.

Example 7.4.1. We consider FOBS given by [81]:

Dαt x(t) = −2x(t) + x(t)2y(t), (7.4.15)

Dβt y(t) = x(t)− x(t)2y(t),

with initial conditions x(0) = 1, y(0) = 1.

Fig. 7.7 presents comparison between exact and approximate solution obtained by the help of BPs

for x(t), y(t) at α = 1, β = 1 when m = 8, 12 and Fig. 7.8 presents comparison between exact solution

and our approximate solution by BPs for (t), y(t) at m = 12 and different values of α and β.

Example 7.4.2. We demonstrate accuracy of the presented numerical technique by considering the FOBS

given by [81]:

Dαx(t) = 0.5− 1.1x(t) + x(t)2y(t), (7.4.16)

Dβy(t) = 0.1x(t)− x(t)2y(t),

with initial conditions x(0) = 0.4, y(0) = 1.5.

108

0.2 0.4 0.6 0.8 1t

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xHtL

0.2 0.4 0.6 0.8 1t

1.025

1.05

1.075

1.1

1.125

1.15

1.175

yHtL

Figure 7.7: The exact solution: (black line) and approximation solutions when α = 1, β = 1 and m = 12

(dotted), m = 8(dashed).

0.2 0.4 0.6 0.8 1t

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xHtL

0.2 0.4 0.6 0.8 1t

1.025

1.05

1.075

1.1

1.125

1.15

1.175

yHtL

Figure 7.8: The exact solution: (black line) and approximation solutions when m = 12 and α = .98, β = 1

(dotted), α = .95, β = .99(dashed), α = 0.9, β = .98(Long-dashed).

Fig. 7.9 demonstrates the exact solution together with the approximate solutions x(t), y(t) for

α, β = 1 and different values of m = 4, 6. Definitely, by increasing the value of m of Bernstein

basis, the approximate values of x(t), y(t) converge to the exact solutions. The approximate solutions

x(t), y(t) together with the exact solution for m = 6 and different values of α, β plotted in Fig. 7.10 we

see that as α approaches 1, the numerical solution converges to exact solution.

0.2 0.4 0.6 0.8 1t

0.45

0.5

0.55

0.6

0.65

0.7

xHtL

0.2 0.4 0.6 0.8 1t

1.15

1.25

1.3

1.35

1.4

1.45

1.5

yHtL

Figure 7.9: The exact solution: (black line) and approximation solutions when α = 1, β = 1 andm = 6 (dotted),

m = 4(dashed).

109

0.2 0.4 0.6 0.8 1t

0.45

0.5

0.55

0.6

0.65

0.7

xHtL

0.2 0.4 0.6 0.8 1t

1.15

1.25

1.3

1.35

1.4

1.45

1.5

yHtL

Figure 7.10: Fig.1. The exact solution: (black line) and approximation solutions when m = 6 and α = 1, β =

.98 (dotted), α = .98, β = .95(dashed).

7.5 Optimal controle problem

In this section, we are interested in the solution of the fractional order OCP:

Minimize:

J(x, u, b) =

∫ b

0L(t, x(t), u(t))dt, (7.5.1)

subjected to the dynamical system:

M1x(t) +M2Dαx(t) = f(t, x(t), u(t)), α ∈ (0, 1], (7.5.2)

x(0) = x0, x(b) = xb, (7.5.3)

where M1,M2 are constants and f, L are multi-variable polynomials of t, x(t), u(t). The scheme is

produced by the help of operational matrices for fractional order integration of Bernstein polynomi-

als(BPs), see for detail [90, 127]. Expending x(t) by Bernstein basis, as follows:

x(t) ≈ CT1 ϕ(t), (7.5.4)

where CT1 = [µ0, µ1, . . . , µm]. Integrating (7.5.4) and using initial condition in (7.5.2) we have

x(t) ≈ CT1 I1ϕ(t) + x0 = CT

1 I1ϕ(t) +XT0 ϕ(t) = HT

1 ϕ(t), (7.5.5)

were XT0 = [x0, x0, . . . , x0] and HT

1 = CT1 I1 +XT

0 . Similary, by the use of (1.1.5) and (7.5.4) we can

have the approximation

Dαx(t) = I1−αx(t) ≈ I1−αCT1 ϕ(t) = HT

2 ϕ(t), (7.5.6)

where HT2 = ΛT

1 I1−α. Expension of u(t) by Bernstein basis yield:

u(t) ≈ CT2 ϕ(t), (7.5.7)

110

where CT2 = [ν0, ν1, . . . , νm]. Thus

J(x, u, T ) ≈∫ b

0L(t,HT

1 ϕ(t), CT2 ϕ(t))dt. (7.5.8)

Using the corollaries 7.2.1, 7.2.1, we can approximate L(t,HT1 ϕ(t), C

T2 ϕ(t)) by:

L(t,HT1 ϕ(t), C

T2 ϕ(t)) ≈ L(H1, C2)ϕ(t), (7.5.9)

where L : R(m+1)×1 × R(m+1)×1 → R1×(m+1). Thus,

J∗(H1, C2) =

∫ b

0L(H1, C2)ϕ(t)dt, (7.5.10)

by the help of (7.5.4)–(7.5.7), the dynamical system (7.5.2), becomes

M1CT1 ϕ(t) +M2HT

2 ϕ(t) = f(t,HT1 ϕ(t), C

T2 ϕ(t)). (7.5.11)

By the help of Lemma 7.2.1, 7.2.1, we can get the approximation

f(t,HT1 ϕ(t), C

T2 ϕ(t)) ≈ F(H1, C2)ϕ(t), (7.5.12)

where F : R(m+1)×1×R(m+1)×1 → R1×(m+1). By the help of (7.5.11), (7.5.12), we have the following

equation:

M1CT1 ϕ(t) +M2HT

2 ϕ(t)−F(H1, C2)ϕ(t) = 0, (7.5.13)

and thus the dynamical system (7.5.2) gets the form

M1CT1 +M2HT

2 −F(HT1 , C

T2 ) = 0. (7.5.14)

Defining, the Lagrangian function as:

L∗[H1, C2, λ] = J∗[H1, C2] + (M1CT1 +M2HT

2 −F(H1, C2))λ+ λ(HT1 ϕ(b)− xb), (7.5.15)

where λ = [C0, C1, . . . , Cm]T , λ are the unknown Lagrange multipliers. For the minimization of the

optimal control problem (7.5.1), (7.5.2) we have the following necessary conditions

∂

∂C2L∗[H1, C2, λ] = 0,

∂

∂H1L∗[H1, C2, λ] = 0,

∂

∂λL∗[H1, C2, λ] = 0, HT

1 ϕ(b)− xb = 0.

