TABLE OF CONTENTS Numerical methods for the solution of differential equations are many and various. So far we have treated only the case of a single differential equation of first order, with one dependent and one independent variable (given an initial condition), but the treatment was fairly rigorous. In this chapter we shall in contrast merely sketch a number of illustrative methods for dealing with more complex problems: initial value problems involving more than one ordinary differential equation, and the related problem of higher order differential equations; boundary value problems; variational methods; and simple eigenvalue problems. Finally, we add a few remarks about partial differential equations. One of the most powerful computational techniques we do not treat at all, namely finite element analysis, because it would take us too far from our main objectives.
The most straightforward extension of the work of the last chapter is to the case where there are several dependent variables, each specified at some initial value of the independent variable. For example, we might wish to consider the two simultaneous equations:
where f, g are given functions of x, y, z. In general, it is convenient to state the problem in vector notation. We have a system of n first-order differential equations for the n dependent variables y 0, , y n ?1, considered as a single vector y, in terms of a single independent variable x, where
