existence theory and numerical solutions of the …

EXISTENCE THEORY AND NUMERICAL SOLUTIONS

OF THE FRACTIONAL ORDER MATHEMATICAL

MODELS

By

Fazal Haq

SUBMITTED IN ACCORDANCE WITH THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

AT

HAZARA UNIVERSITY MANSHERA

KPK, PAKISTAN

2018

c© Copyright by Fazal Haq, 2018


DEPARTMENT OF

MATHEMATICS AND STATISTICS

The undersigned hereby certify that they have read and

recommend to the Faculty of Graduate Studies for acceptance a thesis

entitled “EXISTENCE THEORY AND NUMERICAL SOLUTIONS

OF THE FRACTIONAL ORDER MATHEMATICAL MODELS”

by Fazal Haq in partial fulfillment of the requirements for the degree of

Doctor of Philosophy.

Dated: 2018

External Examiner:

Research Supervisor:Muhammad Shahzad

Examing Committee:

ii


Date: 2018

Author: Fazal Haq

Title: EXISTENCE THEORY AND NUMERICAL

SOLUTIONS OF THE FRACTIONAL ORDER

MATHEMATICAL MODELS

Department: Mathematics and Statistics

Degree: Ph.D. Convocation: February Year: 2019

Permission is herewith granted to Hazara University Manshera to circulate

and to have copied for non-commercial purposes, at its discretion, the above

title upon the request of individuals or institutions.

Signature of Author

THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, ANDNEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BEPRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’S WRITTENPERMISSION.

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iii

My dear parents

iv

Table of Contents

Table of Contents v

List of Tables viii

List of Figures ix

Abstract x

Acknowledgements xii

List of publications from thesis xiii

1 Introduction 1

2 BACKGROUND MATERIALS 7

2.1 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Mittage–Leffler’s function . . . . . . . . . . . . . . . . . . . . . 8

2.1.4 Repeated integrals . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.5 Cauchy result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.6 Fixed point theorem . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Caputo’s approach to fractional calculus . . . . . . . . . . . . 11

2.2.2 Fractional integral . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Caputo’s approach to fractional derivative . . . . . . . . . . . 13

2.3 Differential Equations and Their Coupled Systems . . . . . . . . . . . 16

2.3.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Coupled system of fractional ODEs . . . . . . . . . . . . . . . 17

v

2.4 Laplace Transform and Its Properties . . . . . . . . . . . . . . . . . . 17

2.4.1 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 Properties of Laplace transform . . . . . . . . . . . . . . . . . . 19

2.5 Introduction to Adomian Polynomials . . . . . . . . . . . . . . . . . . 19

3 STUDY OF BOUNDARY VALUE PROBLEMS OF FRACTIONAL OR-

DER via MONOTONE ITERATIVE METHOD 21

3.1 Existence of Positive Solution to a Class of Fractional Differential

Equations with Boundary Conditions . . . . . . . . . . . . . . . . . . 23

3.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Coupled System of Nonlinear Fractional Order Differential Equa-

tion with Non-local Boundary Value Problem . . . . . . . . . . . . . . 36

3.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Monotone Iterative Technique for Boundary Value Problem . . . . . 49

3.3.1 Existence of upper and lower solution by using monotone

sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.2 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . . 56

4 QUALITATIVE STUDY OF PINE WILT DISEASE MODEL 58

4.1 Formulation of Mathematical Model . . . . . . . . . . . . . . . . . . . 60

4.1.1 Compartments in pine trees population: . . . . . . . . . . . . . 60

4.1.2 Compartments in bark beetles population: . . . . . . . . . . . 61

4.1.3 Description of the model . . . . . . . . . . . . . . . . . . . . . . 61

4.1.4 Disease free equilibrium and its stability . . . . . . . . . . . . 62

4.1.5 The endemic equilibrium and its stability . . . . . . . . . . . . 63

4.1.6 Global stability analysis . . . . . . . . . . . . . . . . . . . . . . 67

4.1.7 Numerical simulations and discussion . . . . . . . . . . . . . . 75

4.2 Numerical Solution of a Fractional Order Host Vector Model of a

Pine Wilt Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.1 The Laplace Adomian decomposition method . . . . . . . . . 81

4.2.2 Numerical results and discussion . . . . . . . . . . . . . . . . 85

4.2.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . 90

5 Numerical Analysis of Various Biological Models of Fractional Order 92

5.1 Numerical analysis of Fractional Order Epidemic Model of Child-

hood Diseases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


5.1.2 Numerical results and discussion . . . . . . . . . . . . . . . . . 98


vi

5.2 Numerical Solution of Fractional Order Model of HIV-1 Infection of

CD4+ T-Cells by using LADM . . . . . . . . . . . . . . . . . . . . . . . 102


5.2.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . 106


5.3 Numerical Analysis of Fractional Order Enzyme Kinetics Model by

Laplace Adomian Decomposition Method . . . . . . . . . . . . . . . . 109



5.3.3 Numerical results and discussion . . . . . . . . . . . . . . . . 113

5.3.4 Numerical plot and comparison table . . . . . . . . . . . . . . 117

5.4 Numerical Solution of Fractional Order Smoking Model Via LADM . 117


5.4.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 124


6 Numerical Analysis of Biological Population Model Involving Fractional

Order 132

6.1 LADM for Biological Model . . . . . . . . . . . . . . . . . . . . . . . 133

6.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7 Summary and conclusion 142

Bibliography 145

vii

List of Tables

2.1 Laplace and inverse Laplace transform of some functions. . . . . . . 18

5.1 Numerical solution of proposed model using LADM at classical or-

der α = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.2 Numerical solution of proposed model using RK4 at classical order

α = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.3 Numerical solution of proposed model by using RK4 at classical or-

der α = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.4 Numerical solution of proposed model using LADM at classical or-

der α = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

viii

List of Figures

6.1 Numerical plots of approximate solutions of Example (6.2.1) at var-

ious fractional order and using r = 50, t = 10. . . . . . . . . . . . . . . 136


ious fractional order and using t = 10. . . . . . . . . . . . . . . . . . . 139


ious fractional order and using k = 0.1, t = 10. . . . . . . . . . . . . . 141

ix

Abstract

In last few decades, it has been proved that fractional order differential equations

and their systems are very important in mathematical modeling various phenom-

ena of biological, chemical and physical sciences. In addition to these, a fractional

calculus also contains many applications in various fields of engineering and tech-

nology. For this propose , differential equations of fractional order is the point of

attention in last few years. This project is related with the study of existence theory

and numerical solutions of fractional order differential equations. For this study,

we first review some useful definitions, notations and results from fractional cal-

culus. Also for the study of numerical solutions, we use a power full techniques.

We start our thesis with the study of existence and uniqueness of positive solutions

for simple boundary value problem. Then, we obtain necessary and sufficient con-

ditions for existence of at least three positive solutions for the considered models.

To solve coupled systems of nonlinear fractional differential equations, we dis-

cuss a method which is known as Laplace Adomian decomposition method (LADM).

LADM is an excellent mathematical tool for solving linear and nonlinear differen-

tial equations. This method is a combination of the famous integral transform

known as Laplace transform and the Adomain decomposition method. In this

method, we handle some class of coupled systems of nonlinear fractional order

differential equations. Using the proposed method to obtain successfully an exact

or approximate solution in the form of convergence series. Thus, we can easily ap-

ply LADM to solve a wide class of nonlinear fractional order differential equations.

x

xi

The suggested method is applied without any linearization, discretization and un-

realistic assumptions. It has been proved that LADM is vary efficient and suitable

to solve non-linear problem of physical nature. Some examples are presented to

justify the accuracy and performance of the proposed method.

Acknowledgements

All the praises to Almighty ALLAH, the most merciful and the sovereign power,

who made me able to accomplish this research work successfully. I offer my hum-

blest and sincere words of thanks to his Holy Prophet Muhammad (P.B.U.H) who

is forever a source of guidance and knowledge for humanity.

First of all I would like to thank my supervisorDr. Muhammad Shahzad and

my Co–supervisors Dr. Ghaus ur Rahman for their continuous support through-

out my PhD and for providing me opportunity to work on this project. Special

thank to Dr. Kamal Shah for their helpful suggestion and many useful discussion.

I am also very thankful to my parents for their prayers and support.

Department of Mathematics, Hazara University Fazal Haq

October 25, 2018

xii

List of publications from thesis

• Fazal Haq, Kamal Shah, Muhammad Shahzad, Ghausur Rahman and Yongjin

Li., Computational analysis of complex population dynamical model with

arbitrary order, Complexity, Jan 2018, Article ID 8918541, 8 pages .

• Fazal Haq, Kamal Shah, Ghaus ur Rahman, and Muhammad Shahzad. Nu-

merical solution of fractional order model of childhood disease by using

laplace adomian decomposition method, Discrete dynamics in nature and soci-

ety, 2017 Artical ID 4057089, 7 pages.

• Fazal Haq, Kamal Shah, Ghaus ur Rahman, Muhammad Shahzad, Numeri-

cal solution of fractional order smoking model via laplace Adomian decom-

position method, Alexandria Engineering Journal , 2017 1-9.

• Fazal Haq, Kamal Shah, Ghaus ur Rahman, Muhammad Shahzad, Numer-

ical analysis of fractional order model of HIV-1 infection of CD4+ T-cells,

Computational Methods for Differential Equations, 5(1), 2017 1-11.

• Fazal Haq, Kamal Shah, Ghaus ur Rahman, Muhammad Shahzad, Existence

and Uniqueness of Positive Solution to Boundary Value Problem of Frac-

tional Differential Equations, Sindh Univ. Res. Jour. 48(2), 2016 451-456.

• Yongjin Li, Fazal Haq, Kamal Shah, Muhammad Shahzad, Ghaus ur Rahman,

Numerical analysis of fractional order Pine wilt disease model with bilinear

incident rate, J. Math. Computer Sci, 2017 420-428.

xiii

xiv

• Fazal Haq, Kamal Shah, Ghaus ur Rahman, and Muhammad Shahzad, Cou-

pled system of local boundary value problems of non linear fractional differ-

ential equations, Journal of Mathematical Analysis, 8(2), 2017 51-63.

• Fazal Haq, Kamal Shah, Ghaus ur Rahman, and Muhammad Shahzad. Nu-

merical solution of the fractional order epidemic model of a vector born dis-

ease by Laplace adomian decomposition method, Punjab University Journal of

Mathematics, 49(2), 2017 13-22.

• Fazal Haq, Kamal Shah, Ghaus ur Rahman, and Muhammad Shahzad. ,Ex-

istence results for a coupled systems of Chandrasekhar quadratic integral

equations, Commun. Nonlinear Anal. 3, 2017 15-22.

• Muhammad Shahzad, et al. Computing the Low Dimension Manifold in

Dissipative Dynamical Systems. The Nucleus 53(2), 2016 107-113.

Chapter 1

Introduction

The idea of mathematical modeling of spread of disease was presented for the first

time by Bernoulli in 1766, which gave birth to the start of modern epidemiology.

In the beginning of 20th century Ross and Hamer also presented the modeling

of infectious disease. To explain the behavior of epidemic models, they used the

law of mass action. After that Reed and Frost established Reed-Frost epidemic

model, which give the relationship between the susceptible, infected and immune

individual in a community[19]. Mathematical modeling is a handy tool to under-

stand, how a disease can spread and also it determine various factors that can play

vital role in the spread of a disease. Mathematical models require to unify the

non-decreasing volume of data which is being generated on host pathogen inter-

actions. Several theoretical surveys of population dynamics, evolution of infec-

tious diseases of humans, animals as well as plants are connected with this study

[40, 67].

For Mathematical modeling of real world problems mathematician and researchers

used fractional calculus. Issac Newton (1642 − 1727) and Leibniz (1646 − 1716)

1

2

was introduce fractional calculus. It is the area of Mathematics in which the prop-

erties of fractional order integral and derivatives are discussed. Fractional calculus

originated, when a question was raised that ’could meanings order of derivatives

non-integers instead of integer order n. This was the question when L’Hospital

wrote a latter on 30th September 1695 to Leibniz what would be the result, when

order of the derivative is 12 ? Liebniz replied in his latter.

”It will be a paradox from which useful consequences will be drawn”. For the

first time fractional derivatives introduced by Lacroix (1819) in his paper [56]. Af-

ter this many mathematicians contributed in the development of fractional calcu-

lus such as Fourier (1822), Abel (1823 − 1826), Liouville (1832), Raimman (1847),

Grunwald (1867), Letnikove (1868), Hadamard (1892), Weyel (1917), Erdelyi (1939),

Kober (1940) and Riesz (1949). In last years, the basic work developed by Kiryakove,

Machado, Benchohra and Mainardi etc, since 1974 − 2010 in the area of fractional

calculus was reported in [47] etc. A detailed summary of contribution of fractional

calculus can be found in [64]. De Lacroix used Lagender’s polynomial Γ for a pos-

itive integer p and found the qth derivative of a function u = tp is given by

dqu

dtq =Γ(p + 1)

Γ(p − q + 1)tp−q,

by putting p = 1 , q = 12 get the following [56]

d12 u

dt12

=2√

t√π

.

In 1823 fractional calculus applied by Abel [2] to solve integral equations which

appears in mathematical models of Tautochrone problems, the problems in which

shape of the curve can be determine. Many researchers attempt to define proper

physical and geometrical meaning of fractional calculus but they did not success

3

due to some problems. Afterward Podlubny [58], in 2002 discussed the geomet-

rical interpretation of fractional order derivatives and integrals. Different mathe-

matician provides different definitions of fractional order integrals and derivatives

such as Caputo’s , Reimann Liouville, Hardmard, Weyl, Marchaud and Granwald-

Letnikov definitions etc. Among these, standerd definitions of fractional deriva-

tives and integration are those, that are introduce by Caputo’s, which are mostly

used as compare to Riemann- Liouville, because in some initial and boundary con-

ditions Riemann–Liouville definitions fails to represent geometrical and physical

interpretations.

There are many applications of existence theory for positive solutions of bound-

ary value problems(BVPs) of integer order derivatives, but for BVPs of fractional

order differential equations existence theory is now in the beginning. Recently at-

traction of many researchers is toward the existence theory and positive solutions

[21, 14, 15]. Mathematical models of nonlinear fractional order partial differen-

tial equations (NFODEs) have many applications in science, specially in physics

chemistry, biology and image processing and engineering etc. By using different

methods many researchers are trying to find explicit, exact or approximate solu-

tions. And many of them are interested to solve ordinary differential equations,

partial differential equations and integral equations.

Since serval nonlinear problems can not be solved by direct methods. Most of

the time its exact solution cannot be found. Therefore we need to use numerical

schemes to find approximate solution. With the development of nonlinear sciences

many numerical and analytical methods such as finite element methods, finite

4

difference methods, perturbation, polynomial and non polynomial spline, varia-

tion of parameter, inverse scattering, Homotopy analysis method (HAM), Homo-

topy perturbation methods (HPM), decomposition and many other methods have

been developed and implemented. There are many integral transform methods

[4, 25, 11, 12, 22, 35, 23] to solve partial differential equations, ordinary differential

equations and integral equations. The well known integral transform, available in

literature is Fourier Transform which was named in honor of Jean-Baptiste Joseph

Fourier(1768-1830). Also Mellin Transform was introduced by Robert Hjalmer

Mellin (1854-1933).

In the 1980’s George Adomian formulated a novel powerful scheme for solv-

ing nonlinear functional equations. Afterward, in the literature this method was

known as Adomian Decomposition Method (ADM). The technique is based on the

splitting of a system of differential equations solutions in the form of series of func-

tions. Every term of the associated series is obtained from a polynomial generated

by expansion of an analytic function into a power series. This is an effective tool

for solution of systems of differential equations appearing in physical problems.

Over the last 20 years, the Adomian decomposition approach has been applied to

obtain formal solutions to a wide class of stochastic and deterministic problems in-

volving algebraic, differential, integro-differential, differential delay, integral and

partial differential equations. As compared to other existing numerical schemes,

the main advantage of this method is that it dont require perturbation or liberaliza-

tion for exploring the dynamical behavior of complex dynamical systems. In [26],

the authors show that ADM generally does not converge, when the method is ap-

plied to highly nonlinear differential equation. The coupling of Laplace transform

5

and ADM lead to a efficient technique known as Laplace Adomain decomposition

method (LADM). The LADM has done extensive work to provide approximate so-

lution of nonlinear equations as well as solving differential equations of fractional

order.

The LADM is better than HAM, HPM, VIM, because it needs no parameter

terms to form decomposition equation, no perturbation is needed in the men-

tion methods. The LADM method is simple and does not require or waste ex-

tra memory like Tau-collocation method. Further from ADM, LADM is better as

we applied Laplace transform and then decompose the nonlinear terms in term

of Adomian polynomials, while in Adomian decomposition particular integral is

involved, which often creates difficulties in computation. The main advantage of

this technique is that, it avoids complex transformations like index reductions and

leads to a simple general algorithm. Secondly, it reduces the computational work

by solving only linear algebraic systems. From this method, we obtained a sim-

ple way to control the convergence region of the series solution by using a proper

value of parameters. Certain results show that LADM is very efficient, power full

method to find the approximate solution of nonlinear differential equations. It

is essentially a mathematical tool to solve several problems in ordinary differen-

tial equations and partial differential equations. Wide use of this transform came

about after World War I I. For a function f (t), Laplace transform for all real num-

bers t ≥ 0 is defined as

F(s) =∫ ∞

0e−st f (t)dt, s > 0,

where ”s” is the parameter of the transform. Moreover applications of Laplace

transform are available in literature [18, 6, 55].

6

This thesis is mainly devoted to existence theory of some BVPs. The differen-

tial equations whose solutions are exhibited in the present work model complex

systems. Moreover, the Hyer–Ulam stability of some models also discussed. Fur-

thermore quantitative aspects of dynomical systems is studied in the thesis.

Organization of the thesis as: Chapter 2, contains some useful definitions, spe-

cial functions, some definitions of fractional calculus, differential equations and

their coupled system, some theorems and lemma of fractional calculus, Laplace

transform and their properties. Chapter 3, is devoted existence of positive solution

for class of BVPs and monotone iterative techniques for BVPs. Chapter 4, is related

with pine wilt disease model, stability and numerical simulation of the proposed

model. we present in chapter 5, various biological models, their numerical solu-

tion and convergence analysis. Chapter 6 provide numerical solution of fractional

order population model. Summary and conclusion of the thesis is given in Chapter

7.

Chapter 2

BACKGROUND MATERIALS

This chapter provides some definitions, results and preliminaries which are re-

quired in this thesis. The concerned materials are associated to the basic results,

definitions of fractional calculus, fixed point theory, NFODEs and numerical anal-

ysis.

This chapter is arranged as: We provide in Section 2.1, some basic results of frac-

tional calculus. In Section 2.2, there are contained some basic useful results related

to this work. Section 2.3 is related differential equations and their system. The last

sections 2.4 is related to the introduction and properties of Laplace transform and

Adomain polynomials.

2.1 Special Functions

In this portion, we recall some special functions, as Gamma, Beta and Mittage

Leffler’s functions.

7

8

2.1.1 Gamma function

Euler in 1729 introduced the basic interpretation of the Gamma function. For all

real numbers Gamma function is the generalization of the factorial. Its definition

is given [37]

Γ(w) =∫ ∞

0e−ttw−1dt, w ∈ R

+.

Some cases of Gamma function are given

Γ(w + 1) = wΓ(w) = w!, w ∈ R+

Γ(w) = (w − 1)!, w ∈ N

(2.1.1)

and

Γ(1

2) =

∫ ∞

0e−tt−

12 dt =

√π.

2.1.2 Beta function

Beta function is a special function and defined by the following definite integral

[37]

B(w1, w2) =∫ 1

0tw1−1(1 − t)w2−1dt, w1, w2 ∈ R

+. (2.1.2)

Beta function and Gamma function are related to each other is given by [37]

B(w1, w2) =Γ(w1)Γ(w2)

Γ(w1 + w2), w1, w2 ∈ R

+. (2.1.3)

2.1.3 Mittage–Leffler’s function

A Swedish mathematician defined a function named Mittag–Leffler function [48].

Mittag–Leffler function is the generalization of the ex, this function play an impor-

tant role in fractional order differential and fractional order integral equation. The

9

Mittag–Leffler function in term of power series is defined as:

Eα(t) =∞

∑k=0

tk

Γ(kα + 1), t ∈ C, Re(α) > 0.

The Mittage–Leffler function for one parameter interpolates between exponential

function exand hyperbolic function 11−t is given below:

(a) E0(t) =1

1 − t, (b) E1(x) = ex.

2.1.4 Repeated integrals

For a function f(x) where x > 0, the definite integral from 0 to x, we denote by

I f (x) and is given as

If(x) =∫ x

0f(t)dt.

If we integrate If(x) again, obtained the second integral

I2f(x) =∫ x

0If(t)dt =

∫ x

0

∫ t

0f(s)dsdt.

Similarly the third integration is obtain as

I3f(x) =∫ x

0

∫ t

0

(

∫ s

0f(u)du

)

dsdt. (2.1.4)

Such type of integral is called repeated integral.

2.1.5 Cauchy result

As mentioned earlier that finding for a function f(x) the nth integral is more com-

plicated for large n. Cauchy presented how we can define an integral such as

equation (2.1.4) is given by

Inf(x) =1

(n − 1)!

∫ x

0(x − t)n−1f(t)dt. (2.1.5)

So Cauchy’s result gives n times repeated integral into a single integral.

10

2.1.6 Fixed point theorem

This section contain some fixed point theorems.

Definition 2.1.1. [3] Let W a norm space and T : W → W be an operator. Then T

is called Lipschitzian if there exists a constant K ≥ 0 such that

‖T(x1)− T(x2)‖ ≤ K‖x1 − x2‖, for all x1, x2 ∈ W , (2.1.6)

if K < 1, then T is called a contraction.

Theorem 2.1.1. (Arzela–Ascoli Theorem)[3]

Let D subset of Rn and let N be a subset of C(D). Then N is relatively compact if and

only if it is bounded and equi–continuous.

Definition 2.1.2. For a given mapping T : B → B, where B is a Banach space,

every point u which satisfy the following equation

T(u) = u

is called fixed of T.

Theorem 2.1.2. (Banach fixed point theorem)[3]

Consider a matric space X = (X , d), where X 6= Ø. Suppose X is a complete and let

T : X → X be a contraction on X . Then T has unique fixed point. Let u be a unique point

in X and for any initial value u0 ∈ X , the successive approximation converges to u, if

‖Tu − Tu‖ ≤ k‖u − u‖, for all, u, u ∈ X .

Lemma 2.1.3. [3]Assume that y ∈ C(0, 1) ∩ L(0, 1) with a fractional derivative of order

p > 0, then the general solution of the fractional differential equation cDpy(t) = h(t) of

11

order p > 0 is given by

Ip[cDpy(t)] = Iph(t) + c0 + c1t + c2t2 + ... + cn−1tn−1,

for arbitrary ci ∈ R, i = 0, 1, 2, ..., n − 1.

2.2 Fractional Calculus

Fractional calculus is a free stop and source study. This subject has a long math-

ematical history and over the centuries many mathematicians have built up great

body mathematics fractional order derivatives and integrals.

2.2.1 Caputo’s approach to fractional calculus

Mathematicians like Fourier, Laplace and Euler introduced the theory of fractional

calculus. They have used their own definitions, notation and methodology which

clear the concept of a non-integer order derivative or integral. Grunwald–Letnikov

and Riemann–Liouville definition are the most famous of these definitions that

have been promoted in the field of fractional calculus. In the modeling of real-

world phenomenon with differential equations of fractional order, there was cer-

tain disadvantages of these definitions. Caputo promoted more ‘classic’ definition

to solve his fractional order differential equations using integer order initial condi-

tions [64, 56, 2].

12

2.2.2 Fractional integral

The fractional order integral of order α > 0 for a function f : R+ → R is defined as

[20]

Iαf(t) =1

Γ(α)

∫ t

0(t − τ)α−1 f (τ)dτ. (2.2.1)

This definition is similar to Cauchy result (2.1.5), except, n was replaced by α and

factorial function converted into gamma function using (2.1.1).

Example 2.2.1. Suppose f (x) = x12 , where x > 0 and to evaluate (I

12 f )(x), we using

equation (2.1.5), to obtain

(I12 f )(x) =

Γ(12 + 1)

Γ(12 +

12 + 1)

x12+

12 ,

=Γ(3

2)

Γ(2)x,

=1

2Γ(

1

2)x.

Now using a result Γ(12) =

√π, we get

I12 (x

12 ) =

√π

2x. (2.2.2)

Example 2.2.2. Let we have to find I32 (t2), for this we are going to consider the definition

of fractional integral (2.2.1). Here α = 32 and f (t) = t2, thus we have

I32 (t2) =

1

Γ(32)

∫ t

0(t − τ)

32−1τ2dτ.

Putting τ = tλ, and so that dτ = tdλ in above equation we get

I32 (t2) =

1

Γ(32)

∫ 1

0(t − tλ)

32−1t2λ2tdλ,

=1

Γ(32)

t72

∫ 1

0(1 − λ)

32−1λ2dλ.

13

Comparing the integral with equation (2.1.2) and then using (2.1.3), we get

I32 (t2) =

t72

Γ(32)

B(3

2, 3),

=t

72

Γ(32)

Γ(32)Γ(3)

Γ(32 + 3)

,

=32

105t

72 .

Thus we have I32 (t2) = 32

105 t72 .

Example 2.2.3. To find the half integral of xk, we compare our problem with (2.1.5) and

replacing n by 12 to get

I12 (xk) =

Γ(k + 1)

Γ(12 + k + 1)

x12+k,

where k is any positive real number.

2.2.3 Caputo’s approach to fractional derivative

As earlier discussed that more definitions are available of fractional order deriva-

tives. Among them Caputo definition is most common and known, as it is widely

applicable in real world problems [24]. The Caputo’s derivative of fractional order

of order ε ∈ (n − 1, n) for f (t) is defined as [24]

cD

ε f (t) = In−εD

n f (t), (2.2.3)

where In−ε represent the fractional integral of order n − ε and D denotes the ordi-

nary derivative, further n− 1 is the integer part of α. This definition was presented

by M. Caputo in his book [56], while solving an initial value problem, whose solu-

tion is possible mathematically by Riemann–Lioville definition but useless practi-

cally. Under condition ε → n on the function f (t), the Caputo derivative becomes

a conventional nth derivative of f (t).

14

Example 2.2.4. To find D12 (x

12 ), we using ( 2.2.3 ) with ε = 1

2 ∈ (0, 1] to get

D12 (x

12 ) = I1− 1

2 D(x12 ).

Now using ( 2.2.2 ) we get

D12 (x

12 ) =

√π

2xD(x

12 ),

or

D12 (x

12 ) =

√π

2x(

1

2x−

12 ),

=

√πx

4.

Thus, we conclude

D12 (x

12 ) =

√πx

4.

Example 2.2.5. To find out D32 (t2) using Caputo’s definition of FD, since 1 < ε = 3

2 ≤ 2,

so by using (2.2.3) we have

D32 (t2) = I2− 3

2 D2(t2), (2.2.4)

also D2(t2) = 2, thus

D32 (t2) = 2I

12 (1).

Now using equation (2.2.1)

I12 (1) =

1

Γ(12)

∫ t

0(t − τ)

12−1τ1−1dτ as 1 = τ0

Putting τ = t λ so that dτ = t dλ and hence

I12 (1) =

1

Γ(12)

∫ 1

0(t − tλ)

12−1 · t · λ1−1dλ,

=1

Γ(12)

t12

∫ 1

0(1 − λ)

12−1 · λ1−1dλ.

15

By using equation (2.1.2) and ( 2.1.3) we get

I12 (1) =

2√π

t12 . (2.2.5)

Hence

D32 (t2) =

4√π

t12 .

Example 2.2.6. To find out the half derivative of x, we use Caputo’s definition of fractional

derivative. For ε = 12 ∈ (0, 1], we have

D12 (x) = I1− 1

2 D(x).

Now using the definition of fractional integral and D(x) = 1, we get

I12 (x) = I

12 (1),

and by (2.2.5)

I12 (1) =

2√π

t12 .

Thus

D12 (x) =

2√π

t12 .

Theorem 2.2.1. [56] Let n − 1 < ε ≤ n, then the differential equations of arbitrary order

ε given by cDεy(t) = 0, has solution given by

y(t) =n−1

∑k=0

Cn−1k ,

where Ck ∈ R, k = 0, 1, 2, . . . , n − 1 and n = [ε] + 1.

Theorem 2.2.2. [56] Let n − 1 < ε ≤ n, then the differential equation cDεy(t) = f (t)

has a solution

Iε(cD

εy(t)) = f (t) +n−1

∑i=0

Citi

, where Ci ∈ R, i = 0, 1, 2, ..., n − 1, where n = [ε] + 1.

16

Properties

The following are some properties of Caputo’s definition of fractional derivative:

• Similar to integer-order differentiation, Caputo’s fractional derivative opera-

tor is a linear operation [36] :

Dε(λ f (x) + µg(x)) = λD

ε f (x) + µDεg(x)

where λ and µ are constants.

• For the Caputos derivative we have:

DεC = 0 where C is a constant.

•

Dεxn =

0 for n ∈ N0 and n < ⌈ε⌉Γ(n+1)

Γ(n+1−ε)xn−ε for n ∈ N0 and n ≥ ⌈ε⌉.

(2.2.6)

We use the ceiling function ⌈ε⌉ to denotes the smallest integer greater than or equal

to ε and N0 = {0, 1, 2, · · · } [36]. Recall that for ε ∈ N0, Caputo fractional differen-

tial operator coincides with the usual differential operator of integer order.

2.3 Differential Equations and Their Coupled Systems

In this section, we define both ordinary differential equations (ODEs), partial dif-

ferential equations (PDEs), and their coupled system of fractional order with ex-

amples [26, 27, 70].

17

2.3.1 Mathematical model

The mathematical representation of a real-world phenomena in the form of for-

mula and equations is called mathematical model.

2.3.2 Coupled system of fractional ODEs

A system of FODEs of two or more dependent variables which are interdependent

on each other is called coupled system. For example coupled spring equation for

modeling the motion of two springs. First spring is hanging from a support and

weight of mass m1 is attached to it. And the second spring is attached to this weight

while weight of mass m2 is attached to the bottom of last spring. The governing

model is provided as:

m1dαu

dxα1

= −k1x1 − k2(x1 − x2),

m2dαv

dxα2

= −k2(x2 − x1).

, 0 < α < 1.

We recall a fractional partial differential equations (FPDEs) as:

∂αu(x, t)

∂tα= L(u(x, t)) + N(u(x, t)) + f (x, t), α > 0.

2.4 Laplace Transform and Its Properties

2.4.1 Laplace transform

The Laplace transform is a powerful tool for solving ODEs and PDEs. Laplace

transform reduces ODEs with constant coefficients to an algebraic equation. This

18

technique became popular when Heaviside applied to the solution of an ODE ref-

ereed hereafter as ODE , representing a problem in electrical engineering.

If f (t) is continuous and real valued function defined for all t , 0 < t < ∞ and

is of exponential order , the Laplace transform is then denoted and defined as

L [ f (t); s] = F(s) =∫ ∞

0e−st f (t)dt.

Where “s” is a parameter of transform , s > 0 and “L ” is the operator which trans-

form f (t) into F(s).

Laplace transform has many applications due to which, we need the Laplace in-

verse too. It mean, we have to find the original function f (t) , when the Laplace

transform F(s) is known. Thus if

L [ f (t); s] = F(s) then f (t) = L −1[F(s); t].

The following table is given, that shows the Laplace transform and inverse Laplace

transform of some functions.

f (t) L [ f (t) : s] = F(s) L −1[F(s); t] = f (t)0 0 0

1 1s 1

t 1s2 t

tn n!sn+1 tn

eat 1s−a eat

e−at 1s+a e−at

sin at as2+a2 sin at

cos at ss2+a2 cos at

sinh at as2−a2 sinh at

cosh at ss2−a2 cosh at

Table 2.1: Laplace and inverse Laplace transform of some functions.

19

2.4.2 Properties of Laplace transform

Some famous properties of Laplace transform are as follows.

Linearity property:

If F1(s) and F2(s) are the Laplace transform of f1(t) and f2(t) and C1 , C2 are

constants, then

L [C1 f1(t) + C2 f2(t); s] = C1L [ f1(t); s] + C2L [ f2; s] = C1F1(s) + C2F2(s).

Shifting property:

If eat is multiplied with a function , the transform of the resultant is obtained by

replacing s by (s − a).

L [ f (t); s] = F(s) implies L [eat f (t); s] = F(s − a).

Multiplication by power of t:

If t power any number “n” is multiplied with the function, then the Laplace trans-

form is obtained as (−1) power “n” and “n” times derivative of F(s).

L [tn f (t); s] = (−1)n dn

dsnF(s).

Example:

The laplace transform of 1 is

L [1; s] =∫ ∞

0e−st(1)dt =

1

s.

2.5 Introduction to Adomian Polynomials

Adomian decomposition is a semi analytical method to ODEs and PDEs. The as-

pect of this method is employment of “Adomian Polynomials” which allow for

20

solution convergence of the linear portion of the equation.

The Adomian decomposition method defines the solution u by a series given by

u =∞

∑k=0

uk,

and replacing the nonlinear term by the given series

Qu =∞

∑n=0

An,

where An is Adomian polynomials and is computed as

An =1

n!

[

dn

dλn

(

Q∞

∑i=0

λiui

)

]

λ=0

.

Chapter 3

STUDY OF BOUNDARY VALUEPROBLEMS OF FRACTIONALORDER via MONOTONEITERATIVE METHOD

This chapter contained the existence and uniqueness of solution to a class of highly

nonlinear BVPs of NFODEs. BVPs of fractional order differential equations are

powerful tools to describe various dynamical which are using to describe various

dynamical phenomenons of biology, physics, dynamics, signal and image process-

ing etc. In last few years, the area devoted to investigate BVPs of fractional or-

der differential equations for the existence of solutions is very important. Most

of the papers in literature are devoted to the existence and uniqueness of solu-

tions to the mentioned problems. Because from existence, uniqueness of solutions

one can concluded about the nature of the problem that either well posed or ill

posed. The concerned theory has been explored by many mathematicians, see

[13, 14, 15, 30, 43, 16]. To obtain sufficient conditions for existence and uniqueness

of solutions, one need fixed point theory of classical type which has been applied in

large number of articles, see[64, 56, 2, 47]. For existence and uniqueness sufficient

21

22

conditions are developed. By using Schauder fixed point theorem, we study exis-

tence and uniqueness to a class of fractional differential equation with boundary

conditions. In recent years the area under stood to stability of fractional differ-

ential equations has attracted great attentions . The work was started by Ulam in

1940, which was later conformed by Hyer’s in 1941. For linear and non-linear func-

tion Rassias introduced Hyer’s-Ulam stability during 1982 to 1998. In 1997 Obloza

was the first author who introduce the Hyer’s-Ulam stability for linear differential

equations[34, 57] .

Organization of the chapter: This chapter has two sections. Section3.1 contain

study of existence and uniqueness of a class of highly non-linear BVP of fractional

order differential equations. The proposed model is investigated by means of clas-

sical fixed point theorem for the mentioned requirements. The Ulam–Hyer’s sta-

bility is also established for the class of fractional differential equations.

In Section 3.2 we study existence and uniqueness of positive solution to a cou-

pled system of non-local BVP of NFODEs. We impose some growth conditions on

linear functions explicitly depend on the fractional derivative involved in them.

Then by using classical fixed point theorem such as Laray–Shauder type and Ba-

nach contraction principal. Further in view of the established growth conditions,

we obtain sufficient conditions for existence and uniqueness of positive solution.

23

3.1 Existence of Positive Solution to a Class of Frac-

tional Differential Equations with Boundary Con-

ditions

This section is concerned with the existence of solutions for BVPs with fractional

order differential equations using boundary conditions given by

cD

εu(t) = f (t, u(t), cD

ε−1u(t)), 1 < ε ≤ 2, t ∈ J = [0, 1]

u(0) = δu(1), cD

pu(1) = γ cD

pu(ξ), 0 < p < 1, ξ ∈ (0, 1).(3.1.1)

Where f : J × R × R → R is given continuous function, where D represents Ca-

puto’s fractional order derivative. We use standard fixed point theorem and Ba-

nach fixed point theorem to establish necessary and sufficient conditions for the

existence and uniqueness of positive solution. We present an example for the illus-

tration of the main results.

Definition 3.1.1. The equation (3.1.1) is Hyer–Ulam stable if there exists a real

number c f > 0 such that for each ǫ > 0 and for each solution u ∈ c(J, R) of

the inequality

|cDαu(t)− f (t, u(t), cDαu(t))| ≤ ǫ, t ∈ J,

there exists a solution y ∈ c(J, R) of (3.1.1) with

|u(t)− y(t)| ≤ c f ǫ, t ∈ J.

Theorem 3.1.1. Let f : I × I × R → R is continuous and 1 < ε ≤ 2, then

cD

εu(t) = f (t, u(t),c Dε−1u(t)), t ∈ J

24

with boundary conditions given by u(0) = δu(1) and D pu(1) = γcD pu(ξ),

0 < p < 1, 0 < t < 1 has a solution given by

u(t) =∫ 1

0G(t, s) f (t, u(t),c D

ε−1u(t))ds

where G(t, s) is Green function provided by

G(t, s) =

1

Γε(t − s)ε−1 +

µ

Γε(1 − s)ε−1 − µ

Γ(2 − ε)

1 − γξ1−p

(

γ(ξ − s)ε−p−1

Γ(ε − p)− (1 − s)ε−p−1

Γ(ε − p)

)

+Γ(2 − ε)

1 − γξ1−pt[γ

(ξ − s)ε−p−1

Γ(ε − p − 1)− (1 − s)ε−p−1

Γ(ε − p − 1)], 0 ≤ s ≤ t ≤ 1,

µ

Γε(1 − s)ε−1 − µ

Γ(2 − ε)

1 − γξ1−p

(

γ(ξ − s)ε−p−1

Γ(ε − p)− (1 − s)ε−p−1

Γ(ε − p)

)

,

+Γ(2 − ε)

1 − γξ1−pt[γ

(ξ − s)ε−p−1

Γ(ε − p − 1)− (1 − s)ε−p−1

Γ(ε − p − 1)], 0 ≤ t ≤ s ≤ 1.

(3.1.2)

Proof. Let y ∈ C([0, 1], R), then we solve the following linear BVP of fractional

differential equation given by

cD

εu(t) = y(t), 0 < ξ < 1, 0 < t < 1, 1 < ε ≤ 2, (3.1.3)

subject to boundary conditions u(0) = δu(1) and cD pu(1) = γcD pu(ξ). Thus in

view of theorem (2.2.2), equation (3.1.3) can be written as

Iε cD

εu(t) = Iεy(t) = u(t) = Iεy(t) + c0 + c1t. (3.1.4)

Now using boundary conditions in (3.1.4) and using µ = δ1−δ , we get

c0 = µ (Iεy(1) + c1)

and

cD

pu(t) = [Iε−py(t) +c1t1−p

Γ(2 − p)] ⇒c

Dpu(1) = γc

Dpu(ξ)

25

γ[Iε−py(ξ) +c1

Γ(2 − p)ξ1−p] = [Iε−py(1) +

c1

Γ(2 − p)]

implies that

c1

Γ(2 − p)− γc1ξ1−p

Γ(2 − p)= γIε−py(ξ)− Iε−py(1)

(

1 − γξ1−p

Γ(2 − p)

)

c1 = γIε−py(ξ)− Iε−py(1)

c1 =Γ(2 − ε)

1 − γξ1−p

[

γIε−py(ξ)− Iε−py(1)]

c0 = µ

(

Iεy(1) +Γ(2 − ε)

1 − rξ1−p

(

γIε−py(ξ)− Iε−py(1))

)

using the values of c0, c1 in (3.1.3) we have

u(t) = Iεy(1) + µ

[

Iεy(1) +Γ(2 − ε)

1 − γξ1−p

(

γIε−py(ξ)− Iε−py(1))

]

+Γ(2 − α)

1 − γξ1−p

[


.

(3.1.5)

Thus in view of (3.1.5), our consider class of differential equation can be written as

u(t) =∫ 1

0G(t, s) f (s, u(s), c

Dε−1u(s))ds,

where G(t, s) is Green’s function defined in (3.1.2). Further G(t, s) satisfies the

following condition:

(H) ε − p < 1 then G(t, s) become unbounded but the function t :→∫ 1

0 G(t, s)ds

is continuous on J, So it must attained the supremum value say G∗ given by

G∗ = supt∈J

∫ 1

0|G(t, s)|ds.

26

Theorem 3.1.2. Assume that

(H0) f : J × R × R → R is continuous;

(H1) |g(u)− g(u)| ≤ k|u − u|;

(H2) There exists a constant k > 0 such that t ∈ J and forall u, v, u, v ∈ R hold,

| f (t, u, v)− f (t, u, v)| ≤ k

(

|u − u|+ |v − v|)

.

Then if

max

(

2G∗k,k ((ε − p)Γ(ε − p + 1) + Γ(2 − ε))

(ε − p)Γ(ε − p + 1)

)

< 1,

then the BVP (3.1.1) has a unique solution.

Proof. In order to prove the results, we will show that the operator T : c(J, R) →c(J, R) is defined by

Tu(t) =∫ 1

0G(t, s) f

(

s, u(s), cD

α−1u(s))

ds

has a unique fixed point, where G(t, s) is given by (3.1.2) . By using contraction

mapping techniques, we will show that the fixed point u(t) is the required solution

27

of BVP (3.1.1). Now by using theorem (2.2.1),

cD

ε−1u(t) =cD

ε−1

[

Iεy(t) +Γ(2 − p)

1 − γξ1−p

(

γIε−py(ξ)− Iε−py(t))

]

+Γ(2 − p)

1 − rξ1−pt[

γIε−py(ξ)− Iε−py(t)]

=Iy(t) +Γ(2 − p)

1 − γξ1−p

t2−ε

Γ(3 − ε)

[


=∫ 1

0y(s)ds − Γ(2 − ε)

(1 − γξ1−p)

t2−ε

Γ(3 − ε)Γ(ε − p − 1)

×[

∫ ξ

0(ξ − s)ε−p−1y(s)ds −

∫ 1

0(1 − s)ε−p−1y(s)ds

]

=∫ 1

0y(s)ds − Γ(2 − ε)

(1 − rξ1−p)

t2−ε

Γ(3 − ε)Γ(ε − p − 1)

×[

∫ 1

0(1 − s)ε−p−1y(s)ds − γ

∫ ξ

0(ξ − s)ε−p−1y(s)ds

]

.

Clearly by continuity of f both Tu(t) and cD ε−1Tu(t) are containing on J. By Ba-

nach contraction principle we shall prove that T has a fixed point. Let u, v ∈c(J, R), then for each t ∈ J, we have

|Tu(t)− Tv(t)| ≤∫ 1

0G(t, s)| f ((s, u(s),c D

ε−1u(s))− f (s, v(s),c Dε−1v(s))|ds

≤G∗k(|u − v|+ |cD ε−1u(s)−cD

ε−1v(s)|)

≤G∗k(

‖u − v‖∞, ‖cD

ε−1u −cD

ε−1v‖∞

)

+ G∗k

(

‖u − v‖∞, ‖cD

ε−1u −cD

ε−1v‖∞

)

≤2G∗k‖u − v‖c

28

and

|cD ε−1Tu(t)−cD

ε−1Tv(t)| ≤2k‖u − v‖c + kΓ(2 − p)γξε−pt2−ε

(ε − p)Γ(ε − p − 1)Γ(3 − ε)(1 − γξ1−p)

× ‖u − u‖c

=k‖u − v‖c[2 +Γ(2 − p)A

(ε − p)Γ(ε − p − 1)Γ(3 − ε)],

where A = rξε−p

1−rξ1−p . Which implies that

|cD ε−1Tu(t)−cD

ε−1Tv(t)| = k‖u − v‖c

{2(ε − p)Γ(ε − p − 1) + Γ(2 − p)A

(ε − p)Γ(3 − ε)Γ(ε − p − 1)

}

.

Let(

2G∗k, k2(ε − p)Γ(ε − p − 1)Γ(3 − p) + Γ(2 − p)A

(ε − p)Γ(3 − ε)Γ(ε − p − 1)

)

= d < 1,

so

‖Tu − Tu‖ ≤ d‖u − u‖c.

Hence T has a contraction mapping. Therefore by Banach fixed point theorem the

T has a unique solution.

Theorem 3.1.3. Assume that (H0) and (H2) hold, then there exist

p ∈ C(J, R+) and ψ : [0, ∞) → [0, ∞)

is continuous and non-decreasing functions such that for all t ∈ J, u, v ∈ R

| f (t, u, v)| ≤ p(t)ψ(|v|)for t ∈ J, u, v ∈ R.

(H3) Their exist r > 0 such that

r ≥ maxt∈J

(

G∗p∗ψ(r), p∗ψ(r),2(ε − p)Γ(3 − ε)Γ(ε − p − 1) + AΓ(2 − ε)

(ε − p)(3 − ε)(ε − p − 1)

)

such that p∗ = sup (p(s), s ∈ J). Then BVP (3.1.1) has at least one solution on J with

|u(t)| < r for each t ∈ J.

29

Proof. To prove the theorem we will prove that T satisfies all the conditions of

schauder’s fixed point theorem.

Step 1: First we will prove that T is continuous.

Let us defined a bounded set by D = {u ∈ c(J, R), ‖u‖c ≤ r}, where r is a constant

and r > 0 such that

r ≥ max

(

G∗p∗ψ(r), p∗ψ(r),2(ε − p)Γ(3 − ε)Γ(ε − p − 1) + AΓ(2 − ε)

(ε − p)(3 − ε)(ε − p − 1)r(3 − ε)

)

is closed subset of c(J, R) if (un) is convergent sequence in c(J, R), such that un → u

as n → ∞. Now c(J, R) is Banach space. There exist ρ > 0 such that

‖u‖c ≤ ρ ‖u‖ ≤ ρ for all t ∈ J.

|Tun(t)− Tu(t)| ≤∫ 1

0|G(t, s) f

(

s, un(s),cD

ε−1un(s))

− f(

s, u(s),c Dε−1u(s)

)

|ds.

(3.1.6)

As f (s, u(s), D ε−1) is continuous, then by Lebesgue convergence theorem

limn→∞

∫ 1

0fn(s)ds =

∫ 1

0lim

n→∞fn(s)ds

=∫ 1

0f (s)ds.

30

Therefore if n → ∞, then ‖Tun(t)− Tu(t)‖∞ → 0, similarly

|cD ε−1Tun(t)−cD

ε−1Tu(t)|

≤∫ 1

0| f(

s, un(s),cD

ε−1u(s))

− f(

s, u(s),c Dε−1u(s)

)

|ds

+t2−ε

(ε − p)Γ(ε − p − 1)(1 − rξ1−p)[

∫ 1

0(1 − s)ε−p−1

(

f (s, u(s),c Dε−1u(s)− f (s, u(s),c D

ε−1u(s)))

ds

]

− t2−ε

(ε − p)Γ(ε − p − 1)(1 − rξ1−p)[

r∫ ξ

0(ξ − s)ε−p−1

(

f (s, un(s),cD

ε−1)u(s)− f (s, u(s),c Dε−1u(s)

)

]

ds,

by continuity of f and Lebesgues Dominated convergence theorem we have

‖cD

ε−1Tu(t)−cD

ε−1Tu(t)‖∞ → 0 as n → ∞

‖Tun(t)− Tu(t)‖c → 0 as n → ∞.

T is sequentially continuous. Hence T is continuous.

Step 2: Suppose D ∈ c(J, R) is bounded set, we have to show that T(D) is also

bounded.

for this we have to show that T(D) ⊆ D. Suppose u(t) ∈ D then for all t ∈ J,

|Tu(t)| =∫ 1

0|G(t, s) f (s, u(s),c D

ε−1u(s))|ds

≤G∗∫ 1

0| f (s, u(s), c

Dε−1u(s))|ds

≤G∗P∗ψ(‖u‖c)

≤G∗P∗ψ(r).

31

Further

|cD ε−1Tu(t)| ≤∫ 1

0| f (s, un(s),

cD

ε−1u(s))|ds

+ t2−ε 2(ε − p)Γ(3 − p)Γ(ε − p − 1) + AΓ(2 − ε)

(ε − p)Γ(ε − p − 1)Γ(3 − ε)

×∫ 1

0(1 − s)ε−p−1| f (s, u(s),c D

ε−1u(s)ds|

≤p∗‖ψ‖uc +2(ε − p)

Γ(3 − ε)Γ(ε − p − 1)(ε − p)Γ(ε − p − 1)Γ(3 − ε).

Hence, we have

||Tu(t)|| ≤ max

(

G∗p∗ψ(r), p∗ψ(r)2(ε − p)Γ(ε − p − 1)Γ(3 − ε) + AΓ(2 − p)

Γ(3 − ε)Γ(ε − p − 1)Γ(3 − ε)

)

≤ r,

where

r ≥ max

(

G∗p∗ψ(r), p∗ψ(r)2(ε − p)Γ(ε − p − 1)Γ(3 − ε) + AΓ(2 − p)

Γ(3 − ε)Γ(ε − p − 1)Γ(3 − ε)

)

showing that Tu(t) ∈ D, ∀ u(t) ∈ D. T(D) ⊆ D. So T is bounded.

Step 3: To prove T maps D into an equi-continuous set of c(J, R). For this let

t1, t2 ∈ J, 0 ≤ t1 ≤ t2 ≤ 1 and u(t) ∈ D we have

|Tu(t2)− Tu(t1)| ≤∫ 1

0

(

|G(t2, s)− G(t1, s)|| f (s, u(s),c Dε−1u(s))|ds

)

≤ p∗ψ(‖u‖c)∫ 1

0(|G(t2, s)− G(t1, s)) |ds.

32

If t2 → t1 then |Tu(t2)− Tu(t1)| → 0 and

|TcD

ε−1u(t2)− TcDα−1u(t1)|

=

∣

∣

∣

∣

∫ t2

0f (s, u(s),c D

ε−1u(s))ds −[

t2−p2

Γ(2 − p)

Γ(3 − ε)Γ(ε − p)Γ(ε − p − 1)

]

∫ 1

0(1 − s)ε−p−1 f (s, u(s),c D

ε−1u(s)ds

∣

∣

∣

∣

≤∫ t2

0(1 − s)ε−p−1 f (s, u(s),c D

ε−1u(s))− |∫ t1

0f (s, u(s),c D

ε−1, u(s))ds|

≤ p∗ψ(‖cD

ε−1u(s)‖∞)

(

(t2 − t1)(t2−α

1 − t2−ε2 )Γ(2 − p)

Γ(3 − ε)(ε − p)Γ(ε − p)

(1 − s)ε−p

ε − p

)

,

which implies that

≤(

p∗ψ(‖cD

ε−1u(s)‖∞

(

(t2 − t1) +t2−ε1 − t

ε−p+12 Γ(2 − ε)(1 − s)ε−p

Γ(3 − ε)Γ(ε − p)(ε − p)

))

≤ p∗ψ(‖u‖∞)

(

(t2 − t1) +t2−ε1 − t2−ε

2 Γ(2 − ε)

Γ(3 − ε)Γ(ε − p − 1)

)

.

Now, as t1 → t2, we have

|TcD

ε−1u(t2)− TcD

ε−1u(t1)| → 0,

and ‖Tu(t2 − Tu(t1))‖ → 0.

So T is completely continuous by Arzela–Ascoli theorem. Hence using Shauder

fixed point theorem T has a fixed point u(t) in D, which is the solution of BVP

(3.1) such that |u(t)| < r, ∀ t ∈ J.

Theorem 3.1.4. Under the assumption (H1) and (H2)

u(t) =∫ 1

0G(t, s) f (s, u(s), c

Dε−1u(s))ds (3.1.7)

is Hyers–Ulam stable if there exist c f > 0, for each ǫ > 0 and for each solution u∗ to the

34

3.1.1 Example

Example 3.1.1. To demonstrate our results , we present the following example. Consider

the fractional order BVP given by

D32 u(t) =

1

(40e5t + 1)(1 + |u(t)|+c D12 u(t))

, t ∈ [0, 1]

u(0) =1

2u(1),c D

13 u(1) =

1

3

c

D13 u(

1

3),

(3.1.8)

r =1

3, ξ =

1

3, p = 0, ε =

3

2, δ =

1

2, f (t, u, v) =

1

(40e5t + 1)(1 + |u|+ |v|) .

Let u, v, u, v ∈ R, for all t ∈ J, h(t) = e5tu5(1+e5t)(1+u)

, now

| f (t, u, v)− f (t, u, v| = | 1

40e5t + 1‖ 1

1 + |u|+ |v| −1

1 + |u|+ |v| ‖

≤ 1

41[|u − u|+ |v − v|].

Now

| f (t, u, v)− f (t, u, v| ≤ | 1

41||u − u|+ |v − v| here k =

1

41.

Now

u(t) = I32 y(t) + c0 + c1t (3.1.9)

from equation (3.1.9) we say that

u(0) = c0, c0 =1

2u(1).

Putting t = 0 in equation (3.1.9) we get

u(1) = I32 y(1) + c0 + c1 (3.1.10)

35

so

c0 =1

2[y(1) + c1 (3.1.11)

from Eq.(3.1.10) we say that

1

3u(

1

3) = I

32 y(1) + c0 + c1 (3.1.12)

c1 =3

2I

32 [y(1)− y(

1

3)], c0 =

1

4I

32 y(1)− 3

4I

32 y(

1

3). (3.1.13)

Now u(t) =∫ 1

0 G(t, s) f (s, u(s),c Dα−1u(s))ds, where

G(t, s) =

− (t − s)12 − 3

4(

1

3− s)

12 ,

+3

2t[(1 − s)

12 − (

1

3− s)

12 ] +

1

4(1 − s)

12 , 0 ≤ s ≤ t ≤ 3

4,

1

4(1 − s)

12 − 3

4(

1

3− s)

12

+3

2t[(1 − s)

12 − (

1

3− s)

12 ], 0 ≤ t ≤ s ≤ 1

3,

G∗ = maxt∈[0,1]

|∫ 1

0G(s, s)ds|

≤ 1

Γ 32

∫ 1

0

(

1

4(1 − s)

12 − 3

4(

1

3− s)

12 +

3

2s(1 − s)

12 − 3

2s(

1

3− s)

12 ]

)

ds

G∗ = | 7

11(0.084)| = 0.05, 2G∗k =

2 × 0.005

41≈ 0.00024.