(7.5.16)

From here, we can get C2, H1 and ultimately we can approximate x(t), u(t) from (7.5.5), (7.5.7),

respectively.

111

7.5.1 Applications

Example 7.5.1. We consider the OCP

Minimize J =

∫ 1

0(tu(t)− (α+ 2)x(t))2dt, (7.5.17)

subjected to the dynamical system

x′(t) +Dαx(t) = u(t) + t2, (7.5.18)


x(0) = 0, x(1) =2

Γ(α+ 3), (7.5.19)

the exact solution is given by x(t) = 2tα+2Γ(α+3) , u(t) = 2tα+1

Γ(α+2) .

For m = 7, the approximate solutions of the states functions and the control functions are plotted in

the figures 7.11, fign1.2opc. These figures show that as α→ 1 the approximate solutions get close to the

exact solution. Also, the error in the norm L2(∥f(t)∥L2[a,b] =√∫ b

a |f(t)|2dt) for approximate solutions

obtained are given in tables 1, 2 for x(t), u(t) respectively for different values of approximating terms

m.

Figure 7.11: Plot of x(t) for m = 7 and α = .8, .9, 1 in example 7.5.1.

Table 1. ∥x(t)− xm(t)∥ for different values of α and m in example 7.5.1.

m α = 1 α = 0.9 α = 0.8

7 2.89933 ∗ 10−17 7.5999 ∗ 10−7 1.89168 ∗ 10−6

9 2.41324 ∗ 10−17 1.65651 ∗ 10−7 4.27512 ∗ 10−7

12 1.03578 ∗ 10−17 2.89815 ∗ 10−8 7.7282 ∗ 10−8

112

Figure 7.12: Plot of u(t) for m = 7 and α = .8, .9, 1 in example 7.5.1.

Table 2. ∥u(t)− um(t)∥ for different values of α and m in example 7.5.1.

m α = 1 α = 0.9 α = 0.8

7 1.93584 ∗ 10−16 0.0000120377 0.0000314255

9 3.53488 ∗ 10−16 4.09026 ∗ 10−6 0.0000112193

12 8.26356 ∗ 10−17 1.16833 ∗ 10−6 3.38702 ∗ 10−6

Example 7.5.2. We consider the OCP

Minimize J =

∫ 1

0(u(t)− x(t))2dt, (7.5.20)

subjected to the dynamical system

x′(t) +Dαx(t) = u(t)− x(t) +6tα + 2

Γ(α+ 3), (7.5.21)


x(0) = 0, x(1) =6

Γ(α+ 4), (7.5.22)

the exact solution is given by x(t) = u(t) = 6tα+3Γ(α+4)

In figures 7.13 and 7.14, we can observe plots of the approximate solutions of the states function and

the control function for m = 8 and α = 0.8, 0.9, 1.0 . Moreover, in tables 3 and 4, we can see the errors

in the norm L2 for the obtained solutions.

Table 3. ∥x(t)− xm(t)∥ for different values of α and m in example 7.5.2.

m α = 1 α = 0.9 α = 0.8

4 1.11714 ∗ 10−18 0.0000288778 0.0000575561

8 7.53703 ∗ 10−19 7.17619 ∗ 10−8 1.80442 ∗ 10−7

12 1.06883 ∗ 10−20 3.14699 ∗ 10−9 8.29973 ∗ 10−9

113

Figure 7.13: Plot of x(t) for m = 8 and α = .8, .9, 1 in example 7.5.2.

Figure 7.14: Plot of u(t) for m = 8 and α = .8, .9, 1 in example 7.5.2.

Table 4. ∥u(t)− um(t)∥ for different values of α and m in example 7.5.2.

m α = 1 α = 0.9 α = 0.8

4 5.87904 ∗ 10−18 0.000112289 0.000221961

8 9.3817 ∗ 10−19 2.92602 ∗ 10−7 7.30618 ∗ 10−7

12 7.79996 ∗ 10−20 1.31311 ∗ 10−8 3.42575 ∗ 10−8

By these examples, we see that our numerical technique is highly efficient.

114

Chapter 8

B-Spline functions

In literature, several authors have considered the Fractional order Brusselator system recently. Like as;

Gafiychuk and Datsko [95], investigated stability of the fractional Brusselator system. Wang and Li [94],

proved that the solution of FOBS has a limit cycle using numerical method. Jafari et al. [81], used the

variational iteration method to investigate the approximate solutions of this system.

In this chapter, we discuss the numerical solution of FOBS by the OM of B-Spline functions for the

fractional integration and multiplcation matrix. In section 8.1, we present OM for fractional integration.

In section 8.2, a product matrix is presented. In section 8.3, a numerical technique for the solution of

FOBS is developed which is applied in subsection 8.3.1.

8.1 Operational matrix for fractional integration

The operational matrix of fractional integration of the vector ϕJ can be approximated as:

0Iαx ϕJ(x) ≃ IαϕJ(x), (8.1.1)

where Iα is the (2J + 1) × (2J + 1) operational matrix of fractional integration for linear B-spline

function. We obtain the matrix Iα as follows:

Iα =

∫ 1

0(0Iα

x ϕJ(x))ϕTJ (x)dx =

(∫ 1

0(0Iα

x ϕJ(x))ϕTJ (x)dx

)P−1, (8.1.2)

where

E =

∫ 1

0(0Iα

x ϕJ(x))ϕTJ (x)dx. (8.1.3)

115

In (8.1.3), E is a (2J + 1)× (2J + 1) matrix given by

E =

∫ 10 (0I

αx ϕj,−1(x))ϕ

Tj,−1(x)dx . . .

∫ 10 (0I

αx ϕj,−1(x))ϕ

Tj,2j−1

(x)dx...

. . ....∫ 1

0 (0Iαx ϕj,2j−1(x))ϕ

Tj,−1(x)dx . . .

