Introduction

We have seen that the solution of an equation of the form φ = f + λKφ can, in some circumstances, be expressed as the sum of a series involving all the eigenvalues µn and eigenvectors φn of the operator K. From a theoretical point of view this solution is of undoubted importance, but its practical use is limited by the fact that there are comparatively few operators for which the sequences (µn) and (φn) are known. If these sequences are not known exactly, there remains the prospect of estimating the first N eigenvalues and eigenvectors of K and approximating the solution of φ = f + λKφ by truncating the associated series.

We shall in fact consider approximation methods for the equation φ = f + λKφ in the following chapter. There it will emerge that a knowledge of the dominant eigenvalues of K (that is, its largest positive eigenvalue and its negative eigenvalue of largest magnitude) is of particular value in implementing several of these methods.

In an integral equation arising from a practical problem, λ itself is a quantity of some significance. A simple illustration of this fact was given in Example 1.3, where λ = 1/µ is proportional to the square of the frequency of vibration of a thin rod. The eigenvalues in this case give the natural frequencies of oscillation of the rod, that is, the frequencies at which resonance will occur in certain forced motions of the rod.