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1
Solving Boundary Value Problems and Delay
Differential Equation by Optimal Homotopy
Asymptotic Method
By
Malik Soliman Awwad
Supervisor
Dr. Osama Ababneh
This Thesis was Submitted in Partial Fulfillment of The
Requirement for The Master's Degree in Mathematics
Faculty of Graduate Studies
Zarqa University
May, 2016
i
اإلهـــــــــــــداء
مــــــحيالر نمرحهللا ال م س ب
في نفسي سلي الطريق وغر ن أضاء ى م ـإل
روح التحدي والمثابرة لتحقيق الطموح
يــــأب
المكــان ...ة ـوغيم الحـــنان ..يمة إلى خ
زهرة المدائن إلىبهجة الدنيا ...وقرة العين ...
يـــأم
.هذا الجهد المتواضع د عائلتي الحبيبة ... أهدية أفراإلى بقي
iii
Contents
CHAPTER I INTRODUCTION
1.1 General Introduction 1
1.2 Problem Statement 3
1.3 Research Objective 4
1.4 Thesis Organization 4
CHAPTER II : HAM and OHAM
2.1 Introduction to HAM and OHAM 6
CHAPTER III OPTIMAL HOMOTOPY ASYMPTOTIC METHOD (OHAM)
3.1 Basic idea of OHAM 11
3.2 Numerical Examples 14
3.2.1: Linear Second-order Singular Two-point BVP 14
3.2.2: Second-order Singular Two-point BVP 18
3.2.3: Forth-order linear non-homogenous BVPs 22
3.2.4: higher-order Singular Four-point BVP 29
3.3 Summary 33
iv
CHAPTER IV OHAM FOR DELAY DIFFERANTIL EQUATION
4.1 Introduction 34
4.2 Delay Differential Equations 35
4.3 The basic idea of OHAM for delay differential equation 36
4.5 Numerical Examples 38
4.5.1: nonlinear DDE with unbounded delay 38
4.5.2: Linear DDE with second order 42
4.6 Summary 46
4.7 Conclusion 47
REFERENCES 49
v
LIST OF TABLES
3.1 Comparison of Solution for Example 1: 17
3.2 Comparison of Solution for Example 2: 21
3.3 Comparison of Solution for Example 3: 28
3.4 Comparison of Solution for Example 4: 32
4.1 Comparison of Solution for Example 1: 42
4.2 Comparison of Solution for Example2 : 46
vi
LIST OF ILLUSTRATIONS
3.1 Figure : Comparison between the results obtained using second-order OHAM
approximate analytic solution for Eq. (3.3.16) with the results obtained
numerically using spline with h=1/40 method. 22
3.2 Figure : Comparison between the results obtained using forth-order OHAM
approximate analytic solution for Eq. (3.3.32) with the results obtained
exact solution. 28
3.3 Figure : Comparison between the results obtained using forth-order OHAM
approximate analytic solution for Eq. (3.3.49) with the results obtained
exact solution. 33
vii
LIST OF SYMBOLS
Embedding parameter
Linear operator
Nonlinear operator
Nonzero auxiliary parameter
Nonzero auxiliary function
viii
LIST OF ABBREVIATIONS
ADM Adomian decomposition method
BVPs Boundary value problems
DDEs Delay differential equations
DTM Differential transformation method
Eq./Eqs equation/equations.
Fig./Fig figure/figures
HAM Homotopy analysis method
HPM Homotopy-perturbation method
ODEs Ordinary differential equations
OHAM Optimal homotopy asymptotic method
VIM Variational iteration method
ix
ACKNOWLEDGEMENT
I am truly thankful to Allah for His ultimate blessings and guidance at each
step of this work.
I offer my sincere thanks to my supervisor Dr. Osama Ababneh for his
motivating guidance, beneficial suggestions and support during my work.
I extend thanks to my father and mother, sister and brothers for their help
and support.
I would like to thank the examination committee members for their
cooperation.
x
ABSTRACT
Solving Boundary Value Problems and Delay Differential Equation by Optimal Homotopy
Asymptotic Method
By
Malik Soliman Awwad
Supervisor
Dr. Osama Ababneh
In this thesis, the optimal homotopy asymptotic method (OHAM) is applied to find
approximate solutions of singular two-point boundary value problems(BVPs), higher
order (BVPs) and delay differential equations, comparisons with exact solutions and
spline method were made. The results of equations studied using OHAM solutions were
significantly reliable.