As we know that

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

(α − p)Γ(α − p − 1)Γ(3 − α)

=141

(

2(32)Γ(

32 − 1) + Γ(3 − 3

2) + (0.024)Γ(2))

(32)Γ(

32 − 1)Γ(3 − 3

2)≈ 0.0466

where A = 0.024.

Hence max

(

2G∗k,(2(α − p)Γ(α − p − 1) + Γ(3 − α) + AΓ(2 − p))

(α − p)Γ(α − p − 1)Γ(3 − α)

)

< 1.

36

Hence by Banach contraction theorem the given fractional order BVP (3.1.1) has a unique

positive solution on J ∈ [0, 1]. Also the BVP (3.1.1) is Hyers–Ulam stable by using

Theorem 3.1.4.

3.2 Coupled System of Nonlinear Fractional Order Dif-

ferential Equation with Non-local Boundary Value

Problem

In this section, we study existence and uniqueness of positive solution to a coupled

system of non–local BVP of non linear fractional order differential equations with

non-local BVP of the form

cD

pu(t) + f (t, v(t),c Dµv(t)) = 0 0 < t < 1, 2 < p ≤ 3

cD

qv(t) + g(t, u(t),c Dµu(t)) = 0 0 < t < 1, 2 < q ≤ 3

u(0) = u′(0) = 0, u(1) = u(α), 0 < α < 1,

v(0) = v′(0) = 0, v(1) = v(β), 0 < β < 1,

(3.2.1)

where cD p and cDq are Caputo’s fractional derivatives of order , p, q respectively

and f , g : [0, 1]× R × R → R are given to be continuous nonlinear functions, the

non-linear terms depend on the fractional derivative of the function u, v. Further

p − ν ≥ 1, q − µ ≥ 1. We develop necessary and sufficient conditions for the exis-

tence and uniqueness of positive solutions for the above system by using standard

Banach Fixed point theorem and Leray–Schouder fixed point theorem. We also

present an example to illustrate our main results.

37

Lemma 3.2.1. Let u, v ∈ C[0, 1] and 2 < p, q ≤ 3. Then the fractional BVP(3.2.1) has a

unique solution which is given by

(u(t), v(t)) = (∫ 1

0G1(t, s) f (s, v(s),c D

µv(s))ds,∫ 1

0G2(t, s)g(s, u(s),c D

νu(s))ds),

(3.2.2)

where

G1(t, s) =1

Γ(p)

t2

1 − α2[(1 − s)p−1 − (α − s)p−1]− (t − s)p−1, 0 ≤ s ≤ t ≤ 1, s ≤ α,

t2

1 − α2[(1 − s)p−1 − (α − s)p−1], 0 ≤ t ≤ s ≤ 1, s ≤ α,

t2

1 − α2[(1 − s)p−1 − (t − s)p−1], 0 ≤ s ≤ t ≤ 1, α ≤ s,

t2(1 − s)2

1 − α2, 0 ≤ t ≤ s ≤ 1, α ≤ s,

(3.2.3)

and

G2(t, s) =1

Γ(q)

t2

1 − β2[(1 − s)q−1 − (β − s)q−1]− (t − s)q−1, 0 ≤ s ≤ t ≤ 1, s ≤ β,

t2

1 − β2[(1 − s)q−1 − (β − s)q−1], 0 ≤ t ≤ s ≤ 1, s ≤ β,

t2

1 − β2[(1 − s)q−1 − (t − s)q−1], 0 ≤ s ≤ t ≤ 1, β ≤ s,

t2(1 − s)2

1 − β2, 0 ≤ t ≤ s ≤ 1, β ≤ s.

(3.2.4)

Proof. Applying lemma (2.1.3) to first equation of the system of BVP (3.2.1), we

have

u(t) = c0 + c1t + c2t2 − Ip f (t, v(t),c Dµv(t)) (3.2.5)

using initial conditions u(0) = 0, we have c0 = 0 and

u′(t) = c1 + 2c2t − Ip−1 f (t, v(t),c Dµv(t)), (3.2.6)

38

using u′(0) = 0 in (3.2.6), we get c1 = 0. Hence (3.2.5) becomes

u(t) = c2t2 − Ip f (t, v(t),c Dµv(t)) (3.2.7)

Now by applying boundary conditions u(1) = u(α), we get

c2 −1

Γ(p)(∫ 1

0(1 − s)p−1 f (1, v(1),c D

µv(1))ds

= c2α2 − 1

Γ(p)

(

∫ α

0(α − s)p−1 f (α, v(α),c D

νv(α)

)

ds

which implies that c2 =1

(1 − α2)Γ(p)×

[

∫ 1

0(1 − s)p−1u(1, v(1),c D

νv(1))ds −∫ α

0(α − s)p−1 f (α, v(α),c D

µv(α))ds

]

.

(3.2.8)

Hence (3.2.7) implies that

u(t) =t2

(1 − α2)Γ(p)

[

∫ 1

0(1 − s)p−1 f (1, v(1),c D

µv(1))ds

−∫ α

0(α − s)p−1 f (α, v(α),c D

µv(α))ds

]

− 1

Γ(p)(∫ t

0(t − s)p−1 f (s, v(s),c D

µv(s))ds,

(3.2.9)

which implies that

u(t) =∫ 1

0G1(t, s) f (s, v(s),c D

µv(s))ds. (3.2.10)

Similarly by above fashion, we can get second part of (3.2.2) as

v(t) =∫ 1


νu(s))ds (3.2.11)

where G1(t, s) and G2(t, s) can be obtain as given in (3.2.3) and(3.2.4), which com-

pletes the proof.

Lemma 3.2.2. The Green’s functions Gi(t, s)(i = 1, 2) has the following properties:

39

(P1) Gi(t, s) > 0, (i = 1, 2), for t, s ∈ (0, 1);

(P2) Gi(t, s) ≥ 0, (i = 1, 2), for t, s ∈ [0, 1];

(P3) Gi(t, s), (i = 1, 2) are continuous functions on the unit square for all (t, s) ∈[0, 1]× [0, 1].

On ward in this section, we investigate the existence of positive solution for the

BVP (3.2.1). Let us define

E1 = {u(t) : u(t) ∈ C[0, 1], cD

νu(t) ∈ C(1)[0, 1]}

and

E2 = {v(t) : v(t) ∈ C[0, 1],c Dµv(t) ∈ C(1)[0, 1]}

endowed with the norm

‖u‖E1= max

t∈[0,1]|u(t)|+ max

t∈[0,1]|cDµu(t)|.

For (u, v) ∈ E1 × E2, then ‖(u, v)‖E1×E2= max{‖u‖E1

, ‖v‖E2}. Clearly (E1 ×

E2, ‖(u, v)‖E1×E2) is a Banach space . Writing the system of integral equations ob-

tain in Lemma (3.2.1) as an equivalent system of integral equations.

u(t) =∫ 1

0G1(t, s) f (s, v(s),c D

µv(s))ds,

v(t) =∫ 1


νu(s))ds

(3.2.12)

Further define

T : E1 × E2 → E1 × E2 by

T(u, v)(t) =

(

∫ 1

0G1(t, s) f (s, v(s),c D

µv(s))ds,∫ 1


νu(s))ds

)

implies thatT(u, v)(t) = (T1v(t), T2u(t)) .

(3.2.13)

40

Then the fixed points of operator T coincide with the solution of system (3.2.1). For

further study, the following assumptions are hold:

(A1) The nonlinear functions f , g : [0, 1]× R × R → R are continuous;

(A2) There exists non-negative functions m(t), n(t) ∈ C(0, 1) such that

| f (t, x(t), y(t))| ≤ (m(t) + d1|x|ρ1 + d2|y|ρ2) ,

|g(t, x(t), y(t))| ≤(

n(t) + d′1|x|µ1 + d

′2|y|µ2

)

,

where 0 ≤ µ1, µ2, ρ1, ρ2 < 1 and di, d′i ≥ 0 for i = 1, 2 are real numbers.

Let us use the following notations for simplicity as

A =

(

3 + |(1 − α2)||(1 − α2)|Γ(p + 1)

+4 + Γ(3 − ν)|(1 − α2)|

Γ(3 − ν)|(1 − α2)|Γ(p − ν)

)

,

B =

(

3 + |(1 − β2)||(1 − β2)|Γ(q + 1)

+4 + Γ(3 − µ)|(1 − β2)|

Γ(3 − µ)|(1 − β2)|Γ(q − µ)

)

,

K = maxt∈[0,1]

∫ 1

0|G1(t, s)m(s)ds|, L = max

t∈[0,1]

∫ 1

0|G2(t, s)n(s)ds|

λ1 =

(

4 + Γ(3 − ν)|(1 − α2)|Γ(3 − ν)|(1 − α2)|Γ(p − ν)

)

,

λ2 =

(

4 + Γ(3 − µ)|(1 − β2)|Γ(3 − µ)|(1 − β2)|Γ(q − µ)

)

(3.2.14)

Theorem 3.2.3. Assume that assumptions (A1), (A2) hold, then the BVP (3.2.1) has a

solution.

Proof. Let W is a ball in E1 × E2 defined by

W = {(u(t), v(t)) |(u, v) ∈ E1 × E2, ‖(u, v)‖E1×E2≤ R, t ∈ [0, 1]},

where max{(3Ad1)1

1−ρ1 , (3Ad2)1

1−ρ2 , (3Bd′1)

11−µ1 , (3d

′2)

11−µ2 , 3K, 3L} ≤ R.

(3.2.15)

42

Now from (3.2.9) and using cDνt2 = 2t2−ν

Γ(3−ν), we have

|cDνT1v(t)| =|cDνu(t)|

≤

∣

∣

∣

∣

∣

∣

2t2−ν

Γ(3 − ν)(1 − α2)Γ(p)

1∫

0

(1 − s)p−1 f (s, v(s)cD

νv(s))ds

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

2t2−ν

Γ(3 − ν)(1 − α2)Γ(p)

α∫

0

(α − s)p−1 f (s, v(s)cD

νv(s))ds

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

1

Γ(p − ν)

t∫

0

(t − s)p−ν−1 f (s, v(s)cD

νv(s))ds

∣

∣

∣

∣

∣

∣

≤ 2

Γ(3 − ν)|(1 − α2)|

1∫

0

(1 − s)p−1

Γp(m(s) + d1Rρ1 + d2Rρ2) ds

+2

Γ(3 − ν)|(1 − α2)|

α∫

0

(α − s)p−1

Γ(p)(m(s) + d1Rρ1 + d2Rρ2)ds

+1

Γ(p − ν)

t∫

0

(t − s)p−ν−1(m(s) + d1Rρ1 + d2Rρ2)ds

≤ 4

Γ(3 − ν)|(1 − α2)|

1∫

0

(1 − s)p−1

Γ(p)m(s)ds +

(

d1Rρ1 + d2Rρ2

Γ(p + 1)

)

+1

Γ(p − v)

1∫

0

(1 − s)p−v−1(m(s)ds +

(

d1Rρ1 + d2Rρ2)

Γ(p − ν + 1)

)

≤(

4

Γ(3 − ν)|(1 − α2)|Γ(p)+

1

Γ(p − v)

) 1∫

0

(1 − s)p−1m(s)ds

+

[

4

Γ(3 − ν)|(1 − α2)|Γ(p + 1)+

1

Γ(p − v + 1)

]

+ (d1Rρ1 + d2Rρ2)

≤(

4 + Γ(3 − ν)|(1 − α2)|Γ(3 − ν)|(1 − α2)|Γ(p − ν)

) 1∫

0

(1 − s)p−1m(s)ds

+4 + Γ(3 − ν)|(1 − α2)|

Γ(3 − ν)|(1 − α2)|Γ(p − ν)(d1Rρ1 + d2Rρ2). (3.2.17)

43

Now from (3.2.16) and(3.2.17), we have

‖T1v‖E1≤K +

(

4 + Γ(3 − ν)|(1 − α2)|Γ(3 − ν)|(1 − α2)|Γ(p − ν)

) 1∫

0

(1 − s)p−1m(s)ds

+

(

3 + |(1 − α2)||(1 − α2)|Γ(p + 1)

+4 + Γ(3 − ν)|(1 − α2)|

Γ(3 − ν)|(1 − α2)|Γ(p − ν)

)

(d1Rρ1 + d2Rρ2)

≤K +

(

4 + Γ(3 − ν)|(1 − α2)|Γ(3 − ν)|(1 − α2)|Γ(p − ν)

) 1∫

0

(1 − s)p−1m(s)ds + A(d1Rρ1 + d2Rρ2)

≤K + λ1

1∫

0

(1 − s)p−1m(s)ds + A(d1Rρ1 + d2Rρ2)

≤R3+

R3+

R3

= R

which implies that

‖T1v‖E1≤ R. (3.2.18)

Similarly

‖ T2u ‖E2≤ R. (3.2.19)

Hence

‖T(u, v)‖E2×E1≤ R. (3.2.20)

Thus T : W → W is well defined and T(W) ⊆ W. As f , g, G1, G2 are continuous.

So T is continuous. For completely continuity, let us take

M = maxt∈[0,1]

| f (t, v(t),c Dµv(t)|, N = max

t∈[0,1]| f (t, v(t),c D

µv(t)|, t ≤ τ < 1,

45

Clearly in all of the above when t → τ.

Then right hand side of (3.2.21), (3.2.22), (3.2.23), (3.2.24) tend to zero. Furthermore

t2, τ2, τp, τq, (τ − t)p, (τ − t)q, tp−ν, τp−ν, (τ − t)p−ν, tq−µ, τq−µ, (τ − t)q−µ

are uniformly continuous. Furthermore T is uniformly bounded and continuous,

hence equi-continuous. Hence T is completely continuous. Thus by Schauder fixed

point theorem T has at least one fixed point which is the corresponding solution of

(3.2.1), which completes the proof.

For any u, v, x, y ∈ R and there exist K1, K2 > 0, the following assumption hold:

(A3) | f (t, u, v)− f (t, x, y)| ≤ K1[|u − x|+ |v − y|]and

|g(t, u, v)− g(t, x, y)| ≤ K1[|u − x|+ |v − y|].

Theorem 3.2.4. Under the assumption (A1) and (A3) and if

max

(

3G∗1 K1, K1

4Γ(p − v + 1) + Γ(3 − ν)|(1 − α2)|Γ(p + 1)

Γ(3 − ν)|(1 − α2)|Γ(p + 1)Γ(p − ν + 1)

)

< 1

and

max

(

3G∗2 K2, K2

4Γ(q − µ + 1) + Γ(3 − µ)|(1 − β2)|Γ(q + 1)

Γ(3 − µ)|(1 − β2)|Γ(q + 1)Γ(q − µ + 1)

)

< 1.

Then the problem (3.2.1) under consideration has a unique solutions.

46

Proof. Consider u, u, v, v ∈ R, then

|T1v(t)− T1v(t)| = |u(t)− u(t)|

=

∣

∣

∣

∣

∣

∣

1∫

0

G1(t, s)( f (s, v(s),c Dνv(s))− f (s, v(s),c D

νv(s))ds

∣

∣

∣

∣

∣

∣

≤ maxt∈[0,1]

1∫

0

|G1(t, s)|ds[K1(|v(s)− v(s)|+ |cDνv(s)−cD

νv(s)|)ds]

≤ 3G∗K1[‖v − v‖+ ‖cD

vv −cD

vv‖]

≤ G∗K1[‖v − v‖+ ‖cD

νv −cD

νv‖]

and also

|cDνT1v(t)−cD

νv(t)|

≤

∣

∣

∣

∣

∣

∣

2t2−ν

Γ(3 − ν)|(1 − α2)|Γ(p)

1∫

0

(1 − s)p−1[ f (s, v(s),c Dνv(s))− f (s, v(s),c D

νv(s))]ds

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

2t2−ν

Γ(3 − ν)|(1 − α2)|Γ(p)

α∫

0

(α − s)p−1[ f (s, v(s),c Dνv(s))− f (s, v(s),c D

νv(s))]ds

∣

∣

∣

∣

∣

∣

+

∣

∣

∣

∣

∣

∣

1

Γ(p − v)

t∫

0

(t − s)p−ν−1[ f (s, v(s),c Dνv(s))− f (s, v(s),c D

νv(s))]ds

∣

∣

∣

∣

∣

∣

≤ K1

(

4

Γ(3 − ν)|(1 − α2)|Γ(p + 1)+

1

Γ(p − v + 1)

)

[‖v − v‖+ ‖cD

νv −cD

νv‖]

⇒ ‖T1v − T1v‖E1≤ max

(

3G∗1 K1, K1

4Γ(p − ν + 1) + Γ(3 − ν)|(1 − α2)|Γ(p + 1)

Γ(3 − ν)|(1 − α2)|Γ(p + 1)Γ(p − ν + 1)

)

D [‖v − v‖+ ‖cD

νv −cD

νv‖].

(3.2.25)

47

Similarly

‖T2u − T2u‖E2≤ max

(

3G∗2 K2, K2

4Γ(q − µ + 1) + Γ(3 − µ)|(1 − β2)|Γ(q + 1)

Γ(3 − µ)|(1 − β2)|Γ(q + 1)Γ(q − µ + 1)

)

[‖u − u‖+ ‖cD

µu −cD

µu‖].

(3.2.26)

Thus from (3.2.25) and (3.2.26), we have

‖T(u, v)− T(u, v)‖ ≤ K‖(u, v)− (u, v‖, where K < 1.

Hence T has a unique positive solution, which complete the proof.

3.2.1 Example

Example 3.2.1. Consider the following multi-point BVP

cD

52 u(t) + f (t, v(t),c D

32 v(t)) = 0,c D

52 v(t) + g(t, u(t),c D

32 u(t)) = 0, 0 < t < 1,

u(0) = u′(0) = 0, u(1) = u

(

1

2

)

, v(0) = v′(0) = 0, v(1) = v(1

2),

(3.2.27)

where

f (t, v(t),c D32 v(t)) =

1

(29e2t + 9)(1 + |v(t)|+ |cD 32 v(t)|

,

g(t, u(t),c D12 u(t)) =

cos u(t) +c D12 u(t)

18 + t + t2.

Then, for any u, v, u, v ∈ R, we have

| f (t, u, v)− f (t, u, v)| ≤ 1

38[|u − u|+ |v − v|],

|g(t, u, v)− g(t, u, v)| ≤ 1

18[|u − u|+ |v − v|],

K1 =1

38, K2 =

1

18.

48

G1(t, s) =1

Γ(52)

4t2

3[(1 − s)

32 − (

1 − 2s

2)

32 ]− (t − s)

32 , 0 ≤ s ≤ t ≤ 1, s ≤ 1

24t2

3[(1 − s)

32 − (

1 − 2s

2)

32 ], 0 ≤ t ≤ s ≤ 1, s ≤ 1

2,

4t2

3[(1 − s)

32 − (t − s)

32 ], 0 ≤ s ≤ t ≤ 1,

1

2≤ s,

4t2

3(1 − s)

32 , 0 ≤ t ≤ s ≤ 1,

1

2≤ s,

and

G2(t, s) =1

Γ(52)

4t2

3[(1 − s)

32 − (

1 − 2s

2)

32 ]− (t − s)

32 , 0 ≤ s ≤ t ≤ 1, s ≤ 1

2,

4t2

3[(1 − s)

32 − (

1 − 2s

2)

32 ], 0 ≤ t ≤ s ≤ 1, s ≤ 1

2,

4t2

3[(1 − s)

32 − (t − s)

32 ], 0 ≤ s ≤ t ≤ 1,

1

2≤ s,

4t2

3(1 − s)

32 , 0 ≤ t ≤ s ≤ 1,

1

2≤ s.

Now, we calculate

G∗1 = sup

t∈[0,1]

∫ 1

0|G1(t, s)|ds =

∫ 1

0

4s2(1 − s)33

3Γ(52)

ds

=8

9√

(π)

∫ 1

0s2(1 − s)

32 ds =

128

2835√

π.

Similarly, repeating the same process, we obtain

G∗2 = sup

t∈[0,1]

∫ 1

0|G2(t, s)|ds =

128

2835√

π.

Now by using the above values and putting p = q = 52 , µ = ν = 3

2 , α = β = 12 , we have

a = max{2G∗1 K1, K1

4Γ(p − v + 1) + Γ(3 − ν)|(1 − α2)|Γ(p + 1)

Γ(3 − ν)|(1 − α2)|Γ(p + 1)Γ(p − ν + 1)}

max{0.00134, 0.14071} = 0.14071 < 1.

Similarly we can show that by putting values that

b = max{3G∗2 K2, K2

4Γ(q−µ+1)+Γ(3−µ)|(1−β2)|Γ(q+1)Γ(3−µ)|(1−β2)|Γ(q+1)Γ(q−µ+1)

} < 1.

49

Clearly max(a, b) < 1. Thus by theorem (3.2.4), the BVP (3.2.1) has a unique positive

solution.

3.3 Monotone Iterative Technique for Boundary Value

Problem

This section is related with the investigation of multiple solutions to the following

BVP of NFODEs by using monotone iterative techniques

cD

εu(t) = θ(t, u(t)); t ∈ I ; ε ∈ (1, 2],

u(0) = γu′(0) = 0, u(1) = δu

′(1), γ > 0, δ > 0.

(3.3.1)

Where θ : [0, 1] × R −→ R is continuous function, while cD stands for Caputo

fractional derivative of order ε. The monotone iterative technique combined with

the method of extremal (lower and upper) solutions is a strong tools being used to

find multiple solutions to NFODEs as well as integer order differential equations

and their systems. The monotone iterative technique were used in some papers to

develop conditions for existence of iterative solutions for ordinary and NFODES

(see[44, 49, 66, 39]). By using the aforesaid technique to established some adequate

conditions for existence of iterative solutions to NFDEs, one need a proper differ-

ential inequalities as a comparison results.

Definition 3.3.1. A function v(t) ∈ C2[0, 1], is called a lower solution of the prob-

lem (3.3.1), if

cD

εv(t) + θ(t, v(t)) ≥ 0, t ∈ I , ε ∈ (1, 2],

v(0) ≤ γv′(0), v(1) ≤ δv

′(1).

50

Similarly w ∈ C2[0, 1] is called upper solution of (3.3.1), if

cD

εw(t) + θ(t, w(t)) ≤ 0, t ∈ I , ε ∈ (1, 2],

w(0) ≥ γw′(0), w(1) ≥ δw

′(1).

Theorem 3.3.1. Assume that w ∈ C2[0, 1] attain its minimum at t0 ∈ I , then

(cD

εw)(t0) ≥1

Γ(2 − ε)

[

(ε − 1)t−ε0 (w(0)− w(t0))− t1−ε

0 w′(0)

]

, 1 < ε < 2.

Corollary 3.3.2. Assume that w ∈ C2[0, 1] attain its minimum at t0 ∈ I , and w′(0) ≤

0. Then (cD εw)(t0) ≥ 0, 1 < ε < 2.

This is called the positivity result.

Lemma 3.3.3. Let w(t) ∈ C2[0, 1], µ(t, w) ∈ C([0, 1]× R) and µ(t, w) < 0, ∀t ∈ I .

If w(t) satisfies the inequalities

cD

εw(t) + a(t)w′(t) + µ(t, w)w ≤ 0, t ∈ I ,

w(0)− γw′(0) ≥ 0, w(1)− δw

′(1) ≥ 0,

where a(t) ∈ C[0, 1] and γ, δ ≥ 0, then w(t) ≥ 0, for all t ∈ [0, 1] provided that 1ε−1 6= 0.

Lemma 3.3.4. Let β and α be any upper and lower solutions, respectively of (3.3.1). If

θ(t, u(t)) with respect to u decreasing , then α, β are ordered, i.e α(t) ≤ β(t), for t ∈ [0, 1].

Proof. Consider the lower and upper solution α, β, then (3.3.1) yields

cD

εα(t) + θ(t, α(t)) ≥ 0, t ∈ I ,

α(0) ≤ γα′(0), α(1) ≤ δα

′(1)

51

and

cD

εβ(t) + θ(t, β(t)) ≤ 0, t ∈ I ,

β(0) ≥ γβ′(0), β(1) ≥ δβ

′(1).

Upon subtracting, we get

cD

ε(β − α) + θ(t, β)− θ(t, α) ≤ 0, (3.3.2)

using Mean value theorem from (3.3.2), we have

cD

ε(β − α) +∂θ

∂u(η)(β − α) ≤ 0, (3.3.3)

where η = aα + (1 − a)β, a ∈ [0, 1]. If we put β − α = z,

(3.3.3) yields

cD

εz(t) +∂θ

∂u(η)z ≤ 0,

with z(0) ≤ γz′(0), z(1) ≤ δz

′(1). As θ with respect to u, ∂θ

∂u < 0 is strictly de-

creasing. Using result in lemma 3.3.3, we have z > 0.

Lemma 3.3.5. If θ(t, u) with respect to u is strictly decreasing, then the model(3.2.5)

posses at most one solution.

Proof. Let u, v be two solutions of BVP (3.3.1), then

cD

εu + θ(t, u) = 0 with u(0) = γu′(0), u(1) = δu

′(1),

cD

εv + θ(t, v) = 0 with v(0) = γv′(0), v(1) = δv

′(1),

then on subtraction, we get

cD

ε(u − v) + θ(t, u)− θ(t, v) = 0,

u(0)− v(0) = γ(u′(0)− v

′(0)), u(1)− v(1) = δ(u

′(1)− v

′(1)),

(3.3.4)

52

applying Mean value theorem to (3.3.4), we have

cD

ε(u − v) +∂θ

∂u(u − v) = 0, (3.3.5)

where η = au + (1 − a)v, a ∈ [0, 1]. If we put z = u − v,then (3.3.5) yields

cD

εz(t) +∂θ

∂u(η)z = 0,

with z(0) = γz′(0), z(1) = δz

′(1). Using result in lemma 3.3.3, we have z > 0. But

−z also satisfied (3.3.5), so −z > 0. Therefore z = 0 ⇒ u = v. Hence the solution

is unique.

3.3.1 Existence of upper and lower solution by using monotone

sequences

In this subsection, we construct monotone iterative sequences and their conver-

gence to obtain upper and lower solution of the model(3.3.1). Consider ordered

appears lower and upper solution α and β respectively, then defined a set as

S = [α, β] ={

µ ∈ C2([0, 1]), α ≤ µ ≤ β}

,

as θ(t, u) with respect to u is strictly decreasing, so ∂θ∂u is bounded below in S i.e

there exists a positive constant d such that

−d ≤ ∂θ

∂u(t, η) < 0, ∀η ∈ S. (3.3.6)

Theorem 3.3.6. Consider the BVP (3.3.1) with θ(t, u) satisfies (3.3.6). Let u(0) and v(0)

be initial ordered lower and upper solutions of (3.3.1). Let u(i), v(i), i ≥ 1 be respectively,

the solution of

−cD

εu(i) + cu(i) = cu(i−1) + g(t, u(i−1)), t ∈ I ,

u(i)(0) = u(i)0 ≤ u(i−1)(0), u(i)(1) = u

(i)1 ≤ u(i−1)(1),

(3.3.7)

53

and

−cD

εv(i) + cv(i) = cv(i−1) + g(t, v(i−1)), t ∈ I ,

v(i)(0) = v(i)0 ≥ v(i−1)(0), v(i)(1) = v

(i)1 ≥ v(i−1)(1),

(3.3.8)

then we have

(1) The sequence u(i), i > 0, for (3.3.1) is increasing sequence of lower solution.

(2) The sequence v(i), i > 0, for (3.3.1) is decreasing sequence of upper solution. More-

over,

(3) u(i) ≤ v(i), ∀ i ≥ 1.

