ABSTRACT

In this thesis, the important fundamental properties of numerical methods for ordinary differential equations are investigated. This involves the derivation of Adams Bashforth methods, order and stability conditions and other basic concepts that are required to understand the methods.

TABLE OF CONTENTS

Title Page                                                                                                                    i

Declaration                                                                                                                  ii

Certification                                                                                                                iii

Dedication                                                                                                                  iv

Acknowledgement                                                                                                      v

Table of Contents                                                                                                       vi

List of figures                                                                                                             ix

List of tables                                                                                                               x

Abstract                                                                                                                      xi

CHAPTER 1: Introduction                                                     1

1.1       Background of Study                                                                                     1

1.2       Statement of the Problem                                                                               2

1.3       Aim and objectives                                                                                         3

1.4       Significance of Study                                                                                     3

1.5       Scope and Limitation                                                                                      4

CHAPTER 2: LITERATURE REVIEW                                                                   6

CHAPTER 3: RESEARCH METHOD                                                                     8

3.1       Step Size                                                                                                         8

3.2       Mesh Size                                                                                                        8

3.3       One Step Methods                                                                                          8

3.4       Linear Multistep Methods (LMMs)                                                                9

3.5       General K-Step Linear Multistep Method                                                      9

3.6       Lipschitz Condition                                                                                        9

3.7       Existence and Uniqueness of Solution                                                           10

3.8       Basis                                                                                                                10

3.9       Basis Function                                                                                                10

3.10     Theorem                                                                                                          11

3.11     One Step Method                                                                                           12

3.12     The Euler Scheme                                  12                                                                                                                         

3.13     Runge-Kutta Scheme                                                                                      12

3.14     Linear Multistep Method (LMM)                                                                   12

3.15     Families of Linear Multistep Method.                                                            13

3.16     Adams-Bashforth                                                                                           13

3.17     Adams – Moulton                                                                                           13

3.18     Backward Differentiation Formula                                                               13

3.19  Chebyshev Polynomials                                                                                     14

3.20     Derivation of Euler’s method                                                                                     15

CHAPTER 4: ANALYSIS OF METHODS                                                             18

4.1.      Introduction                                                                                                    18

4.2       Derivation of Adams Bashforth Method, when k = 2                                               19

4.3       Derivation of Adams Bashforth Method, when k = 3                                               23

4.4       Derivation of Adams- Bashforth Method                                                      26

4.5       Derivation of Adams-Bashforth Method                                                       29

4.6       Consistency                                                                  32

4.7       Stability                                                                                                           32

4.8       Order and Error Estimation                                                                            38

4.9       Implementation                                                                                               40

4.10     Numerical Solution                                                                            42                                                                   

4.11      Exact solution                                                                                                            42

Chapter 5: Summary, Conclusion and Recommendation                                           48

5.1       Recommendation                                                                                            48

REFERENCE

LIST OF FIGURES

Figure 4.1: The stability region for Adams Bashforth method for k=3                                  35

Figure 4.2: The Stability region for Adams Bashforth method for k=4                                 36

Figure 4.3: The Stability region for Adams Bashforth method for k=5                                 36

Figure 4.4: The Stability region for Adams Bashforth method for k=5                                 37

LIST OF TABLES

Table 4.1Adams Bashforth Method for k=1                                                              43

Table 4.2: Adams Bashforth method for   K =2                                                         43

Table 4.3: Adams Bashforth method for   K =3                                                         44

Table 4.4: Adams Bashforth method   K= 4                                                              44

Table 4.5: Adams Bashforh method for K =5                                                45

Table 4.6                                                                                       45

Table 4.7:Error from Reformulated scheme                                                               46

Table 4.8: Adam Bashforth Methods for k=3, 4, and 5                                             47

Table 4.9: Error from Adams Bashforth methods for k=3,4 and 5                            47

CHAPTER 1

Introduction

1.1       Background of Study

Numerical methods are truly a crucial part of solving differential equations which cannot be neglected since the late 18^th century. Numerical methods for solving differential equations have been developed continuously by many Mathematicians Euler’s et al.,(1895).

