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Computational methods for linear integral equations. (English) Zbl 1023.65134
Boston, MA: Birkhäuser. xviii, 508 p. EUR 120.56/net; sFr. 190.00 (2002).
As stated in its preface, this book “is designed as a new source of classical as well as modern topics on the subject of the numerical computation of linear integral equations…[discussing…] the underlying theory of integral equations and numerical analysis of numerical integration and convergence…”. It is primarily written for graduate students.
Its contents is: Chapter 1: Introduction (43 pages): Classification, function spaces, Neumann series and resolvents, the Fredholm alternative. Chapter 2: Eigenvalue problems (45 pages): Linear symmetric equations, residual methods, degenerate kernel methods. Chapter 3: Equations of the second kind (15 pages): Taylor, quadrature and block methods.
Chapter 4: Classical methods for second-kind Fredholm equations (20 pages). Chapter 5: Variational methods (15 pages). Chapter 6: Iteration methods (30 pages). Chapter 7: Singular equations (40 pages): Fredholm and Volterra equations with (weakly) singular kernels, equations with Hilbert kernels, finite part singular equations. Chapter 8: Weakly singular equations (40 pages): Taylor series methods (!), product integration and splines methods.
Chapter 9: Cauchy singular equations (35 pages). Chapter 10: Sinc-Galerkin methods (35 pages): Sinc-Galerkin and Sinc collocation, application to single-layer and double-layer problems. Chapter 11: Equations of the first kind 50 pages). Chapter 12: Inversion of Laplace transforms (40 pages).
This is followed by an appendix of four sections describing quadrature rules, orthogonal polynomials, Whittaker’s cardinal function, and singular integrals, and by a extensive bibliography.
The book consists essentially of a listing of numerical methods for solving various kinds of integral equations. In spite of the mathematical machinery (functional analysis) introduced in Sections 1.3-1.5, the authors do not discuss or analyze the convergence properties of the methods. But more serious is the fact that, with very few exceptions, the presentation in this book reflects the state of the art in the numerical treatment of integral equations of about 1975. The results contained in many of the fundamental books and papers of the last twenty years appear to have had no impact and are not mentioned in the bibliography.
Typical examples are the monographs by F. Chatelin [Spectral approximation of linear operator (1983; Zbl 0517.65036)], by S. Prössdorf and B. Silbermann [Numerical analysis for integral and related operator equations (1991; Zbl 0763.65102)], and by G. Vainikko [Multidimensional weakly singular integral equations (1993; Zbl 0789.65097)], on projection methods for integral equations and on the numerical analysis of (singular) integral equations; the papers by R.S. Anderssen, F. Natterer, H. W. Engl, and many others, and the book by A. Kirsch [An introduction to the mathematical theory of inverse problems (1996; Zbl 0865.35004)], on Fredholm integral equations of the first kind; and the work by, e.g., W. Wendland, J. Saranen, I. Sloan, J. Elschner and I. Graham on numerical methods for boundary integral equations. This is complemented by many mathematical inaccuracies (e.g. in (1.2.10) where the kernel \(k(x,s)\) is complex; in Section 4.6, in the description, and comparison, of Galerkin and collocation methods; the important notion of a compact (integral) operator is not introduced – although the book is intended for graduate students; etc.). Although Galerkin and collocation methods are described in a number of places, the reader is told neither about the general framework nor about the superconvergence properties of these important (projection) methods.

65R20 Numerical methods for integral equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
45D05 Volterra integral equations
45Exx Singular integral equations
44A10 Laplace transform
65R10 Numerical methods for integral transforms
45C05 Eigenvalue problems for integral equations
45B05 Fredholm integral equations