Proof. Clearly the proof of (2) and the proof of (1) are similar, so first prove (1),

we need to prove that

(i) u(i), u(i−1) > 0 for each i > 1, and

(ii) u(i) is a lower solution for each i ≥ 1.

To prove (i), we using induction procedure, from (3.3.7) with i = 1, we have

−cD

εu(1) + cu(1) = cu(0) + g(t, u(0)). (3.3.9)

Since u(0) is a lower solution,

cD

εu(0) + cu(i) + g(t, u(0)) ≥ 0. (3.3.10)

Adding (3.3.9) with (3.3.10), we obtain

cD

ε(u(1) − u(0))− c(u(1) − u(0)) ≤ 0.

Let u(1) − u(0) = z, the z satisfies

cD

ε(z)− cz ≤ 0, z(0) ≥ γz′(0), z(1) ≥ δz

′(1).

54

Since −c < 0, by positivity result in lemma 3.3.3, we have z > 0 or u(0) < u(1), and

the result is true for i = 1. Now suppose that the result is true for n ≤ i and prove

for n = i + 1.

From (3.3.7), we have

−cD

ε(u(i+1) − u(i)) + c(u(i+1) − u(i)) = c(u(i)− u(i−1)) + g(t, ui)− g(t, u(i−1)).

let z = u(i+1) − u(i), applying mean value theorem and and using induction hy-

pothesis u(i+1) < u(i), we obtain

cD

ε(z) + cz = c(u(i) − u(i−1)) +∂g

∂u(ζ)(u(i) − u(i−1))

≥ c(u(i) − u(i−1))− c(u(i) − u(i−1)) = 0.

Or cD ε(z)− cz ≤ 0,which gives z ≥ 0, by the positivity result 3.3.3. Hence u(i) ≤u(i+1) and the result is proved for n = i + 1. Therefore u(i+1) ≤ u(i) for all i ≥ 1,

which proves (i).

To prove (ii) subtracting g(u, u(i) from both side of (3.3.7) and using mean value

theorem, we get

cD

εu(i) + g(t, u(i)) = c(u(i) − u(i−1)) + g(t, u(i))− g(t, u(i−1))(

c +∂g

∂u(ζ)

)

(u(i) − u(i−1)) ≥ 0.

So u(i), i > 1 is a lower solution of model (3.3.1). Hence proved (ii). The proof

of (3) clear from Lemma(3.3.3), since by (1) and (2) , u(i) and v(i) are lower and

upper solutions respectively.

For the convergence results we provide the following theorems.

Theorem 3.3.7. Consider the BVP presented in (3.3.1), with g(u, t) conditions (3.3.6).

Let u(i) and v(i), i ≥ 0 ≥ be stated in Theorem (3.3.6). Then the sequences u(i) and

v(i), i ≥ 0, converge uniformly to u∗ and v∗ respectively, with u(i) ≤ v(i).

55

Proof. Asu(i) is bounded above by v(0) and monotonically increasing sequence, it

converge to say u∗. Similarly, the sequence w(i) is bounded below by u(0) and

monotonically decreasing, and it converge to say v∗. The sequences ui and v(i)

are sequences of continuous functions defined on the compact [0, 1], therefore by

Dini’s theorem [21], the convergence is uniform. Since by Theorem (??), u(i) ≤v(i), ∀ i ≥ 0, we have

u∗ = limi→∞

u(i) ≤ limi→∞

v(i) = v∗.

Theorem 3.3.8. If the boundary conditions in (3.3.7) and (3.3.8) are the same as in (3.3.1),

i.e., u(i)(0) = v(i)(0) and u(i)(1) = v(i)(1), i ≥ 1, then u∗ = v∗ = w, where z is the

local solution to (3.3.1).

Proof. We shall prove that u∗ = v∗ by showing that u∗ and v∗ are solution to (3.3.1)

and by lemma 3.3.5, followed. From (3.3.7), we have

−cD

εu(i) + cu(i) = cu(i−1) + g(t, u(i−1)). (3.3.11)

Applying the operator jβ for 1 ≤ β ≤ 2, (3.3.11) yields

−u(i) + c(i)0 + c

(i)1 t + cjβu(i) = cjβu(i−1) + jβg(t, u(i−1)).

Taking the limit and u(i) → u∗ and the continuity of g, we have

−u∗ + c∞0 + c∞

1 t + cjβu∗ = cjβu∗ + jβg(t, u∗), (3.3.12)

where c∞0 = limi→∞u(i)(0) = a and c∞

1 = limi→∞(ui)′(0) = a. Applying cD ε to

(3.3.12), using theorem 2.2.1, and denoting cD εti = 0, for i = 0.1, (3.3.12) reduces

to

−cD

εu∗ + cu∗ = cu∗ + g(t, u∗),

56

or cD εu∗ + g(t, u∗) = 0 which means that u∗ is the solution of the problem (3.3.1).

Since u(i)(0) = v′i(0) and v(i)(1) = δv

′i(1), i ≥ 1, u∗(0) = γu

′i(0) and u∗(1) =

δu′i(1), and it show that u∗ is a solution of problem (3.3.1). The same result applied

to v(i) show that v∗ is a solution of (3.3.1). Conclusion is that u∗ = v∗ = w, the

uniqueness of the solution of the problem.

3.3.2 Illustrative examples

we present the following examples for demonstration of the aforesaid established

theory.

Example 3.3.1. Consider the following fractional order BVP

cD

1.75 = (u3 − 5); t ∈ I ,

u(0) = 0.5u′(0), u(1) = 0.5u

′(1).

(3.3.13)

From the above system(3.3.13), we see θ(t, u) = −u3 + 5 and let lower and upper solution

be α(0)(t) = 0, β(0)(t) = 1. Then θ(t, u) is decreasing as

−3 ≤ ∂θ

∂u= −3u2

< 0, ∀ u ∈ [0, 1], with d = 3.

These extremal solutions can be computed from taking limit of monotone iterative se-

quences which can be developed .

Example 3.3.2. Consider the following fractional order BVP

cD

1.5u = u exp(u)− 6; t ∈ I ,

u(0) = 0.333u′(0), u(1) = 0.333u

′(1).

(3.3.14)

From the above system(3.3.14), we see

θ(t, u) = −u exp(u) + 6

57

and let α(0)(t) = 0, β(0)(t) = 1 be lower and upper solution respectively, then θ(t, u) is

decreasing with

−3e2 ≤ ∂θ

∂u= exp(u)(−u + 1) < 0, ∀ u ∈ [0, 1].

Thus the BVP (3.3.14) has an extremal solutions.

Chapter 4

QUALITATIVE STUDY OF PINEWILT DISEASE MODEL

This chapter is devoted to study the qualitative aspects of pine wilt disease model

while incorporating convex incidence rate. Pine Wilt disease (PWD), produced

by the pinewood nematode is a threatical disease, because it kills usually effected

trees with in a few weeks. In eastern Asia it caused major losses in coniferous trees,

particularly it spread suddenly in Japan, South Korea, China and in western Eu-

rope. Around the world in the quarantine list of many countries PWD is one of the

major pests [63]. As PWD originated in north America while in 1905 for the first

time PWD was observed in Japan. After that the disease spread to China, Korea,

Mexico and Portugal, in the process it become a major threat to ecological side,

which affected the economy of the forest industry. Basically Pine wilt kills Scots

pine, beside this many other pine species such as jack, Austrian, mogo and red

pines are occasionally destroyed by the pine wilt. In living trees, PWD disturb the

xylem conduits it effects the cortical and xylem resin canals. Then they can move

to reach all parts of the steam and branches through the resin canals rapidly. In

infected fine trees both virulent fine trees PWD and a virulent PWD produce some

58

59

localized embolism in the xylem. Afterward the virulent PWD disturb the steam

and water potential to the leaf suddenly decreases till it destroy the trees [8]. When

a susceptible tree effect from PWD, there is no treatment but just a standard tech-

niques can be used for the protection. The best technique is to separate non-native

pine species in area in a range from planting where the normal temperature exceed

from 200C. In the area of non-native trees, sufficient supply of water in dry season

can be another technique for the protection. Another way is that to increase the

rate of mortality of insects in timber through the specialized treatment, remove the

damaged branches of affected trees or remove the dead trees.

Mathematical tools is a way to study the dynamical aspects of PWD. Different

control techniques can be used to analyze the spread of the disease. On pest-tress

dynamics several models have been developed. The authors of [40], [67] explored

some results for the transmission of Pine Wild Disease models. While incorpo-

rating nonlinear incidence rate K. S. Lee et. al. investigated the global stability

analysis of host vector PWD model [41]. Recently Awan et. al. in [1], explored the

stability analysis of pine wilt while incorporating the periodic use of insecticides.

The authors showed that epidemic level of infected vectors do not depend upon

the saturation while assuming the transmission through mating. We propose host

vector model for pine wilt disease with convex incident rate and will discuss the

qualitative aspects of the model. Significance of convex incidence rate is that when

the infected population I grow, the possibility of a single infected individual to

transfer the disease further to other potential individuals increase. This kind of

effect is usually associated with some form of community effect or cooperation. In

mathematical epidemiology incidence is a measure which shows the probability of

60

occurrence of a given medical condition in a population for a particular time inter-

val. Incident rate of the transmission of the disease play a major role in the study

of the mathematical epidemiology. Moreover, in medicine, public health trans-

mission is passing of a pathogen causing communicable disease from one infected

host individual or group to a particular individual or group.

4.1 Formulation of Mathematical Model

The present section is devoted to formulation of mathematical model governing

the proposed phenomena. It represents the transmission of disease induced by

vectors. The threshold quantity will be used to help in exploring the dynamics of

the model. It has been observed that during the maturation feeding of infected

vectors the transmission of nematodes occur in pine trees bark beetles. When

pine sawyers came out from infected pine trees it possess pinewood nematode

Bursephelendus. In some situation the beetles catch infection directly during mat-

ing. We suppose the following hypothesis on various compartments of the popu-

lations in the environment.

4.1.1 Compartments in pine trees population:

The population of trees is divided into three subclasses, susceptible host, exposed

and infected trees. The description is given as;

Susceptible pine trees St(H) : It denotes those individuals in pine trees that have

capability to catch infection viz nematode and can exude oleoresin. It plays a role

of physical barrier to beetle oviposition and beetles cannot oviposit on them.

61

Exposed pine trees Et(H):Those infected pine trees that posses the potential for

oleoresin exudation.

Infected pine trees It(H): Those pine trees such that beetles can oviposit on them.

4.1.2 Compartments in bark beetles population:

Since there is no concept of exposed and recovered population in Bark beetles

therefore the whole population is divided into two subclasses. (i) The suscepti-

ble adults St(V), beetles which don’t have pine wild nematode. (ii) The infected

beetles It(V), which posses pine wild nematode.

4.1.3 Description of the model

If Mt(H) and Mt(V) represents the total population of pine trees and beetles re-

spectively then, in light of the divisions of both populations we can express it math-

ematically as, Mt(H) = St(H) + Et(H) + It(H) and Mt(V) = St(V) + It(V). The

rate at which new induction enter to compartments of trees and beetles are aH and

bV respectively. While the death rate for trees is represented by µ1 and µ2 for bee-

tles. The term αIt(V)St(H) shows the single contact that leads to infection while

the new infective come into being from double exposures at a rate mαIt(V)2St(H).

The possibility of natural death rate of pine wild as well as of beetles are incorpo-

rated in the model. The parameter γ is used to represent the rate at which grown

up beetles carry the pine wild nematode when they come out from dead trees. The

levels at which the infections saturate in trees and beetles are denoted by m and n

respectively.

Now incorporating all the above mentioned parameters the compact mathematical

62

form obtained by the aforementioned biological model is given as:

dSH

dt= aH − αSH IV(1 + mIV)− µ1SH,

dEH

dt= αSH IV(1 + mIV)− (β + µ1)EH

dIH

dt= βEH − µ1 IH

dSV

dt= bV − γIHSV(1 + nIH)− µ2SV

dIV

dt= γIHSv(1 + nIH)− µ2 IV

(4.1.1)

Studying the nature of compartments and their dependency upon each other we

reduce the system (4.1.1) to the following form, while removing SH and SV recep-

tively.

dEH

dt= αφIV(1 + mIV)

(

aH

µ1− IH − EH

)

− (β + µ1)EH

dIH

dt= βEH − µ1 IH

dIV

dt= γIH(1 + nIH)

(

bV

µ2− IV

)

− µ2 IV

(4.1.2)

The feasible region for the system (4.1.2) is given as

Ξ = [(Et(H), It(H), It(V)) ∈ R3+ |0 ≤ aH

µ10 ≤ It(H) ≤ aH

µ2,

Et(H) ≥ 0, It(H) ≥ 0, It(V) ≥ 0]

4.1.4 Disease free equilibrium and its stability

Inspecting the system (4.1.2) one can show that the proposed systems of differen-

tial equations (4.1.2) exhibit disease free equilibrium point, given by E0 = (0, 0, 0).

The dynamic of disease are presented by studying the basic reproductive number

R0 = aHbVφαβγ/µ21µ2

2(β + µ1). Using Theorem 2 in [69], it is not difficult to prove

the following results.

63

Theorem 4.1.1. If R0 < 1, the disease free equilibrium E0 of the model (3.2.6) is locally

asymptotically stable, and is unstable if R0 > 1.

Theorem 4.1.2. If R0 < 1, the disease free equilibrium E0 of the model (3.2.6) is globally

asymptotically stable.

4.1.5 The endemic equilibrium and its stability

This subsection is devoted to the existence and stability of the disease present equi-

librium point. In the literature it is known as endemic equilibrium point. Moreover

if R0 > 1, then the endemic equilibrium E∗ = (E∗H, I∗H, I∗V) is given as

E∗H =

µ1

βI∗H,

I∗V =γI∗H(1 + nI∗H)bV

µ2

(

γI∗H(1 + nI∗H) + µ2

)

(4.1.3)

and It(H)∗ is the zero of the following cubic equation

A1 I3H + A2 I2

H + A3 IH + A4 = 0,(4.1.4)

with

A1 = f n2µ2γI3H,

A2 = f n(γ + µ2)− mβγbvn(µ1 + β)− knγ

A3 = f γ − mβγbv(β + µ1)− kγ

A4 = f µ2 + mβγbvβah

µ1(n + 1)− kµ2

(4.1.5)

Here f = αφβγbVµ2, g = (µ1 + µ2)(µ1 + β) and d = mβγbV . Using Descart’s rule

of sign it is obvious to show that A1 > 0 if A3 > 0 we get A2 < 0, now if A4 < 0,

then there exists 3 positive real roots of (4.1.4), and if A4 > 0, then there exists 1

negative and 2 positive real roots of (4.1.4). This is summarized below.

64

Theorem 4.1.3. The system (4.1.2) always has the infection-free equilibrium E0. If R0 >

1, then the system (4.1.2) has a unique endemic equilibrium point E∗ = (E∗H, I∗H, I∗V),

defined by equation (4.1.4) and (4.1.5).

To explore the stability analysis of the disease present equilibrium point the

approach given in [46, 54] will be used. After linearzing the system (4.1.2), we get

the following matrix.

J(E∗) =

−αφI∗H(1 + mI∗V)− (β + µ1) −αφI∗V(1 + mI∗V) αφ(1 + 2mI∗V)(aHµ1

− E∗H − I∗H)

β −µ1 0

0 γ(1 + 2nI∗H)(bVµ2

− I∗V) −γI∗H(1 + nI∗H)− µ2

(4.1.6)

from the Jacobian matrix J(E∗), the second compound matrix is given by

J[2](E∗) =

a 0 c

d e f1

0 β g1

(4.1.7)

where

• a = −αφI∗H(1 + mI∗V)− (β + 2µ1);

• c = −αφ(1 + 2mI∗V)(aHµ1

− E∗H − I∗H);

• d = γ(1 + 2nI∗H)(bVµ2

− I∗V)

• e = −αφI∗V(1 + mI∗V)− αφ(1 + 2mI∗V)(aHµ1

− E∗H − I∗V)− (β + µ1);

65

• f1 = −αφI∗H(1 + mI∗V);

• g1 = −γI∗H(1 + nI∗H)− µ2 − µ1.

To study local stability at disease present equilibrium point the following result

will be used [40].

Lemma 4.1.4. Let M be a 3 × 3 real matrix. If tr(M), det(M) and detM[2] are all nega-

tive, then all eigen values of M have negative real part.

Theorem 4.1.5. If R0 > 1, then endemic equilibrium E∗ of the model (3.2.6) is locally

asymptotically stable

Proof. From the Jacbian matrix J(E∗), we have

tr(M) = − [αφI∗H(1 + mI∗V) + (β + µ1) + µ1 + γI∗H(1 + nI∗H) + µ2] < 0

because

aH

µ1− E∗

H − I∗H =µ1 + βE∗

H

αφI∗V(1 + mI∗V),

E∗H =

µ1

βI∗H,

bV

µ2− I∗V =

µ2 I∗VγI∗H(1 + nI∗H)

(4.1.8)

from (4.1.8), it is easy to check that

γI∗H(1 + nI∗H)(bV

µ2− I∗V)× αφ(1 + 2mIV)(

aH

µ1− E∗

H − I∗H)

=(1 + 2nIH)(1 + 2mIV)µ1µ2(β + µ1)

β(1 + nIH)(1 + mIV)

66

thus

det(J(E∗)) =∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

−αφI∗H(1 + mI∗V)− (β + µ1) −αφI∗V(1 + mI∗V) αφ(1 + 2mI∗V)(aHµ1

− E∗H − I∗H)

β −µ1 0

0 γ(1 + 2nI∗H)(bVµ2

− I∗V) −γI∗H(1 + nI∗H)− µ2

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

=− (αφI∗V(1 + mI∗V) + (β + µ1)) (µ1γI∗H(1 + nI∗H) + µ1µ2)

− β[αφI∗V(1 + mI∗V)(γIH(1 + nIH) + µ2))+

γ(1 + 2nIH)(bV

µ2− I∗V)αφ(1 + 2mIV)(

aH

µ1− E∗

H − I∗H)]

=− (αφI∗V(1 + mI∗V) + (β + µ1)) (µ1γI∗H(1 + nI∗H) + µ1µ2)

− β

[

αφI∗V(1 + mI∗V)(γIH(1 + nIH) + µ2)) +(1 + 2nIH)(1 + 2mIV)µ1µ2(β + µ1)

β(1 + nIH)(1 + mIV)

]

=− αφI∗V(1 + mI∗VγI∗H(1 + nI∗H)− αφI∗V(1 + mI∗V)µ1µ2

− (β + µ1)µ1γIH(1 + nIH)− µ1µ2(β + µ1)

− βαφI∗V(1 + mI∗V)(γIH(1 + nIH) + µ2)) +(1 + 2nIH)(1 + 2mIV)µ1µ2(β + µ1)

(1 + nIH)(1 + mIV).

=− αφI∗V(1 + mI∗VγI∗H(1 + nI∗H)− αφI∗V(1 + mI∗V)µ1µ2 − (β + µ1)µ1γIH(1 + nIH)

− βαφI∗V(1 + mI∗V)(γIH(1 + nIH) + µ2)) + µ1µ2(β + µ1)

×[

1 − (1 + 2nIH)(1 + 2mIV)µ1µ2(β + µ1)

(1 + nIH)(1 + mIV)

]

< 0. (4.1.9)

If we assume that

• a11 = −αφI∗H(1 + mI∗V)− (β + 2µ1);

• a13 = −αφ(1 + 2mI∗V)(aHµ1

− E∗H − I∗H);

• a21 = γ(1 + 2nI∗H)(bVµ2

− I∗V);

67

• a22 = −αφI∗V(1 + mI∗V)− αφ(1 + 2mI∗V)(aHµ1

− E∗H − I∗V)− (β + µ1);

• a23 = −αφI∗H(1 + mI∗V);

• a33 = −γI∗H(1 + nI∗H)− µ2 − µ1.

Now computing directly the determinant of J[2](E∗), we can get

J[2](E∗) =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

a11 0 a13

a21 a22 a23

0 β a33

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

=− (αφI∗H(1 + mI∗V) + (β + 2µ1))(αφI∗V(1 + mI∗V) + αφ(1 + 2mI∗V)

×[

aH

µ1+ E∗

H + I∗V) + (β + µ1))(γI∗H(1 + nI∗H) + µ2 + µ1

]

− γ(1 + 2nI∗H)(bV

µ2− I∗V)β(αφ(1 + 2mI∗V)(

aH

µ1+ E∗

H + I∗H)) < 0.

4.1.6 Global stability analysis

This part of the thesis is devoted to, the global stability analysis of the proposed

model for both disease free as well as disease present equilibrium points. The

results suggested by Castillo Chavez [19] is applied to prove the global asymp-

totically stable at disease free equilibrium. While to show that the model (4.1.1)

is globally asymptotic stability at disease present equilibrium, the geometrical ap-

proach is used as main tool [45]. Prior to prove global stability analysis first we

discuss some result about uniform persistent of the proposed model.

68

Lemma 4.1.6. The system (4.1.2) is uniformly persistent if the basic reproductive number

is greater than unity.

First we provide a brief analysis of the Castillo Chavez method and geometri-

cal approach to prove the global stability of the model (4.1.1) at disease free equi-

librium and disease preset equilibrium. Therefor, using the method of Castillo

Chavez [19], initially the proposed model (4.1.1) is reduced into the following two

subsystems given by

dχ1

dt= G(χ1, χ2),

dχ2

dt= H(χ1, χ2). (4.1.10)

In the system (4.1.10), χ1 and χ2 represent the number of susceptible and infected

individuals that is, χ1 = (St(H), St(V)) ∈ R2 and χ2 = (Et(H), It(V)) ∈ R2. The

disease free equilibrium is denoted by E0 and define as E0 = (χ01, 0). Thus the

existence of the global stability at disease free equilibrium point depends on the

following two conditions

C1. If dχ1dt = G(χ1, 0), χ0

1 is globally asymptotically stable.

C2. H(χ1, χ2) = Bχ1 − H(χ1, χ2), where H(χ1, χ2) ≥ 0 for (χ1, χ2) ∈ ∆.

At second condition B = Dχ2 H(χ01, 0) is an M-matrix that is the off diagonal entries

are positive and ∆ is the feasible region. Then the following statement hold.

Lemma 4.1.7. The equilibrium point E0 = (χ0, 0) of the system (13) is said to be globally

asymptotically stable, if the aforementioned conditions C1 and C2 are satisfied.

Lemma 4.1.8. Let U is simply connected and the condition (C1) − (C2) are satisfied,

then the unique equilibrium x∗ of equation (4.1.11) is globally asymptotically stable in U,

if q < 0.

69

Similarly to prove the global stability of the model (1) at endemic equilibrium

E1, we use the geometrical approach [45]. According to this method, we will inves-

tigate sufficient condition through which the endemic equilibrium point is globally

asymptotically stable. Assume a map f : D ⊂ Rn → Rn such that every solution

x(t, x0) of the first order differential equation

x = f (x)

is expounded uniquely via initial condition x(0, x0). Here, U ⊂ Rn is an open

set simply connected and f ∈ C1(U). Consider a solution to equation (4.1.11) is

f (x∗) = 0 and for x(t, x0) the following hypothesis hold

HP-1 The domain D of f is simply connected set;

HP-2 There exist a subset K of D which is compact absorbing;

HP-3 The set D contains a unique solution of equation (4.1.11) i.e x ∈ D.

The solution x∗ is said to be globally asymptotically stable in U, if it is locally

asymptotically stable and all trajectories in U converges to the equilibrium x∗. For

n ≥ 2, a condition satisfied for f , which precludes the existence of non-constant

periodic solution of equation (4.1.11) known is Bendixson criteria. The classical

Bendixson criteria div f (x) < 0 for n = 2 is robust under C1 [45]. Furthermore a

point x0 ∈ U is wandering for equation (4.1.11), if there exist a neighborhood N of

x0 and τ > 0, such that N ∩ x(t, N) is empty for all t > τ. Thus the following global

stability principle established for autonomous system in any finite dimension.

Lemma 4.1.9. If the conditions (C1)− (C2) and Bendixson criterion satisfied for equation

70

(4.1.11) that is robust under C1 local perturbation of f at all non equilibrium, non wan-

dering point for equation (4.1.11). Then x∗ is globally asymptotically stable in U provided

it is stable.

Suppose a non-singular

(

n2

)

×(

n2

)

matrix valued function P : x 7→ P(x) such

that it belongs to C1 space and a vector norm on RM, M =

(

n2

)

. Further assume

that P−1 exist and is continuous for x ∈ K. Now define a quantity define, such that

q = limt→∞ sup sup1

t

∫ t

0[µ(B(x(s, x0)))]ds, (4.1.11)

where B = Pf P−1 + PJ[2]P−1 and J[2] is the second additive compound matrix of

the Jacobian matrix J, that is J(x) = U f (x). Let ℓ(B) be the Lozinski measure of

the matrix B with respect to the norm ‖.‖ in Rn [50] defined by

ℓ(B) = limx→0|I + Bx| − 1

x. (4.1.12)

Hence if q < 0, which shows that the presence of any orbit that give rise to a simple

closed rectifiable curve, such Suppose a vector norm | · | on Euclidean space R3.

In order to prove the global asymptotic stability of the proposed model (4.1.1) at

endemic equilibrium E1, we present the following results.

Consider a vector norm ‖.‖ in space and a three by three matrix valued mapping

P(x) = diag(

1, Et(H)It(V)

, Et(H)It(V)

)

. The function P is non-singular and belongs to C1

class. The Linearized form of system (4.1.5) about disease present equalibrium

point E∗ gives the following variational matrix

J =

A11 A12 A13

β −µ1 0

0 C33 −(1 + nIH)− µ2

,

71

A11 = −αφIV(1 + mIV)− (β + µ1)

A12 = −αφIV(1 + mIV)

A13 =

(

aH

µ1− EH − IH

)

(αφ + 2αφmIV)

C33 =

(

bV

µ2− IV

)

(γ + 2γnIH).

The second additive compound matrix of J is denoted by J|2| and define as

J|2| =

J11 0 J13

J21 J22 J23

0 β J33

, (4.1.13)

where

• J11 = −αφIV(1 + mIV)− (β + µ1)− µ1;

• J13 = −(

aHµ1

− EH − IH

)

(αφ + 2αφmIV) ;

• J21 =(

bVµ2

− IV

)

(γ + 2γnIH);

• J22 = −αφIV(1 + mIV)− (β + µ1)− γIH(1 + nIH)− µ2;

• J23 = −αφIV(1 + mIV);

• J33 = −µ1 − γIH(1 + nIH)− µ2.

Now choose a function P(χ) = P(EH, IV) = diag{EHIV

, EHIV

, EHIV}, which implies

that P−1

(χ) = diag{ IVEH

, IVEH

, IVEH

}, then taking the time derivative, that is P f (χ), we

get

P f (χ) = diag

{

EH

IV− EH

˙IV

I2V

,EH

IV− EH

˙IV

I2V

,EH

IV− EH

˙IV

E2H

}

. (4.1.14)

Now P f (P)−1 = diag{

˙EHIV

− ˙IVIV

,˙EH

IV− ˙IV

IV,

˙EHIV

− ˙IVIV

}

and B = P f (P)−1 + PJ|2|(P)−1,

72

which can be written as

B =

(

b11 b12

b21 b22

)

, (4.1.15)

where

b11 =EH

EH−

˙IV

IV− αIV(1 + mIV)− (β + µ1)− µ1,

b12 =(

0 ,−(

aHµ1

− EH − IH

) (

αϕ + ϕαmIV

) )

,

b21 =

(

bVµ2

− IV

) (

γ + 2nγIH

)

0

,

b22 =

(

y11 y12

β y22

)

,

with y11 =˙EH

IV− ˙IV

IV− αϕIV

(

1 + mIV

)

− (β + µ1) − γIH(1 + nIH) − µ2, y12 =

−αϕIV(1 + mIV) and y22 =˙EH

IV− ˙IV

IV−(

µ1 + µ2 + γIH

(

1 + η IH

))

.

