EXISTENCE THEORY AND NUMERICAL SOLUTIONS OF THE …
Transcript of 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
HAZARA UNIVERSITY MANSHERA
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
HAZARA UNIVERSITY MANSHERA
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
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My dear parents
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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
5.1.1 The Laplace Adomian decomposition method . . . . . . . . . 94
5.1.2 Numerical results and discussion . . . . . . . . . . . . . . . . . 98
5.1.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . 101
vi
5.2 Numerical Solution of Fractional Order Model of HIV-1 Infection of
CD4+ T-Cells by using LADM . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.1 The Laplace Adomian decomposition method . . . . . . . . . 103
5.2.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Numerical Analysis of Fractional Order Enzyme Kinetics Model by
Laplace Adomian Decomposition Method . . . . . . . . . . . . . . . . 109
5.3.1 The Laplace Adomian decomposition method . . . . . . . . . 110
5.3.2 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . 113
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
5.4.1 The Laplace Adomian decomposition method . . . . . . . . . 118
5.4.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . 129
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
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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
6.2 Numerical plots of approximate solutions of Example (6.2.2) at var-
ious fractional order and using t = 10. . . . . . . . . . . . . . . . . . . 139
6.3 Numerical plots of approximate solutions of Example (6.2.3) at var-
ious fractional order and using k = 0.1, t = 10. . . . . . . . . . . . . . 141
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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
[
γIε−py(ξ)− Iε−py(1)]
.
(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)
[
γIε−py(ξ)− Iε−py(1)]
=∫ 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
33
following inequalities
|u∗(t)−∫ 1
0G(t, s) f (s, u∗(s), c
Dε−1u∗(s))ds| ≤ ǫ,
there exists a solution (x∗) of (3.1.7)such that
|u∗(t)− x∗(t)| ≤ c f ǫ,
where c f =1
1−G∗kd , 1 6= G∗kd.
Proof. In view of theorem 3.1.2,
let for a unique solution u∗ ∈ c(J, R) of (3.1.7) , we have for each ǫ > 0, the in-
equality
|u∗(t)−∫ 1
0G(t, s) f (s, u∗(s), c
Dε−1u∗(s))ds| ≤ ǫ.
To show that there exists a solution x∗(t) of (3.1.7)such that
|u∗(t)− x∗(t)| ≤ c f ǫ.
Consider
|u∗(t)− x∗(t)| =|Tu∗(t)− Tx∗(t)|
=|∫ 1
0G(t, s) ( f (s, u∗(s), c
Dεu∗(s))− f (s, x∗(s),c D
εx∗(s))) |ds
≤|∫ 1
0G(t, s) ( f (s, u∗(s), c
Dεu∗(s))− f (s, x∗(s),c D
εx∗(s))) |ds
+ |u∗(t)−∫ 1
0G(t, s) f (s, u∗(s), c
Dεu∗(s))|ds
≤ε +∫ 1
0G(t, s)[k[|u∗
s − x∗s |+ |cD εu∗(s)−cD
εx∗(s)|]]ds, as G∗kd > 0,
≤ε + G∗kd‖u∗ − x∗‖which implies that ‖u∗ − x∗‖ ≤ ǫ
1 − G∗kd= εc f .
Therefore, (3.1.7) is Hyers–Ulam stable.
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
0G2(t, s)g(s, u(s),c D
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
0G2(t, s)g(s, u(s),c D
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
0G2(t, s)g(s, u(s),c D
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)
41
Now, we have to prove that T : W → W is completely continuous operator. Con-
sider
|T1v(t)| =|∫ 1
0G1(t, s) f (s, v(s),c D
µv(s))ds|
≤∫ 1
0|G1(t, s)m(s)|ds + (d1Rρ1 + d2Rρ2)
∫ 1
0|G1(t, s)|ds
×∫ 1
0|G1(t, s)m(s)|ds + (d1Rρ1 + d2Rρ2)
(
t2
|1 − α2|
)
[∫ 1
0
(1 − s)p−1
Γ(p)ds
+∫ α
0
(α − s)p−1
Γ(p)ds] + (d1Rρ1 + d2Rρ2)
∫ t
0
(t − s)p−1
Γ(p)ds, as using t ≤ 1
≤∫ 1
0|G1(t, s)m(s)|ds + 3(d1Rρ1 + d2Rρ2)[
1 + αp
|1 − α2|Γ(p + 1)+
1
Γ(p + 1)]
≤∫ 1
0|G1(t, s)m(s)|ds + 3A(d1Rρ1 + d2Rρ2). (3.2.16)
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,
44
then we have
|T1v(t)− T1v(τ)| =|u(t)− u(τ)|
=
∣
∣
∣
∣
∣
∣
1∫
0
(G1(t, s)− G1(τ, s)) f (s, v(s),c Dνv(s))ds
∣
∣
∣
∣
∣
∣
≤M
1∫
0
|G1(t, s)− G1(τ, s)|ds
≤M
(
t2 − τ2
|1 − α2|
)
1∫
0
(1 − s)p−1
Γ(p)ds −
α∫
0
(α − s)p−1
Γ(p)ds
+
1
Γ(p)
t∫
0
(t − s)p−1 − (τ − s)p−1ds −τ∫
t
(τ − s)p−1ds
≤ M
Γ(p + 1)
[
t2 − τ2
|1 − α2| + tp − τp − (τ − t)p
]
. (3.2.21)
Furthermore
|cDνT1v(t)−cD
νT1v(τ)|
≤ M
[
2(tp−ν − τp−ν)
Γ(3 − ν)|1 − α2|Γ(p + 1)+
tp−ν − τp−ν − (τ − t)p−ν
Γ(p − ν + 1)
]
.(3.2.22)
Similarly
|T2u(t)− T2u(τ)| ≤ N
Γ(q + 1)
[
t2 − τ2
|1 − β2| + tq − τq − (τ − t)q
]
(3.2.23)
and
|cDµT2u(t)−cD
µT2u(τ)|
≤ N
Γ(q − µ + 1)
[
2(tv−µ − τv−µ) + Γ(3 − v)|1 − β2| (tq−µ − τq−µ − (τ − t)q−µ)
Γ(3 − µ)|1 − β2|
]
.
(3.2.24)
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
trees population at the given various initial values.
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
trees population at the given various initial values.
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
trees population at the given various initial values.
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
trees population at the given various initial values.
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.
5.1.1 The Laplace Adomian decomposition method
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)
5.1.2 Numerical results and discussion
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
different fractional values of αk for k = 1, 2, 3..
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
different fractional values of αk for k = 1, 2, 3..
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.
5.1.3 Convergence analysis
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
5.2.1 The Laplace Adomian decomposition method
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 .
5.2.3 Convergence analysis
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
5.3.1 The Laplace Adomian decomposition method
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.
5.3.2 Convergence analysis
For the convergence analysis of the enzyme kinetic model see theorem (4.2.1).
5.3.3 Numerical results and discussion
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.
5.4.1 The Laplace Adomian decomposition method
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.
5.4.3 Convergence analysis
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],
Dαt v(w, z, t) = (D2
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]
Dαt v(w, z, t) = (D2
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.
Dαt v(w, z, t) = (D2
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.
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