Linear Integral Equations

1. Front Matter

   Pages I-XI

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2. Normed Spaces

   • Rainer Kress

   Pages 1-12

3. Bounded and Compact Operators

   • Rainer Kress

   Pages 13-24

4. The Riesz Theory

   • Rainer Kress

   Pages 25-35

5. Dual Systems and Fredholm Theory

   • Rainer Kress

   Pages 36-49

6. Regularization in Dual Systems

   • Rainer Kress

   Pages 50-57

7. Potential Theory

   • Rainer Kress

   Pages 58-81

8. Singular Integral Equations

   • Rainer Kress

   Pages 82-107

9. Sobolev Spaces

   • Rainer Kress

   Pages 108-131

10. The Heat Equation

    • Rainer Kress

    Pages 132-141

11. Operator Approximations

    • Rainer Kress

    Pages 142-153

12. Degenerate Kernel Approximation

    • Rainer Kress

    Pages 154-167

13. Quadrature Methods

    • Rainer Kress

    Pages 168-183

14. Projection Methods

    • Rainer Kress

    Pages 184-205

15. Iterative Solution and Stability

    • Rainer Kress

    Pages 206-220

16. Equations of the First Kind

    • Rainer Kress

    Pages 221-242

17. Tikhonov Regularization

    • Rainer Kress

    Pages 243-258

18. Regularization by Discretization

    • Rainer Kress

    Pages 259-269

19. Inverse Scattering Theory

    • Rainer Kress

    Pages 270-288

20. Back Matter

    Pages 289-301

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I fell in love with integral equations about twenty years ago when I was working on my thesis, and I am still attracted by their mathematical beauty. This book will try to stimulate the reader to share this love with me. Having taught integral equations a number of times I felt a lack of a text which adequately combines theory, applications and numerical methods. Therefore, in this book I intend to cover each of these fields with the same weight. The first part provides the basic Riesz-Fredholm theory for equa­ tions of the second kind with compact opertors in dual systems including all functional analytic concepts necessary for developing this theory. The second part then illustrates the classical applications of integral equation methods to boundary value problems for the Laplace and the heat equation as one of the main historical sources for the development of integral equations, and also in­ troduces Cauchy type singular integral equations. The third part is devoted to describing the fundamental ideas for the numerical solution of integral equa­ tions. Finally, in a fourth part, ill-posed integral equations of the first kind and their regularization are studied in a Hilbert space setting. In order to make the book accessible not only to mathematicans 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.

• Hilbert space
• Integral calculus
• Integral equation
• Riesz-Fredholm theory
• calculus
• differential equation
• functional analysis
• numerical methods
• numerical solutions
• x integral equations

• Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, Fed. Rep. of Germany

  Rainer Kress

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