APPLICATION OF NUMERICAL METHOD IN SOLVING SECOND ORDER DIFFERENTIAL EQUATION

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APPLICATION OF NUMERICAL METHOD IN SOLVING SECOND ORDER DIFFERENTIAL EQUATION

CHAPTER ONE

INTRODUCTION

Background of the study

Differential equations play a role in the modelling of almost every scientific discipline. However, it is relatively rare for a differential equation to have a solution that can be written in terms of elementary functions. Usually, the only information about the solution is that it is known to exist and to be unique, on theoretical grounds, and that it can be approximated more or less accurately using computational techniques. In this review paper, we will consider some aspects of numerical methods for the solution of initial value problems in systems of ordinary differential equations. There are two standard forms for expressing such problems. The first of these is

Here the solution y is assumed to be a differentiable function on an interval [x0,x] to a finite dimensional Euclidean space IRN. The formulation (1) is very general and includes, for example, second and higher order differential equations; these are easily recast in this way. By introducing an additional variable, if necessary, which always remains exactly equal to x, it is possible to reformulate the general problem as an ‘autonomous’ system of equations. This is the second standard form.

Computer software for solving ordinary differential equations exists for both formulations and there is no advantage to either, except that many problems are most naturally expressed in the non-autonomous form (1) rather than autonomous form (2). However, for many theoretical investigations, the autonomous form is to be preferred. We will see an example of this in Section 3 where the exposition is much simpler and more transparent than would have been possible for the non-autonomous formulation.

Statement of the problem

There may have been previous researches in this subject. This work gives further explanations and analysis in application of numerical method in solving second order differential equation

Objectives of the study
To understand the impact of application of numerical method in solving second order differential equation
To understand the relationship between application of numerical method and solving second order differential equation

Research questions
What is the impact of application of numerical method in solving second order differential equation
What is the relationship between application of numerical method and solving second order differential equation