∫ 10 (0I

αx ϕj,2j−1(x))ϕ

Tj,2j−1

(x)dx

, (8.1.4)

And we have 0Iαx ϕj,k(x) as follows:

0Iαx ϕj,k(x) = 0Iα

x

(2∑

i=0

(2

i

)(−1)i(2Jx− (k + i))+

)

=

0, x ≤ k2J,

2−Jα(2Jx−k)α+1

Γ(α+2) , k2J

≤ x < k+12J,

2−Jα((2Jx−k)α+1−2(2Jx−(k+1))α+1)Γ(α+2) , k+1

2J≤ x < k+2

2J,

2−Jα((2Jx−k)α+1−2(2Jx−(k+1))α+1+(2Jx−(k+2))α+1)Γ(α+2) , x ≥ k+2

2J.

8.2 Operational matrix of multiplication

It is always necessary to assess the product of ϕ(t) and ϕ(t)T , which is called the product matrix for

linear B-spline function. The operational matrices for the product C is given by

CTϕJ(t)ϕJ(t)T ≃ ϕJ(t)

T C, (8.2.1)

where C is an (m+ 1)× (m+ 1) matrix. So we have

CTϕJ(x)ϕJ(x)T = cTϕJ(x)[ϕj,−1(x), ϕj,0(x), . . . , ϕj,2j−1(x)] (8.2.2)

=

2j−1∑i=−1

ciϕj,i(x)ϕj,−1(x),2j−1∑i=−1

ciϕj,iϕj,0(x), . . . ,2j−1∑i=−1

ciϕj,iϕj,2j−1(x)

,Now, we approximate the product of ϕj,i(x)ϕj,k(x) in terms of ϕj,i for i, k = −1, . . . , 2j − 1, as

ϕj,i(x)ϕj,k(x) ≈ eTk,iϕJ , (8.2.3)

where ek,i = [e0k,i, e1k,i, . . . , e

mk,i]

T , then we obtain the components of the vector of ek,i

ejk,i = P−1

∫ 1

0ϕj,i(x)ϕj,k(x)ϕJ(x)dx (8.2.4)

= P−1

∫ 10 ϕj,i(x)ϕj,k(x)ϕj,−1(x)dx∫ 10 ϕj,i(x)ϕj,k(x)ϕj,0(x)dx

...∫ 10 ϕj,i(x)ϕj,k(x)ϕj,2j−1(x)dx

, (8.2.5)

116

Thus we obtain finally

2j−1∑i=−1

ciϕj,i(x)ϕj,k(x) ≈2j−1∑i=−1

ci

2j−1∑n=−1

ek,in ϕj,n

=2j−1∑n=−1

ϕj,n

2j−1∑i=−1

ciek,in

= ϕTJ

∑2j−1i=−1 cie

k,i−1∑2j−1

i=−1 ciek,i0

...∑2j−1i=−1 cie

k,i2j−1

= ϕTJ [ek,−1, ek,0, . . . , ek,2j−1]C

= ϕTJ Ck+2C, (8.2.6)

where Ck+2(k = −1, 0, . . . , 2j − 1) is an (2J + 1) × (2J + 1) matrix that has vectors ek,i(i =

−1, 0, . . . , 2j − 1) for each column. Therfore we obtain the operational matrix of product C = Ck+2C.

Corollary 8.2.1. If y(t) = CTϕJ(t), consequently we can get the approximate function for yk(t), using

BPs as follows:

yk(t) = ϕJ(t)T Ck, (8.2.7)

where Ck = Ck−1C and C is (2j+1)×(2j+1) operational matrix of product using B–spline functions.

8.3 Fractional order Brusselator system

Here, we are interested in getting the numerical solution of the nonlinear fractional Brusselator system:

Dαt x(t) = a− (µ+ 1)x(t) + x2(t)y(t), (8.3.1)

Dβt y(t) = µx(t)− x2(t)y(t),


x(0) = c1, y(0) = c2, (8.3.2)

provided that a > 0, µ > 0, α, β ∈ (0, 1], and c1, c2 are constants. Moreover, Dα,Dβ represent Caputo’s

derivative of order α, β respectively. For this, we give a numerical technique for the solutions of FOBS,

based on the operational matrices (OMs) of functions approximations, OM for multiplications and OM

for fractional order integration. By the proposed method the FOBS system (8.3.1), will be replaced in a

system of algebraic equations. For this purpose, we take a start from functions approximation (1.2.15),

as under:

Dαx(t) ≃ CT1 ϕJ(t), (8.3.3)

117

CT1 = [c−1, c0, . . . , c2j−1]

T and ϕJ as defined in (1.2.17). Using initial condition (8.3.4) and (1.1.4), we

approximate x(t) by

x(t) = 0Iαt Dα

t x(t) + x(0) ≈ (CT1 Iα + dT1 )ϕJ(t) = CTϕJ(t), (8.3.4)

where dT1 = [x0, x0, . . . , x0] and C = CT1 Iα + dT1 . Similarly, approximating

Dαy(t) ≈ CT2 ϕJ(t), (8.3.5)

where CT2 = [g−1, g0, . . . , g2j−1]

T and ϕJ as defined in (1.2.17). Using initial condition (8.3.2) and

(1.1.4), we approximate y(t) by

y(t) = 0Iαt D

βt y(t) + y(0) ≈ (CT

2 Iβ + dT2 )ϕJ(t) = GTϕJ(t), (8.3.6)

where dT2 = [y0, y0, . . . , y0] and G = CT2 Iβ + dT2 . Using (8.2.7), (8.3.6), we are approximating the

terms x2(t), x2(t)y(t):

x2(t) = x(t)x(t) ≈ CTϕJ(t)ϕTJ (t)C ≈ ϕTJ (t)CC = CT CTϕJ(t), (8.3.7)

and

y(t)x2(t) ≃ GTϕJ(t)ϕTJ (t)CC ≃ ϕTJ (t)GCC = CT CT GTϕJ(t). (8.3.8)

By the help of (8.3.3), (8.3.7), (8.3.8), the FOBS (8.3.1), (8.3.2), gets the form:

CT1 ϕJ(t) = ATϕJ(t)− (µ+ 1)CTϕJ(t) + CT CT GTϕJ(t), (8.3.9)

CT2 ϕJ(t) = µGTϕJ(t)− CT CT GTϕJ(t). (8.3.10)

Finally, we get the following algebraic system:

CT1 −AT + (µ+ 1)CT − CT CT GT = 0, (8.3.11)

CT2 − µGT + CT CT GT = 0, (8.3.12)

solving this system for C1, C2 we can approximate the solutions x(t) and y(t) from (8.3.4), (8.3.6)

respectively.

