Block Method for Numerical Integration of Initial Value Problems in Ordinary Differential Equations

Block Method for Numerical Integration of Initial Value Problems in Ordinary Differential Equations

ABSTRACT

A self-starting six-step ten order block method with three off-grids points has been derived for solving Ordinary Differential Equations (ODE) using interpolation and collocation procedures. Multiple numerical integrators were arrived at.

These integrators are combined into a single block matrix equations. Our method was found to be zero stable, consistent and convergent. The numerical examples considered showed that our method gave better accuracy than some existing methods. The stability analysis shows that our method is A-stable and of order 10.

TABLE OF CONTENTS

Title Page ii

Certification iii

DEDICATION IV
Acknowledgment v

Table of Content vi

Abstract ix

CHAPTER ONE

1.1 Introduction 1

1.2 Differential Equation 2

1.2.1 Types of Differential Equation 3

1.2.2 Ordinary Differential Equations 3

1.2.3 Partial Differential Equation 3

1.3 Initial Value Problems 4

1.4 Objective of the Study 4

1.5 Method of Research 4

CHAPTER TWO

LITERATURE REVIEW

Introduction 6
Existing Numerical Methods 7
2.2a Definitions Associated with one step Methods 7

2.3 Runge Kutta Methods 9

2.3a Classes of Runge Kutta Methods 9

2.4 Linear Multistep Method 10

2.4b Linear Multistep family 12

2.4c Adams Methods 12

2.4d Backward Differentiation Formula (BDF) 13

2.5 The Hybrid Method 14

2.6a Parallel Block Method 17

2.6b Parallel Block Implicit Method 19

2.6c Parallel Block Predictor- Corrector (PBPC) Method 21

2.6d Parallel Block Methods For Stiff Equation 23

2.6e Block Runge Kutta Method 25

CHAPTER THREE

Development of the Method

3.1 Introduction 27

CHAPTER FOUR

STABILITY ANALYSIS OF THE METHOD

4.0 Preamble 51

4.1 The Basic Properties of the Method 51

4.5 Consistency 52

4.6 Convergence: 52

4.7 Zero Stability 53

4.8 Region of Absolute Stability 57

CHAPTER FIVE

Implementation and Numerical Results

5.0 Introduction 59

5.1 Algorithm 59

5.2 Numerical Experiment and Results 59

CHAPTER ONE

INTRODUCTION

PREAMBLE

The development of numerical integration formula for stiff problems has attracted great attention in the past. It is also important to note that mathematical model of physical situation, in kinetic chemical controls, and electric theory often leads to stiff Ordinary Differential Equations.

Stiffness in common terms refers to a difficult and severe situation than usual. One of the basic problems that are in the solution of stiff problem is that of numerical stability.

In this thesis, emphasis will be on the solution of initial value problem of Ordinary Differential Equation.

1.1 We seek a solution on the range, where a and b are finite, and we assume that f satisfies the continuous which guarantee that the problem have a unique continuous differentiable solution which we shall denote by

Various researchers have discussed and published literature in the solution of ordinary differential equation of (1.1) amongst other are Lie and Norsett (1980), Serisena (1989). These researchers have developed various numerical methods to solve such initial value problems.

In the course of this work, we shall examine various existing methods which will act as motivation in developing our scheme.

1.2 Differential Equation

Differential equation is a mathematical equation which relates a function with it derivatives. The function usually represents physical quantities, while the derivatives represent their rate of change and the equation defines a relationship between the two. Because such relationships are extremely common, differential equation plays a pivotal role in many disciplines including Engineering.

In Mathematics, differential equations are studied from several directions mostly concerned with their solution, the set of functions that satisfy the equation.

Differential equation first came into existence with the invention of Calculus by Newton and the three kinds of differential equation according to Newton are;

He solved these given equations and others using infinite series and discus the non uniqueness of solution.

Other examples of differential equation are Bernouli (1965), differential equations, which have the form , Euler-Lagrange equation (1750), which he uses in the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.

1.2.1 TYPES OF DIFFERENTIAL EQUATIONS

Differential Equations can be divided into various types. Apart from the description of the properties of the equation itself. These classes of differential equations can help inform the choice of approach to solutions.

1.2.2 ORDINARY DIFFERENTIAL EQUATIONS

This is an equation containing a function of one independent variable and its derivatives. Linear differential equation can be added and multiplied by coefficients Ordinary Differential Equations that lack additive solution are known as non linear, and solving them is more intricate.

Ordinary Differential Equations can be used to describe a wide variety of phenomenon such as sound, heat, electrostatics e.t.c.

1.2.3 PARTIAL DIFFERENTIAL EQUATION

A Partial Differential Equation is a differential equation that contains multi-derivative function and their partial derivatives. They are used to formulate problem involving function of several variable, and are either solved by hand or used to create a relevant computer model.

1.3a INITIAL VALUE PROBLEMS

Initial value problem is an Ordinary Differential Equation together with a specified value called the initial condition of the unknown function at a given point in the domain of the solution.

1.3b LINEAR MULTI STEP METHOD

A linear multistep method is a computational methods for determining the numerical solution of initial value problems of ordinary differential equations which form a linear relation between .

The general formula is given as

(1.2)

Where is the numerical solution of the initial value problems

1.4 OBJECTIVES OF THE STUDY

The overall aim of the study is to modify polynomial basis function

The specific objectives are as follows:

To derive an order ten continuous hybrid block method.

To investigate the stability properties, characteristics, consistency and the nature of convergence of the integrator constructed.

To implement the integrators and compare it with existing methods.

Apply MAPLE programming for the implementation of the integrator and compare their performance with some established results.

1.5 METHOD OF RESEARCH

In carrying out our research satisfactorily, we intend to:

Adopt and modify polynomial basis function of the form

Use interpolation and collocation procedure to choose interpolating point S

Determine coefficients by taking the derivative of, given as and substituting into.

Collocate at various point where

Investigating and analyze the stability of our derived method to ascertain if it satisfies the A-stable condition.

Experimenting the derived formulas by using appropriate MAPLE algorithm to solve some IVPs.

Compare result of our formulas with that of some existing formulas with the aim of ascertaining level of accuracy.