It frequently happens that the end product of a calculation or piece of analysis is one or more equations, algebraic or differential (or an integral), which cannot be evaluated in closed form or in terms of available tabulated functions. From the point of view of the physical scientist or engineer, who needs numerical values for prediction or comparison with experiment, the calculation or analysis is thus incomplete. The present chapter on numerical methods indicates (at the very simplest levels) some of the ways in which further progress towards extracting numerical values might be made.

In the restricted space available in a book of this nature it is clearly not possible to give anything like a full discussion, even of the elementary points that will be made in this chapter. The limited objective adopted is that of explaining and illustrating by very simple examples some of the basic principles involved. The examples used can in many cases be solved in closed form anyway, but this ‘obviousness’ of the answer should not detract from their illustrative usefulness, and it is hoped that their transparency will help the reader to appreciate some of the inner workings of the methods described.

The student who proposes to study complicated sets of equations or make repeated use of the same procedures by, for example, writing computer programmes to carry out the computations, will find it essential to acquire a good understanding of topics hardly mentioned here.

In this and the following chapter the solution of differential equations of the types typically encountered in physical science and engineering is extended to situations involving more than one independent variable. Only linear equations will be considered.

The most commonly occurring independent variables are those describing position and time, and we will couch our discussion and examples in notations which suggest these variables; of course the methods are not restricted to such cases. To this end we will, in discussing partial differential equations (p.d.e.), use the symbols u, v, w, for the dependent variables and reserve x, y, z, t for independent ones. For reasons explained in the preface, the partial equations will be written out in full rather than employ a suffix notation, but unless they have a particular significance at any point in the development, the arguments of the dependent variables, once established, will be generally omitted.

As in other chapters we will concentrate most of our attention on second-order equations since these are the ones which arise most often in physical situations. The solution of first-order p.d.e. will necessarily be involved in treating these, and some of the methods discussed can be extended without difficulty to third- and higher-order equations.

The method of ‘separation of variables’ has been placed in a separate chapter, but the division is rather arbitrary and has really only been made because of the general usefulness of that method.