8.3.1 Applications

Below we use the presented approach in order to solve fractional fractional order Brusselator system.

Example 8.3.1. We consider fractional order Brusselator system given by [81]:

Dαt x(t) = −2x(t) + x(t)2y(t), (8.3.13)

Dβt y(t) = x(t)− x(t)2y(t),

with initial conditions x(0) = 1, y(0) = 1.

118

0.2 0.4 0.6 0.8 1t

0.4

0.6

0.8

xHtL

0.2 0.4 0.6 0.8 1t

1.025

1.05

1.075

1.1

1.125

1.15

1.175

yHtL

Figure 8.1: The exact solution (black line) and approximation solutions when α = 1, β = 1 and J = 5 (dotted)

and J = 4 (dashed).

0.2 0.4 0.6 0.8 1t

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xHtL

0.2 0.4 0.6 0.8 1t

1.025

1.05

1.075

1.1

1.125

1.15

1.175

yHtL

Figure 8.2: The exact solution (black line) and approximation solutions when J = 5 and α = 0.98, β = 1

(dotted) and α = 0.95, β = 0.98 (dashed).

In Figure 8.1 presents comparison between exact and approximate solution obtained by the help of

the linear B-spline function for x(t), y(t) at α = 1, β = 1 when J = 4, 5 and Figure 8.2 presents

comparison between exact solution and our approximate solution for x(t), y(t) at J = 5 and different

values of α and β.

Example 8.3.2. We demonstrate accuracy of the presented numerical technique by considering the FOBS

given by [81]:

Dαx(t) = 0.5− 1.1x(t) + x(t)2y(t), (8.3.14)

Dβy(t) = 0.1x(t)− x(t)2y(t),

with initial conditions x(0) = 0.4, y(0) = 1.5.

Figure 8.3 demonstrates the exact solution together with the approximate solutions x(t), y(t) for

α, β = 1 and different values of J = 4, 5. Definitely, by increasing the value of J of B-spline basis, the

approximate values of x(t), y(t) converge to the exact solutions. The approximate solutions x(t), y(t)

together with the exact solution for J = 5 and different values of α, β plotted in Figure 8.4 we see that

as α approaches 1, the numerical solution converges to exact solution.

119

0.2 0.4 0.6 0.8 1t

0.45

0.5

0.55

0.6

0.65

0.7

xHtL

0.2 0.4 0.6 0.8 1t

1.15

1.25

1.3

1.35

1.4

1.45

1.5

yHtL

Figure 8.3: The exact solution: (black line) and approximation solutions when α = 1, β = 1 and J = 5 (dotted),

J = 4(dashed).

0.2 0.4 0.6 0.8 1t

0.45

0.5

0.55

0.6

0.65

0.7

xHtL

0.2 0.4 0.6 0.8 1t

1.15

1.25

1.3

1.35

1.4

1.45

1.5

yHtL

Figure 8.4: The exact solution (black line) and approximation solutions when J = 5 and α = 0.98, β = 0.99

(dotted) and α = 0.97, β = 0.98 (dashed).

Nomenclature

In this theses we have used the following abbreviations:

FDEs for fractional differential equations, ES for existence of solutions, EUS for existence and unique-

ness of solutions, BVP for boundary value problem, HFDEs for hybrid fractional differential equations,

CSFDIEs for coupled system of fractional differential integral equations, FTE for fractional Telegraph

equation, FPDE for fractional partial differential equation, OM for operational matrix, OMBPs opera-

tional matrices for Bernstein polynomials, OCP for optimal controle problem, FOBS for fractional order

Brusselator system, LF for local fractional, LFSE for local fractional Schrodinger equation, SEM for

series expension method, LFLT for Yang laplace transform, LFLVIM for local fractional laplace vari-

ational iteration method, FVTFM for Fractal Vehicular Traffic flow model and LFWEs for the local

fractional wave equations. Other abbreviations have explained in the relevant work.

We declare that the thesis is our original work and before the submission of the final copy, Professor

Dr Rahmat Ali Khan has done the proof reading.

120

Bibliography

[1] I. Podlubny “Fractional differential equations”, Academic Press, San Diego, 1999.

[2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo “Theory and applications of fractional differential

equations” North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam, 2006.

[3] D. D. Ganji, A. Sadighi “Application of He’s Homotopy-Perturbation method to nonlinear coupled

systems of reaction diffusion equations”, Int. J. Nonlinear Sci. Numer. Simul., 7(4), 2006.

[4] V. Gafiychuka, B. Datskob, V. Meleshkob, D. Blackmorec “Analysis of the solutions of coupled

nonlinear fractional reaction–diffusion equations”, Chaos, Solitons & Fractals, 41(3), 1095–1104

(2009).

[5] V. S. Ertrka, S. Momani “Solving systems of fractional differential equations using differential trans-

form method” J. Comput. Appl. Math., 215, 142 –151 (2008).

[6] R. Hilfer(Ed) “Application of fractional calculus in physics, World scientific publishing Co. Singa-

pore, 2000.

[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo “ Theory and applications of fractional differential

equations, North-Holland Mathematics Studies,Amsterdame, 24 2006.

[8] K. S. Miller and B. Ross “An introduction to the fractional calculus and fractional differential equa-

tions, Wiley, 1993.

[9] K. B. Oldhalm and J. Spainer “The fractional calculus, Academic Press, New York, 1974.

[10] J. Sabatier, O. P. Agrawal, J. A. Tenreiro and Machado, Adv. Frac. Calc., Springer, 2007.

[11] M. El-Shahed “On the existence of positive solutions for a boundary value problem of fractional

order”, Thai J. Math., 5, 143-150 (2007).

121

[12] M. Benchohra, J. R. Graef and S. Hamani “ Existence results for boundary value problems with

nonlinear fractional differential equations, Appl. Anal., 87, 851–863(2008).

[13] B. Ahmad, J. J. Nieto “Existence of solutions for nonlocal boundary value problems of higher-order

nonlinear fractional differential equations, Abstr. Appl. anal., art. ID 494720, 9pp. (2009).

[14] B. Ahmad, J. J. Nieto “Existence results for a coupled system of nonlinear fractional differential

equations with three point boundary conditions, Comput. Math. Appl., 58, 1838-1843 (2009).

