## ABSTRACT

In this thesis, we have developed two new iterative schemes for solving nonlinear equations.

The two schemes have been constructed from Taylor’s series expansion and Adomian

decomposition method. The two schemes have been compared with other existing iterative

methods using one way analysis of variance (ANOVA). They are found to be efficient and

better than some of the existing schemes. The results show that Newton-Raphson method and

New scheme 1 have more advantage with a maximum of seven iterations each, while new

scheme 2 has nine. Basto et al. and Abbasbany have equal number of thirteen iterations each.

The Adomian has sixteen iterations. Thirty numerical examples are given and solved to justify

the efficiency of the new iterative schemes.

## TABLE OF CONTENTS

COVER PAGE

FLY PAGE

TITLE PAGE

DECLARATION…………………………………………………………………………………………………….. i

CERTIFICATION ………………………………………………………………………………………………….. ii

DEDICATION ……………………………………………………………………………………………………… iii

ACKNOWLEDGEMENT ………………………………………………………………………………………. iv

ABSTRACT ……………………………………………………………………………………………………………v

TABLE OF CONTENTS……………………………………………………………………………………….. vi

CHAPTER ONE : GENERAL INTRODUCTION …………………………………………………….1

1.1 Introduction ……………………………………………………………………………………………………….1

1.2 Motivation for this study ………………………………………………………………………………………5

1.3 Problem studied in the thesis …………………………………………………………………………………6

1.4 Aim and objectives ……………………………………………………………………………………………..6

1.5 Limitation of the study …………………………………………………………………………………………7

1.6 Definitions of term ………………………………………………………………………………………………7

1.7 Theorems used in the study …………………………………………………………………………………..8

1.8 Outline of the thesis …………………………………………………………………………………………….9

CHAPTER TWO : LITERATURE REVIEW ………………………………………………………… 10

2.1 Introduction …………………………………………………………………………………………………….. 10

2.2 Generalizations of Newton’s method …………………………………………………………………… 11

2.3 The Adomian decomposition method …………………………………………………………………… 14

2.4 Studies based on Adomian decomposition method …………………………………………………. 16

2.5 Studies based on Homotopy perturbation method…………………………………………………… 19

CHAPTER THREE : CONSTRUCTION OF THE NEW SCHEMES ……………………… 22

3.1 Introduction …………………………………………………………………………………………………….. 22

3.2 The present work ……………………………………………………………………………………………… 23

3.3 Construction of the new schemes ………………………………………………………………………… 26

3.3.1 New iterative scheme 1 …………………………………………………………………………………… 26

3.3.1.1 Convergence analysis for new the scheme 1 …………………………………………………….. 29

3.3.2 New iterative scheme 2 …………………………………………………………………………………… 34

vii

3.3.2.1 Convergence analysis for new scheme 2 …………………………………………………………. 36

CHAPTER FOUR : ANALYSIS OF RESULTS ……………………………………………………… 41

4.1Introduction ……………………………………………………………………………………………………… 41

4.2 Thirty Examples of Different Nature ……………………………………………………………………. 42

Table 4.1Comparison between Number of Iterations for Thirty Different Examples …………. 43

4.3 Summary of Results Obtained for Some Solved Examples ………………………………………. 44

4.4 Results obtained from ANOVA ………………………………………………………………………….. 47

CHAPTER FIVE : ……………………………………………………………………………………………….. 51

SUMMARY, CONCLUSION AND RECOMMENDATIONS ………………………………….. 51

5.1 Summary ………………………………………………………………………………………………………… 51

5.2 Conclusion………………………………………………………………………………………………………. 52

5.3 Recommendations ……………………………………………………………………………………………. 52

References ……………………………………………………………………………………………………………. 54

Appendix ……………………………………………………………………………………………………………… 57

1

## CHAPTER ONE

GENERAL INTRODUCTION

1.1 Introduction

Solving nonlinear equations is an important part of numerical analysis. In recent years,

interests have grown considerably in developing effective iterative methods (IM) for

computing solutions for large systems in science and engineering. The development of faster

and more robust IM and preconditions which can be efficiently mapped to a variety of

problems is of fundamental importance in that it will be of great assistance to scientists and

engineers throughout many disciplines. In numerical analysis this approach is in contrast

to direct methods which attempt to solve the problem by a finite sequence of operations.

Numerical analysis assumes this task, and with it the limitations of practical calculations.

Numerical answers are usually tentative and, at best, known to be accurate only to within

certain bounds. IM are often useful even for linear problems involving a large number of

variables, where direct methods would be prohibitively expensive. Intuitively iterative

methods keep on improving upon subsequent iterations. With iteration methods, the

operational cost can often be reduced.

