solving boundary value problems and delay differential equation by optimal homotopy...

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

ii

COMMITTEE DECISION

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





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


exact solution. 28

3.3 Figure : Comparison between the results obtained using forth-order OHAM


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)


. (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)


(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)


(3.2.29)

and

(3.2.30)

Thus, the values of the convergent control parameters are obtained in the following form

.


(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)


(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)


(3.2.38)


(3.2.39)


(3.2.40)

When applying eq. (3.1.8) for , the third-order problem is defined as:

24

(3.2.41)


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


26


(3.2.45)

and

(3.2.46)


The approximate solution (3.2.43) now becomes

(3.2.47)

27


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)


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:

, .


(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)


(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


the least square method can be applied as

3.2.60)

And

(3.2.61)


31


. (3.2.62)


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)


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:


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


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)


.

(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


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.

References

Alomari, A. K., Noorani, M. S. M. & Nazar, R. 2009. Solution of delay

48

differential equation by means of homotopy analysis method. Acta Applicandae

Mathematicae 108: 395–412.

Amore, P. & Fernndez, F. M. 2005. Exact and approximate expressions for the period

of anharmonic oscillators. European Journal of Physics 26: 589–601.

Anakira, N. Ratib, A. K. Alomari, and Ishak Hashim, Application of optimal homotopy

asymptotic method for solving linear delay differential equations, The 2013 UKM

FST Postgraduate Colloquium. AIP Conf. Proc. 1571,1013-1019 ( 2013).

Baker, C. H., Paul, C. A. H. & Will´e, D. R. 1995. Issues in the numerical

solution of evolutionary delay differential equations. Advances in Computational

Mathematics 3 (3): 171–196.

Bataineh, A. S., Noorani, M. S. M. & Hashim, I. 2008. Approximate solutions

of singular two-point BVPs by modified homotopy analysis method. Physics

Letters A 372: 4062–4066.

Busenberg, S. & Tang, B. 1994. Mathematical models of the early embryonic cell

cycle: the role of mpf activation and cyclin degradation. Journal of Mathematical

Biology 32: 573–596.

Campbell, D. K. 1992. Nolinear science: the next decade. New York Massachusetts:

MIT press: Massachusetts: MIT press.

Ebaid, A. 2011. A new analytical and numerical treatment for singular two-point

boundary value problems via the Adomian decomposition method. Journal of

Computational and Applied Mathematics 235: 1914–1924.

Ebaid, A. & Aljoufi, M. D. 2012. Exact solutions for a class of singular two-point

49

boundary value problems using Adomian decomposition method. Applied

Mathematical Sciences 6: 6097–6108.

Esmaeilpour, M. & Ganji, D. D. 2010. Solution of the Jeffery-Hamel flow problem

by optimal homotopy asymptotic method. Computers and Mathematics with

Applications 59: 3405–3411.

Evans, D. J. & Raslan, K. R. 2005. The Adomian decompostion method for solving

delay differential equation. International Journal of Computer Mathematics 82:

49–54.

Evans, D. J. & Raslan, K. R. 2005. The Adomian decompostion method for solving

delay differential equation. International Journal of Computer Mathematics 82:

49–54.

Ezzati, R., and K. Shakibi. "Using Adomian’s Decomposition and Multiquadric Quasi-

Interpolation Methods for Solving Newell–Whitehead Equation." Procedia Computer

Science 3 (2011): 1043-1048.

Glass, L. & Mackey, M. C. 1979. Pathological conditions resulting from instabilities in

physiological control systems. Annals of the New York Academy of Sciences 316:

214–235.

Golbabai, A., Fardi, M. & Sayevand, K. 2013. Application of the optimal homotopy

asymptotic method for solving a strongly nonlinear oscillatory system.

Mathematical and Computer Modelling 58: 1837–1843.

Haq, S. & Ishaq, M. 2012. Solution of strongly nonlinear ordinary differential equations

arising in heat transfer with optimal homotopy asymptotic method. International

Journal of heat and Mass Transfer 55: 5737–5743.

50

He, J. H. 1997. A new approach to nonlinear partial differential equations.

Communications in Nonlinear Science and Numerical Simulation 2: 230–235.

He, J. H. 1999a. Homotopy perturbation technique. Computer Methods in Applied

Mechanics and Engineering 178: 257–262.

He, J. H. 2000. A new perturbation technique which is also valid for large parameters.

Journal of Sound and Vibration 229: 1257–1263.

He, J. H. 2000b. Variational iteration method for autonomous ordinary differential

systems. Applied Mathematics and Computation 114: 115–123.

He, J. H. 2005. Homotopy perturbation method for bifurcation of nonlinear problems.

International Journal of Nonlinear Sciences and Numerical Simulation 6:

207–208.

Karakoc, F. & Bereketolu, H. 2009. Solutions of delay differential equations by using

differential transform method. International Journal of Computer Mathematics

86: 914–923.

Kanth, ASV Ravi, and Y. N. Reddy. "Cubic spline for a class of singular two-point

boundary value problems." Applied Mathematics and Computation 170.2 (2005):

733-740.

