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Linear Integral Equations (3rd Edition) by Rainer Kress



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Linear Integral Equations (3rd Edition) written by Rainer Kress , Institut f¨ur Numerische und Angewandte, Georg-August-Universit¨at G¨ottingen, G¨ottingen, Germany

This book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter.
For this third edition in order to make the introduction to the basic functional analytic tools more complete the Hahn–Banach extension theorem and the Banach open mapping theorem are now included in the text. The treatment of boundary value problems in potential theory has been extended by a more complete discussion of integral equations of the first kind in the classical Holder space setting and of both integral equations of the first and second kind in the contemporary Sobolev space setting. In the numerical solution part of the book, the author included a new collocation method for two-dimensional hypersingular boundary integral equations and a collocation method for the three-dimensional Lippmann-Schwinger equation. The final chapter of the book on inverse boundary value problems for the Laplace equation has been largely rewritten with special attention to the trilogy of decomposition, iterative and sampling methods
This book is an excellent introductory text for students, scientists, and engineers who want to learn the basic theory of linear integral equations and their numerical solution.
This is a good introductory text book on linear integral equations. It contains almost all the topics necessary for a student. The presentation of the subject matter is lucid, clear and in the proper modern framework without being too abstract.
In order tomake the book accessible not only to mathematicians but also to physicists and engineers I have planned it as self-contained as possible by requiring only a solid foundation in differential and integral calculus and, for parts of the book, in complex function theory. Some background in functional analysis will be helpful, but the basic concepts of the theory of normed spaces will be briefly reviewed, and all functional analytic tools which are relevant in the study of integral equations will be developed in the book. Of course, I expect the reader to be willing to accept the functional analytic language for describing the theory and the numerical solution of integral equations. I hope that I succeeded in finding the adequate compromise between presenting integral equations in the proper modern framework and the danger of being too abstract.
An introduction to integral equations cannot present a complete picture of all classical aspects of the theory and of all recent developments. In this sense, this book intends to tell the reader what I think appropriate to teach students in a twosemester course on integral equations. I am willing to admit that the choice of a few of the topics might be biased by my own preferences and that some important subjects are omitted.

Title: Linear Integral Equations
Edition: 3rd Edition
Author(s): Rainer Kress
Publisher:
Series: Applied Mathematical Sciences
Year: 2014
Pages: 427
Type: PDF
Language: English
ISBN: 978-1-4614-9592-5,978-1-4614-9593-2
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Linear Integral Equations (3rd Edition) written by Rainer Kress cover the following topics.
1. Introduction and Basic Functional Analysis
2. Bounded and Compact Operators
3. Riesz Theory
4. Dual Systems and Fredholm Alternative
5. Regularization in Dual Systems
6. Potential Theory
7. Singular Boundary Integral Equations
8. Sobolev Spaces
9. The Heat Equation
10. Operator Approximations
11. Degenerate Kernel Approximation
12. QuadratureMethods
13. ProjectionMethods
14. Iterative Solution and Stability
15. Equations of the First Kind
16. Tikhonov Regularization
17. Regularization by Discretization
18. Inverse Boundary Value Problems
Problems
References
Index

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