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 MULTISTEP 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:
1.5 METHOD OF RESEARCH
In carrying out our research satisfactorily, we intend to: