SOLUTION OF FIRST ORDER DIFFERENTIAL EQUATION USING NUMERICAL NEWTON’S INTERPOLATION AND LAGRANGE

CHAPTER ONE

1.0 INTRODUCTION

1.1 BACKGROUND OF STUDY

Differential equation is one of the major areas in mathematics with series of method and solutions. A differential equation as for example u(x) = Cos(x) for 0 <x< 3 is written as an equation involving some derivative of an unknown function u (E.W Weisstein, 2004). There is also a domain of the differential equation (for the example 0 <x< 3).

In reality, a differential equation is then an infinite number of equations, one for each x in the domain. The analytic or exact solution is the functional expression of u or for the example case u(x) = sin(x) + c where c is an arbitrary constant. This can be verified using Maple and the command dsolve(diff(u(x),x)=cos(x));. Because of this non uniqueness which is inherent in differential equations we typically include some additional equations. For our example case, an appropriate additional equation would be u(1) = 2 which would allow us to determine c to be 2 − sin(1) and hence recover the unique analytical solution u(x) = sin(x)+2 − sin(1). Here the appropriate Maple command is dsolve(diff(u(x),x)=cos(x),u(1)=2);. The differential equation together with the additional equation(s) are denoted a differential equation problem.

Note that for our example, if the value of u(1) is changed slightly, for example from 2 to 1.95 then also the values of u are only changing slightly in the entire domain. This is an example of the continuous dependence on data that we shall require: A well-posed differential equation problem consists of at least one differential equation and at least one additional equation such that the system together have one and only one solution (existence and uniqueness) called the analytic or exact solution (Joshn Wiley, 1969); to distinguish it from the approximate numerical solutions that we shall consider later on. Further, this analytic solution must depend continuously on the data in the (vague) sense that if the equations are changed slightly then also the solution does not change too much. The study in this regard wishes to determine the solution of first order differential equation using numerical Newton’s interpolation and Lagrange.

1.2 STATEMENT OF RESEARCH PROBLEM

What really instigated the study was due to the need to solve first order differential equations using numerical approaches. Most of the researches on numerical approach to the solution of ordinary differential equation tend to adopt other methods such as Runge Kutta method, and Euler’s method; but none of the study has actually combined the newton’s interpolation and Lagrange method to solve first order differential equation. However the study will try to use Newton’s interpolation and Lagrange to solve the problems below:

Find the polynomial interpolating the points.