According to some historians of Mathematics, the study of differential equations began in 1675, when German Mathematician; Gottfried Leibniz wrote the following equation:

The attempt to solve physical problems led gradually to mathematical models involving an equation in which a function and its derivatives play important roles. However, the theoretical development of this new branch of mathematics, Ordinary differential equations (ODE) has its origin rooted in a small number of mathematical problems. These problems and their solutions led to independent disciplines with the solutions of such equations an end in itself; however, involving derivatives of only one independent variable are called ordinary differential equations and may be classified as either initial value problem (IVP) or boundary value

Problem (BVP).

The search for general methods of integrating differential equations began when Isaac Newton (1642-1727) classified first order differential equation into three classes.

Firstly, ordinary derivative of one dependent variable with respect to a single independent variable example

Secondly,

Ordinary derivative of one or more dependent variables with respect to a single independent variable example;

1.2       STATEMENT OF THE PROBLEM

The methods of solving higher order initial value problems (IVPs) of general Ordinary differential equations by reduction order method has been reported to have difficulties in computation, complication in writing computer programs and resultant wastage of time.

Linear multistep methods implemented by the predictor-Corrector mode have been found to be very expensive to implement in times of the number of function evaluations per step, the predictors often have lower order of accuracy than the correctors especially when all the step and off step points are used for collocation and interpolation, Asymptotically correct interpolation based spatial error by Tom Arsenault.

In view of the aforementioned problems, this work is motivated by the need to address the setbacks associated with the existing methods, by considering the Adams Bashforth methods. The methods are less expensive, in terms of the number of functions evaluated per step. Also efficient in terms of accuracy and error term, possess better rate of convergence and very easy to program resulting in economy of computer time.

1.3       AIM AND OBJECTIVES

The aim of this work is to derive Adams Bashforth methods using Chebyshev polynomial as basis function. To accomplish this aim, the following objectives were outlined:

i.                    To derive  Adams Bashforth methods using Chebyshev polynomials

ii.                  To implement the methods without the tedious process of developing predictor separately.

iii.                To analyse basic properties of the methods which include order, consistency zero stability, convergence and region of absolute stability.

iv.                To test the performance of the methods for accuracy and efficiency.

1.4       SIGNIFICANCE OF STUDY

Conventional methods of solving higher order initial value problems (IVP) of general ordianery differential equation (ODE) by reduction order method has been reported to have difficulties such as computational burden, complication in writing computer programs and was of computer time (Awoyemi, 1992) and inability of the method to utilize additional information associated with a specific ODEs such as the Oscillatory nature of the solution (Vigo-Aguiar and Ramos, 2006).

Equivalently, linear multistep methods implemented by the predictor-corrector mode have been found to be very expensive to implement interms of the number of function evaluations per step.

Hence the Adams Bashforth method would be less expensive in terms of the number of functions evaluation per step; highly efficient in terms of accuracy and error terms: flexible in change of step size, posses better rate of convergence and weak stability properties, and very easy to program resulting in economy of computer time.

1.5       SCOPE AND LIMITATION

This work is limited to the following:

1.      Only continuously differentiable functions in the interval of integration were  considered

2.      The basis function considered in this work is the Chebyshev polynomial

3.      The research work adopted only continuous Adams Basthforth method

.

The desire to obtain more accurate approximate solutions to mathematical models, arising from science, engineering, and even social sciences in the form of ordinary differential equations which do not have analytical solution has led many scholars to propose several different numerical methods.

In this chapter, some of the many contributions available in the literature are reviewed. Specifically those numerical methods for the solution of ordinary differential equations (ODE).

In most applications, ordinary differential equations (ODEs) is solved by reduction to an equivalent system of first order ordinary differential equations of the form

For any appropriate numerical method to be employed to solve the resultant system.

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