1
CHAPTER I
1.1 GENERAL INTRODUCTION
There are many types of differential equation used for the modeling of real life
phenomena. For example, ordinary differential equation (ODEs) is used in the fields of
science and engineering, and in particular they are used at a large scale in classical
mechanics so as to model problems emerging in these fields. These differential
equations can be classified in many ways.One of those is the boundary value problem
(BVP) resulting from differential equations along with a set of additional restrictions.
Another type of differential equations where the derivative of the unknown function at a
certain time is given in terms of the values of the function at a previous time is called
delay differential equation (DDEs).Lately;(DDEs) have been used to investigate
biological models. In various fields of science, there are few phenomena occurring
linearly. Most problems are basically nonlinear and described by nonlinear differential
equations. If the problem is linear, the corresponding set of differential equations is also
linear and it can be solved without any mathematical difficulties. If the problem under
consideration is nonlinear, the obtained set of differential equation is in general
nonlinear. The way to solve the nonlinear equations has been a main focus, especially to
give analytical research expressions of them (Liao 1997a).
Many researchers have shown a great deal of interest in the approximate analytical
solution. One of the well-known techniques is the perturbation technique, which is used
to solve the nonlinear problems (Liao 2003). Nevertheless, all perturbation techniques
depend on the assumption that a small parameter must exist (Liao 1997b) because the
small parameter plays a significant role in the perturbation technique, and it does not
2
only determine the accuracy of the perturbation approximations, but also the validity of
the perturbation method. In fact, and based on the above argument, the small parameter
significantly restricts the application of the perturbation method. Moreover, many of the
nonlinear problems exist in the fields of science and engineering, and they do not
include any small parameters particularly the nonlinear problems of strong nonlinearity.
Therefore, it is highly important to develop and improve some nonlinear analytical
methods that do not rely on small parameters. Recently, Marinca et al.(2008,2009) and
Marinca and Herisanu (2008) have been the first to propose a type of approximate
analytic method which requires no small parameter.This method is known as the
optimal homotopy asymptotic method (OHAM) and it is used to obtain approximate
analytic solution of nonlinear problems of thin film flow of a fourth-grade fluid down a
vertical cylinder. In their work, this method was to understand the behavior of nonlinear
mechanical vibration of an electrical machine. The same method was also used by
Marinca et al. (2008,2009) and Marinca and Herisanu to obtain the nonlinear equations
solution, arising in the steady state flow of a fourth-grade fluid past a porous plate and
nonlinear equations solution, arising in heat transfer. In many research papers,the
effectiveness, generalization and reliability of this method were proved and solutions of
currently important applications in science and engineering were obtained by several
authors (Ali et al. 2010; Esmaeilpour & Ganji 2010; Golbabai et al. 2013; Haq & Ishaq
2012; Hashmi et al. 2012a, 2012b; Khan et al. 2013; Mabood et al. 2014b; Marinca &
Ene 2014a; Nadeem et al. 2014). Therefore, the OHAM can overcome the foregoing
restrictions and limitations of perturbation techniques due to the fact that OHAM
provides an efficient numerical solution of better accuracy compared with the
approximate analytical methods at the same order of approximation requiring minimal
calculation and avoidance of physically unrealistic assumption. Furthermore, the
3
OHAM very frequently provides an appropriate way that controls and adjusts the
convergence region of the series solution using the auxiliary convergence-control
function involving several optimal convergence-control parameters to ensure
a fast convergence of the solutions.
1.2Problem statement
The study of nonlinear problem is so important in areas of physics and
engineering. This is because most phenomena of the world are basically nonlinear
(Campbell 1992; Liao 2003) and described by nonlinear equations. Recently, the
approximate analytical methods and numerical ones have been applied to solve
nonlinear problems. After the develop of supercomputers, it is easy now to come up
with the solution of the linear problems. Nevertheless, it is very difficult to solve
nonlinear problems by numerical or analytical methods. Despite being subject to fast
development, nonlinear analysis techniques do not fully meet the demands of
mathematicians and engineers. To obtain an exact or approximate analytical solution for
linear and nonlinear differential equations, various methods have been applied like the
perturbation method (Amore & Fernndez 2005; He 200; 2002b; 200a; Mickens 1996;
Nayfeh 1985), the homotopy analysis method (Alomari et al. 2009), the modified
homotopy analysis method (Bataineh et al. 2008), the homotopy perturbation method
(He 1997; 1999a;2005), the variational iteration method (Khader 2013; Liu et al. 2013;
Lu 2007a, 2007b; Noor &Mohyud-Din 2007b; Rangkuti & Shakeri & Dehghan 2008)
and the Adomoan decompsiton method (Ebaid 2001; Ebaid & Aljoufi 2012; Evans &
Raslan 2005; Saeed & Rahman 2010) and so on. Some of these techniques apply
transformation to reduce the equations into more simple equations or even a system of
equations while some other techniques offer the solution in the form of series that
4
converges to the exact solution. Besides, some other techniques which employ a trial
function in an iterative scheme converging quickly. The concept of homotopy from
topology and conventional perturbation methods in HPM, HAM and OHAM were
combined to suggest a general analytic solution. Therefore, these techniques are
independent of the availability of a small parameter in the present problem and therefore
defeat the drawbacks of conventional perturbation methods. However, OHAM is the
most generalized kind of HAM and HPM because it uses an auxiliary function which is
more general. In the present work, a wide class of differential equation will be solved by
OHAM.
1.3 Research objectives
The objectives of this research are:
1. To present a general framework of the OHAM for solving singular two-point BVPs
and higher order of linear and nonlinear BVPs in order to determine the accuracy and
the effectiveness of OHAM.
2. To apply a new algorithm based on OHAM for finding exact or approximate analytic
solution of linear and nonlinearofDDEs.
1.4 THESIS ORGANIZATION
This thesis is organized and presented in three chapters. In the Chapter Ia
general introduction, problem statement and research objective are given, in addition the
thesis organization.
5
Chapter II explores the nature and utility of the basic idea of (HAM) and how it was
derived from the early ( HAM ), which was proposed by Liao in 1992 in his PHD
dissertation.
Chapter III this chapter contain the basic idea of (OHAM) and another development was
made by another researcher. Moreover, contain applications of OHAM for solving
singular two-point boundary value problem (BVPs) and higher order (BVPs).
In chapter IV, the OHAM is investigated to provide approximate solutions for delay
differential equations (DDEs), the accuracy of this procedure is tested through two
examples. Numerical comparison with exact solution were made .
6
CHAPTER II
MAH and OHAM
2.1 Basic Idea of HAM
In topology, two continuous functions from one topological space to another are
known as homotopic (greek homos = identical and topos = place) if one can be
continuously deformed into the other and such a deformation is known as a homotopy
between the two functions. Formally, a homotopy between two continuous functions f
and g from a topological space to a topological space is defined to be a continuous
function from the product of the space with the unit interval
to such that, for all points in and .
The theory of homotopy, that came up with application in differential geometry, can be
dated back to 1900 by Poincar'e work (Wu &Cheung 2009). With the emergence of
modern computers, the theory has been used for a class of numerical techniques to solve
nonlinear equations, i.e. using the homotopy method to solve finite difference
approximations to nonlinear two-point boundary value problems (Chow et al. 1978;
Watson 1980). They found zeros of maps for homotopy methods that are constructive
with probability one.
The fundamental idea of the homotopy solution aims at mapping an initial
approximation to the exact solution via a homotopy function compized an auxiliary
operator and an embedding parameter as . A series of problems are derived by
7
gradually varying the embedding parameter as and solving recursively, using an
iterative technique for a numerical solution.
The Optimal Homotopy Asymptotic Method (OHAM) was presented firstly by Marinca
et al. (2008, 2009) and Marinca and Herisanu (2008), aiming at solving nonlinear
problems without depending on a small parameter. It can be noted that the HAM and
HPM are special cases affiliated to OHAM. An advantages, of OHAM is that it does not
require the identification of the curve and it is also parameter free.
In OHAM, the control and adjustment of the convergence region are provided in a
convenient way. Moreover, the OHAM has been built on convergence criteria similar to
HAM but it differs from it in that its level of flexibility is greater than that of HAM
(Iqbal et al. 2010). This method is successfully applied by Marinca et al.(2008,2009)
and Marinca and Herisanu (2008,2010,2011) to problems in mechanics, and has also
shown its effectiveness and accuracy.
Recently, there has been a great deal of interest in OHAM. The method was
successfully applied to a large amount of equations which have applications in applied
sciences. The solution of stagnation point flows with heat transfer analysis, and Couette
and Poiseuille flows for fourth grade fluid which were obtained using OHAM by Shah
et al. (2010a,b). OHAM is used by Ullah et al.(2013,2014b) for the solution of boundary
layer problems with heat transfer. Moreover, they used it to obtain approximate solution
of the coupled Schrdinger-KdV equation. The technique was also used for the solution
of nonlinear Volterra integral equation of the first kind by Khan et al. (2014). Mabood
et al. (2014b) used OHAM to compute the solution of two-dimensional incompressible
laminar boundary layer flow over a flat plate (Blasius problem). Numerical solution of
the second order initial value problems of Bratu-type via optimal homotopy asymptotic
8
method is obtained by Darwish and Kashkari (2014). To be able to describe the basic
idea of OHAM, it is intended to describe some analytic techniques and its history of
development and modification in a brief manner.
Liao (1992) was the first one who suggested the early version of OHAM, which
appeared in his PhD dissertation. Consider the following nonlinear differential equation
(2.1.1)
where symbolizes a nonlinear operator and is an unknown function, Liao (1992)
used the concept of homotopy in topology to build up a one-parameter equation family
in the embedding parameters so-called the zeroth-order deformation
equation as follows:
(2.1.2)
Where refers to the linear operator and for an initial guess. At and
we have and respectively. So, as
embedding parameter increase from 0 to1, the solution zeroth-order
deformation equation (2.1.2) various form the initial guess to exact solution
of the original nonlinear equation (2.1.1), so that Eq. (2.1.2) is called the zeroth-order
deformation equation. Since the embedding parameter has no physical meaning, one
can construct such kind of zeroth-order deformation equation, no matter whether there
exist small or large parameters or not.
On the contrary, the early HAM presented above cannot provide a convenient way to
adjust the convergence region and rate of approximation series of nonlinear equations in
general. To overcome this limitation, a nonzero convergence-control parameter
9
was introduced by Liao (1997b) to construct such a two-parameter family of equations,
i.e. the zeroth-order deformation method
(2.1.3)
It is obvious that the corresponding homotopy series solution is not only depend on the
embedding parameter but also the convergence-control parameter . It is significantly
important to indicate that the convergence-control parameter ħ can adjust and control
the convergence region and rate of homotopy series solution. Actually, provides
appropriate way to guarantee the convergence of the homotopy series solution. Thus the
use of the embedding parameter is indeed a great progress; more degree of freedom
implies bigger possibility to obtain better approximations. Thus Liao (1999) made more
degree of freedom through the use of a zeroth-order deformation equation in more
general form:
(2.1.4)
where and are the so-called deformation function satisfying
(2.1.5)
Whose Taylor series
= ( )= (2.1.6)
are convergent . Thus, the generalized zeroth-order deformation equation (2.1.7)
gives us high level of freedom that is more possibility that help us ensure the
convergence of homotopy series solution. Actually, the zeroth-order deformation
equation (2.1.7) can be further generalized, as shown by Liao (2003, 2004).
10
Yabushita et al. (2007) suggested an optimization method. Yabushita et al based on the
squared residual of the governing equations to determine the optimal values of the
convergence control parameters in the frame of HAM.
, (2.1.7)
The nonlinear Eq. (3.1.1), where Gives the mth-order HAM solution.
In 2008, Akyildiz and Vajravelu (2008) suggested to use the optimal convergence
control parameter determined by the minimum of the squared residual of the governing
equation. Marinca et al. (2008, 2009) and Marinca and Herisanu (2008) suggested the
so-called optimal homotopy asymptotic method based on the homotopy equation
(2.1.8)
where the optimal value of , ( =1,2,3, …) is determined by the minimum of squared
residual of governing equations.
Let
(2.1.9)
Denote the squared residual of governing equation (2.1.1) at the mth-order of
approximation. Then, one has to solve a set of nonlinear algebraic equations
(2.1.10)
so as to obtain the mth-order approximation.
11
CHAPTER III
OPTIMAL HOMOTOPY ASYMPTOTIC METHOD
3.1 BASIC IDEA OF OHAM
We review the basic principles of OHAM as expounded by Ghoreishi et al. (2011);
Idrees et al. (2012) and Marinca and Herisanu (2008).
Consider the following differential equation and boundary condition
(3.1.1)
Where is the linear operator, nonlinear operator, is an unknown function,
denotes an independent variable, is a known function and is a boundary
operator.An equation known as a deformation equation is constructed
(3.1.2)
where is an embedding parameter, is a nonzero auxiliary function for
and is an unknown function. For and it holds that
and respectively.
Hence, as varies from to the solution varies from to the solution
where, is obtained from (3.1.2) for .
12
, ( . (3.1.3)
the auxiliary function is choosing in the form
(3.1.4)
where , , , ... are the convergent control parameters which can be determined
later.
For solution is expanded in Taylor’s series about and given:
(3.1.5)
Substituting (3.1.4) and (3.1.5) into (3.1.2) and equating the coefficients of the like
powers of equal to zero, gives the linear equations as described below:
The zeroth order problem is given by (3.1.3) and the firstand second order problems are
given by the (3.1.6) and (3.1.7),respectively:
+ = (3.1.6)
(3.1.7)
The general governing equations for are given by
,B( (3.1.8)
13
Where , and is the coefficient of in the
expansion of about the embedding parameter q.
