The aim of this volume is to give an introduction to the principal problems and methods in the numerical solution of integral equations, together with some theoretical background, and a number of applications. The basic methods are treated in some detail, and recent developments are also discussed and compared, with full lists of references for further study.
Part I (Chapters 1-5) deals with mathematical preliminaries, in the theory of integral equations, in numerical analysis, function spaces and approximation theory. Part II (Chapters 6 - 18) considers the main numerical methods used for the different types of integral equations, treating general classes of problems rather than particular cases. Chapter 6 gives the direct approach to the numerical solution of Fredholm equations of the second kind, using quadrature formulae. Chapters 7, 8 and 9 consider methods based on expansions of various kinds, with the solution obtained by variational or Galerkin methods, or by collocation. Chapter 10 applies quadrature and variational methods to the solution of the eigenvalue problem. Chapters 11 and 12 deal with Volterra equations of the first and second kinds, with a discussion of the stability of step-by-step methods. Chapter 13 considers the problem of Fredholm equations of the first kind, and discusses the ill-conditioned nature of the problem and the associated numerical difficulties. Chapter 14 surveys the wide field of integro-differential equations, with a summary of recent theoretical work and a brief description of some particular examples. Chapters 15, 16 and 17 treat the general problem of nonlinear integral equations, with a study of the convergence of iterative methods, and error estimation. Part II ends with a discussion in Chapter 18 of library programs for the practical solution of certain classes of integral equations.