Khader, M. M. 2013. Numerical and theoretical treatment for solving linear and

nonlinear delay differential equations using variational iteration method. Arab

Journal of Mathematical Sciences 19: 243–256.

Khan, N., Mahmood, T. & Hashmi, M. S. 2013. OHAM solution for thin film flow of

a third order fluid through porous medium over an inclined plane. Heat Transfer

51

Research 44: 719–731.

Liao, S. J. 1997b. A kind of approximate solution technique which does not depend

upon small parameters–II. an application in fluid mechanics. International Journal

of Non–Linear Mechanics 32: 815–822.

Liao, S. J. 2003. Beyond perturbation: Introduction to the Homotopy Analysis

method .Florida: Chapmman & Hall /CRC Press, Boca Raton.

Lu, J. 2007a. Variational iteration method for solving two-point boundary value

problems. Journal of Computational and Applied Mathematics 207: 92–95.

Lu, J. 2007b. Variational iteration method for solving a nonlinear system of

second-orderboundary value problems. Computers and Mathematics with

Applications 54:1133–1138.

Liu, H., Xiao, A. & Su, L. 2013. Convergence of variational iteration method for

second-order delay differential equations. Journal of Applied Mathematics 2013:

Article ID 634670, 9 pages.

Lv, C.&Yuan, Z. 2009. Stability analysis of delay differential equation models of HIV-1

therapy for fighting a virus with another virus. Journal of Mathematical Analysis

and Applications 352: 672–683.

Mabood, F. E., Khan, W. A. & Ismail, A. I. 2014b. Solution of nonlinear boundary

layer equation for flat plate via optimal homotopy asymptotic method. Heat

Transfer-Asian Research 43: 197–203.

Marinca, V. & Herisanu, N. 2008. Application of optimal homotopy asymptotic

method for solving nonlinear equations arising in heat transfer. International

Communications in Heat and Mass Transfer 35: 710–715.

52

Marinca, V., Herisanu, N. & Nemes, L. 2008. Optimal homotopy asymptotic method

with application to thin film flow. Central European Journal of Physics 6:

648–653.

Marinca, V., Herisanu, N., Bota, C. & Marinca, B. 2009. An optimal homotopy

asymptotic method applied to the steady flow of a fourth-grade fluid past a

porous plate. Applied Mathematics Letters 22: 245–251.

Marinca, V. & Herisanu, N. 2014a. On the flow of a Walters-type B’ viscoelastic

fluid in a vertical channel with porous wall. International Journal of Heat and

MassTransfer 79: 146–165.

Martin, A. & Ruan, S. 2001. Predator-Prey models with delay and prey harvesting.

Journal of Mathematical Biology 43 (3): 247–267.

Mickens, R. E. 1996. Oscillations in Planar Dynamic Systems. World Scientific.

Nadeem, S., Mehmood, R. & Akbar, N. S. 2014. Optimized analytical solution for

oblique flow of a Casson-nano fluid with convective boundary conditions.

International Journal of Thermal Sciences 78: 90–100.

Nayfeh, A. H. 1985. Problemes in Pertubation. John Wiley and Sons.

Noor, M. A. & Mohyud-Din, S. T. 2007b. Variational iteration technique for solving

higher order boundary value problems. Applied Mathematics and Computation

189 (2): 1929–1942.

Rangkuti, Y. M. & Noorani, M. S. M. 2012. The exact solution of delay differential

equations using coupling variational iteration with taylor series and small term.

Bulletin of Mathemaatics 4: 1–15.

Raslan, K. R. & Sheer, Z. F. A. 2013. Comparison study between differential transform

53

method and Adomian decomposition method for some delay differential

equations. International Journal of Physical Sciences 8: 744–749.

Patel, N. K., Das, P. C. & Parbhu, S. S. 1982. Optimal control of systems described by

delay differential equations. International Journal of Control 36: 303–311.

Saeed, R. K. & Rahman, B. M. 2010. Adomian decomposition method for solving

system of delay differential equation. Australian Journal of Basic and Applied

Sciences 4: 3613–3621.

Sedaghat, S., Ordokhani, Y. & Dehghan, M. 2012. Numerical solution of the

delay differential equations of pantograph type via Chebyshev polynomials.

Communications in Nonlinear Science and Numerical Simulation 17:

4815–4830.

Shakeri, F.&Dehghan, M. 2008. Solution of delay differential equations via a homotopy

perturbation method. Mathematical and Computer Modelling 48: 486–498.

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. )

55

حل مسائل القيم الحدية والمعادالت التفاضلية المتأخرة بطريقة

الهوموتوبي ذات التقارب المثالي

عدادإ

مالك سليمان عواد

شرافإ

د. اسامه عبابنة

لمتطلبات الحصول على درجة الماجستير في الرياضيات لرسالةتمت هذه ا

كلية الدراسات العليا

جامعة الزرقاء

2016مايو ،

solving boundary value problems and delay differential equation by optimal homotopy...

Documents