[15] C. Yuan, D. Jiang and X. Xu “Singular positone and semipositone boundary value problems of

nonlinear fractional differential equations Math. Probl. Engineering, 2009, Article ID 535209, 17

pp. (2009).

[16] M. Rehman and R. A. Khan “Existence and uniqueness of solutions for multi-point boundary value

problems for fractional differential equations, Appl. Math. Letters, 23, 1038-1044 (2010).

[17] X. Dou, Y. Li, P. Liu “Existence of solutions for a four point boundary value problem of a nonlinear

fractional diffrential equation”, Open Math. 31, 359-372 (2011).

[18] C. S. Goodrich “Existence and uniqueness of solutions to a fractional difference equation with

non-local conditions” Comput. Math. Appl., 61, 191-202(2011).

[19] R. A. Khan and M. Rehman “Existence of multiple positive solutions for a general system of

fractional differential equations, Commun. Appl. Nonlinear Anal, 18, 25-35(2011).

[20] R. A. Khan, M. Rehman and N. Asif “Three point boundary value problems for nonlinear fractional

differential equations”, Acta Math. Sci., 31(4B), 1-10(2011).

[21] M. Rehman and R.A. Khan “A note on boundary value problems for a coupled system of fractional

differential equations”, Comput. Math. Appl, 61, 2630–2637 (2011).

[22] M. Rehman, R.A. Khan and J. Henderson “Existence and uniqueness of solutions for nonlinear

fractional differential equations with integral boundary conditions, Fract. Dyn.Syst, 1 , 29-43(2011).

[23] X. Su “Boundary value problem for a coupled system of nonlinear fractional differential equations,

Appl. Math. Lett., 22, 64–69(2009).

[24] R. A. Khan “Three-point boundary value problems for higher order nonlinear fractional differential

equations, J. Appl. Math. Informatics, 31, 221–228(2013).

122

[25] Q. Zhang, S. Chen and J. Lu “Upper and lower solution method for fourth-order four point bound-

ary value problems”, J. Comput. Appl. Math., 196, 387-393 (2006).

[26] M. Benchohra, N. Hamidi and J. Henderson “Fractional differential equations with anti-periodic

boundary conditions, Numerical Funct. Anal. and Opti., 34, 404-414 (2013).

[27] Z. Bai and H. Lu “Positive solutions for boundary value problem of nonlinear fractional differential

equation”, J. Math. Anal. Appl., 311(2), 495–505, (2005).

[28] X. Xu, D. Jiang and C. Yuan “Multiple positive solutions for the boundary value problem of a

nonlinear fractional differential equation. Nonlinear Anal., 71, 4676–4688 (2009).

[29] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equa-

tions. Electron. J. Differ. Equat., 2006;36:1–12.

[30] S. Liang and J. Zhang, Positive solutions for boundary value problems of nonlinear fractional

differential equation. Nonlinear Anal., 2009, 71(55), 45–50 (2009).

[31] A. H. Sallo, A. S. Hasan “Existence of solutions for nonlinear fractional differential equation with

integral boundary conditions”. Int. J. Eng. Sci., 2, 18–22 (2013).

[32] R. Agarwal, M. Meehan, D. O’Regan “Fixed point theory and applications”, Cambridge University

Press, Cambridge, 2001.

[33] R. A. Khan, H. Khan “Existence of solution for a three point boundary value problem of fractional

differential equation”, J. Fract. Cal. Appl., 5(1), 156–164 (2014).

[34] W. Yang “Positive solutions for boundary value problems involving nonlinear fractional q-

difference equations”, Differ. Equ. Appl., 5 (2013), 205–219 (2013).

[35] Y. Zhao, H. Chen, Q. Zhang “Existence results for fractional q-difference equations with nonlocal

q-integral boundary conditions”, Adv. Differ. Equ., 2013, 48, 15 pages(2013).

[36] Y. Zhao, H. Chen, Q. Zhang “Existence and multiplicity of positive solutions for nonhomogeneous

boundary value problems with fractional q-derivative”, Bound. Value Probl., 2013, 103, 15 pages

(2013) .

[37] Y. Zhao, G. Ye, H. Chen “Multiple positive solutions of a singular semipositone integral boundary

value problem for fractional q-derivatives equation”, Abstr. Appl. Anal., 2013, Art. ID 643571, 12

pages (2013).

123

[38] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang “Spectral Methods in Fluid Dynamic”,

Prentice-Hall, Englewood Cliffs, NJ, 1988.

[39] M. El-Shahed and M. F. Al-Askar, “Positive solutions for boundary value problem of nonlinear

fractional q-difference equation”, ISRN Math. Anal. 2011, Article ID 385459 (2011).

[40] R. A. C. Ferreira “Nontrivial solutions for fractional q-difference boundary value problems”, Elec-

tron. J. Qual. Theory Differ. Equ., 70, 1-10 (2010).

[41] B. Ahmad and S. K. Ntouyas “Fractional q-difference hybrid equations and inclusions with Dirich-

let boundary conditions”, Adv. Differ. equ., 2014: 199(2014).

[42] B. Ahmad, S. K. Ntouyas, A. Alsaedi and H. Al-Hutami “Nonlinear q-fractional differential equa-

tions with nonlocal and sub-strip type boundary conditions”, Electron. J. Qual. Theory Differ. Equ.,

26, 112 (2014).

[43] G. Chai and S. Hu “Existence of positive solutions for a fractional high order three-point boundary

value problem” Adv. Differ. equ., 2014:90 (2014).

[44] W. A. Al-Salam, “Some fractional q-integrals and q-derivatives”, Proc. Edinburgh Math. Soc.,

15(2), 135–140 (1966/1967).

[45] R. P. Agarwal “Certain fractional q-integrals and q-derivatives”, Proc. Cambridge Philos. Soc., 66,

365–370 (1969).

[46] R. Finkelstein, E. Marcus “Transformation theory of the q-oscillator”, J. Math. Phys., 36, 2652-

2672 (1995).

[47] P. G. O. Freund, A. V. Zabrodin “The spectral problem for the q-Knizhnik-Zamolodchikov equation

and continuous q-Jacobi polynomials”, Commun. Math. Phys., 173, 17-42 (1995).

[48] X. Li, Z. Han, S. Sun and P. Zhao “Existence of solutions for fractional q-difference equation with

mixed nonlinear boundary conditions”, Adv. Differ. Equ., 2014:326 (2014).

