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Integralgleichungen. Theorie und Numerik. (Integral equations. Theory and numerics). (German) Zbl 0681.65099
Leitfäden der Angewandten Mathematik und Mechanik, 68; Teubner Studienbücher Mathematik. Stuttgart: B. G. Teubner. 374 S. DM 38.00 (1989).
This book is based on lecture courses given by the author at the universities of Bochum and Kiel. It represents a very comprehensive introduction to the numerical treatment of standard Volterra and Fredholm integral equations, as well as of singular integral equations of Cauchy type and of various types of integral equations arising in the boundary integral method.
While the focus of the book is on the numerical analysis of these equations, it also provides relevant aspects of their theory. The reader is assumed to have a knowledge of elementary Analysis and Numerical Analysis but not necessarily of Functional Analysis: in analogy to the historical development of Functional Analysis the chapters on Fredholm integral equations yield ample motivation for the reader to fill existing gaps. Moreover, the author largely avoids using Sobolev spaces; thus, integral operators (especially in the last two chapters) are not discussed in full generality.
The following is a brief description of the book’s contents.
Chapter 1 (24 pages): Mathematical tools (fundamentals, mostly without proofs, from Analysis, Functional Analysis, and Numerical Analysis (numerical quadrature; linear systems in \({\mathbb{R}}^ n)).\)
Chapter 2 (17 pages): Volterra integral equations (elementary theory; numerical methods: direct quadrature and collocation methods).
Chapter 3 (18 pages): Theory of linear Fredholm integral equations (compact integral operators in Banach spaces; existence and uniqueness of solutions).
Chapter 4 (98 pages): Numerical analysis of linear Fredholm equations (kernel approximations; general projection methods; collocation and Galerkin methods; the Nyström method; comparison of various methods; eigenvalue problems; comprehensive convergence analyses).
Chapter 5 (48 pages): Multigrid methods for linear Fredholm equations (applications and analysis of multigrid techniques; the chapter contains numerous ALGOL-like procedures).
Chapter 6 (16 pages): Abel’s integral equations (mostly theory: inversion of Abel-type equations in C(I)).
Chapter 7 (52 pages): Singular (Cauchy-type) integral equations (theory of Cauchy-type integrals and integral equations; approximation on closed curves; multigrid methods; application to the Dirichlet problem for Laplace’s equation.
Chapter 8 (53 pages): The boundary integral method (theory of integral equations of the first and second kind); the single and double layer potentials; boundary integral method for other PDEs (Helmholtz equation, biharmonic equation).
Chapter 9 (25 pages): The boundary element method (this chapter is a good introduction to the numerical realization of the boundary integral method of Chapter 8).
The book is is concisely but clearly written and has many positive features: frequent examples and problems (often intended as additional remarks without proofs), a good index, and a detailed list of contents are just some of these. It is perhaps regrettable that, in a book of this breadth, there was no space for complementing its material with comments on the historical development of the theory and numerical analysis of the various types of integral equations.
Reviewer: H.Brunner

65R20 Numerical methods for integral equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
45-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral equations
45B05 Fredholm integral equations
45Exx Singular integral equations
45D05 Volterra integral equations
65F10 Iterative numerical methods for linear systems
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35C15 Integral representations of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation