In this book, numerical methods are presented and analyzed for the solution of integral equations of the second kind, especially Fredholm integral equations. Major additions have been made to this subject in the twenty years since the publication of my survey [39], and I present here an up-to-date account of the subject. In addition, I am interested in methods that are suitable for the solution of boundary integral equation reformulations of Laplace's equation, and three chapters are devoted to the numerical solution of such boundary integral equations. Boundary integral equations of the first kind that have a numerical theory closely related to that for boundary integral equations of the second kind are also discussed.

This book is directed toward several audiences. It is first directed to numerical analysts working on the numerical solution of integral equations. Second, it is directed toward applied mathematicians, including both those interested directly in integral equations and those interested in solving elliptic boundary value problems by use of boundary integral equation reformulations. Finally, it is directed toward that very large group of engineers needing to solve problems involving integral equations. In all of these cases, I hope the book is also readable and useful to well-prepared graduate students, as I had them in mind when writing the book.

During the period of 1960–1990, there has been much work on developing and analyzing numerical methods for solving linear Fredholm integral equations of the second kind, with the integral operator being compact on a suitable space of functions.