[49] B. Ahmad, S. K. Ntouyas and I. K. Purnaras “Existence results for nonlocal boundary value prob-

lems of nonlinear fractional q-difference equations”, Adv. Difference Equ., 2012:140 (2012).

[50] X. Li, Z. Han, S. Sun and H. Lu “Boundary value problems for Fractional q-difference equations

with nonlocal conditions”, Adv. Differ. Equ., 2014:57 (2014).

124

[51] H. A. Gawad, A. Aldailami “On q-dynamic equations modelling and complexity”, Appl. Math.

Model., 34(3), 697-709 (2010).

[52] C. Field, N. Joshi, F. Nijhoff “q-difference equations of KdV type and Chazy-type second-degree

difference equations” J. Phys. A, Math. Theor., 41(33), Article ID332005 (2008).

[53] W. Abdi “Certain inversion and representation formulae for q-Laplace transforms” Math. Z., 83,

238-249 (1964)

[54] L. Abreu “Sampling theory associated with q-difference equations of the Sturm–Liouville type”,

J. Phys. A 38(48), 10311-10319 (2005).

[55] M. H. Annaby, Z. S. Mansour “q-Fractional calculus and equations, lecture notes in mathematics

2056”, Springer–Verlag, Berlin, 2012.

[56] G. A. Anastassiou “On right fractional calculus”, Chaos, Solitons and fractals, 42(1), 365–376

(2009).

[57] M. Houas, Z. Dahmani “New results for a coupled system of fractional differential equations”,

Facta Universitatis, Ser. Math. Inform., 28(2), 133–150(2013).

[58] R. A. Khan, A. Khan, A. Samad, H. Khan “On existence of solutions for fractional differential

equations with P–Laplacian operator”, J. Fract. Cal. Appl., 5(2), 28–37 (2014).

[59] M. Rehman, R. A. Khan “Positive solutions to toupled system of fractional differential equations”,

Int. J. Nonlinear Sci., 10(1), 96–104 (2010).

[60] M. Li, Y. Liu “Existence and uniqueness of positive solutions for a coupled system of nonlinear

fractional differential equations”, Open J. Appl. Sci., 3, 53–61 (2013).

[61] S. K. Ntouyas, M. Obaid “A coupled system of fractional differential equations with nonlocal

integral boundary conditions”, Adv. Differ. Equ. 2012, 130 (2012).

[62] B. Ahmad, S. K. Ntouyas, A. Alsaedi “Existence results for a system of coupled hybrid fractional

differential equations”, Sci. W. J. 2014, Article ID 426438, 6 pages (2014).

[63] M. A. E. Herzallah, D. Baleanu “On Fractional order hybrid differential equations”, Abs. Appl.

Anal., 2014, Article ID 389386, 7 pages (2014).

[64] B. Ahmad, S. K. Ntouyas “Initial value problems for hybrid Hadamard fractional differential equa-

tions”, Elect. J. Differ. Equ., 2014, No. 161, 1-8 (2014).

125

[65] K. R. Hedrih “Fractional order hybrid system dynamics”, PAMM. Proc. Appl. Math. Mech., 13,

25–26(2013).

[66] B. C. Dhage “Quadratic perturbations of periodic boundary value problems of second order ordi-

nary differential equations”, Differ. Equ. Appl., 2, 465-486 (2010).

[67] Y. Gupta, M. Kumar “Anthology of spline based numerical techniques for singularly perturbed

boundary value problems”, Int. J. Pure Appl. Math., 74(4), 437–453(2012).

[68] M. A. Krasnoselskii “Topological Methods in the Theory of Nonlinear Integral Equations”, Perg-

amon Press, 1964.

[69] T. A. Burton, “A fixed point theorem of Krasnoselskii”, Appl. Math. Lett., 11 (1998), 83–88 (1998).

[70] B. C. Dhage, “A nonlinear alternative in Banach algebras with applications to functional differential

equations”, Nonlinear Funct. Anal. Appl., 8, 563–575 (2004).

[71] B. C. Dhage, V. Lakshmikantham “Basic results on hybrid differential equations”, Nonlinear Anal:

Hybrid Syst., 4, 414–424(2010).

[72] M. Inc, A. Akgul and A. Kilicman “Numerical solutions of the second order one dimensional

Telegraph equation based on reproducing Kernel Hilbert space method”, Abstr. Appl. Anal., 2013,

Article ID 768963, 13 pages 2013.

[73] R. A. Khan and H. Khalil “A new method based on legendre polynomials for solution of system

of fractional order differential equations”, Int. J. Comput. Math., DOI: 10.1080/00207160, Article ID

880781, (2014).

[74] M. Alipour and D. Rostamy “Solving Nonlinear Fractional Differential Equations by Bernstein

Polynomials Operational Matrices”, J. Math. Comput. Sci., 5 (3), 185–196(2012).

[75] R.L. Bagley, P.J. Torvik “A theoretical basis for the application of fractional calculus to viscoelas-

ticity”, J. Rheol. 27, 201–210 (1983).

[76] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo “Fractional calculus models and numerical meth-

ods (Series on Complexity, Nonlinearity and Chaos)”, World Scientific, 2012.

[77] M. Bhatti, P. Bracken “Solutions of differential equations in a Bernstein polynomial basis”, J.

Comput. Appl. Math., 205, 272–280 (2007).

126

[78] T.S. Chow “Fractional dynamics of interfaces between soft-nanoparticles and rough substrates”,

Phys. Lett. A., 342, 148–155 (2005).

[79] E.H. Doha, A.H. Bhrawy, M.A. Saker “Integrals of Bernstein polynomials: An application for the

solution of high even-order differential equations”, Appl. Math. Lett., 24, 559–565 (2011).

[80] H. Jafari “An introduction to fractional differential equations”, Mazandaran University Press,

2013.

[81] H. Jafari, K. Sayevand, H. Tajadodi, D. Baleanu “Homotopy analysis method for solving Abel

differential equation of fractional order”, Cen. Eur. J. Phys., 11 (10), 1523– 1527 (2013).

[82] K. Maleknejad, E. Hashemizadeh, B. Basirat “Computational method based on Bernstein oper-

ational matrices for nonlinear Volterra-Fredholm-Hammerstein integral equations”, Commun. Non-

linear Sci. Numer. Simul., 17 (1), 52–61 (2012).