Let (a1, a2, a3) be a vector in R3 and its norm ‖.‖ defined by

‖a1, a2, a3‖ = max{‖a1‖, ‖a2‖+ ‖a3‖}. (4.1.16)

Moreover, assume ℓ(B) to be the Lozinski measure with respect to the norm given

by (4.1.16) and described by Martin et. al. [50], then we choose

ℓ(B) ≤ sup{g1, g2} = sup{ℓ(B11) + ‖B12‖, ℓ(B22) + ‖B21‖}, (4.1.17)

where ‖ B12 ‖ and ‖ B21 ‖ are matrix norms, then

g1 = ℓ(B11) + ‖B12‖, g2 = ℓ(B22) + ‖B21‖, (4.1.18)

where

ℓ(B11) =EH

EH−

˙IV

IV− αIV(1 + mIV)− (β + µ1)− µ1

73

,

‖B12‖ =

(

EH + IH − aHµ1

)

(α + 2αφmIV) , if EH + IH >aHµ1

;

0 other wise;(4.1.19)

‖B21‖ =

(

bVµ2

− IV

)

(γ + 2γnIH) , if bVµ2

> IV ;

0 other wise;(4.1.20)

Case 1: When EH + IH >aHµ1

and bVµ2

> IV

Putting the values of ℓ(B11), ‖B12‖, ℓ(B22), ‖B21‖ in (4.1.18) we have the the fol-

lowing

.

g1 =EH

EH−

˙IV

IV− αIV(1 + mIV)− (β + µ1)− µ1 +

(

EH + IH − aH

µ1

)

(α + 2αφmIV)

g2 =EH

IV−

˙IV

IV−(

µ1 + µ2 + γIH

(

1 + η IH

))

+

(

bV

µ2− IV

)

(γ + 2γnIH)

Using the 1st and 3rd equation of the system (3.2.6) we estimate g1 and g2 as

g1 ≤ EH

EH− µ1 − {αφIV(1 + mIV) + β + µ1 − µ2}

g2 ≤ EH

EH− µ1 + υ.

where υ =(

bVµ2

− IV

)

(γ + 2γnIH)− γIH(1 + nIH).

Now by (4.1.17) we have

µ(B) ≤ EH

EH− µ1 + max

[

− {αφIV(1 + mIV) + β + µ1 − µ2} , υ]

.

In view of the bounds of the state variable one may obtain that if

υ ≤ µ1 and µ2 <

(

αφ

(

aH

µ1

)(

1 +

(

maH

µ1

))

+ β + µ1

)

, (4.1.21)

74

then

µ(B) ≤ EH

EH+ Z.

Here,

Z = max

{

µ2 − αφ

(

aH

µ1

)(

1 +

(

maH

µ1+ β + µ1

))

, υ − µ1

}

< 0

Hence,

1

tlog

EH(t)

EH(0)− Z ≥ 1

t

∫ t

0µ(B)ds.

Therefore, the Bendixson criterion is verified.

Case 2: When EH + IH <aHµ1

and bVµ2

< IV .

Using system (3.2.6), g1 and g2 can be estimated as;

g1 ≤ EH

EH− µ1 − αφIv(1 + mIv)− (β + µ1) + µ2

(4.1.22)

g2 ≤ EH

EH− µ1 − γIH(1 + nIH)

(4.1.23)

Hence

µ(B) ≤ EH

EH− µ1 + max {µ2 − (β + µ1)− αφIv(1 + mIv),−γIh(1 + nIh)}(4.1.24)

Using the bounds of compartment variables and the relations

µ1 ≤ γIH(1 + IH) and µ2 < β + µ1 + αφIV(1 + mIv)

(4.1.25)

we have

µ(B) ≤ EH

EH+ Z (4.1.26)

75

where Z = max {µ2 − β − µ1 − αφIV(1 + mIV), µ1 + γIH(1 + nIH)} < 0. Conse-

quently, we obtain

1

tlog

EH(t)

EH(0)− Z ≥ 1

t

∫ t

0µ(B)ds.

Therefore, for case 2 the Bendixson criterion is verified. Now we summarize the

preceding discussion as;

Theorem 4.1.10. If R0 > 1, then the system (4.1.2) exhibit unique endemic equilibrium

point E∗. Moreover, it is globally asymptotically stable, provided that the relation (4.1.21)

holds true.

Theorem 4.1.11. If R0 > 1, then the system (4.1.2) have unique endemic equilibrium

point E∗, which is globally asymptotically stable, if the relation (4.1.25) is true.

4.1.7 Numerical simulations and discussion

In the current work, we investigate the dynamical behavior of PWD model with

convex incidence rate. Qualitative analysis of the proposed model has been pro-

vided in the previous section. The suggested model has two equilibria which are

the disease-free equilibrium and the endemic equilibrium. We have investigated

the behavior of the model close to each equilibrium. Further, threshold value of the

relative basic reproductive number R0 required for the determination of the spread

of infection has been computed. Incorporating convex incidence rate the present

work describes PWD nametodes model. From mathematical point of view we have

explored the model for fourset vectors pests with pine wild namedates. Using the

threshold quantity we have studied the global stability analysis of the proposed

model. It has shown that the disease free steady-state is globally asymptotically

76

stable in the feasible region whenever the basic reproductive number, R0 ≤ 1

which results in the extinction of disease. Moreover, for R0 > 1 the endemic equi-

librium exist and is unique, in addition it is shown that the disease persist at dis-

ease present equilibrium point with the condition that if the disease exist initially.

This is due to the fact that the equilibrium point is also globally asymptotically

stable. To carry out the numerical simulations of different classes of the proposed

model, we assign the values to the parameters involved in the considered problem

as

αH = 10.7, bV = 10.8, β = 0.062, γ = 0.0009, µ1 = 0.02, µ2 = 0.0619,

m = 0.0003, n = 0.001, φ = 0.2, α = 0.000761.

The respective simulations for different compartments are given in the given Fig-

ures 1 − 5 respectively. It is to be noted that we have used Nonstandard Finite

Difference (NSFD) scheme introduced by Mickens[51] in 1989. Since standard fi-

nite difference (SFD) schemes suffers from instability of different kinds. To over-

come these instabilities NSFD method is the best choice, for further detail about

the method see [52].

77

time t (Days)0 50 100 150 200 250

SH

40

60

80

100

120

140

160

180

SH

(0)=155

SH

(0)=40

SH

(0)=45

SH

(0)=50

Figure. 1 Numerical plots of the susceptible compartment of pine trees SH in pine

trees population at the given various initial values.

time t (Days)0 50 100 150 200 250

EH

20

40

60

80

100

120

140

160

EH

(0)=45

EH

(0)=60

EH

(0)=155

EH

(0)=40

Figure 2. Numerical plots of the susceptible compartment of pine trees EH in pine


78

time t (Days)0 50 100 150 200 250

I H

0

50

100

150

200

250

300

350

IH

(0)=60

IH

(0)=45

IH

(0)=50

IH

(0)=155

Figure 3. Numerical plots of the susceptible compartment of pine trees IH in pine


time t (Days)0 50 100 150 200 250

SV

20

30

40

50

60

70

80

SV

(0)=40

SV

(0)=50

SV

(0)=60

SV

(0)=45

Figure 4. Numerical plots of the susceptible compartment of pine trees SV in pine


79

time t (Days)0 50 100 150 200 250

I V

40

60

80

100

120

140

160

IV

(0)=50

IV

(0)=155

IV

(0)=40

IV

(0)=60

Figure 5. Numerical plots of the susceptible compartment of pine trees IV in pine


Studying the nature of the proposed model it has shown that the basic reproduc-

tive number completely determine various dynamics of the model. The global

asymptotic stability analysis has expounded at disease free equilibrium point. It

has been shown that the disease extinct if the threshold quantity is less than unity.

Moreover, local as well as global stability has been expounded at disease present

equilibrium point using the techniques described in [19] and [45]. Numerical re-

sults are represented graphically to illustrate theoretical discussions.

80

4.2 Numerical Solution of a Fractional Order Host Vec-

tor Model of a Pine Wilt Disease

This section considers a fractional order epidemic model for the spread of the pine

wilt disease. Few mathematical models have been developed pest-tress dynamics,

Lee et.al [47] and shi [58] studied PWD transmission. In 2014 K. k. S and A. Lashari

determined the global stability of a host vector model for pine-wilt disease with

non linear incident rate [4].

We proposed a fractional order host vector model for pine wilt disease with

convex incident rate and find its numerical solution. To obtain an approximate so-

lution of the system of non-linear fractional differential equations for the proposed

model LADM is used. Numerical results show that LADM is efficient and accurate

method for solving fractional order pine wilt disease model given as

cD

αSh = ah − kSh Iv(1 + αIv)− µ1Sh,

cD

αEh = kSh Iv(1 + αIv)− (β + µ1)Eh,

cD

α Ih = βEh − µ1 Ih,

cD

αSv = bv − γIhSv(1 + ξ Ih)− mu2Sv,

cD

α Iv = γIhSv(1 + ξ Ih)− µ2 Iv.

(4.2.1)

with given initial condition Sh(0) = n1, Eh(0) = n2, Ih(0) = n3, Sv(0) = n4, Iv(0) =

n5, where cDα, 0 < α < 1 is the Caputo’s factional derivative of order α.

81

4.2.1 The Laplace Adomian decomposition method

In this section, we discuss the general procedure of the model (4.2.1) with given

initial conditions. Applying Laplace transform on both side of the model (4.2.1) as

L {cD

αSh} = L {ah − kSh Iv(1 + αIv)− µ1Sh},

L {cD

αEh} = L {kSh Iv(1 + αIv)− (β + µ1)Eh},

L {cD

α Ih} = L {βEh − µ1 Ih},

L {cD

αSv} = L {bv − γIhSv(1 + ξ Ih)− mu2Sv},

L {cD

α Iv} = L {γIhSv(1 + ξ Ih)− µ2 Iv}

(4.2.2)

which implies that

sαL {Sh} − sαSh(0) = L {ah − kSh Iv(1 + αIv)− µ1Sh},

sαL {Eh} − sαEh(0) = L {kSh Iv(1 + αIv)− (β + µ1)Eh},

sαL {Ih} − sα Ih(0) = L {βEh − µ1 Ih},

sαL {Sv} − sαSv(0) = L {bv − γIhSv(1 + ξ Ih)− mu2Sv},

sαL {Iv} − sα Iv(0) = L {γIhSv(1 + ξ Ih)− µ2 Iv}.

(4.2.3)

Now using initial conditions , we have

L {Sh} =n1

s+

1

sαL {ah − kSh Iv(1 + αIv)− µ1Sh},

L {Eh} =n2

s+

1

sαL {kSh Iv(1 + αIv)− (β + µ1)Eh},

L {Ih} =n3

s+

1

sαL {βEh − µ1 Ih},

L {Sv} =n4

s+

1

sαL {bv − γIhSv(1 + ξ Ih)− µ2Sv},

L {Iv} =n5

s+

1

sαL {γIhSv(1 + ξ Ih)− µ2 Iv}.

(4.2.4)

82

Assuming that the solutions, Sh(t), Eh(t), Ih(t), Sv(t), Iv(t) in the form of infinite

series given by

Sh(t) =∞

∑n=0

Sn, Eh(t) =∞

∑n=0

En, Ih(t) =∞

∑n=0

In, Sv(t) =∞

∑n=0

Vn, Sv(t) =∞

∑n=0

Vn.

(4.2.5)

and the nonlinear terms are involved in the model are Sh(t)Iv(t), Sh(t)I2v(t), Sv(t)Ih(t)

are decompose by Adomian polynomial as

Sh(t)Iv(t) =∞

∑n=0

Bn,

Sh(t)I2v(t) =

∞

∑n=0

pn,

Ih(t)Sv(t) =∞

∑n=0

Qn.

(4.2.6)

Where Bn, pn, Qn are Adomian polynomials defined as

Bn =1

Γ(n + 1)

dn

dtn

[

n

∑k=0

λkShkn

∑k=0

λk Ivk

]

|λ=0,

pn =1

Γ(n + 1)

dn

dtn

[

n

∑k=0

λkShkn

∑k=0

λk I2v k

]

|λ=0,

Qn =1

Γ(n + 1)

dn

dtn

[

n

∑k=0

λk Ihkn

∑k=0

λkSvk

]

|λ=0 .

(4.2.7)

using (4.2.5),(4.2.6) in model (4.2.4), we get

83

L (S0h) =

n1

s, L (E0

h) =n2

s, L (I0

h) =n3

s,

L (S0v) =

n4

s, L (I0

v) =n5

s

L (S1h) =

1

sαL {ah − kB0 − kαP0 − µ1S0

h}

L (E1h) =

1

sαL {kB0 + kαP0 − (β + µ1)E0

h}

L (I1h) =

1

sαL {βE0 − µ1 I0

h}

L (S1v) =

1

sαL {bv − γ(Q0 + ξP0)− µ2S0

v}

L (I1v) =

1

sαL {γ(Q0 + ξP0)− µ2 I0

v}

L (S2h) =

1

sαL {ah − kB1 − kαP1 − µ1S1

h}

L (E2h) =

1

sαL {kB1 + kαP1 − (β + µ1)E1

h}

L (I2h) =

1

sαL {βE1 − µ1 I1

h}

L (S2v) =

1

sαL {bv − γ(Q1 + ξP1)− µ2S1

v}

L (I2v) =

1

sαL {γ(Q1 + ξP1)− µ2 I1

v}...

L (Sn+1h ) =

1

sαL {ah − kBn − kαPn − µ1Sn

h}

L (En+1h ) =

1

sαL {kBn + kαPn − (β + µ1)En

h}

L (In+1h ) =

1

sαL {βEn − µ1 In

h }

L (Sn+1v ) =

1

sαL {bv − γ(Qn + ξPn)− µ2Sn

v}

L (In+1v ) =

1

sαL {γ(Qn + ξPn)− µ2 In

v }.

(4.2.8)

84

To study mathematical behavior of the of the solution of Sh, Eh, Ih, Sv, Iv we use

different values of α. For the solution we take inverse transform of (4.2.8), we have

S0h = n1, E0

h = n2, I0h = n3, S0

v = n4, I0v = n5

S1h = (ah − kn1n5 − kαn1n2

5 − µn1)tα

Γ(α + 1),

E1h = (kn1n5 + kαn1n2

5 − (β + µ)n2)tα

Γ(α + 1),

I1h = (βn2 − µ1n3)

tα

Γ(α + 1),

S1v = (bv − γ(n3n4 + ξn1n2

5)− µ2S1v)

tα

Γ(α + 1),

I1v = (γ(n3n4 + ξn1n2

5)− µ2n5)tα

Γ(α + 1),

S2h = (ah − kαn1n5)

tα

Γ(α + 1)− k[n1(γ(n3n4 + ξn1n2

5)− µ2n5))

+ µ1(ah − n1n5 − kαn1n25 − µn1)]

t2α

Γ(2α + 1)

−(

kα2n1n5γ(n3n4 + ξn1n25)− µ2n5) + kn5(ah − kn1n5 − kαn1n2

5 − µ1n1)) t2α

Γ(2α + 1),

E2h = k

[

n1(γ(n3n4 + ξn1n25)− µ2n5) + kn5(ah − kn1n5 − kαn1n2

5 − µ1n1)] t2α

Γ(2α + 1)

+(

kα2n1n5(γ(n3n4 + ξn1n25)− µ2n5)− (β + µ1)(βn2 − µ1n3)

) t2α

Γ(2α + 1)

+ kαn1n5tα

Γ(α + 1),

I2h = (β(kn1n5 + kαn1n2

5 − (β + µ1)n2)− µ1(βn2 − µ1n3))t2α

Γ(2α + 1),

S2v = (bv − ξγn1n5)

tα

Γ(α + 1)

− γ[n3(bv − γ(n3n4 + ξn1n25)− µ2n4) + n4(βn2 − µ1n3)

+ 2ξn1n5(γ(n3n4 + ξn1n25)− µ2n5)− µ2n4]

t2α

Γ(2α + 1),

I2v = γξn1n5

tα

Γ(α + 1)+ γ(n3(bv − γ(n3n4 + ξn1n2

5)) + n4(βn2 − µ1n3)

+ 2ξ2n1n5(γ(n3n4 + ξn1n25)− µ2n5)− µ2(γ(n3n4 + ξn1n2

5)− µ2n5))t2α

Γ(2α + 1).

(4.2.9)

85

which can be written after simplification for three terms using the values. Finally

we get the solution in the form of infinite series is given by

Sh(t) = S0h + S1

h + S2h + S3

h + ...,

Eh(t) = E0h + E1

h + E2h + E3

h + ...,

Ih(t) = I0h + I1

h + I2h + I3

h + ...,

Sv(t) = S0v + S1

v + S2v + S3

v + ...,

Iv(t) = I0v + I1

v + I2v + I3

v + ...,

(4.2.10)

4.2.2 Numerical results and discussion

The LADM provide us an analytical solution in the form of infinite series. For

numerical results we consider the following values for parameters. Thus the first

few terms of LADM solution is Sh, Eh, Ih and Sv, Iv are calculated. We calculated

the first three terms of the series solution of the system 4.2.1.

Two of them as given as

n1 = 20, n2 = 15, n3 = 10, n4 = 10, n5 = 5, β = 0.02, ξ = 0.7, γ = 0.002,

k = 0.1, µ1 = 0.03, µ2 = 0.04, α = 0.05,

ah = 12, bv = 6,

86

S0h = 20, E0

h = 15, I0h = 10, S0

v = 10, I0v = 5

S1h = 7.6500

tα

Γ(α + 1), E1

h = 3tα

Γ(α + 1), I1

h = 0, S1v = 5.06000

tα

Γ(α + 1).

I1v = 0.08800

tα

Γ(α + 1),

S2h = 12

tα

Γ(α + 1)− 2.4559

t2α

Γ(2α + 1),

E2h = 2.0764

t2α

Γ(2α + 1)+ 0.5000

tα

Γ(α + 1),

I2h = 0.0600

t2α

Γ(2α + 1),

S2v = 6

tα

Γ(α + 1)− 0.8177

t2α

Γ(2α + 1)

I2v = 0.4153

t2α

Γ(2α + 1)

S3h = 12.6100

tα

Γ(α + 1)− 0.2583

t3α+1

Γ(3α + 2)+ 0.1604

t3α

Γ(3α + 1)− 0.2583

t3α+1

Γ(3α + 2),

E3h = −0.5063

t3α

Γ(3α + 1)+ 0.2583

t3α+1

Γ(3α + 2)+ 2.2500

t2α

Γ(2α + 1),

I3h = 0.0397

t3α

Γ(3α + 1),

S3v = 6

tα

Γ(α + 1)− 0.7800

t2α

Γ(2α + 1)+ 0.0175

t3α

Γ(3α + 1)− 0.0523

t3α+1

Γ(3α + 2),

I3v = 0.5400

t2α

Γ(2α + 1)− 0.0183

t3α

Γ(3α + 1)− 0.0523

t3α+1

Γ(3α + 2).

(4.2.11)

87

Thus solutions after three terms becomes

Sh(t) = 20 + 9.6500tα

Γ(α + 1)− 5.0659

t2α

Γ(2α + 1)− 0.2583

t3α+1

Γ(3α + 2)− 0.0979

t3α

Γ(3α + 1).

Eh(t) = 15 + 3tα

Γ(α + 1)+ 2.326

t2α

Γ(2α + 1)− 0.36017

t3α

Γ(3α + 1)+ 0.2585

t3α+1

Γ(3α + 2).

Ih(t) = 10 + 0.0600t2α

Γ(2α + 1)+ 0.0397

t3α

Γ(3α + 1).

Sv(t) = 10 + 7.7000tα

Γ(α + 1)− 1.5900

t2α

Γ(2α + 1)+ 0.0175

t3α

Γ(3α + 1)− 0.0523

t3α+1

Γ(3α + 2)

Iv(t) = 5 + 0.8800tα

Γ(α + 1)+ 0.5953

t2α

Γ(2α + 1)− 0.0318

t3α

Γ(3α + 1)− 0.02553

t3α+1

Γ(3α + 2).

(4.2.12)

Thus the series solution of system (4.2.2) given at α = 1, we obtain the approximate

solution of the three terms given as

Sh(t) = 20 + 9.6500t − 2.532950000t2 − 0.2673333334t3 − 0.1076250000t4

Eh(t) = 15 + 3t + 2.163450000t2 − 0.6002833335t3 + .1292500000t4

Ih(t) = 10 + 0.3000000000t2 + 0.6616666668t3

Sv(t) = 10 + 7.7000t − 0.7950000000t2 + 0.2916666667t3 − 0.2179166667t4

Iv(t) = 5 + 0.8800t + 0.4776500000t2 − 0.5300000001t3 − 0.2179166667t4.

(4.2.13)

Similarly we get the following system of series for α = 0.95

Sh(t) = 20 + 10.29985198t0.95 − 2.772258141t1.90 − 0.3217178657t2.85 − 0.1345657362t3.85

Eh(t) = 15 + 3.061597344t0.95 + 2.367848507t1.90 − 0.7224010206t2.85 + .1352859681t3.85

Ih(t) = 10 + 0.3283434108t1.90 + 0.7962717749t2.85

Sv(t) = 10 + 8.06342433t0.95 − 0.8701100386t1.90 + 0.351001412t2.85 − 0.2724656602t3.85

Iv(t) = 5 + 0.8980685542t0.95 + 0.5227774339t1.90 − 0.6378197089t2.85 − 0.272465660t3.85.

(4.2.14)

88

Now for α = 0.85, we get the following series

Sh(t) = 20 + 11.47041659t0.85 − 3.279566530t1.70 − 0.4565570400t2.55 − 0.2071031637t3.55

Eh(t) = 15. + 3.172551336t0.85 + 2.801152099t1.70 − 0.1025175493t2.55 + 0.1477665802t3.55

Ih(t) = 10 + 0.3884284960t1.70 + 0.1130007138t2.55

Sv(t) = 10 + 8.71805288t0.85 − 1.029335514t1.70 + 0.498113977t2.55 − 0.4193378034t3.55

Iv(t) = 5 + 0.9306150586t0.85 + 0.6184429037t1.70 − 0.9051442563t2.55 − 0.419337803t3.55.

(4.2.15)

Similarly, the solution after three terms for α = 0.75, we can write:

Sh(t) = 20 + 12.43726523t0.75 − 3.810837349t1.50 − 0.6292029485t2.25 − 0.3117650521t3.25

Eh(t) = 15 + 3.264195756t0.75 + 3.254922546t1.50 − 0.1412843055t2.25 + .1607227815t3.25

Ih(t) = 10 + 0.4513516669t1.50 + 0.1557316525t2.25

Sv(t) = 10 + 9.25875496t0.75 − 1.196081917t1.50 + 0.6864745386t2.25 − 0.6312548286t3.25

Iv(t) = 5 + 0.9574974218t0.75 + 0.7186270790t1.50 − 0.124742230t2.25 − 0.631254828t3.25.

(4.2.16)

89

0 2 4 6 820

30

40

50

60

70

80

90

0.750.850.951.0

t (Time in Weeks)

Sh

(a)

((

0 2 4 6 8

100

200

300

400

500

600

0.750.850.951.0

t(Time in weeks)

Eh

(b)

0 2 4 6 810

11

12

13

14

15

0.750.850.951.0

Ih

t (Time in weeks)(e)0 2 4 6 8

5

10

15

20

25

30

Iv

t (Time in weeks)

0.750.850.951.0

(d)

0 2 4 6 810

20

30

40

50

60

70

80

90

0.750.850.951.0

t(Time in weeks)

Sv

(c)

Fig. (1) Plot various compartments of proposed model (4.2.1) at different values

of fractional order α.

90

0 1 2 3 415

20

25

30

35

40

45

50

LADMRK4

t (Time in weeks)

Eh

(b)0 1 2 3 4

10

11

12

13

14

t (Time in weeks)

Sh

LADMRK4

(a) 0 1 2 3 4

10

20

30

40

50

t (Time in weeks)

Ih

LADMRK4

(c)

0 1 2 3 4

20

21

22

23

24

25

26

27

28

Sv

t (Time in weeks)

LADMRK4

(d) 0 1 2 3 410

10.5

11

11.5

12

12.5

13

13.5

14

14.5

Iv

t (Time in weeks)

LADMRK4

(e)

Fig. (2) Plot various compartments of proposed model (4.2.1) comparing LADM

with RK4.

4.2.3 Convergence analysis

The series solution (4.2.12) is rapidly convergent series and converges uniformly

to the exact solution. To check the convergence of the series (4.2.12), we use tech-

niques, (see [18]). For sufficient conditions of convergence of this method, we give

the following theorem by using idea [6, 55].

Theorem 4.2.1. Let X and Y be two Banach spaces and T : X → Y be a contractive

91

nonlinear operator such that

for all x, x′ ∈ X, ‖T(x)− T(x

′)‖ ≤ k‖x − x

′‖, 0 < k < 1.

The by use of Banach contraction principle T has a unique fixed point x such that Tx = x,

where x = (Sh, Eh, Ih, Sv, Iv). The series given in (4.2.12) can be written by applying

ADM as:

xn = Txn−1, xn−1 =n−1

∑i=1

xi, n = 1, 2, 3, ...,

and assume that x0 = x0 ∈ Br(x) where Br(x) = x′ ∈ X : ‖x

′ − x‖ < r, then, we have

(i) xn ∈ Br(x);

(ii) limn→∞

xn = x.

Proof. : For (i), using mathematical induction for n = 1, we have

‖x0 − x‖ = ‖T(x0)− T(x)‖ ≤ k‖x0 − x‖.

Let the result is true for n − 1, then

||x0 − x|| ≤ kn−1||x0 − x||.

we have

||xn − x|| = ||T(xn−1)− T(x)|| ≤ k||xn−1 − x|| ≤ kn||x0 − x||.

Hence using (i) we, have

||xn − x|| ≤ kn||x0 − x|| ≤ knr < r

which implies that xn ∈ Br(x).

(ii) Since ||xn − x|| ≤ kn||x0 − x|| and as limn→∞ kn = 0.

So, we have limn→∞ ||xn − x|| = 0 ⇒ limn→∞ xn = x.

Chapter 5

Numerical Analysis of VariousBiological Models of Fractional Order

In last few years researchers of mathematics, physics, chemistry, engineering and

computer science focused on the applications of fractional differential equations,

because fractional differential equations have many applications in in applied na-

ture and science. The efficient applications of fractional calculus are found in

physics, chemistry, biology, image processing, fluid mechanics and photography

etc. In mathematical modeling fractional order operator used as a global operator

for both integral and derivatives and provide greater degree of freedom as com-

pare to classical order which does not provide greater degree of freedom and also

a local operator. It is almost difficult to find the exact solution of FDEs due to

the complexity of fractional calculus arising in these equations. However approxi-

mate solutions of these complex equations can be easily find by different numerical

methods. For example in [71, 62, 29] the Variational iteration method, Homotopy

analysis method and Adomian decomposition method has been developed and

obtained approximate solution with a good efficiency for a class of fractional dif-

ferential equations. In last few years LADM method has been applied to solve

92

93

some fractional order coupled system both ODEs and PDEs [59, 17, 60, 61, 9].