(3.1.9)
has been observed by previous researchers that the convergence of the series (6) is
dependent upon the auxiliary constants , … . If it is convergent at , one
has
Substituting (3.1.10) into (3.1.2), the general problem, resultsin the following residual
g(x) . (3.1.11)
If , then will be the exact solution. For nonlinear problems, generally this will
not be the case.
For determining , a and b are chosen such that the optimum values for
are obtained using the method of least squares
(3.1.12)
where g(x) . is the residual and
(3.1.13)
With these constants, one can get the approximate solution of order
3.2 NUMERICAL EXAMPLES
14
3.2.1 Example1: Consider the second order homotopy linear BVPs equation taken from
Kanth, Ravi and Reddy (2005).
(3.2.1)
The exact solution of this problem in the case is given by
To use the basic idea of OHAM formulated and according to eq. (3.1.1), we define the
linear and nonlinear operators in the following form
.
(3.2.3)
Now, apply eq. (3.1.3) when , it gives the zeroth-order problem as follow:
. (3.2.4)
The solution of eq. (2.3.4) is given by
. (3.2.5)
From eq. (3.2.6), the first-order problem is
(3.2.6)
This has the following solution
15
(3.2.7)
According to eq. (3.2.7), the second-order problem is
th BCs(3.2.8)wi
And has the solution
(3.2.9)
By applying equation (3.2.8) for = 3, the third-order problem is defined as:
(3.2.10)
And has the following solution
(3.2.11) Substituting eq. (3.2.5),
(3.2.7), (3.2.9) and (3.2.11) yields the third-order OHAM approximation solution for
( ) for eq. (3.2.1)
(3.2.12)now, on the domain between and , we use the method of least
squares to obtain the unknown convergent constant in Eq. (3.2.12)
16
(3.2.13)
The least square method can be applied as
(3.2.14)
Thus, the values of the convergent control parameters are obtain in the following form
The approximate solution (3.2.12) now become
(3.2.15)
Values of form cod of mathematica
Table 3.1: comparison between the OHAM solution and spline solution together with the
exact solution for example 1
17
From this table it can be seen that the result obtained by using three order OHAM
solutions is nearly identity to the exact solution.
3.2.2 Example 2: Consider the second order linear non-homogenous BVPs form Kanth,
Ravi and Reddy (2005).
18
(3.2.16)
The exact solution of this problem in this case is given by
. (3.2.17)
According to eq. (3.1.1), the linear and nonlinear operators are defined as follow:
(3.2.18)
order problem as follow:-, it gives the zeros.3) when 3.1Now, apply eq. (
(3.2.19)
The solution of eq. (3.2.19) is given by
. (3.2.20)
From eq. (3.1.6), the first-order problem is
(3.2.21)
And has the solution
(3.2.22)
From eq. (3.1.7), the second-order problem is
19
and has the solution
(3.2.24)
when , and by applying eq. (3.1.8), the third-order problem become
(3.2.25)
and has the solution
(3.2.26)
By substitution these values of the convergent control parameters in equation (3.2.27),
the third-order approximation become
20
(3.2.27)
Now, on the domain between and , we use the method of least squares to
obtain the unknown convergent constant in Eq.(3.2.27)
(3.2.28)
The least square method can be applied as
(3.2.29)
and
(3.2.30)
Thus, the values of the convergent control parameters are obtained in the following form
.
The approximate solution (3.2.27) now become
(3.2.31)
Table 3.2 : exact and approximate solution using OHAM for example 2
21
0.700
0.900
1.00
Figure 3.1:Exact and approximate solution using OHAM for example 2
Table 3.2 and Fig.3.1 cite a comparison between the approximate solution obtained by
three-order OHAM approximate solution and spline solution. From this comparison one
can see a good agreement between the exact solution and the OHAM solution. Moreover,
22
the absolute error between the approximate and the exact solution presented which proves
the accuracy of the method.
3.2.3 Example 3: Consider the following forth order linear non-homogenous BVPs.
With BCs, . (3.2.32)
The exact solution of this problem in the case is given by
(3.2.33)
To use the basic ideas of OHAM formulated in chapter 3 and according to eq. (3.1.1), we
define the linear and nonlinear operators in the following form
(3.2.34)
Now, apply eq.(3.2.32) when , it gives the zeroth-order problem as follow:
(3.2.35)
The solution of eq. (3.2.35) is given by (3.2.36)
(2.3.36)
From eq.(3.1.6), the first-order problem is
23
(3.2.37)
and has the solution
(3.2.38)
From eq. (3.1.7), the second-order problem is
(3.2.39)
and has the solution
(3.2.40)
When applying eq. (3.1.8) for , the third-order problem is defined as:
25
(3.2.42)
by substation these values of the convergent control parameters in equation (3.2.43), the
third-order approximation becomes
Now. on the domain between and , we use the method of least squares to
obtain the unknown convergent constant in Eq.(3.2.43)
26
The least square method can be applied as
(3.2.45)
and
(3.2.46)
Thus, the values of the convergent control parameters are obtained in the following form
The approximate solution (3.2.43) now becomes
(3.2.47)
27
Table 3.3 : exact and approximate solution using OHAM for example 3
solution
Figure 3. 2: exact and approximate solution using OHAM for example 3
28
The obtained result with are presented and displayed in Table 3.3 and Fig.3.2
demonstrate that this method provides highly accurate solution with reportable low
error.
3.2.4 Example 4: Consider the following forth order linear BVPs example:
,
(3.2.48)
The exact solution of this problem in the case is given by
To use the basic ideas of OHAM formulated in chapter 3 and according to eq. (3.1.1), we
define the linear and nonlinear operators in the following form
(3.2.50)
Now, apply eq.(3.2.47) when , it gives the zeroth-order problem as follow:
, .
The solution of eq. (3.1.6) is given by
(3.1.51)
29
From eq. (3.2.6) the first-order problem is
(3.2.52)
which has the following solution
(3.2.53)
From eq. (3.1.7), the second-order problem is
(3.2.54)
And has the following solution
When k=3, and by applying eq. (3.1.8), the third-order problem become
With BCs, (3.1.56)
The solution of equation (3.2.56) is given below
30
.
(3.2.57)
By substitution these values of the convergent control parameters in eq. (3.2.58)
third order approximation become
(3.2.58)
Now, on the domain between and , we use the method of least squares to
obtain the unknown convergent constant in Eq.(3.2.58)
the least square method can be applied as
3.2.60)
And
(3.2.61)
Thus, the values of the convergent control parameters are obtained in the following form
31
The approximate solution (3.2.58) now become
. (3.2.62)
Table 3.4 : exact and approximate solution using OHAM for example 4
32
Exact
OHAM
1.0 0.5 0.0 0.5 1.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
x
u
Figure 3.3 : exact and approximate solution using OHAM for example 4
Example 4 is regulated in Table 3.4 and Fig.3.3 which show high accuracy of OHAM,
thatproves and demonstrate the capability and reliability of the OHAM.
3.3SUMMARY
OHAM has been applied successfully to obtain approximate analytical solution of
singular boundary value problem and higher order boundary value problem. The
practicality and effectively of OHAM have been illustrated through various examples.
This shows that the method is efficient and reliable from singular two points boundary
value problems and higher-order boundary value problem.
33
CHAPTER IV
OHAM FOR DELAY DIFFERENTIAL EQUATION
4.1 INTRODUCTION
Delay differential equation (DDEs) is a kind of differential equation used for modeling
many real-life phenomena in science and engineering. A lot of problems in the fields of
physics, biological modeled, control system, medical and biochemical can be modeled
by DDEs. Modern studies in various fields have shown that DDEs play an important
role in explaining many different phenomena. For example, in physiology, Glass and
Mackey (1979) applied time delays to many physiological models. Patel et al. (1982)
introduced an iterative scheme for the optimal control systems described by DDEs with
a quadratic cost functional. Busenberg and Tang (1994) created model for cell cycle by
delay equations. In recent years, Lv and Yuan (2009) used DDEs to design models as
HIV-1 therapy where a virus fights another virus. In engineering, pure delays are often
used to investigate the effects of transmission, transportation, and inertial phenomena.
In biology, they can be used to model gestation, maturation, transcription, and
numerous cell-cycle phenomena.
Over the past years, numerous researchers paid great attention to the studying of DDEs.
Therefore, they solved them by numerical methods and approximation approaches. for
example, Evans and Raslan (2005).used the ADM to compute an approximation to the
solution of the DDEs. Shaki and Dehghan (2008) used the HPM to obtain approximate
solution for this initial value problem of DDEs Alomari et al. (2009) obtain the
algorithm of approximate analytical solution to find exact or approximate solution for
the linear , nonlinear and system of DDE via HAM. Karakoc and Bereketolu (2009)
34
Presented DTM for solving delay differential equation. Sedaghat et al. (2012).Proposed
a numerical scheme using shifted chebshev polynomials to solve the DDEs of
pantograph type.Rangkuti and Noorani (2012) employed the VIM to find the exact
solution of DDEs either Taylor series .Raslan and Sheer (2013) proposed numerical
methods based on the DTM and ADM for the approximate solution of DDEs .Khader
(2013)used the VIM for solving linear and nonlinear DDEs Systems of DDEs now play
a fundamental role in all fields of science especially in the biological science (e.g.,
population dynamics and epidemiology).Baker et al. (1995) contains reference for
several application areas. The manner in which the properties of systems of DDEs differ
from those of systems of ODEs has been an active area of research by martin & Ruan
(2001). Comparison of the framework with the exact one is made. In this chapter, the
applicability of DDEs, General framework of the OHAM solution was given. Various
examples of linear, nonlinear and system of initial value problem of DDEs presented to
demonstrate the efficiency and the capability of the framework.
4.2 DELAY DIFFERENTIAL EQUATIONS
In mathematics, a delay differential equation (DDEs) isa type of differential
equations in which the derivative of the unknown function at a certain time is given in
terms of the values at an earlier time . In this thesis, delay differential equations are
considered, DDEs in the form:
(4.2.1)
Where is the delay function, n and I (n denoted the orders of the
derivative.
35
4.3 The basic idea of OHAM for delay differential equation
To simplify and illustrate the fundamental idea of OHAM for DDE (Anakira et al.
2015), the following differential equation is considered,
. (4.3.1)
Where is linear operators and is nonlinear operators containing delay function,
is unknown functions, denotes an independent variable, is known functions
and is the delay functions.
According to OHAM, a homotopy map can be constructed
which satisfies
(4.3.2)
where is an embedding parameter, is a nonzero auxiliary function
for , and is an unknown function. Obviously, when and
it holds that and respectively. Thus, as varies
from 0 to 1, the solution approach from to where is the initial guess
that satisfies the linear operator and the initial conditions
(4.3.3)
Next, we choose the auxiliary function in the form
(4.3.4)
where are constants which can be determined later.
36
To get an approximate solution, expand in Taylor's series about in the
following manner,
(4.3.5)
By substituting (4.3.5) into (4.3.3)and equating the coefficient of similar powers of ,
we obtain the following linear equations. Define the vectors
Where The zeroth-order problem is given by (4.3.3), the first and second-
order problems are given as
(4.3.6)
(4.3.7)
The general governing equations for are given as
(4.3.8)
Where and is the coefficient of in the
expansion of about the embedding parameter .
(4.3.9)
37
It has been observed that the convergence of the series (4.3.5) depends on the auxiliary
constant If the series convergent at then
(4.3.10)
The result of the th-order approximation is given as
(4.3.11)
Substituting (4.3.10) into (4.3.1) yields the following residue
(4.3.12)
If then will be the exact solution. Generally such a case will not arise for
nonlinear problems, but we can minimize the functional
(4.3.13)
where and are endpoints of the given problems. The unknown convergence control
parameters can be calculated from the system of equations
(4.3.14)
With these constants are known, the approximate solution of order is well determined.
4.5 NUMERICAL EXAMPLES
In this part, two examples with a known exact solution are presented in order to
demonstrate the accuracy and effectiveness of this algorithm.
38
4.5.1 Example 1: Consider the following nonlinear delay differential equation with
unbounded delay studied by Karakoc and Bereketoglu (2009).
(4.5.1)
With the exact solution
(4.5.2)
According to Eq. (3.3.1), the linear operator is considerd as
. (4.5.3)
and the nonlinear operator as
Now, apply Eq. (4.3.3) When to give the zeroth-order problem as :
(4.5.5)
The solution of the zeroth order problem is:
(4.5.6)
The first-order problem which is obtained from Eq. (4.3.6) is given as :
(4.5.7)
and has the solution
39
The second-order deformation is given by Eq. (4.3.7)
(4.5.9)
The solution of equation (4.5.9) is given by :
(4.5.10)
The third-order problem is obtained from (4.5.8) when as:
and has the solution
using Eqs. (4.5.6), (4.5.8), (4.5.10) and (4.5.12) into Eq. (4.3.11) for , the third-
order OHAM approximate solution for Eq. (4.5.1) is given as follow
(4.5.13)
40
Substituting the OHAM approximate solution of the third-order (4.5.13) into Eq.
(4.3.12) yields the residual
(4.5.14)
The leastsquare method can be formed as
and
. (4.5.16)
Thus, the following optimal values of the convergent control parameters are
obtained as follow:
.
by substituting these values in Eq. (4.5.13), the OHAM approximate solution of third-
order is obtained in the form
41
Table 4.1 : exact and approximate solution using OHAM for example 1
Absolute error |exact-OHAM| solution
Example 4.5.2Consider the following linear delay differential equation studied by
Karakoc and Bereketoglu (2009).
With the exact solution
42
According to Eq. (4.3.1), the linear and nonlinear operatorsare defined as follows
Now, apply Eq. (4.3.3) When to give the zeroth-order problem as :
(4.5.21)
The solution of the zeroth order problem is:
By applying Eq. (4.3.6), the first-order problem is giving in the following form
which has the solution
According to Eq. (4.3.7), the second-or1der problem is giving in the following form
(4.5.25)
The solution of equation (4.5.9) is given by:
43
The third-order problem is obtained from (4.3.8) when as:
(4.5.27)
and has the solution
.
(4.5.28)
Using Eqs. (4.5.22), (4.5.24), (4.5.26) and (4.5.28) into Eq. (4.3.11) for , the third-
order OHAM approximate solution of the third-order (3.5.13) into Eq. (3.5.1)
Substituting the OHAM approximate solution of the third-order (4.5.13) into Eq.
(4.3.12) yields the residual
(4.5.30)
The leastsquare method can be formed as
and
44
Thus, the following optimal values of the convergent control parameters are
obtained as follow:
.
by considering these values in Eq. (4.5.13), the OHAM approximate solution of third-
order is obtained in the form
(4.5.34)
45
Table 4.2 : exact and approximate solution using OHAM for example 2
solution
The numerical result obtained by OHAM approximation are summarized in Table 4.1
and Table 4.2. These results show the high accuracy of the approximate solutions
obtained by OHAM.
4.6 SUMMARY
In this chapter, OHAM is successfully employed in order to obtain an approximate
solution for DDE. This procedure was tested in two examples and was seen to produce a
satisfactory result. The OHAM solution has a good agreement with the exact solution,
which indicates that OHAM is efficient and feasible method for DDE.
46
4.7 CONCLUTION
CONCLUSIONS
The work presented on this thesis focused largely on solving ODEs by using OHAM.
The OHAM is a relatively new procedure to provide approximate analytical solution to
linear and nonlinear problems. The OHAM could be considered as one of the new
techniques affiliated to the general classification of perturbation methods. The OHAM
revolves around exact solvers for linear differential equations and approximate solvers
for nonlinear equations. It is a useful tool for scientists and applied mathematicians,
because it gives instant and obvious symbolic terms of analytic solutions, as well as
numerical solutions to both linear and nonlinear differential equations without
linearization or discretization. The effectiveness and validity of our procedure, which
does not imply the presence of a small parameter in the equation, depends on the
construction and determination of the auxiliary functionH(q), combined with an
appropriate way to optimally control the convergence of the series solution throughout
several convergent control parameters Ci’s, which are optimally determined such that
H(0) = 0 and H(q) 0 for q 0. When q increases from 0 to 1, the solution Φ(x, q)
changes from the initial approximation (x) to the solution u(x).
Chapter I contains the introduction to the topic of study with present status of the
problem and a brief-chapter wise summary.
Chapter II contains the historical background was introduced.
Chapter III, the basic idea of OHAM with applied successfully to solve singular two
point -BVPs and higher-order BVPs. Comparisons between the exact solution and
spline solution reveal that the OHAM is very effective and convenient.
47
In Chapter IV the OHAM solution for delay differential equations (DDEs) is presented.
Two examples of linear initial value problems of DDEs are considered to show and
demonstrate the efficiency and power of this procedure that enable us to find accurate
and approximate solutions for wide classes of DDEs.
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حل مسائل القيم الحدية والمعادالت التفاضلية المتأخرة بطريقة الهوموتوبي
ذات التقارب المثالي
54
عدادإ
مالك سليمان عواد
شرافإ
د. اسامه عبابنة
الملخص
المعادالت التفاضلية الخطية والغير خطية تستخدم لوصف مجموعة كبيرة من الظواهر الطبيعية
ة والهندسة, ومن خالل هذه الظواهر, شكلة الظواهر الغير ئوعلوم الحياة وعلوم األرض والبي
اهر الطبيعية هي غير خطية لذلك اخذت مساحة وم الظخطية األهتمام األكبر خصوصا أن معظ
واسعه جدا من البحث والدراسة خالل السنوات األخيرة الماضية, ومن أهم الطرق التي وجدت
( التي لعبة دور كبير في حل المعادالت الطبيعية OHAMلحل مثل هذه المعادالت هي طريقة )
الخطية والغير خطية.
( لحل دقيق لبعض المعاالت مثل معادلة OHAMوفي هذه الرسالة قمنا باستخدام طريقة ال )
(bessel. )