[83] D. Rostamy, M. Alipour, H. Jafari, and D. Baleanu “Solving multi-term orders fractional differen-

tial equations by operational matrices of BPs with convergence analysis”, Roman. Reports Phys., 65

(2), 334–349 (2013).

[84] S. A. Yousefi, M. Behroozifar “Operational matrices of Bernstein polynomials and their applica-

tions”, Int. J. Syst. Sci., 41(6), 709–716 (2010).

[85] V. K.Srivastava, M. K. Awasthi, R. K. Chaurasia, M. Tamsir “The Telegraph equation and its

solution by reduced differential transform method”, Model. Simul. Eng., Article ID 746351, 6 pages

2013.

[86] M. Garg, P. Manohar, S. L. Kalla “Generalized differential transform method to Space-Time frac-

tional Telegraph equation”, Int. J. Differ. Equ., Article ID 548982, 9 pages (2011).

[87] E. Kreyszig “Introductory functional analysis with applications”, New York: John Wiley and Sons.

Inc. 1978.

[88] D. Baleanu, M. Alipour and H. Jafari “The Bernstein operational matrices for solving the fractional

quadratic Riccati differential equations with the Riemann-Liouville derivative”, Abstr. Appl. Anal.,

2013, Article ID 461970, 7 pages (2013)

[89] M. Alipour and D. Rostamy “Bernstein polynomials for solving Abel’s integral equation”, J. Math.

Comput. Sci., 3 (4), 403 – 412 (2011).

127

[90] M. Alipour and D. Rostamy “Bernstein polynomials operational matrices for solving time varying

fractional optimal control Problems”, J. Math. Comput. Sci., 6, 292 – 304 (2013).

[91] R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto “Fractional calculus and continuous-time finance

III. The diffusion limit, in Mathematical Finance”, Trends Math., 171–180 (2001).

[92] G. Chen, X. Yu, Chaos Control “Theory and Applications”, Springer-Verlag, Berlin, Germany,

2003.

[93] M. S. Tavazoei, M. Haeri “Chaos control via a simple fractional order controller”, Phys. Lett. A.,

372, 798–807, (2008).

[94] Y. Wang, C. Li “Does the fractional Brusselator with efficient dimension less than 1 have a limit

cycle”, Phys. Lett. A, 363 (5-6), 414–419(2007).

[95] V. Gafiychuk, B. Datsko “Stability analysis and limit cycle in fractional system with Brusselator

nonlinearities”, Phys. Lett. A, 372 (29), 4902–4904 (2008).

[96] S. Bhalekar and V. D. Gejji “Solving fractional order Logistic equation using a new iterative

method”, Int. J. Differ. Equ., 2012, Article ID 975829, 12 pages, (2012) doi:10.1155/2012/975829

[97] H. Jafari, A. Kadem, D. Baleanu “Variational iteration method for a fractional order Brusselator

system”, Abstr. Appl. Anal., 2014, Article ID 496323, 6 pages (2014).

[98] H. Jafari, H. Tajadodi “Fractional order optimal control problems via the operational matrices of

Bernstein polynomials”, U.P.B. Sci. Bull., Series A, 76 (3), 115–128 (2014).

[99] J. C. Goswami, A. K. Chan “Fundamentals of wavelets: theory, algorithms, and applications”,

John Wiley & Sons, Inc., 1999.

[100] K. Maleknejad, K. Nouri, M. Sahlan “Convergence of approximate solution of nonlinear

Fredholm–Hammerstein integral equations”, Commun. Nonlinear Sci. Numer. Simul., 15, 1432–1443

(2010).

[101] N. Xu, W. Liu “Iterative solutions for a coupled system of fractional differential integral equations

with two point boundary conditions”, Appl. Math. Comput., 244, 903–911 (2014).

[102] Z. Wei, Q. Li, J. Che “Initial value problems for fractional differential equations involving

Riemann–Liouville Sequential fractional derivative”, J. Math. Anal. Appl., 367, 260–272 (2010).

128

[103] G. Wang, D. Baleanu, L. Zhang “Monotone iterative method for a class of nonlinear fractional

differential equations”, J. Fract. Calc. Appl. Anal., 2012, 15(2), 244–252 (2012).

[104] B. C. Dhage “On a fixed point theorem in Banach algebras with applications”, Appl. Math. Lett.,

18, 273–280 (2005).

[105] B. B. Mandelbrot “The Fractal Geometry of Nature”, Freeman, New York, 1982.

[106] G. Calcagni “Geometry and field theory in multi–fractional space time”, J. High Energy Phys.,

65(1), 1–65 (2012).

[107] F. B. Adda, J. Cresson “About non-differentiable functions”, J. Math. Anal. Appl., 263, 721–737

(2001).

[108] K. M. Kolwankar, A. D. Gangal, Fractional differentiability of non differentiable functions and

dimensions, Chaos, (6), 505–513 (1996).

[109] A. K. Golmankhareh, D. Baleanu, New derivatives on the fractal subset of real line, Entropy,

18(1) (2016).

[110] Y. Zhao, D. Baleanu, C. Cattani, D. F. Cheng, X. J. Yang “Local fractional Discrete wavelet

Transform for solving signals on Cantor sets”, Math. Prob. Eng., 2013, Article Number: 560932

(2013).

[111] A. Parvate, A. D. Gangal, “Calculus on fractal subsets of real line”, I. Formulation. Fractals,

17(1), 53–81 (2009).

[112] A. M. Yang, J. Li, H. M. Srivastava, G. N. Xie, X. J. Yang “Local fractional Laplace Variational

Iteration Method for solving linear partial differential equations with local Fractional Derivative”,

Discrete. Dyn. Nat. Soc., Article ID 365981, 8 pages (2014).

[113] S. Xu, X. Ling, C. Cattani, G. N. Xie, X. J. Yang, Y. Zhao “Local fractional Fourier series so-

lutions for non-homogeneous Heat equations arising in fractal heat flow with local fractional deriva-

tive”, Adv. Mechanical Eng., Article ID 514639, 5 pages 2014.

[114] Y. Zhang, D. Baleanu, X. J. Yang “On a local fractional wave equation under fixed entropy arising

in fractal hydrodynamics”, Entropy, 16(12), 6254–6262 (2014).

[115] H. M. Srivastava, K. Golmankhaneh, D. Baleanu, X. J. Yang “Local fractional Sumudu transform

with application to IVPs on Cantor Sets”, Abs. Appl. Anal., Article No: 620529 (2014).

129

[116] Y. Li, Y. Zhao, G. N. Xie, D. Baleanu, X. J. Yang, K. Zhao “Local fractional Poisson and

Laplace equations with applications to electrostatics in Fractal Domain”, Adv. Math. Phys., Article

No: 590574 (2014).

[117] B. B. Mandelbrot “Fractal geometry: what is it, and what does it do?”, Proc. R. Soc. Lond. A.,

1989.

[118] V. Christianto, and B. A. Rahul “Derivation of Proca equations on Cantor sets: A local fractional

approach”, Bull. Math. Sci. Appl., 3, 75–87 (2014).

[119] Y. J. Hao, H. M. Srivastava, H. Jafari and X. J. Yang “Helmholtz and diffusion equations associ-

ated with local fractional derivative operators involving the Cantorian and Cantor–Type Cylindrical

Coordinates”, Adv. Math. Sci. Appl., 2013, Pages 5, Article ID 754248, (2013)

[120] A. Carpinteri, A. Sapora “Diffusion problems in fractal media defined on Cantor sets”, Z. Angew.

Math. Mech., 90, 203–210 (2010).

[121] A. M. Yang, X. J. Yang, Z. B. Li “Local fractional expansion method for solving wave and

diffusion equations on Cantor sets”, Abstr. Appl. Anal., 2013, Article ID 351057, 5 Pages (2013).

[122] W. H. Su, X. J. Yang, H. Jafari, D. Baleanu “Fractional complex transform method for wave

equations on Cantor sets within local fractional differential operator”, Adv. Differ. Equ., 2013, 97

(2013).

[123] L. F. Wang, X.J. Yang, D. Baleanu, C. Cattani, Y. Zhao “Fractal dynamical model of Vehicu-

lar traffic flow within the local fractional Conservation Laws”, Abstr. Appl. Anal., 2014, Article ID

635760, 5 pages, (2014).

[124] A. Babakhani, V. Daftardar Gejji “On calculus of local fractional derivatives” , J. Math. Anal.

Appl., 270, 66–79 (2002).

[125] Y. Chen, Y. Yan, K. Zhang “On the local fractional derivative”, J. Math. Anal. Appl., 362, 17–33

(2010).

[126] D. Guo V. Lakshmikantham, Nonlinear problems in abstract Cones, Academic press, Orland,

1988.

[127] D. Rostamy, H. Jafari, M. Alipour, C. M. Khalique “Computational method based on Bern-

stein operational matrices for multi order fractional differential equations”, Filomat, 28(3), 591–601

(2014).

130

[128] Z. Han, H. Lu, S. Sun, D. Yang “Positive solution to boundary value problem of p-Laplacian

fractional differential equations with a parameter in the boundary”, Elect. J. Differ. Equ., 213, 1–14

(2012).

[129] M. Lakestani, M. Dehghan, S. Irandoust-pakchin “The construction of operational matrix of frac-

tional derivatives using B-spline functions”, Commun. Nonlinear Sci. Numer. Simulat., 17, 1149–1162

(2012).

[130] D. Baleanu, R. P. Agarwal, H. Khan, R. A. Khan and H. Jafari “On the existence of solution for

fractional differential equations of order 3 < δ1 ≤ 4”, Adv. Differ. Equ., 2015:362 (2015).

[131] D. Baleanu, H. Khan, H. Jafari, R. A. Khan and M. Alipour “On existence results for solutions of

a coupled systemof hybrid boundary value problems with hybrid conditions”, Adv. Differ. Equ., 2015:

318, (2015).

[132] H. Jafari, D. Baleanu, H. Khan, R. A. Khan and A. Khan “Existence criterion for the solutions of

fractional oder p– Laplacian boundary value problems”, Bound. Value Prob., 2015, 164, (2015).

[133] D. Baleanu, H. Khan, H. Jafari and R. A. Khan “On the exact solution of wave equations on

Cantor sets”, Entropy, 17, 6229–6237 (2015).

[134] D. Baleanu, H. Jafari, H. Khan and S. J. Johnston “Results for Mild solution of fractional coupled

hybrid boundary value problems”, Open Math., 13, 601–608, (2015).

[135] H. Khan, H. Jafari, R. A. Khan, H. Tajadodi and S. J. Johnston “Numerical solutions of the

nonlinear fractional–order Brusselator system by Bernstein polynomials”, Sci. World J., 2014, ID

257484, pp7, (2014).

[136] H. Khan, M. Alipour, H. Jafari and R. A. khan “Approximate analytical solution of a coupled

system of fractional partial differential equations by Bernstein polynomials”, Int. J. Appl. Comput.

Math., DOI 10.1007/s40819-015-0052-8 (2015).

[137] R. A. Khan and H. Khan “ Existence of solution for a three point boundary value problem of

fractional differential equation”, J. Frac. Calc. Appl., 5(1), 156–164 (2014).

[138] R. A. Khan and H. Khan, “On existence of solution for multi point boundary value problem”, J.

Frac. Calc. Appl., 5(2), 121–132, (2014).

131

[139] H. Khan, M. Alipour, R. A. Khan, H. Tajadodi and A. Khan “On approximate solution of frac-

tional order Logistic equations by operational matrices of Bernstein polynomials”, J. Math. Comput.

Sci., 14, 222–232 (2015).

[140] H. Khan, R. A. Khan and M. Alipour “Existence and uniqueness results for coupled system of

fractional qdifference equations with boundary conditions” J. Comput. Complex. Appl., 1(2) 79–88

(2015).

[141] M. Alipour, R. A. Khan, H. Khan “Applications of local fractional series expansion method to

fractal Vehicular traffic flow model, (Accepted in Sindh University Research Journal, Pakistan) Ac-

cepted

[142] D. Baleanu, H. Jafari, R. A. Khan and H. Khan “On approximations of local fractional

Schrodinger equations on Cantor sets, (Accepted in Sindh University Research journal, Pakistan)

Accepted

[143] M. Alipour, R. A. Khan and H. Khan “Computational method Based on Bernstein polynomials

for solving a fractional optimal control problem”, Journal of Punjab University Accepted

[144] Positive solution for Boundary value problems of nonlinear fractional differential equations, Elec-

tronic Journal of Differential Equations, Vol. 2006(2006)

132

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