8.1 INTRODUCTION

In Chapters 5 and 6, we have discussed several analytical methods for solving P.D.E. s. These methods cannot be used to solve all P.D.E. s. For example, the method of separation of variables can generally be applied only to linear homogeneous equations with homogeneous boundary conditions. Under favorable conditions, non-homogeneity can be transformed to homogeneity via auxiliary functions as shown in Section 5.7.

Many equations cannot be solved exactly and we have to be satisfied with approximate solutions. Even some of the exact solutions given in Chapter 5 are in reality approximate solutions. For instance, the infinite Fourier series solution obtained in Chapter 5 has the appearance of an exact solution, but, in many cases, we can sum only a finite number of terms. The solution is then an approximate solution. Also, the Fourier coefficients are expressed as integrals and some of them cannot be evaluated analytically, and we have to integrate them numerically.

Using computers, numerical methods are the most appropriate methods of solving some P.D.E. s. In this chapter, we shall extend the method of finite differences described in Chapter 7 to P.D.E. s. We shall also consider the method of finite elements.

8.2 FINITE DIFFERENCES

For simplicity, we consider the unknown function u to be a function of two variables x and y. We divide the xy-plane into a grid consisting of (n m) rectangles with sides ?x=h and ?y=k as shown in Figure 8.2 1. We denote the value of u(x _i, y _j)...