Mathematical Methods for the Physical Sciences

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.