This book

The theory of differential equations is an essential ingredient of any undergraduate course in Mathematics and the majority of Numerical Analysis undergraduate courses introduce the student, at an early stage, to the numerical solution of differential equations. For some reason the theory, and, perhaps more so, the numerical solution, of integral equations are deferred to a later stage: in some sense integral equations must be felt to be either more advanced or of less practical interest than differential equations. This reflects the situation in practical calculations and probably in turn helps to perpetuate it; we turn more readily to a differential formulation of a problem than to an integral formalism. Yet the theory of linear nonsingular integral equations is at least as well developed as that of differential equations and it is in many respects rather simpler. The corresponding operators are bounded rather than unbounded, leading to a very straightforward existence theory (the Fredholm theory); perhaps, as one consequence of this, there is a much tighter link between the theory and practice of integral equations than is the case with differential equations. Most of the convergence proofs are constructive in nature and all or nearly all of the constructions have been used as the basis of algorithms for the numerical solution of the underlying equations (although not always with any great success!).

Some years ago I regularly gave a traditional course on metric spaces to second-year special honours mathematics students. I was then asked to give a watered-down version of the same material to a class of combined honours students (who were doing several subjects, including mathematics, at a more general level) but, to put it mildly, the course was not a success. It was impossible to motivate students to generalise real analysis when they had never understood it in the first place and certainly could not remember much of it. It was also counter-productive to start the course by revising real analysis because that convinced the students that this was ‘just another analysis course’ and their interest was lost for evermore.

So when I gave the course again the following year I decided to turn the material inside out and to start with the applications (namely the use of contractions in solving a wide range of equations). This meant that the first chapter was a revision of some iterative techniques used to obtain approximations to solutions of equations. This immediately captured the interest of the class: they enjoyed using their calculators and writing programs to solve the equations. Some of the ideas were entirely new to them; for example using iteration to solve an equation with constraints, or solving a differential equation by iterating with an integral and obtaining a sequence of functions.

The second and third chapters were more traditional but the big difference was that the need for distance, function space, closed set, and so on, had been anticipated and motivated. Another difference was that, having approached the subject via iteration, it was then natural to define all the concepts in terms of sequences: hence closed sets (rather than open ones) formed the basis of the approach.

For most students the fourth chapter was the highlight of the course. It consisted of the contraction mapping principle and the use of its algorithmic proof in solving equations.