This book provides a modern treatment of the solution of integral equations. It covers second-kind Fredholm and Volterra types, in particular, and it examines first-kind and eigenvalue problems to a lesser extent. There is enough background theory in the opening chapters to make it self-contained as a graduate text or research reference. Sections indicating the computer results on applying the various methods make interesting comparisons of available software. Other works are adequately referenced.
The opening chapters deal with background material covering the properties of L2[a,b], linear operators on this space, and quadrature methods. In Chapter 4, the Nystrom method is expounded with particular reference to convergence and error estimates using various families of quadrature rules. This is followed by a discussion of methods for second-kind Volterra equations and eigenvalue problems. Various expansion methods by approximation in subspaces of L2 are thoroughly covered, and the Galerkin method is explained in some detail. An interesting chapter on numerical comparisons of various codes in timing and accuracy follows, and the concluding chapters cover singular equations, first-kind equations, and integrodifferential equations.
The text, at times, becomes tedious because of the necessary detail required for some of the error analyses. Of course, for a graduate course, some of this detail can be easily skipped since the principles of the methodology for computing solutions are easily separated from the detail. For the researcher who needs the error analysis details, there are adequate exposition and examples. The precision of the computers used for the numerical work should have been stated, especially when discussing numerical convergence, as some examples looked as though they had “reached roundoff error.” First-kind equations are not well covered, which is perhaps intentional. But more emphasis on the proper way to pose these problems is needed. Overall, this book should prove a welcome addition to the literature on the numerical solution of integral equations; it brings together much of the research of the last ten years.
