Linear Integral Equations

1. Front Matter

   Pages i-xiv

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2. Normed Spaces

   • Rainer Kress

   Pages 1-14

3. Bounded and Compact Operators

   • Rainer Kress

   Pages 15-27

4. Riesz Theory

   • Rainer Kress

   Pages 28-38

5. Dual Systems and Fredholm Alternative

   • Rainer Kress

   Pages 39-54

6. Regularization in Dual Systems

   • Rainer Kress

   Pages 55-66

7. Potential Theory

   • Rainer Kress

   Pages 67-93

8. Singular Integral Equations

   • Rainer Kress

   Pages 94-124

9. Sobolev Spaces

   • Rainer Kress

   Pages 125-151

10. The Heat Equation

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    Pages 152-162

11. Operator Approximations

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    Pages 163-176

12. Degenerate Kernel Approximation

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    Pages 177-196

13. Quadrature Methods

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    Pages 197-217

14. Projection Methods

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    Pages 218-246

15. Iterative Solution and Stability

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    Pages 247-264

16. Equations of the First Kind

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    Pages 265-289

17. Tikhonov Regularization

    • Rainer Kress

    Pages 290-307

18. Regularization by Discretization

    • Rainer Kress

    Pages 308-319

19. Inverse Boundary Value Problems

    • Rainer Kress

    Pages 320-346

20. Back Matter

    Pages 347-367

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In the ten years since the first edition of this book appeared, integral equations and integral operators have revealed more of their mathematical beauty and power to me. Therefore, I am pleased to have the opportunity to share some of these new insights with the readers of this book. As in the first edition, the main motivation is to present the fundamental theory of integral equations, some of their main applications, and the basic concepts of their numerical solution in a single volume. This is done from my own perspective of integral equations; I have made no attempt to include all of the recent developments. In addition to making corrections and adjustments throughout the text and updating the references, the following topics have been added: In Sec­ tion 4.3 the presentation of the Fredholm alternative in dual systems has been slightly simplified and in Section 5.3 the short presentation on the index of operators has been extended. The treatment of boundary value problems in potential theory now includes proofs of the jump relations for single-and double-layer potentials in Section 6.3 and the solution of the Dirichlet problem for the exterior of an arc in two dimensions (Section 7.6). The numerical analysis of the boundary integral equations in Sobolev space settings has been extended for both integral equations of the first kind in Section 13.4 and integral equations of the second kind in Section 12.4.

• Boundary value problem
• Integral Equations
• Integral calculus
• Integral equation
• Linear Integral Equations
• Sobolev space
• calculus
• compactness
• functional analysis
• logarithm
• minimum
• numerical methods

• Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, Germany

  Rainer Kress

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