×

Numerical analysis for integral and related operator equations. (English) Zbl 0763.65102

In this comprehensive monograph the highly competent authors have compiled many methods and results that have been developed and found in the last two decades. Their principal interest is in describing and analyzing various approximation methods (e.g. splines, Galerkin, collocation, finite section) for singular integral equations, certain classes of convolution integral equations (Mellin, Wiener-Hopf), pseudo- differential equations of elliptic type.
Drawing their main motivation from the field of boundary element methods, they treat such equations on open and on closed curves in the complex plane, thereby making use of standard transformations of such curves to an interval or to a circle. The authors’ treatise is essentially self- contained. They do not only give numerical schemes but also a lot of basic theory concerning the equations treated (the relevant functional analysis, operator theory, Banach algebras, approximation theory, Fourier analysis, the method of symbols, etc.).
Let some of the numerical methods be cited in a few words (taken from the preface): approximation methods for Fredholm equations, functions of shifts and finite sections, polynomial approximation of singular equations on \([-1,1]\), spline approximation of convolution equations, polynomial approximation of singular equations on the unit circle, spline approximation and quadrature methods on a simple smooth closed curve, polynomial approximation for singular equations in Zygmund-Hölder spaces, Mellin techniques in numerical analysis, spline approximation of pseudodifferential equations, spline approximation and quadrature methods on an interval.
The style of the book is concise, however, the concepts, methods and theorems are clearly described and stated, most of the proofs are carried out (or references are given for). Each chapter is closed by helpful “Notes and comments”, and, finally, there is an extensive bibliography of 22 pages.
Thus, this monographs will be of great value on one hand for mathematicians interested or working in (numerical) analysis and applications of integral equations, on the other hand for physicists and engineers who want to see what mathematics can do.

MSC:

65R20 Numerical methods for integral equations
65J10 Numerical solutions to equations with linear operators
65N38 Boundary element methods for boundary value problems involving PDEs
45L05 Theoretical approximation of solutions to integral equations
45-02 Research exposition (monographs, survey articles) pertaining to integral equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
45Exx Singular integral equations
47A50 Equations and inequalities involving linear operators, with vector unknowns

Citations:

Zbl 0763.65103
PDFBibTeX XMLCite