Introduction

This chapter attempts to describe a few methods for finding approximate values of a solution of a non-linear initial value problems. By no means, it is not an exhaustive study but can be looked upon as an introduction to numerical methods. Again, we stress that no attempt is made towards a deeper analysis. A question may arise: why one needs numerical methods for differential equations. Probably because, differential equations play an important role in many problems of engineering and science. This is so because the differential equations arise in mathematical modelling of many physical problems. The use of the numerical methods have become vital in the absence of explicit solutions. Normally, numerical methods have two major roles:

1. the amicability of the method for easy implementation on a computer;
2. the method allows us to deal with the analysis of error estimates.

In this chapter, we do not enter into the aspect of error analysis. For the present, we deal with some of the numerical methods to find approximate value of the solution of initial value problems (IVP's) on finite intervals. We also mention that no effort is made to study the boundary value problems. Let us consider an initial value problem

$$y^\prime = f(x,y), \; a \le x \le b, \; y(a) = y_0$$ (14.1.1)

where $f: [a, \; b] {\longrightarrow}{\mathbb{R}}, \; a, b$ and $y_0$ are prescribed real numbers and $\vert b - a\vert < \infty$ . By Picard's Theorem 7.6.8, we know that under certain conditions, the initial value problem () has a unique solution. Our aim is to determine $y$ on the interval $[a, \; b]$ . Let us also assume that our aim is to determine $y$ at $b$ . To do so, we divide the interval $[a, \; b]$ into $n$ equal parts

$$a, a+h, a+2h, \ldots, a + n h = b, \hspace{.15in} {\mbox{where }} h = \displaystyle \frac{b-a}{n}.$$

we know the value of $y$ at $x=a$ . with this data and with the use of (), we determine the approximate value of $y$ at $a + h$ , denoted by $y_1$ . We repeat this process till we reach the point $x = a + nh = b$ . The details are given in the ensuing sections.

The number $h$ is called the step size and $h$ depends on $a, b$ and $n$ . For $0 \le i \le n$ , the value of $y$ at $a + i h$ is $y(a + i h)$ and its approximate value is denoted by $y_i$ . In this chapter, we develop methods for determining an approximate value of $y(b)$ which is denoted by $y_b$ . This is achieved by evaluating the approximate values $y_1, y_2, \ldots, y_{n-1}$ . Normally $x_0 = a$ , is called the INITIAL VALUE, $x_1 = a + h$ is called the FIRST STEP and $x_j = a + j h, \; 2 \le j \le n$ are called the $j^{\mbox{th}}$ steps. The approximate value $y_j$ is called the approximation of the function $y$ at the $j^{\mbox{th}}$ step. The method which uses $y_{j-1}$ to determine $y_j$ is called a SINGLE STEP METHOD. If the method uses more than one value of $y_i, \; 1 \le i \le j-1$ is called a MULTI-STEP METHOD.

In the sequel, we deal with some simple single step methods to find the approximate value of the solution of (). In these methods, our stress is on the use of computers for numerical evaluation. In other words, the implementation of the methods by using computers is one of our present aims. Each method will be pictorially represented by what is called a flow chart. Flow chart is a pictorial representation which shows the sequence of each step of the computation. Usually the details of both the input and the termination of the method is indicated in the flow chart. In short, a flow chart is a chart that shows the flow of the computation including the start and the termination of the method.