In this chapter we operate Laplace transform method, which is a powerful tech-

niques in engineering and applied mathematics. With the help of this method we

transform fractional differential equations into algebraic equations, then solved

this algebraic equations by LADM [5]. Furthermore we successfully applied LADM

to solve different coupled system of fractional differential equation [31, 32, 33]. We

organize this chapter as:

5.1 Numerical analysis of Fractional Order Epidemic

Model of Childhood Diseases

the fractional order Susceptible-Infected-Recovered (SIR) epidemic model of child-

hood disease is considered. LADM is used to compute an approximate solution of

the system of NFODEs. We obtain the solutions of fractional differential equations

in the form of infinite series. The series solution of the proposed model converges

rapidly to its exact value. The obtained results are compared with the classical case.

Childhood diseases are most serious infectious diseases. Measles, poliomyelitis

and rubella are famous among them [7, 68]. Measles is a highly infectious disease,

caused by respiratory infection by a morbilli virus. These diseases normally affects

the children, because child population is more prone to the disease as compared

to the adults [7]. Therefore, the population can be divided into two major classes;

pre-mature and mature populations. Pre-mature population takes a constant time

to become mature, which is known as maturation delay. In disease dynamics, a

disease cannot spread instantaneously rather it will take some time in the body

94

which is called latent period for the particular disease. For the control of child-

hood disease vaccination is a significant strategy being used all over the world.

A universal effort to extend vaccination rang to all children began in 1974, when

the World Health Organization (WHO) founded the Expanded Program on Immu-

nization [68]. The authors [7], published (SIR) model in the following form,

dS

dt= (1 − p)π − βSI − πS,

dI

dt= βSI − (γ + π)I,

dR

dt= pπ + γI − πR.

(5.1.1)

The description of the preceding model is given below.

The model shows that vaccination is 100 percent efficient and the natural death rate

µ is unequal. Therefore the total population size N is not constant. The birth rate is

represented by π while the rate of mortality of the childhood disease is very low.

The parameter, p represents a fraction of vaccinated population at birth, where

0 < p < 1 and considered that the rest of population is susceptible. A susceptible

individual suffers from the disease through a contact with infected individuals at

rate β. Infected individuals recover at a rate γ.


Using the Caputo fractional derivative the system (5.1.1) gets the following form

cD

αS(t) = (1 − p)π − βSI − πS,

cD

α I(t) = βSI − (γ + π)I,

cD

αR(t) = pπ + γI − πR,

(5.1.2)

95

where α ∈ (0, 1] while π, γ, µ, β are positive parameters and the given initial con-

ditions are S(0) = N1, I(0) = N2, R(0) = N3. To solve the system (5.1.2), we use

LADM [65]. Moreover, the obtained solution will be compared with the integer

order derivative case. Furthermore, we use Laplace transformation to convert the

system of differential equations into a system of algebraic equations. Then, the al-

gebraic equations are used to obtain the required solution in form of series. We will

discuss the procedure for solving model (5.1.2) with given initial conditions. Ap-

plying Laplace transform on both side of the model (5.1.2), we obtain the following

system

L {cD

αS(t)} = L {(1 − p)π − βSI − πS},

L {cD

α I(t)} = L {βSI − (γ + π)I},

L {cD

αR(t) = L {pπ + γI − πR},

(5.1.3)

or

sαL {S(t)} − sα−1S(0) = L {(1 − p)π − βSI − πS},

sαL {I(t)} − sα−1 I(0) = L {βSI − (γ + π)I},

sαL {R(t)} − sα−1R(0) = L {pπ + γI − πR}.

(5.1.4)

Using the initial conditions (5.1.4), we obtain the form,

L {S(t)} =S0

s+

[

1

sαL {(1 − p)π − βSI − πS}

]

,

L {I(t)} =I0

s+

[

1

sαL {βSI − (γ + π)I}

]

,

L {R(t)} =R0

s+

[

1

sαL {pπ + γI − πR}

]

.

(5.1.5)

Assuming that the solutions S(t), I(t), R(t) in the form of infinite series given by

S(t) =∞

∑k=0

Sk(t), I(t) =∞

∑k=0

Ik(t), R(t) =∞

∑k=0

Rk(t), (5.1.6)

96

while the nonlinear term S(t)I(t) is decomposed as follow

S(t)I(t) =∞

∑k=0

Ak(t), (5.1.7)

where each Ak is the Adomian polynomials defined as

Ak =1

Γ(k + 1)

dk

dλk

[ k

∑j=0

λjSj(t)k

∑j=0

λj Ij(t)

]∣

∣

∣

∣

λ=0

. (5.1.8)

The first three polynomials are given by

A0 = S0(t)I0(t),

A1 = S0(t)I1(t) + S1(t)I0(t),

A2 = 2S0(t)I2(t) + 2S1(t)I1(t) + 2S2(t)I0(t).

Substituting (5.1.6) and (5.1.7) into (5.1.5) results in

L

{ ∞

∑k=0

Sk(t)

}

=S0

s+

[

1

sαL

{

(1 − p)π − β∞

∑k=0

Ak(t)− π∞

∑k=0

Sk(t)

}

]

,

L

{ ∞

∑k=0

Ik(t)

}

=I0

s+

[

1

sαL

{

β∞

∑k=0

Ak(t)− (γ + π)∞

∑k=0

Ik(t)

}

]

,

L

{ ∞

∑k=0

Rk(t)

}

=R0

s+

[

1

sαL

{

pπ + γ∞

∑k=0

Ik(t)− π∞

∑k=0

Rk(t)

}

]

.

(5.1.9)

Matching the two sides of (5.1.9) yields the following iterative algorithm:

L (S0) =N1

s,

L (S1) =(1 − p)π

sα− β

sαL {A0} −

π

sαL {S0},

L (S2) =(1 − p)π

sα− β

sαL {A1} −

π

sαL {S1},

...

L (Sk+1) =(1 − p)π

sα− β

sαL {Ak} −

π

sαL {Sk}, k ≥ 1,

(5.1.10)

97

L (I0) =N2

s,

L (I1) =β

sαL {A0} −

γ + π

sαL {I0},

L (I2) =β

sαL {A1} −

γ + π

sαL {I1},

...

L (Ik+1) =β

sαL {Ak} −

γ + π

sαL {Ik}, k ≥ 1,

(5.1.11)

and

L (R0) =N3

s,

L (R1) =pπ

sα+

γ

sαL {I0} −

π

sαL {R0},

...

L (Rk+1) =pπ

sα+

γ

sαL {Ik} −

π

sαL {Rk}, k ≥ 1.

(5.1.12)

98

Taking Laplace inverse of (5.1.10), (5.1.11) and (5.1.12) and considering first three

terms, we get

S0 = N1,

S1 = (1 − p)πtα

Γ(α + 1)− β(N1N2 + N1π)

tα

Γ(tα + 1)− (1 − p)π2 t2α

Γ(2α + 1),

S2 = (1 − p)πtα

Γ(α + 1)− β2(N2

1 N2 + β(γ + π)N1N2)t2α

Γ(t2α + 1),

− β(N1N2 + N1π)(βN2π)t2α

Γ(α + 1)− (1 − p)π2(βN2 + π)

t3α

Γ(3α + 1),

I0 = N2,

I1 = βN1N2tα

Γ(tα + 1)− (γ + π)N2

tα

Γ(tα + 1),

I2 = βN1N2 (βN1N2 − (γ + π)− (γ + π)βN1N2 − (γ + π)N2)t2α

Γ(2α + 1)

− βN2(βN1N2 + πN1)t2α

Γ(2α + 1)−(

(1 − p)π2 − (βN2 + π)) t3α

Γ(3α + 1),

R0 = N3,

R1 = (pπ − γN2 − γN3)tα

Γ(α + 1)− pπ

t2α

Γ(2α + 1),

R2 = pπtα

Γ(α + 1)+ γ(βN1N2 − (γ + π) + (πγN2 − π2N3))

t2α

Γ(2α + 1)− pπ2 t3α

Γ(3α + 1).

(5.1.13)


Using N1 = 1, N2 = 0.5, N3 = 0, µ = 0.4, β = 0.8, γ = 0.03, p = 0.9 and π = 0.4,

the LADM provides us an approximate solution in the form of infinite series. Thus

99

we calculate the first four terms of (3.2.6) to obtain

S(t) = 1 + 0.4400tα

Γ(α + 1)− 0.0868

t2α

Γ(2α + 1)+ 0.0490

t3α

Γ(3α + 1),

I(t) = 0.2 + 0.2340tα

Γ(α + 1)+ 0.0923

t2α

Γ(2α + 1)+ 0.0080

t3α

Γ(3α + 1),

R(t) = 1.0800tα

Γ(α + 1)− 0.3240

t2α

Γ(2α + 1)+ 0.0336

t3α

Γ(3α + 1).

(5.1.14)

For α = 1, the equation (5.1.14) attains the form,

S(t) = 1 + 0.4400t − 0.1278000t2 + 0.0081666668t3,

I(t) = 0.2 + 0.2340t + 0.0465000t2 + 0.0003333668t3,

R(t) = 1.0800t − 0.324000t2 + 0.0366000t3.

Similarly, we get the following system for α = 0.95;

S(t) = 1 + 0.44903t0.95 − 0.1398742t1.90 + 0.0098280395t2.85

I(t) = 0.2 + 0.23880453t0.95 + 0.050899322t1.90 + 0.000449848t2.85

R(t) = 1.1021503t0.95 − 0.35461087t1.90 + 0.0400435362t2.85.

Now for α = 0.85, one can obtain the following system,

S(t) = 1 + 0.4653075t0.85 − 0.16547042t1.70 + 0.01394913t2.55,

I(t) = 0.2 + 0.2474402t0.85 + 0.06020641t1.70 + 0.00078925t2.55,

R(t) = 1.14211848t0.85 − 0.41950277t1.70 + 0.05738273t2.55.

Similarly, the solution after three terms for α = 0.75, is calculated as;

S(t) = 1 + 0.44787487t0.75 − 0.1922758t1.50 + 0.0122128708t2.55,

I(t) = 0.2 + 0.254607t0.75 + 0.0699959508t1.50 + 0.00133334t2.55,

R(t) = 1.1751104t0.75 − 0.4874598003t1.50 + 0.790818664t2.55.

100

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

11

12

i(t)

t (Time in weeks)

1.00.950.850.75

Fig. (1) Plot of approximate solutions of susceptible class (s) corresponding to

different fractional values of αk for k = 1, 2, 3..

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

10

s(t)

t (Time in weeks)

1.00.950.850.75

Fig. (2) Plot of approximate solutions of infected class (i) corresponding to


0 2 4 6 8 10

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

r(t)

t (Time in weeks)

1.00.950.850.75

Fig. (3) Plot of approximate solutions of removed class (r) corresponding to


101

From the graphical results it is clear that the result obtained by using LADM is

very efficient. It also shows that the presented method can predict the behavior of

the variables accurately for the region under consideration. It is also clear that the

efficiency of this method can be dramatically increased by increasing the terms.

Fractional order derivative provides a greater degree of freedom as compared in

the Figure 1 to integer order derivative. The dynamics of various compartments

of the model has been shown in the Figure 1. In addition, we give a comparison

of RK4 and LADM in Table 1 and 2 for α = 1, which shows that both the methods

agree for short interval of time. The proposed method is better than RK4 method

as it needs no extra predefined parameter which controls the method.


For the convergence analysis of the childhood disease model see theorem (4.2.1).

102

5.2 Numerical Solution of Fractional Order Model of

HIV-1 Infection of CD4+ T-Cells by using LADM

This section present a fractional order HIV-1 infection model of CD4+ T-cell to an-

alyze the effect of the changing the average number of the viral particle N with

initial conditions of the proposed model. LADM is applying to check the ana-

lytical solution of the problem. Then obtain a solutions of the fractional order

HIV-1 model in the form of infinite series. The concerned series rapidly converges

to its exact value. Moreover comparing our results with the results obtained by

Runge-Kutta method in case of integer order derivative. Thus the fractional order

epidemic model is given by

cD

α1 T = β − kVT − dT + bT′,

cD

α2 T′= kVT − (b + δ)T

′,

cD

α3V = NδT′ − cV,

(5.2.1)

with given initial conditions, T(0) = T0, T′(0) = T

′0, V(0) = V0, where 0 < αi ≤ 1

for i = 1, 2, 3. The initial conditions are independent on each other and satisfy the

relation N(0) = T + T′+ V where N is the total number of the individuals in the

population. T, T′

and V denote the uninfected CD4+ cells, infected CD4+ T-cells

and free HIV virus particles in the blood respectively, d is the natural death rate

, k represent rate of infection T-cells, δ represents death rate of infected T-cells, b

represent rate of those infected cells which return to uninfected class, c represent

death rate of virus and N is the average number of viral particles produced by an

infected cells.

103


In this section, we discuss the general procedure of the model (5.2.1) with given

initial conditions. Applying Laplace transform on both side of the model (5.2.1) as,

L {cD

α1 T} = L {β − kVT − dT + bT′},

L {cD

α2 T′} = L {kVT − (b + δ)T

′},

L {cD

α3V} = L {NδT′ − cV},

(5.2.2)

which implies that

sα1L {T} − sα1−1T(0) = L {β − kVT − dT + bT′},

sα2L {T′} − sα2−1T

′(0) = L {kVT − (b + δ)T

′},

sα3L {V} − sα3−1V(0) = L {NδT′ − cV}.

(5.2.3)

Now using initial conditions and taking inverse Inverse Laplace transform in model

(5.2.3), we have

T = T0 +L−1

[

1

sα1L {β − kVT − dT + bT

′}]

,

T′= T

′0 +L

−1

[

1

sα1L {kVT − (b + δ)T

′}]

,

V = V0 +L−1

[

1

sα1L {NδT

′ − cV}]

.

(5.2.4)

Assuming that the solutions, T, T′, V in the form of infinite series given by

T =∞

∑n=0

Tn, T′=

∞

∑n=0

T′n, V =

∞

∑n=0

Vn (5.2.5)

and the nonlinear term VT involved in the model is decompose by Adomian poly-

nomial as

VT =∞

∑n=0

Pn, (5.2.6)

104

where Pn are Adomian polynomials defined as

Pn =1

Γ(n + 1)

dn

dλn

[ n

∑k=0

λkVk

n

∑k=0

λkTk

]

|λ=0 . (5.2.7)

Using (5.2.6),(5.2.7) and (5.2.5) in model (5.2.4), we have

L (T0) =T0

s, L (T

′0) =

T′0

sL (V0) =

V0

s

L (T1) =β

sα1+1+

−k

sα1L {P0} −

d

sα1L {T0}+

b

sα1L {T

′0}

L (T′1) =

−k

sα2L {P0} −

(b + δ)

sα2L {T

′0}, L (V1) =

Nδ

sα3L {T

′0} −

c

sα3L {V0}

L (T2) =−k

sα1L {P1} −

d

sα1L {T1}+

b

sα1L {T

′1}

L (T′2) =

−k

sα2L {P1} −

(b + δ)

sα2L {T

′1}, L (V2) =

Nδ

sα3L {T

′1} −

c

sα3L {V1}

...

L (Tn+1) =−k

sα1L {Pn} −

d

sα1L {Tn}+

b

sα1L {T

′n}

L (T′n+1)

=−k

sα2L {Pn} −

(b + δ)

sα2L {T

′n}, L (Vn+1) =

Nδ

sα3L {T

′n} −

c

sα3L {Vn}

(5.2.8)

105

Taking laplace inverse of (5.2.8) on both side, we get

T0 = T0, T′0 = T

′0, V0 = V0,

T1 =tα1

Γ(α1 + 1)+ (−kV0T0 − T0 + T

′0)

tα1

Γ(α1 + 1), T

′1 = (kV0T0 − (b + δ)T

′0)

tα2

Γ(α2 + 1)

V1 = (δNT′0 − cV0)

tα3

Γ(α3 + 1),

T2 = (−kV0T0 − T0 + T′0)(−kV0 − d)

t2α1

Γ(2α1 + 1)− kT0(δNT0 − cV0)

tα3+α1

Γ(α3 + α1)

− k(δNT0 − cV0)tα3+2α1

Γ(α3 + 2α1 + 1)+ b(kV0T0 − (b + δ)T

′0)

tα2+α1

Γ(α2 + α1 + 1),

T′2 = kV0(−kV0T0 − T0 + T

′0)

tα2+α1

Γ(α2 + α1 + 1)+ k(T0δNT

′0 − cV0)T0

tα3+α2

Γ(α3 + α2 + 1)

+ (δNT′0 − cV0)

α3 + α2

Γ(α3 + α2 + 1)+ (δNT0 − cV0)

tα3+α1

Γ(α3 + α1 + 1)

− (b + δ)(kV0T0 − (b + δ)T′0)

t2α2

2α2 + 1,

V2 = δN(kV0T0 − (b + δ)T′0)

tα3+α2

Γ(α3 + α2 + 1)− c(δNT

′0 − cV0)

t2α3

Γ(2α3 + 1).

(5.2.9)

On the above fashion, we can obtain the remaining terms similarly. Finally, we get

the solution in the form of infinite series is given by

T(t) = T0 + T1 + T2 + T3 + ... =∞

∑i=0

Ti,

T′(t) = T

′0 + T

′1 + T

′2 + ... =

∞

∑i=0

T′i

V(t) = V0 + V1 + V2 + V3 + ... =∞

∑i=0

Vi.

(5.2.10)

106

5.2.2 Numerical simulation

In this section we find numerical simulation of the considered problem (5.2.1), us-

ing values of the parameter

N = 100, δ = 0.16, k = 0.0024, V0 = 10, T′0 = 20, T0 = 70, b = 0.2, c = 3.4, β =

1, α1 = α2 = α3 = α, then after some simplification, we can write terms of System

(5.2.9) as

T0 = 70, T′0 = 20, V0 = 10,

T1 =tα1

Γ(α1 + 1)− 51.68

tα1

Γ(α1 + 1), T

′1 = −5.52

tα2

Γ(α2 + 1), V1 = 286

tα3

Γ(α3 + 1),

T2 = 1.36t2α1

Γ(2α1 + 1)− 48.04

tα3+α1

Γ(α3 + α1)− 182.44

tα3+2α1

Γ(α3 + 2α1 + 1)− 2.60

tα2+α1

Γ(α2 + α1 + 1),

T′2 = −1.24

tα2+α1

Γ(α2 + α1 + 1)+ 50.73

tα3+α2

Γ(α3 + α2 + 1)− 29.20

tα3+α2

Γ(α3 + α2 + 1)

− 17.20α3 + α1

Γ(α3 + α1 + 1)+ 1.11

t2α2

Γ(2α2 + 1),

V2 = −0.88tα3+α2

Γ(α3 + α2 + 1)− 972.40

t2α3

Γ(2α3 + 1).

(5.2.11)

107

After, three terms the solutions become

T = 70 +tα1

Γ(α1 + 1)− 51.68

tα1

Γ(α1 + 1)+ 1.36

t2α1

Γ(2α1 + 1)

− 48.04tα3+α1

Γ(α3 + α1)− 182.44

tα3+2α1

Γ(α3 + 2α1 + 1)

− 2.60tα2+α1

Γ(α2 + α1 + 1),

T′= 20 − 5.52

tα2

Γ(α2 + 1)− 1.24

tα2+α1

Γ(α2 + α1 + 1)

+ 50.73tα3+α2

Γ(α3 + α2 + 1)− 29.20

tα3+α2

Γ(α3 + α2 + 1)

− 17.20α3 + α1

Γ(α3 + α1 + 1)+ 1.11

t2α2

Γ(2α2 + 1),

V = 10 + 286tα3

Γ(α3 + 1)− 0.88

tα3+α2+1

Γ(α3 + α2 + 1)− 972.40

t2α3

Γ(2α3 + 1).

(5.2.12)

0 1 2 3 4 5

20

25

30

35

40

t (Time in weeks)

T(t)

1.00.850.75

(a)

0 0.1 0.2 0.310

15

20

25

30

35

40

45

50

t (Time in weeks)

T’(t)

(b)

1.00.850.75

0 0.1 0.2 0.310

15

20

25

30

35

40

45

50

t (Time in weeks)

V(t)

(c)

1.00.850.75

Fig. (1) The plot shows the dynamics of T(t), T′(t) and V(t) for various values of

αi(i = 1, 2, 3) = 1 via LADM.

108

0 0.1 0.2 0.3

8.7

8.8

8.9

9

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10

t (Time in weeks)

T(t)

LADMRK4

(a)0 0.1 0.2 0.3

18.7

18.8

18.9

19

19.1

19.2

19.3

19.4

19.5

19.6

19.7

19.8

19.9

20

t (Time in weeks)

T’(t)

LADMRK4

(b)

0 0.1 0.2 0.310

15

20

25

30

35

40

45

50

t (Time in weeks)

V(t)

LADMRK4

(c)

Fig. (2) The comparison between the solutions via RK4 and LADM method at

classical orderαi(i = 1, 2, 3) = 1 .


For the convergence analysis of the proposed model see theorem (4.2.1).

109

5.3 Numerical Analysis of Fractional Order Enzyme

Kinetics Model by Laplace Adomian Decomposi-

tion Method

Enzymes are one of the important aspects of all the biochemical process, they work

as a catalyst in most reaction taking place in living organism. In this section, the

enzyme kinetics model of fractional order derivative is considered. Using Laplace

transformation and ADM analytical (approximate) solution of the model will be

investigated. The solutions obtained by this method will be in form of infinite se-

ries, which will converge to its exact value very rapidly. Furthermore, using the

numerical Runga–Kutta method the obtained results will be compared for integer

order derivative. The behavior of the obtained solution is also presented graphi-

cally. Assume the following fractional order enzyme kinematic model.

cD

αs(t) = k2c − k1es,

cD

αe(t) = k3c + k2c − k1es,

cD

αc(t) = −k3c − k2c + k1es,

cD

α p(t) = k3e,

(5.3.1)

with the given initial condition s(0) = N1, e(0) = N2, c(0) = N3, p(0) = N4, where

cDα, 0 < α < 1 is the Caputo derivative of fractional order.

In model (5.3.1), the initial conditions are independent of each other and satisfy the

relation M(0) = s(t) + e(t) + c(t) + p(t), where M is the total number of the items

in a chemical reaction.

110


This section is devoted to, discuss the general procedure for solving the model

(3.2.6) with given initial conditions. Applying Laplace transform on both side of

the model (5.3.1) as

L {cD

αs(t)} = L {−k1es + k2c},

L {cD

αe(t)} = L {−k1es + k2c + k3c},

L {cD

αc(t)} = L {k1es − k2c − k3c},

L {cD

α p(t)} = L {k3e}

(5.3.2)

which implies that

sα1L {s(t)} − sα−1s(0) = L {−k1es + k2c},

sα2L {e(t)} − sα−1e(0) = L {−k1es + k2c + k3c},

sα3L {c(t)} − sα−1c(0) = L {k1es − k2c − k3c},

sα4L {p(t)} − sα−1 p(0) = L {k3e}.

(5.3.3)

Now using initial conditions and taking inverse Laplace transform of system (5.3.3),

we have

s(t) = s0 +L−1

[

1

sαL {−k1es + k2c}

]

,

e(t) = e0 +L−1

[

1

sαL {−k1es + k2c + k3c}

]

,

c(t) = c0 +L−1

[

1

sαL {k1es − k2c − k3c}

]

,

p(t) = p0 +L−1

[

1

sαL {k3e}

]

.

(5.3.4)

111

Using the values of initial conditions in (5.3.4), we get

s(t) = N1 +L−1

[

1

sαL {−k1es + k2c}

]

,

e(t) = N2 +L−1

[

1

sαL {−k1es + k2c + k3c}

]

,

c(t) = N3 +L−1

[

1

sαL {k1es − k2c − k3c}

]

,

p(t) = N4 +L−1

[

1

sαL {k3e}

]

.

Assuming that the solutions s(t), e(t), c(t), p(t) can be expressed in the form of

infinite series given by

s(t) =∞

∑n=0

sn, e(t) =∞

∑n=0

en, c(t) =∞

∑n=0

cn,

p(t) =∞

∑n=0

pn

(5.3.5)

and the nonlinear terms involved in the model, e(t)s(t) is decomposed by Ado-

mian polynomial,

e(t)s(t) =∞

∑n=0

An. (5.3.6)

Where An are Adomian polynomials defined as

An =1

Γ(n + 1)

dn

dλn

[

n

∑k=0

λkek

n

∑k=0

λksk

]

|λ=0 . (5.3.7)

112

Using (5.3.5), (5.3.6) in model (5.3.4), we get

L (s0) =N1

s, L (e0) =

N2

s, L (c0) =

N3

s, L (p0) =

N4

s,

L (s1) =1

sαL {−k1e0s0 + k2c0}, L (e1) =

1

sαL {−k1e0s0 + k2c0 + k3c0},

L (c1) =1

sαL {k1e0s0 − k2c0 − k3c0}, L (p1) =

1

sαL {k3e0},

L (s2) =1

sαL {−k1e1s1 + k2c1}, L (e2) =

1

sαL {−k1e1s1 + k2c1 + k3c1},

L (c2) =1

sαL {k1e1s1 − k2c1 − k3c1}, L (p2) =

1

sαL {k3e1},

...

L (sn+1) =1

sαL {−k1ensn + k2cn}, L (en+1) =

1

sαL {−k1ensn + k2cn + k3cn},

L (cn+1) =1

sαL {k1ensn − k2cn − k3cn}, L (pn+1) =

1

sαL {k3en}.

(5.3.8)

To study mathematical behavior of the solution vector, (s(t), e(t), c(t), p(t)) we use

different values of α in (0, 1]. For the solution, taking inverse Laplace transform of

113

(5.3.8), we have

s0 = N1, e0 = N2, c0 = N3, p0 = N4,

s1 = (−k1N1N2 + k2N3)tα

Γ(α + 1), e1 = (−k1N1N2 + k2N3 + k3N3)

tα

Γ(α + 1),

c1 = (k1N1N2 − (k2 + k3)N3)tα

Γ(α + 1), p1 = (k3N4)

tα

Γ(α + 1),

s2 = −k1N2(−k1N1N2 + k2N3)t2α

Γ(2α + 1)− k1N1(−k1N1N2 + (k2 + k3)N3)

t2α

Γ(2α + 1)

+ k2(k1N1N2 − (k2 + k3)N3)t2α

Γ(2α + 1),

e2 = −k1N2(−k1N1N2 + k2N3)t2α

Γ(2α + 1)− k1N1(−k1N1N2 + (k2 + k3)N3)

t2α2

Γ(2α2 + 1)

+ k2(k1N1N2 − (k2 + k3)N3)t2α

Γ(2α + 1)+ k3(k1N1N2 − (k2 + k3)N3)

t2α

Γ(2α + 1),

c2 = k1N2(−k1N1N2 + k2N3)t2α

Γ(2α + 1)+ k1N1(−k1N1N2 + (k2 + k3)N3)

t2α

Γ(2α + 1)

− k2(k1N1N2 − (k2 + k3)N3)t2α

Γ(2α + 1)− k3(k1N1N2 − (k2 + k3)N3)

t2α

Γ(2α + 1),

p2 = k3(−k1N1N2 + (k2 + k3)N3)t2α

Γ(2α + 1).

(5.3.9)

In the same fashion, we can compute the other terms.


For the convergence analysis of the enzyme kinetic model see theorem (4.2.1).


The Laplace Adomian-decomposition method provide an analytical solution in

the form of infinite series. For numerical results we consider the following values

114

for parameters. Thus the first few terms of LADM solution for s(t), e(t), c(t) and

p(t) are calculated. We calculated the first three terms of the series solution of the

system 5.3.1 by taking the following values

N1 = 10, N2 = 1, N3 = 0, N4 = 0, k1 = 0.073, k2 = 0.1, k3 = 0.03.

s0 = 10, e0 = 1, c0 = 0, p0 = 0,

s1 = −0.7tα

Γ(α + 1), e1 = −0.7

tα

Γ(α + 1), c1 = 0.7

tα

Γ(α + 1), p1 = 0.03

tα

Γ(α + 1).

s2 = 0.6090t2α

Γ(2α + 1), e2 = 0.6300

t2α

Γ(2α + 1),

c2 = 0.4480t2α

Γ(2α + 1), p2 = −0.0210

t2α

Γ(2α + 1),

s3 = −0.9039t3α

Γ(3α + 1)+ 0.0686

t3α+1

Γ(3α + 2),

e3 = 0.8905t3α

Γ(3α + 1)− 0.8820

t3α+1

Γ(3α + 2),

c3 = 0.8865t3α

Γ(3α + 1)− 0.8820

t3α+1

Γ(3α + 2),

p3 = 0.018865t3α

Γ(3α + 1).

(5.3.10)

Thus solutions after three terms becomes

s(t) = 10 − 0.7tα

Γ(α + 1)+ 0.6090

t2α

Γ(2α + 1)− 0.9093

t3α

Γ(3α + 1)+ 0.0686

t3α+1

Γ(3α + 2),

e(t) = 1 − 0.7tα

Γ(α + 1)+ 0.63

t2α

Γ(2α + 1)+ 0.8905

t3α

Γ(3α + 1)+ 0.0686

t3α+1

Γ(3α + 2),

c(t) = 0.7tα

Γ(α + 1)+ 0.4480

t2α

Γ(2α + 1)+ 0.8865

t3α

Γ(3α + 1)− 0.8820

t3α+1

Γ(3α + 2),

p(t) = 0.03tα

Γ(α + 1)− 0.0210

t2α

Γ(2α + 1)+ 0.018865

t3α

Γ(3α + 1).

(5.3.11)

115

By taking α = 1 in system (5.3.11), we obtain the approximate solution after three

terms as

s(t) = 5 − 12.08t + 36.96500000t2,

e(t) = 4 − 9.88t + 34.25500000t2,

c(t) = 4 + 9.88t + 1.805000000t2,

p(t) = 3 + 1.65t − 2.715000000t2.

(5.3.12)

Similarly, we get the following system of series solution by taking α = 0.95 in

system (5.3.11)

s(t) = 5 − 12.32803197t0.95 + 40.45738060t1.90,

e(t) = 4 − 10.08286059t0.95 + 37.49134512t1.90,

c(t) = 4 + 10.08286059t0.95 + 1.975532855t1.90,

p(t) = 3 + 1.683878539t0.95 − 2.971507868t1.90.

(5.3.13)

In the same line taking α = 0.85, we get the following series solution of system

(5.3.1)

s(t) = 5 − 12.77480671t0.85 + 47.86086452t1.70,

e(t) = 4 − 10.44826907t0.85 + 44.35206044t1.70,

c(t) = 4 + 10.44826907t0.85 + 2.337044784t1.70,

p(t) = 3 + 1.744903235t0.85 − 3.515277889t1.70.

(5.3.14)

Similarly, the solution after three terms for α = 0.75 is given by

s(t) = 5 − 13.14382824t0.75 + 55.61404789t1.50

e(t) = 4. − 10.75008469t0.75 + 51.53683783t1.50

c(t) = 4 + 10.75008469t0.75 + 2.715632529t1.50

p(t) = 3 + 1.795307666 ∗ t0.75 − 4.084732586t1.50.

(5.3.15)

116

One can observe from Figure 1, that at the concentration of the substrate s, enzyme

substrate complex c and product e and the product p depend on the fractional or-

der derivative. For instance, the concentration of s decrease slowly from its initial

value of the concentration s(0) to and converges to zero at classical order α = 1

and decreases slowly to zero at fractional values i.e at α = 0.9, 0.85, 0.75 etc. But

in the case of product c, e and p increase repidly for classical order derivative and

increase slowly in the case of fractional order derivative as shown in Figure 1.

From the plot we concluded that fractional derivative provide greater degree of

freedom and can be varied to get various responses of the concentration of various

compartments in the model (3.2.6). Next we also compare the solution obtained

through our proposed method with that of RK4 method for the given values of

time at classical order α = 1. From the Tables 5.3 and 5.4, one can observe that

the proposed method gives approximately the same solution as received by RK4

method. Thus the proposed method is an excellent method which need no prior

defined step size and also the implementation and computation of the method is

easy for the numerical solutions of fractional order models.

117

5.3.4 Numerical plot and comparison table

0 1 2 3 40

1

2

3

4

5

6

t (Time in weeks)

c(t)

0 1 2 3 4

1

.5

2

.5

3

t (Time in weeks)

e(t)

0 1 2 3 4

1

1.5

2

2.5

3

t (Time in weeks)

e(t)

0 1 2 3 4

1

1.5

2

2.5

3

t (Time in weeks)

e(t)

0 1 2 3 4

4

5

6

7

8

9

10

t (Time in weeks)

s(t)

(a)

1.00.950.850.75

0 1 2 3 40

1

2

3

4

5

6

t (Time in weeks)

c(t)

(c)

1.00.950.850.75

0 1 2 3 40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

t (Time in weeks)

p(t)

1.00.950.850.75

(d)

Frame 005 16 Jan 2017 No Dataset

0 1 2 3 4

1

1.5

2

2.5

3

t (Time in weeks)

e(t)

(b)

1.00.950.850.75

Fig. (1) Plot of various compartments of proposed model (3.2.6) at different

values of fractional order α.

.

5.4 Numerical Solution of Fractional Order Smoking

Model Via LADM

The section deals with the dynamics of giving up a coupled system smoking model

of fractional order. Smoking is one of the major cause of health problems around

118

the globe. We study approximate solution of the concerned model with the help

of Laplace transformation. The solution of the model will be obtained in form of

infinite series which converges rapidly to its exact value. Moreover, we compare

our results with the results obtained by Runge-Kutta method. Some plots are pre-

sented to show the reliability and simplicity of the method. Thus the new fractional

order model is given by

cD

α1 P(t) = bN(t)− β1L(t)P(t)− (d1 + µ)P(t) + τQ(t),

cD

α2 L(t) = β1L(t)P(t)− β2L(t)S(t)− (d2 + µ)L(t),

cD

α3S(t) = β2L(t)S(t)− (γ + d3 + µ)S(t),

cD

α4 Q(t) = γS(t)− (τ + d4 + µ)Q(t),

cD

α5 N(t) = (b − µ)N(t)− (d1P(t) + d2L(t) + d3S(t) + d4Q(t)),

(5.4.1)

with given initial condition P(0) = n1, L(0) = n2, S(0) = n3, Q(0) = n4, N(0) =

n5, where cDα 0 < αi ≤ 1 for i = 0, 1, 2 is the Caputo’s derivative of fractional

order and α show fractional time derivative.

In model (5.4.1) the initial conditions are independent of each other and sat-

isfy the relation M(0) = P(t) + L(t) + S(t) + Q(t) + N(t) where M is the total

population.


The present section is devoted to the general procedure of the model (5.4.1) with

given initial conditions. Applying Laplace transform to both sides of the model

119

(5.4.1) we have

L {cD

α1 P(t)} = L {bN(t)− β1L(t)P(t)− (d1 + µ)P(t) + τQ(t)},

L {cD

α2 L(t)} = L {β1L(t)P(t)− β2L(t)S(t)− (d2 + µ)L(t)},

L {cD

α3S(t)} = L {β2L(t)S(t)− (γ + d3 + µ)S(t)},

L {cD

α4 Q(t)} = L {γS(t)− (τ + d4 + µ)Q(t)},

L {cD

α5 N(t)} = L {(b − µ)N(t)− (d1P(t) + d2L(t) + d3S(t)

+ d4Q(t))}

(5.4.2)

which implies that

sα1L {P(t)} − sα1−1P(0) = L {bN(t)− β1L(t)P(t)− (d1 + µ)P(t) + τQ(t)},

sα2L {L(t)} − sα2−1L(0) = L {β1L(t)P(t)− β2L(t)S(t)− (d2 + µ)L(t)},

sα3L {S(t)} − sα3−1S(0) = L {β2L(t)S(t)− (γ + d3 + µ)S(t)},

sα4L {Q(t)} − sα4−1Q(0) = L {γS(t)− (τ + d4 + µ)Q(t)},

sα5L {N(t)} − sα5−1N(0) = L {(b − µ)N(t)− (d1P(t) + d2L(t) + d3S(t)

+ d4Q(t))}.

(5.4.3)

120

Now using initial conditions and taking inverse inverse Laplace transform to sys-

tem (5.4.3), we have

P(t) = P0 +L−1

[

1

sα1L {bN(t)− β1L(t)P(t)− (d1 + µ)P(t) + τQ(t)}

]

,

L(t) = L0 +L−1

[

1

sα2L {β1L(t)P(t)− β2L(t)S(t)− (d2 + µ)L(t)}

]

,

S(t) = S0 +L−1

[

1

sα3L {β2L(t)S(t)− (γ + d3 + µ)S(t)},

]

Q(t) = Q0 +L−1

[

1

sα4L {γS(t)− (τ + d4 + µ)Q(t)},

]

N(t) = N0 +L−1

[

1

sα5L {(b − µ)N(t)− (d1P(t) + d2L(t) + d3S(t) + d4Q(t))}

]

.

(5.4.4)

Using the values of initial condition in (5.4.4), we get

P(t) = n1 +L−1

[

1

sα1L {bN(t)− β1L(t)P(t)− (d1 + µ)P(t) + τQ(t)}

]

,

L(t) = n2 +L−1

[

1

sα2L {β1L(t)P(t)− β2L(t)S(t)− (d2 + µ)L(t)}

]

,

S(t) = n3 +L−1

[

1

sα3L {β2L(t)S(t)− (γ + d3 + µ)S(t)},

]

Q(t) = n4 +L−1

[

1

sα4L {γS(t)− (τ + d4 + µ)Q(t)},

]

N(t) = n5 +L−1

[

1

sα5L {(b − µ)N(t)− (d1P(t) + d2L(t) + d3S(t) + d4Q(t))}

]

.

Assuming that the solutions, P(t), L(t), S(t), Q(t), N(t) in the form of infinite series

given by

P(t) =∞

∑n=0

Pn, L(t) =∞

∑n=0

Ln, S(t) =∞

∑n=0

Sn,

Q(t) =∞

∑n=0

Qn, N(t) =∞

∑n=0

Nn.

(5.4.5)

121

and the nonlinear terms involved in the model are L(t)P(t), L(t)S(t) are decom-

posed by Adomian polynomials as,

L(t)P(t) =∞

∑n=0

An, L(t)S(t) =∞

∑n=0

En, (5.4.6)

where An and En are Adomian polynomials are given by,

An =1

Γ(n + 1)

dn

dλn

[

n

∑k=0

λkLk

n

∑k=0

λkPk

]

|λ=0, En =1

Γ(n + 1)

dn

dλn

[

n

∑k=0

λkLk

n

∑k=0

λkSk

]

|λ=0,

(5.4.7)

using (5.4.5),(5.4.6) in model (5.4.4), we get

L (P0) =n1

s, L (L0) =

n2

s, L (S0) =

n3

s, L (Q0) =

n4

s, L (N0) =

n5

s,

L (P1) = (bN0(t)− β1A0 − (d1 + µ)P0(t) + τQ0(t))1

sα1+1,

L (L1) = (β1A0 − β2E0 − (d2 + µ)L0(t))1

sα2+1,

L (S1) = β2E0 − (γ + d3 + µ)S0(t))1

sα3+1,

L (Q1) = (γ(S0)− (τ + d4 + µ)Q0(t))1

sα4+1,

L (N1) = (β1 − µ)N0 − (d1P0 + d2L0 + d3S0 + d4Q0)1

sα5+1,

L (p2) = (bN1(t)− β1A1 − (d1 + µ)p1(t) + τQ1(t))1

sα1+1,

L (L2) = (β1A1 − β2E1 − (d2 + µ)L1(t))1

sα2+1,

L (S2) = β2E1 − (γ + d3 + µ)S1(t))1

sα3+1,

L (Q2) = (γ(S1)− (τ + d4 + µ)Q1(t))1

sα4+1,

L (N2) = (β1 − µ)N1 − (d1P1 + d2L1 + d3S1 + d4Q1)1

sα5+1,

...

(5.4.8)

122

L (Pn+1) = (bNn(t)− β1An − (d1 + µ)pn(t) + τQn(t))1

sα1+1,

L (Ln+1) = (β1An − β2En − (d2 + µ)Ln(t))1

sα2+1,

L (Sn+1) = β2En − (γ + d3 + µ)Sn(t))1

sα3+1,

L (Qn+1) = (γ(Sn)− (τ + d4 + µ)Qn(t))1

sα4+1,

L (Nn+1) = (β1 − µ)Nn − (d1Pn + d2Ln + d3Sn + d4Qn)1

sα5+1.

123

Taking Laplace inverse transform of (5.4.8), we obtain

P0 = n1, L0 = n2, S0 = n3, Q0 = n4, N0 = n5,

P1 = (bn5 − β1n2n2 − (d1 + µ)n1 + τn4)tα1

Γ(α1 + 1),

L1 = (β1n1n2 − β2n2n3 − (d2 + µ)n2)tα2

Γ(α2 + 1),

S1 = β2n2n3 − (γ + d3 + µ)n3)tα3

Γ(α3 + 1),

Q1 = (γ(n3)− (τ + d4 + µ)n4)tα4

Γ(α4 + 1),

N1 = (β1 − µ)n5 − (d1n1 + d2n2 + d3n3 + d4n4)tα5

Γ(α5 + 1),

P2 = b((b − µ)n5 − (d1n1 + d2n2 + d3n3 + d4n4))tα1+α5

Γ(α1 + α5 + 1)− β1[n2(bn5 − β1n2n2

− (d1 + µ)n1 + τn4)t2α1

Γ(2α1 + 1)+ n1(β1n1n2 − β2n2n3 − (d2 + µ)n2(t))

tα2+α1

Γ(α2 + α1 + 1)]

− (d1 + µ)(b1n5 − β1n1n2 − (d1 + µ)n1 + τn4)tα1

Γ(α1 + 1)

+ τ(γn3 − (τ + d4 + µ)n4)α4

Γ(α4 + 1).

L2 = β1[n2(bn5 − β1n2n2 − (d1 + µ)n1 + τn4)tα1+α4

Γ(α1 + α4 + 1)+ n1(β1n1n2 − β2n2n3−

(d2 + µ)n2(t))t2α2

Γ(2α2 + 1)]− β2[n2(β2n2n3 − (γ + d3 + µ)n3)

tα3+α2

Γ(α3 + α2 + 1)

+ n3(β1n1n2 − β2n2n3 − (d2 + µ)n2(t))t2α2

Γ(2α2 + 1)]

− (d2 + µ)[(β1n1n2 − β2n2n3 − (d2 + µ)n2)]t2α2

Γ(2α2 + 1).

S2 = β2[n2(β2n2n3 − (γ + d3 + µ))t2α3

Γ(α3 + 1)

+ (β1n1n2 − β2n2n3 − (d2 + µ)n2)n3tα2+α3

Γ(α2 + α3 + 1)

− ((γ + d3 + µ)(β2n2n3 − (γ + d3 + µ))n3t2α3

Γ(2α3 + 1)]

Q2 = γ(β2n2n3 − (γ + d3 + µ))n3tα4+α3

Γ(α4 + α3 + 1)

− (τ + d4 + µ)(γn3 − (τ + d4 + µ))t2α4

Γ(2α4 + 1).

(5.4.9)

124

N2 = (b − µ)(b − µ)n5t2α5

Γ(2α5 + 1)− [d1(bn5 − βn1n2 − (d1 + µ)n1 + τn4)

tα1+α5

Γ(α1 + α5 + 1)

+ d2(β1n1n2 − β2n2n3 − (d2 + µ)n2)tα5+α2

Γ(α5 + α2 + 1)

+ d3(β2n2n3 − (γ + d3 + µ))n3tα5+α3

Γ(α5 + α3 + 1)+ d4(γn3 − (τ + d4 + µ)n4)

tα5+α4

α5 + α4 + 1].

Repeating in the same fashion, we can compute the remaining terms in the same

way. Here, we have computed only three terms for the required approximate so-

lution of the proposed model (5.4.1).

5.4.2 Numerical simulations

After simplification for three terms using the following values

n1 = 20, n2 = 40, n3 = 60, n4 = 80, n5 = 200, d1 = 0.33, d2 = 0.44, d3 = 0.55,

d4 = 0.66, µ = 0.05, b = 0.1, β1 = 0.01, β2 = 0.001, γ = 0.99, τ = 0.2,

α1 = α2 = α3 = α4 = α5 = α.

System (5.4.9) can be written as

P0 = 20, L0 = 40, S0 = 60, Q0 = 80, N0 = 200,

P1 = 2.4tα

Γ(α + 1), L1 = −14.4

tα

Γ(α + 1), S1 = −93.00

tα

Γ(α + 1),

Q1 = −13.2tα

Γ(α + 1), N1 = −191.50

tα

Γ(α + 1)

P2 = −20.9500t2α

Γ(2α + 1), L2 = −16.88

t2α

Γ(2α + 1).

S2 = −99.9600t2α

Γ(2α + 1), Q2 = −79.87

t2α

Γ(2α + 1).

N2 = 65.4820t2α

Γ(2α + 1).

(5.4.10)

125

Thus solution after three terms is given as:

P(t) = 20 + 2.4tα

Γ(α + 1)− 20.9500

t2α

Γ(2α + 1),

L(t) = 40 − 14.4tα

Γ(α + 1)− 16.88

t2α

Γ(2α + 1),

S(t) = 60 − 93.00tα

Γ(α + 1)− 99.9600

t2α

Γ(2α + 1),

Q(t) = 80 − 13.2tα

Γ(α + 1)− 79.87

t2α

Γ(2α + 1),

N(t) = 200 − 191.50tα

Γ(α + 1)+ 65.4820

t2α

Γ(2α + 1).

(5.4.11)

In particular for taking α = 1, the solution of proposed model(5.4.1) after three

terms is given by

P(t) = 20 = 2.4t − 10.975t2,

L(t) = 40 − 14.4t − 8.44t2,

S(t) = 60 − 93t − 49.98t2,

Q(t) = 80 − 13.2t − 39.94t2,

N(t) = 200 − 191.50t + 32.7410t2.

(5.4.12)

126

0 0.5 1

10

15

20

25

30

35

40

P(t)

t (time in weeks)

α=1.0α=0.85α=0.75

Fig. (1) The plot shows the population of potential smokers for different values of

αi, (i = 1, 2, 3, 4).

0 0.5 1

10

15

20

25

30

35

40

L(t)

t (time in weeks)

α=1.0α=0.85α=0.75

Fig. (2) The plot shows the population of occasional smokers for different values

of αi, (i = 1, 2, 3, 4).

127

0 0.1 0.2 0.3 0.415

20

25

30

35

40

45

50

55

60

S(t)

t (time in weeks)

α=1.0α=0.85α=0.75

Fig. (3) The plot shows the population of smokers for different values of

αi, (i = 1, 2, 3, 4).

0 0.25 0.5 0.75 1

30

40

50

60

70

80

Q(t)

t (time in weeks)

α=1α=0.85α=0.75

Fig. (4) The plot shows the population of quit smokers for different values of

αi, (i = 1, 2, 3, 4).

128

0 0.5 10

25

50

75

100

125

150

175

200

N(t)

t (time in weeks)

α=1.0α=0.85α=0.75

Fig. (5) The plot shows the dynamics of total population for different values of

αi, (i = 1, 2, 3, 4).

From the Figures 1-5, one can observe that fractional order smoking model has

more degree of freedom and therefore can be varied to get various responses of the

different compartments of the proposed model (5.1.2). Another interesting point to

be noted that as we have assumed comparatively small initial values therefore we

have used small interval of time. For larger time interval, the initial data should be

taken large so that the concerned population may not be negative and vice versa.

In Figure 6, we have compared the obtained solutions after three terms with the

RK4 method using classical order α = 1. We see that for the given values of pa-

rameters by using integer order the solutions obtained via LADM are closely agree

with that of RK4 at the given value of time.

129

0 0.5 1 1.5

2

4

6

8

10

12

14

16

18

20

t (time in weeks)

P(t)RK4

LADM

(a) 0 0.5 1 1.5

0

5

10

15

20

25

30

35

40

t (time in weeks)

L(t)

RK4LADM

(b)0 0.1 0.2 0.3 0.4 0.5

10

20

30

40

50

60

t (time in weeks)

S(t)

RK4LADM

(c)

0 0.5 1

10

20

30

40

50

60

70

80

t(time in weeks)

Q(t)

RK4LADM

(d)0 0.5 1

0

25

50

75

100

125

150

175

200

t(time in weeks)

N(t)

RK4LADM

(e)

Fig. (6) The comparison plots of the dynamics of potential smokers, occasional

smokers, smokers, quite smokers and total population for αi = 1, (i = 1, 2, 3, 4)

using RK4 and LADM method.


For the convergence analysis of the smoking model see theorem (4.2.1).

130

Time(week) S(t) I(t) R(t)t=0 1.00000 0.20000 0.00t=0.1 0.57945 0.23226 0.29235t=0.2 0.34554 0.22790 0.49526t=0.3 0.22200 0.17660 0.50051t=0.4 0.15708 0.13220 0.63007t=0.5 0.12364 0.10420 0.65919t=0.6 0.10530 0.06465 0.70740t=0.7 0.19586 0.04311 0.70311t=0.8 0.10088 0.02750 0.75970t=0.9 0.08861 0.01640 0.74071

Table 5.1: Numerical solution of proposed model using LADM at classical orderα = 1.

Time(week) S(t) I(t) R(t)t=0 1.00000 0.20000 0.00t=0.1 0.57735 0.24227 0.30235t=0.2 0.35667 0.22792 0.50527t=0.3 0.23300 0.18672 0.64052t=0.4 0.16808 0.14221 0.73009t=0.5 0.13374 0.10423 0.78910t=0.6 0.11546 0.07485 0.82784t=0.7 0.10581 0.05317 0.85318t=0.8 0.10088 0.03756 0.86971t=0.9 0.09853 0.02646 0.88047

Table 5.2: Numerical solution of proposed model using RK4 at classical order α =1.

131

Time(week) s(t) e(t) c(t) p(t)t=0 10.00000 1.00000 0.00000 0.00000t=0.1 10.9365 0.92981 0.07019 0.00289t=0.2 10.8756 0.86499 0.13501 0.00558t=0.3 10.8215 0.80517 0.19483 0.00809t=0.4 10.7784 0.75001 0.24999 0.01042t=0.5 10.7451 0.69919 0.30081 0.01259t=0.6 10.7225 0.65239 0.34761 0.01462t=0.7 10.7116 0.60933 0.39067 0.01651t=0.8 10.7132 0.56972 0.43028 0.01828t=0.9 10.7283 0.53331 0.46669 0.01993

Table 5.3: Numerical solution of proposed model by using RK4 at classical orderα = 1.

Time(week) s(t) e(t) c(t) p(t)t=0 10.00000 1.00000 0.00000 0.00000t=0.1 10.07101 0.9333 0.0724 0.0031t=0.2 10.13820 0.8738 0.1502 0.0064t=0.3 10.20182 0.8224 0.2344 0.0100t=0.4 10.26210 0.7800 0.3262 0.0139t=0.5 10.31927 0.7475 0.4268 0.0180t=0.6 10.37353 0.7258 0.5373 0.0225t=0.7 10.42509 0.7159 0.6593 0.0272t=0.8 10.47415 0.7188 0.7941 0.0323t=0.9 10.52088 0.7352 0.9433 0.0378

Table 5.4: Numerical solution of proposed model using LADM at classical orderα = 1.

Chapter 6

Numerical Analysis of BiologicalPopulation Model InvolvingFractional Order

Now a days in most of the research areas, the use of fractional differential equa-

tions has been increased due to wide range of it’s applications of in the real life

problems. In different scientific and engineering categories such as chemistry, me-

chanics and physics applications of fractional calculus can be found. It can be

also used in control theory, optimization theory, image processing, economics etc.

[64, 58]. Furthermore with the help of fractional derivative, we can model nonlin-

ear oscillation of earth quick. Fractional derivatives can also reduce the deficiency

of traffic flow arising in the fluid dynamic fluid flow.

Non-linear system of equations are very important for mathematicians, engineers

and physicists, because most physical system are non-linear in nature. The solu-

tion of nonlinear equation are difficult and applicable in most branches of sciences.

Exact solution of the nonlinear evolution equation plays an important role in the

study of non-linear problems. For the exact solution there are many approaches

such as Hirota,s method, Darboux transformation and Painleve expansions.

132

133

Currently, more alternative numerical methods can be used for solving both linear

and non linear problems of physical intrust. HPM [28], ADM[31, 32, 33], HAM[38]

and many other methods have been applied to solve both non-linear and linear

problems. We have solved the following model:

Dαt v(w, z, t) = (D2

w +D2z )v

2(w, z, t) + f (v(w, z, t)), (6.0.1)

with initial condition as

v(w, z, 0) = f0(w, z),

where ∂∂t = Dt,

∂2

∂w2 = D2w, ∂2

∂z2 = D2z . Further, v represents density of the population

and f denotes supply of the population due to birth and death rate.

6.1 LADM for Biological Model

This section present the general procedure of the LADM for solving the Eq. (6.0.1)

with given initial conditions. Applying Laplace transform operator to model (6.0.1)

we get,

L {Dαt v} = L {(D2

w +D2z )v

2 − krv(a+b) + kva}, t > 0, w, z ∈ R, 0 < α ≤ 1, (6.1.1)

with the given initial condition

v(w, z, 0) = f0(w, z).

From definition of Laplace transform on both side of Eq.(6.1.1), we have

sαL {v(w, z, t)} − sα−1v(w, z, 0) = L {(D2

w +D2z )v

2 − krv(a+b) + kva}. (6.1.2)

134

Upon using given initial conditions yields that

L {v(w, z, t)} =1

sf0(w, z) +

1

sαL {(D2

w +D2z )v

2 − krv(a+b) + kva}.

Therefore, Equation(6.1.2), can be written as after decomposing nonlinear terms

interms of Adomian polynomials and considering the unknown solutions v =

∑∞n=0 vn, we get

L

{ ∞

∑n=0

u(x, y, t)

}

=1

sf0(w, z) +

1

sαL

{

(D2w +D

2z )

∞

∑n=0

Pn − kr∞

∑n=0

Qn + k∞

∑n=0

Tn

}

.

Comparing terms on both side, we get

L {v0(w, z, t)} =1

sf0(w, z),

L {v1(w, z, t)} =1

sαL {(D2

w +D2z )P0 − krQ0 + kv0},

L {v2(w, z, t)} =1

sαL {(D2

w +D2z )P1 − krQ1 + kv1},

L {v3(w, z, t)} =1

sαL {(D2

w +D2z )P2 − krQ2 + kv2}

...