In this thesis, we study iterative methods for solving nonlinear equations, f x 0, where

f x is any continuously differentiable real valued function. The iterative methods we try to

develop for this class of equations will require knowledge of initial guess for desired roots of

the equation.

2

Traub (1964), Numerical methods for solving nonlinear equations are divided into two

categories; the interval methods and the initial point methods.. Bisection method is an

example of interval methods. The initial point methods use one or more initial points as the

starting values to find the approximate solution using recurrence relation. In this study, we

concentrate mainly on one point initial methods. The major disadvantage of these methods

however, is that their convergences are not guaranteed and the choice of initial guess requires

some insight. These methods are however, usually faster than the interval methods. Secant

method and Newton method are examples of initial point methods. There are several methods

for solving nonlinear equations and here we introduce a few of them.

Newton or Newton-Raphson method is the most widely used method for finding roots of an

equation. According to Traub (1964), it begins with an initial approximation, 0 x and

generates a sequence of successive iterates

k k0 x converging quadratically to simple roots.

In Secant Method, which is a variant of Newton-Raphson’s method it use finite difference to

approximate the derivative of the function y = f (x) close to the root by the line (secant) and

requires two initial points 1 1 , n n x f and n n x , f , where n n f f x .Taking the point of

intersection of this line with the x-axis as the subsequent iterate. We get

, 1,2

1

1

1

f n

f f

x x x n

n n

n n

n where xn1 and xn are two consecutive iterates. Since a secant

line is defined using two points on the graph of f (x), as opposed to a tangent line that requires

information at only one point on the graph, we need two initial approximations 0 x and 1 x .

Traub (1964), the method has a super linear convergence.

3

The Bisection Method tries to decrease the size of the interval in which a solution exist. If

the function f xsatisfies < 0, 0 0 f a f b then the equation starts with one sign at 0 a and

ends with the opposite sign at 0 b , and if 0 0 = 0 f a f b , then either 0 a or 0 b or both are roots

of f(x) = 0. This method consists of finding midpoint 0 a and 0 b . If 1 2 0 0

m 1 a b is the

midpoint of this interval, then the root will lie either in the interval 0 1 a ,m or in the interval

1 0 m ,b provided that 0 1 f m . If 0 1 f m , then 1 m is the required root. Repeating this

procedure, we obtain the bisection method

, 0,1,

2

1

1 m a b a n n n n n .

Where 1 , 1 an an mn if f an f mn1 < 0 (1.1)

and

bn1 mn1,bn if f mn 1 f bn < 0

We take the midpoint of the last interval as an approximation to the root. Traub (1964), if f (x)

is continuous in the interval [a, b] which contains the root, the method converges.

Hafiz and Bahgat (2012), several iterative methods have been developed to solve nonlinear

algebraic equations and the system of nonlinear equations. These methods have been

improved using Taylor polynomials, homotopy perturbation method and Adomian

decomposition methods.

4

The homotopy perturbation method (HPM) was developed for solving nonlinear systems, He

(1999). HPM linearizes any given problem (converting it to a series of linear equations). The

method gives a rapid convergence of the solution and only a few iterations lead to accurate

result. In contrast to the traditional perturbation methods, this method does not require a small

parameter in an equation. In this method, a homotopy with an imbedding parameter p ∈ [0, 1]

is constructed, and the imbedding parameter is considered as a “small parameter”. Li (2009),

when p=0, the system of equation usually reduces to a sufficiently simplified form, which

normally admits a rather simple solution. As p increases to 1, the system under goes a

sequence of deformations, the solution of each is close to that at the previous stage of

deformation. When p=1, the system takes the original form of the equation and the final stage

of deformation gives the desired solution.

Adomian (1984), developed a new method known as the Adomian decomposition method

(ADM) for solving functional equations of any kind: ordinary differential equations (ODEs),

algebraic, partial differential equations (PDEs), integral equations, etc. The ADM breaks any

given problem into linear and nonlinear parts. The linear operator representing the linear

portion of the equation is inverted and the inverse operator is then applied to both sides of the

equation, before applying the initial or boundary conditions. The term that contains the

independent variable alone is taken as the initial approximation. The unknown function is

then decomposed into a series whose components are to be determined. The components are

determined in terms of polynomials called Adomian polynomials whose successive terms are

determined using a recurrent relation.

5

Traub (1964), classified one-point iterative methods into,

(i) One-point methods without memory

(ii) One-point methods with memory

If the value of the root is ascertained by using new data only at one point and no previous data

is used, then it is called one-point iterative method without memory. An example of one-point

iterative methods without memory is the Newton-Raphson method. Hence, if n1 x is estimated

by new data at n x and no previous data is used, we have n n x x 1 , then is called onepoint

iteration function without memory.

If the value of the root is ascertained by using new data at one point and by using the previous

data at either one or more than one points, then the iterative method is called one-point

method with memory. Secant method is an example of one point iterative method with

memory. If n1 x is estimated by new data at n x and the previous data at n n m x x , , 1 , we have

n n n n m x x x x , , , 1 1 and is called one-point iteration function with memory.

1.2 Motivation for this study

Many methods and algorithms have been developed to solve problems of nonlinear algebraic

equations over the years. Despite these efforts, no single algorithm is capable of solving any

and all nonlinear problems. Depending on the system and the degree of nonlinearity, one

solution scheme may be preferred over another. To keep up with recent computational

challenges in the field of numerical analysis, it is imperative to develop new schemes that are

capable of taking advantage of the latest advances in numerical analysis. This is what

6

motivates us to undertake this study. Advances such as Adomian decomposition method,

Basto et al. method.

1.3 Problem studied in the thesis

In our present work, we have tried to develop two new iterative schemes which are based on

Taylor’s series and Adomian method. The first scheme truncates the Taylor’s series after the

third term while the second scheme truncates the series after the fourth term. Moreover in

both schemes, it is assumed that

1.

f x

f x

1.4 Aim and objectives

The aim of the study is to develop and analyse new iterative schemes for solving nonlinear

equations.

The objectives of the study are

(i) To review iterative schemes between 1998 and 2012 which have been

developed from Adomian decomposition method, Homotopy perturbation method

and variants of Newton-Raphson’s method for solving nonlinear equations.

(ii) To develop new schemes that could compete with previous schemes and

probably have further advantages.

(iii)To compare the new schemes with the existing known iterative schemes.

7

1.5 Limitation of the study

This work is limited to initial point methods and specifically, only to one point iterative

methods. Moreover, for those problems whose second derivatives are far away from the first

derivatives, the assumption

1

f x

f x is a limitation.

1.6 Definitions of term

(i) Let n x x be the truncation error in the nth iterate where x is the required

root. If there exists a number p 1 and a constant c 0 such that

c

x x

x x

p

n

n

n

1 lim ,

then p is called the order of convergence of the method, Burden and Faires (2011) .

Note that;

If p 1, we say that n x is linearly convergent.

If p >1, we have super linear convergence.

If p 2 , we have quadratic convergence.

8

1.7 Theorems used in the study

We now state the following theorems without proofs, which will be used in this study. The

proofs can be found in any standard analysis text book such as Burden and Faires (2011).

Theorem 1.7.1: Mean value Theorem

If f C[a,b] (whereC[a,b] is the space of continuous functions on[a,b]) and f xis

differentiable ina,b, then there exists a point ca,bsuch that f b f a b af c.

Theorem 1.7.2: Fixed Point Theorem

If g C[a,b]and gx[a,b]for all x[a,b],then g has a fixed point in [a,b]. If in addition,

gxexists on a,b and a positive constant k < 1exists with gx k , for all xa,b, then

the fixed point in [a,b] is unique.

Theorem 1.7.3: Rolle’s Theorem

Suppose f C[a,b] and f is differentiable on a,b. If f a f b, then a number c in

a,b exists with f c 0.

Theorem 1.7.4: Extreme Value Theorem

Suppose f C[a,b] and , [ , ] 1 2 c c a b with 1 2 f c f x f c , for all x[a,b]. In

addition, if f is differentiable on a,b, then the numbers 1 c and 2 c occur either at the

endpoints of [a,b] or where f 0 .

9

Theorem1.7.5: Taylor’s Theorem

Suppose f Cn[a,b], (where Cn[a,b]is the space of n-times continuously differentiable

functions on[a,b]) and that f n1 exists on[a,b], with [ , ] 0 x a b . Then for every x[a,b] ,

there exists a number x between xo and x with

x x R x

n

f x f x f x x x f x x x f x n

n

n

0

2 0

0

0

0 0 0 2! !

Where

1

0

1

1 !

n

n

n x x

n

R x f x

(called the remainder term or truncation error).

1.8 Outline of the thesis

The present thesis is structured as follows:

Chapter 1, which is the present chapter constitutes the general introduction, the aims and

objectives of the study and limitations of the study. It also contains some basic definitions and

theorems that are important to the study being done.

Chapter 2 is a review of previous work that is related to the work under study.

Chapter 3 is a description of the Adomian method and how it is used to develop the two new

iterative methods.

Chapter 4 contains analysis of numerical results to illustrate the effectiveness of the new

methods.

Chapter 5 contains summary, conclusion and recommendations. It also contains discussion of

potential areas for further work.

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