L {vn+1(w, z, t)} =1

sαL {(D2

w +D2z )Pn − krQn + kvn}, n ≥ 1.

(6.1.3)

After taking Laplace inverse transform of the systems (6.1.3) of equations, we get

v0(w, z, t) = f0(w, z),

v1(w, z, t) = L−1

[

1

sαL {(D2

w +D2z )P0 − krQ0 + kv0}

]

,

v2(w, z, t) = L−1

[

1

sαL {(D2

w +D2z )P1 − krQ1 + kv1}

]

,

...

vn+1(w, z, t) = L−1

[

1

sαL {(D2

w +D2z )Pn − krQn + kvn}

]

, n ≥ 1.

135

6.2 Applications

In this section we present some application of LADM to solve biological popula-

tion model.

Example 6.2.1. Consider the following fractional order biological model [27],


w +D2z )v(w, z, t)− rv2(w, z, t) + v(w, z, t), (6.2.1)

with given initial condition

v(w, z, 0) = exp

(

1

2

√

r

2(w + z)

)

.

Upon using the proposed method on Equation(6.2.1) and comparing terms of both side and

then taking Laplace inverse transform, we get

v0(w, z, 0) = exp

(

1

2

√

r

2(w + z)

)

L {v1(w, z, t)} =1

sαL {(D2

w +D2z − r)P0 + v0)}

v1(w, z, t) = exp

(

1

2

√

r

2(w + z)

)

tα

Γ(α + 1),

v2(w, z, t) = exp

(

1

2

√

r

2(w + z)

)

t2α

Γ(2α + 1),

v3(w, z, t) = exp

(

1

2

√

r

2(w + z)

)

t3α

Γ(3α + 1)

...

and so on.

136

−10 −5 0 5 10

−10

0

100

2

4

6

8

x 1017

xy (a)

u(x

, y,1

0)

−10−5

05

10

−10

0

100

1

2

3

x 1017

xy

u(x

,y,1

0)

(b)

−100

10

−10

0

100

5

10

x 1017

yx

(c)

u(x

,y,10)

α = 0.85

α = 1.0

α = 0.95

−10 −5 0 5 10−10

010

0

2

4

x 1017

yx

(d) u(x

,y,10)

α = 0.75

Figure 6.1: Numerical plots of approximate solutions of Example (6.2.1) at various frac-tional order and using r = 50, t = 10.

The series solution is provided as

v = exp

(

1

2

√

r

2(w + z)

)

+ exp

(

1

2

√

r

2(w + z)

)

tα

Γ(α + 1)

+ exp

(

1

2

√

r

2(w + z)

)

t2α

Γ(2α + 1). . .

v = exp

(

1

2

√

r

2(w + z)

) [

1 +tα

Γ(α + 1)+

t2α

Γ(α + 1)+

t3α

Γ(3α + 1). . .

]

.

In closed form, the solution is given by

v(w, z, t) = exp

(

1

2

√

r

2(w + z)

)

Eα(tα), (6.2.2)

which is the exact solution. Putting α = 1 in Equation(6.2.2), we get solution as

v(w, z, t) = exp

(

1

2

√

r

2(w + z)− t

)

. (6.2.3)

This is the classical solution. Next we showing sufficient conditions of convergence for

137

LADM using theorem (4.2.1), T is the strict contraction mapping. Therefore

‖v0 − v‖ =

∥

∥

∥

∥

exp(1

2

√

r

2(w + z))[1 − exp(−t)]

∥

∥

∥

∥

‖V1 − v‖ =

∥

∥

∥

∥

exp(1

2

√

r

2(w + z))[1 + t − exp(−t)]

∥

∥

∥

∥

≤∥

∥

∥

∥

exp(1

2

√

r

2(w + z))[1 − exp(−t)]

∥

∥

∥

∥

∥

∥

∥

∥

1 − t

1 − exp(−t)

∥

∥

∥

∥

.

Because for t ∈ [0, 10], we have

∥

∥

∥

∥

1 − t1−exp(−t)

∥

∥

∥

∥

≤ k = 0.634 < 1, it follow that

‖V1 − v‖ ≤ k

∥

∥

∥

∥

exp(1

2

√

r

2(w + z))

∥

∥

∥

∥

= k‖v0 − v‖

‖V2 − v‖ =

∥

∥

∥

∥

exp(1

2

√

r

2(w + z))[1 + t − t2

2− exp(−t)]

≤∥

∥

∥

∥

exp(1

2

√

r

2(w + z))[1 + t − exp(−t)]

∥

∥

∥

∥

∥

∥

∥

∥

1 − t2

2− t

1 − exp(−t)

∥

∥

∥

∥

.

For t ∈ [0, 10], we have

∥

∥

∥

∥

1 − t

1− t22 −exp(−t)

∥

∥

∥

∥

≤ 0.201 < k,

thus ‖V2 − v‖ ≤ k2‖v0 − v‖.

‖V3 − v‖ =

∥

∥

∥

∥

exp(1

2

√

r

2(w + z))[1 + t − t2

2− t3

6− exp(−t)]

≤∥

∥

∥

∥

exp(1

2

√

r

2(w + z))[1 + t − t2

2− exp(−t)]

∥

∥

∥

∥

∥

∥

∥

∥

1 +t3

6− t

1 − exp(−t)

∥

∥

∥

∥

.

For t ∈ [0, 10], we have

∥

∥

∥

∥

1 − t

1− t22 + t3

6 −exp(−t)

∥

∥

∥

∥

≤ 0.071 < k,

thus ‖V3 − v‖ ≤ k3‖v0 − v‖.

...

‖Vn − v‖ = kn‖v0 − v‖.

Therefore, limn→∞ ‖Vn − v‖ = limn→∞ kn‖v0 − v‖ = 0, that is

v(w, z, t) = limn→∞

Vn = exp

(

1

2

√

r

2(w + z)− t

)

.

138

Example 6.2.2. Consider the following fractional order biological model[27]


w +D2z )v

2(w, z, t) + v(w, z, t), (6.2.4)

with given initial condition

v(w, z, 0) =√

sin w sinh z.

Applying Laplace transform both side, we have

L {D2t v} = L {(Dw

2 +Dz2 ) + vn}

sαL {v(w, z, t)} − sα−1v(w, z, 0) = L {(D2

w +D2z )v

2n(w, z, t) + vn(w, z, t))}

L {vn(w, z, t)} =1

sv(w, z, 0) +

1

sαL {(D2

w +D2z )v

2n + vn)}

L {v0(w, z, t)} =1

s

√sin w sinh z

L {v1(w, z, t)} =1

sαL {(D2

w +D2z )v

20(w, z, t) + v0(w, z, t))}

L {v2(w, z, t)} =1

sαL {(D2

w +D2z )v

21(w, z, t) + v1(w, z, t))}

L {v3(w, z, t)} =1

sαL {(D2

w +D2z )v

22(w, z, t) + v2)(w, z, t)}

...

(6.2.5)

After using Laplace inverse transform on both side of Eq.(6.2.6), we have

v0(w, z, 0) =√

sin w sinh z

v1(w, z, t) =√

sin w sinh ztα

Γ(α + 1)

v2(w, z, t) =√

sin w sinh zt2α

Γ(2α + 1)

v3(w, z, t) =√

sin w sinh zt3α

Γ(3α + 1)

...

139

−10−5

05

10

−10−5

05

10−2

−1

0

1

2

x 107

xy(a) −10

−50

510

−10−5

05

10−3

−2

−1

0

1

2

3

x 107

xy

α = 0.75α = 0.85

Figure 6.2: Numerical plots of approximate solutions of Example (6.2.2) at various frac-tional order and using t = 10.

and so no. Now the series solution of problem (6.2.4) is given by

v(w, z, t) = v0(w, z, 0) + v1(w, z, t) + v2(w, z, t) + . . .

v(w, z, t) =√

sin w sinh z +√

sin w sinh ztα

Γ(α + 1)+√

sin w sinh zt2α

Γ(2α + 1)

+√

sin w sinh zt3α

Γ(3α + 1). . .

v(w, z, t) =√

sin w sinh z

[

1 +tα

Γ(α + 1)+

t2α

Γ(α + 1)+

t3α

Γ(3α + 1). . .

]

The closed form is given by

v(w, z, t) =√

sin w sinh zEα(tα)

Considering α = 1, we get classical solution as

v(w, z, t) =√

sin w sinh zet.

140

Example 6.2.3. Consider the following fractional order biological model.


w +D2z )v(w, z, t) + kv(w, z, t), (6.2.6)

corresponding to the initial condition

v(w, z, 0) =√

wz.

Applying Laplace transform both side, we have

L {D2t v(w, z, t)} = L {(Dw

2 +Dz2 )v

2n(w, z, t) + kvn(w, z, t)}

sαL {v(w, z, t)} − sα−1v(w, z, 0) = L {(D2

w +D2z )v

2n(w, z, t) + vn(w, z, t)}

L {vn(w, z, t)} =1

sv(w, z, 0) +

1

sαL {(D2

w +D2z )v

2n(w, z, t) + vn(w, z, t)}

L {v0(w, z, t)} =1

s

√wz

L {v1(w, z, t)} = k1

sαL {(D2

w +D2z )v

20(w, z, t) + v0(w, z, t)}

L {v2(w, z, t)} = k1

sαL {(D2

w +D2z )v

21(w, z, t) + v1(w, z, t)}

L {v3(w, z, t)} = k1

sαL {(D2

w +D2z )v

22(w, z, t) + v2(w, z, t)}

...

After using Laplace transform on both sides, we have

v0(w, z, t) =√

wz

v1(w, z, t) = k√

wztα

Γ(α + 1)

v2(w, z, t) = k2√

wzt2α

Γ(2α + 1)

v3(w, z, t) = k3√

wzt3α

Γ(3α + 1)

...

141

05

10

05

100

5000

10000

15000

xy (a)

0

5

10

02

46

810

0

1

2

x 104

yx

05

10

0

5

100

2

4

x 104

yx

(c)

05

10

0

5

100

5

10

x 104

(d)

α = 1

α = 0.95

α = 0.75

α = 0.85

Figure 6.3: Numerical plots of approximate solutions of Example (6.2.3) at various frac-tional order and using k = 0.1, t = 10.

and so on. The series solution of the problem (6.2.6), is given by

v(w, z, t) = v0(w, z, 0) + v1(w, z, t) + v2(w, z, t) + . . .

v(w, z, t) =√

wz + k√

wztα

Γ(α + 1)+ k2

√wz

t2α

Γ(2α + 1)+ h3

√wz

t3α

Γ(3α + 1). . .

v(w, z, t) =√

wz

[

1 + ktα

Γ(α + 1)+ k2 t2α

Γ(α + 1)+ k3 t3α

Γ(3α + 1). . .

]

.

Hence closed form of the solution is given by

v(w, z, t) =√

wzEα(ktα).

Considering α = 1, we get classical solution as

v(w, z, t) =√

wzet.

From the numerical plots of the considered examples, we see that the procedure is

efficient to obtain approximate or exact solutions to fractional order partial differ-

ential equations corresponding to various fractional order.

Chapter 7

Summary and conclusion

This thesis consists of seven chapters. Chapter 1 provide introduction of fractional

calculus. Chapter 2 is related with some basic definitions, special functions, frac-

tional calculus, differential equations, coupled systems and Laplace transform as

well as their properties. Chapter 3 contain existence theory of positive solution

to a class of fractional differential equations with boundary conditions. With the

help of standard fixed point theorem of Schauder and Banach contraction type,

we developed successfully sufficient for existence and uniqueness of positive so-

lutions highly class of nonlinear fractional differential equations with boundary

conditions. We make some remarks about the use of fractional derivatives and

fixed point theorem.

Section 3.4 is devoted to monotone iterative technique, which has been used for

the existence of extremal solutions to a class of arbitrary order differential equa-

tions. The concerned problem has been studied under some boundary conditions.

The established results demonstrated by suitable examples.

In Chapter 4 we have successfully built up mathematical analysis along with a

suitable numerical simulation for the dynamics of host vector PWD model with

142

143

convex incidence rate. Further, extinction of disease and existence, uniqueness en-

demic equilibrium has been shown in Section (4.1). Also we computed the basic

reproductive number R0, for the existence of endemic equilibrium. In addition it

has been shown that the disease persist at disease present equilibrium point with

the condition that if the disease exist initially. This is due to the fact that the equilib-

rium point is also globally asymptotically stable. The whole theatrical results have

been demonstrated by providing numerical simulations at different initial values

of the compartments of the suggested model.

In Chapter 5, we have presented numerical solution of different mathematical

models of fractional order. We discussed in Section (5.1), the numerical solution of

fractional order childhood model. In Section (5.2) numerical solution of fractional

order model of HIV-1 Infection of CD4+ T-Cells is studied while in Section (5.3)

numerical analysis of fractional order enzyme kinetics model is given. Moreover

Section (5.4) shows the numerical solution of fractional order smoking model via

LADM.

In Chapter 6 the LADM has been applied successfully to obtain the exact solution

of the generalized fractional order biological models with given initial conditions.

From the plots given in Figures, we observe that the procedure is efficient to obtain

the approximate or exact solutions to fractional order partial differential equations.

In the current situation we have come across with the exact solution for the corre-

sponding nonlinear FPDEs.

In [10], the authors showed that ADM generally does not converge, when the

method is applied to highly nonlinear differential equation. Our proposed method

144

is better than HAM, HPM, VIM, because it needs no parameter terms to form de-

composition equation, and no perturbation is required as needed in the mention

methods. Our proposed method is simple and does not require or waste extra

memory like Tau-collocation method. Further from direct ADM, our method is

better as we applied Laplace transform and then decompose the nonlinear terms

in term of Adomian polynomials, while in Adomian decomposition particular in-

tegral is involved, which often creates difficulties in computation. The main ad-

vantage of this technique is that, it avoids complex transformations like index re-

ductions and leads to a simple general algorithm. Secondly, it reduces the com-

putational work by solving only linear algebraic systems. From this method, we

obtained a simple way to control the convergence region of the series solution by

using a proper value of parameters. The results showed that LADM is very ef-

ficient, power full method to find the analytical solution of nonlinear differential

equations.

Bibliography

[1] Awan, A. U., Ozair, M., Din, Q., and Hussain, T. 2016. Stability analysis of

pine wilt disease model by periodic use of insecticides. Journal of biological

dynamics, 10 (1): 506–524.

[2] Agarwal, R., Belmekki, M., and Benchohra, M. 2009. A survey on semilinear

differential equations and inclusions involving Riemann-Liouville fractional

derivative. Advances in Difference Equations, (1), 981728.

[3] Agarwal, R. P., Meehan, M., and O’Regan, D. 2001. Fixed point theory and

applications. 141. Cambridge university press.

[4] Adomian, G. 1988. A review of the decomposition method in applied mathe-

matics. Journal of mathematical analysis and applications, 135 (2): 501–544.

[5] Adomian, G. 2013. Solving frontier problems of physics: the decomposition

method. Springer Science and Business Media, 60.

[6] Abdelrazec, A., and Pelinovsky, D. 2011. Convergence of the Adomian de-

composition method for initial-value problems. Numerical Methods for Par-

tial Differential Equations, 27 (4): 749–766.

[7] Arafa, A. A. M., Rida, S. Z., and Khalil, M. 2012. Solutions of fractional order

model of childhood diseases with constant vaccination strategy. Math. Sci.

Lett, 1(1): 17–23.

145

146

[8] Awan, A. U., Ozair, M., Din, Q., and Hussain, T. 2016. Stability analysis of

pine wilt disease model by periodic use of insecticides. Journal of biological

dynamics, 10 (1): 506–524.

[9] Awawdeh, F., Adawi, A., and Mustafa, Z. 2009. Solutions of the SIR models

of epidemics using HAM. Chaos, Solitons and Fractals, 42 (5): 3047–3052.

[10] Arafa, A. A. M., Rida, S. Z., and Mohamed, H. 2011. Homotopy analysis

method for solving biological population model. Communications in Theo-

retical Physics, 56(5): 797.

[11] Belgacem, F. B. M., Karaballi, A. A., and Kalla, S. L. 2003. Analytical inves-

tigations of the Sumudu transform and applications to integral production

equations. Mathematical problems in Engineering, (3): 103-118.

[12] Belgacem, F. B. M., and Karaballi, A. A. 2006. Sumudu transform fundamental

properties investigations and applications. International Journal of Stochastic

Analysis,: 1–23.

[13] Bai, Z. 2010. On positive solutions of a nonlocal fractional boundary value

problem. Nonlinear Analysis: Theory, Methods and Applications, 72(2): 916–

924.

[14] Benchohra, M., Hamani, S., and Ntouyas, S. K. 2009. Boundary value prob-

lems for differential equations with fractional order and nonlocal conditions.

Nonlinear Analysis: Theory, Methods and Applications, 71(7): 2391–2396.

[15] Bai, Z., and L, H. 2005. Positive solutions for boundary value problem of non-

linear fractional differential equation. Journal of Mathematical Analysis and

Applications, 311(2): 495–505.

147

[16] Babakhani, A., and Daftardar-Gejji, V. 2002. On calculus of local fractional

derivatives. Journal of Mathematical Analysis and Applications, 270(1): 66–

79.

[17] Biazar, J. 2006. Solution of the epidemic model by Adomian decomposition

method. Applied Mathematics and Computation, 173(2): 1101–1106.

[18] Biazar, J., and Aminikhah, H. 2009. Study of convergence of homotopy per-

turbation method for systems of partial differential equations. Computers and

Mathematics with Applications, 58(11), 2221–2230.

[19] Castillo-Chavez, C., Feng, Z., and Huang, W. 2002. Mathematical Approaches

for Emerging and Re-Emerging Infectious Diseases: An Introduction, IMA

Volumes in Mathematics and Its Applications, 125.

[20] Demirci, E., Unal, A., and alp, N. 2011. A Fractional Order SEIR Model with

Density Dependent Death Rate Abstract full Text. Hacettepe Journal of Math-

ematics and Statistics, 40(2): 287–296.

[21] Debnath, L. 2003. Recent applications of fractional calculus to science and en-

gineering. International Journal of Mathematics and Mathematical Sciences,

(54): 3413–3442.

[22] Elzaki, T. M. 2011. The new integral transform Elzaki Transform. Global Jour-

nal of Pure and Applied Mathematics, 7(1): 57–64.

[23] Elzaki, T. M. 2011. On the Connections between Laplace and Elzaki trans-

forms. Advances in Theoretical and Applied Mathematics, 6(1): 1-11.

[24] El-Shahed, M., and Alsaedi, A. 2011. The fractional SIRC model and influenza

A. Mathematical Problems in Engineering, ID 490378 page 9.

148

[25] Farah, T. M. E. 2009, Existence and Uniqueness of Solution for Composite

Type Equation, Journal of Science and Technology, 10(2): 215–220.

[26] Fay, T. H., and Graham, S. D. 2003. Coupled spring equations. International

Journal of Mathematical Education in Science and Technology, 34(1): 65–79.

[27] GarcOlivares, A. 2003. Analytic solution of partial differential equations with

Adomian’s decomposition. Kybernetes, 32(3): 354–368.

[28] Ganji, D. D. 2006. The application of He’s homotopy perturbation method to

nonlinear equations arising in heat transfer. Physics letters A, 355(5): 337–341.

[29] Hashim, I., Abdulaziz, O., and Momani, S. 2009. Homotopy analysis method

for fractional IVPs. Communications in Nonlinear Science and Numerical

Simulation, 14(3): 674–684.

[30] Haq, F., Shah, K., Rahman, G., and Shahzad, M. 2016. Existence and unique-

ness of positive solution to boundary value problem of fractional differ-

ential equations. Sindh University Research Journal-SURJ (Science Series),

48(2):451–456.

[31] Haq, F., Shah, K., Khan, A., Shahzad, M., and Rahman, G. 2017. Numerical so-

lution of fractional order epidemic model of a vector born disease by Laplace

Adomian decomposition method. Punjab University Journal of Mathematics,

49(2): 13–22.

[32] Haq, F., Shah, K., Rahman, G. U., and Shahzad, M. 2017. Numerical analysis

of fractional order model of HIV-1 infection of CD4+ T-cells. Computational

Methods for Differential Equations, 5(1): 1–11.

[33] Haq, F., Shah, K., ur Rahman, G., Shahzad, M. 2017. Numerical solution of

fractional order smoking model via laplace Adomian decomposition method.

Alexandria Engineering Journal, xxx–xxx.

149

[34] Jung, S. M. 2006. HyersUlam stability of linear differential equations of first

order, II. Applied Mathematics Letters, 19(9): 854–858.

[35] Khan, Z. H. and Khan, W. A. 2008. N-transform-properties and applications.

NUST journal of engineering sciences, 1(1): 127–133.

[36] Khader, M. M., Hendy, A. S. 2012. The approximate and exact solutions of

the fractional-order delay differential equations using Legedre seudospectral

method. International Journal of Pure and Applied Mathematics, 74(3): 287–

297.

[37] Kreyszig, E. 2010. Advanced engineering mathematics. John Wiley and Sons.

[38] Kheiri, H., Alipour, N., and Dehghani, R. 2011. Homotopy analysis and

Homotopy Padthods for the modified Burgers-Korteweg-de Vries and the

Newell-Whitehead equations,5(1), 33-50.

[39] Ladde, G. S., Lakshmikantham, V., and Vatsala, A. S. 1985. Monotone iterative

techniques for nonlinear differential equations, 27: Pitman Publishing.

[40] Lee, K. S., and Kim, D. 2013. Global dynamics of a pine wilt disease transmis-

sion model with nonlinear incidence rates. Applied Mathematical Modelling,

37(6): 4561–4569.

[41] Lee, K. S., and Lashari, A. A. 2014. Global stability of a host-vector model

for pine wilt disease with nonlinear incidence rate. In Abstract and Applied

Analysis, 2014. Hindawi.

[42] Leung, K., Shizgal, B. D., and Chen, H. 1998. The Quadrature Discretization

Method (QDM) in comparison with other numerical methods of solution of

the FokkerPlanck equation for electron thermalization. Journal of mathemat-

ical chemistry, 24(4): 291–319.

150

[43] Li, C. F., Luo, X. N., and Zhou, Y. 2010. Existence of positive solutions of

the boundary value problem for nonlinear fractional differential equations.

Computers and Mathematics with Applications, 59(3): 1363–1375.

[44] Lin, L., Liu, X., and Fang, H. 2012. Method of upper and lower solutions for

fractional differential equations. Electron. J. Differential Equations, 100: 1–13.

[45] Li, M. Y., and Muldowney, J. S. 1996. A geometric approach to global-stability

problems. SIAM Journal on Mathematical Analysis, 27(4): 1070–1083.

[46] Li, M. Y., and Muldowney, J. S. 1995. Global stability for the SEIR model in

epidemiology. Mathematical biosciences, 125(2): 155–164.

[47] Machado, J. T., Kiryakova, V., and Mainardi, F. 2011. Recent history of frac-

tional calculus. Communications in Nonlinear Science and Numerical Simu-

lation, 16(3): 1140–1153.

[48] Mittag-Leffler, G. M. 1903. Sur la nouvelle fonction Ea (x). CR Acad. Sci. Paris,

137(2): 554–558.

[49] McRae, F. A. 2009. Monotone iterative technique and existence results for frac-

tional differential equations. Nonlinear Analysis: Theory, Methods and Ap-

plications, 71(12): 6093–6096.

[50] Martin Jr, R. H. 1974. Logarithmic norms and projections applied to linear dif-

ferential systems. Journal of Mathematical Analysis and Applications, 45(2):

432–454.

[51] Mickens, R. E. 2000. Applications of nonstandard finite difference schemes.

World Scientific.

[52] Mickens, R. E. 1989. Exact solutions to a finite-difference model of a nonlinear

reaction-advection equation: Implications for numerical analysis. Numerical

Methods for Partial Differential Equations, 5(4): 313–325.

151

[53] Mickens, R. E. 2010. A SIR-model with square-root dynamics: An NSFD

scheme. Journal of Difference Equations and Applications, 16(2-3): 209–216.

[54] Muldowney, J. S. 1990. Compound matrices and ordinary differential equa-

tions. The Rocky Mountain Journal of Mathematics, 857–872.

[55] Naghipour, A., and Manafian, J. 2015. Application of the Laplace Adomian

decomposition and implicit methods for solving Burgers equation. TWMS

Journal of Pure and Applied Mathematics, 6(1): 68–77.

[56] Oldham, K., and Spanier, J. 1974. The fractional calculus theory and applica-

tions of differentiation and integration to arbitrary order 111.

[57] Obloza, M. 1993. Hyers stability of the linear differential equation. Rocznik

Naukowo-Dydaktyczny. Prace Matematyczne, 13(13): 259–270.

[58] Podlubny, I. 2001. Geometric and physical interpretation of fractional integra-

tion and fractional differentiation. Fractional Calculus and Applied Analysis,

(4): 367−386.

[59] Rida, S. Z., Arafa, A. A. M., and Gaber, Y. A. 2016. Solution of the fractional

epidimic model by L-ADM. Journal of Fractional Calculus and Applications,

7(1): 189–195.

[60] Rafei, M., Ganji, D. D., and Daniali, H. 2007. Solution of the epidemic model

by homotopy perturbation method. Applied Mathematics and Computation,

187(2): 1056–1062.

[61] Rafei, M., Daniali, H., and Ganji, D. D. 2007. Variational iteration method

for solving the epidemic model and the prey and predator problem. Applied

Mathematics and Computation, 186(2): 1701–1709.

152

[62] Ray, S. S., and Bera, R. K. 2004. Solution of an extraordinary differential equa-

tion by Adomian decomposition method. Journal of Applied Mathematics,

(4): 331–338.

[63] Ryss, A. Y., Kulinich, O. A., and Sutherland, J. R. 2011. Pine wilt disease: a

short review of worldwide research. Forestry Studies in China, 13(2): 132–

138.

[64] Samko, S. G., Kilbas, A. A., and Marichev, O. I. 1993. Fractional integrals and

derivatives. Theory and Applications, Gordon and Breach, Swetzerland, (44).

[65] Shah, K., Khalil, H., and Khan, R. A. 2016. Analytical solutions of fractional

order diffusion equations by natural transform method. Iranian Journal of

Science and Technology, Transactions A: Science, 1–12.

[66] Shah, K., and Khan, R. A. 2018. Iterative scheme for a coupled system of

fractional-order differential equations with three-point boundary conditions.

Mathematical Methods in the Applied Sciences, 41(3): 1047–1053.

[67] Shi, X., and Song, G. 2013. Analysis of the mathematical model for the spread

of pine wilt disease. Journal of Applied Mathematics.

[68] Singh, H., Dhar, J., Bhatti, H. S., and Chandok, S. 2016. An epidemic model

of childhood disease dynamics with maturation delay and latent period of

infection. Modeling Earth Systems and Environment, 2(2): 61–79.

[69] Van den Driessche, P., and Watmough, J. 2002. Reproduction numbers and

sub-threshold endemic equilibria for compartmental models of disease trans-

mission. Mathematical biosciences, 180(1-2): 29–48.

[70] Wang, Q. 2006. Numerical solutions for fractional KdV-Burgers equation by

Adomian decomposition method, Applied Mathematics and Computation,

182: 1048–1055.

153

[71] Yang, S., Xiao, A., and Su, H. 2010. Convergence of the variational iteration

method for solving multi-order fractional differential equations. Computers

and Mathematics with Applications, 60(10): 2871–2879.

existence theory and numerical solutions of the …

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