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.
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ISBN-13
978-3-642-97146-4 (9783642971464)
DOI
10.1007/978-3-642-97146-4
Schweitzer Klassifikation
1. Normed Spaces.- 1.1 Convergence and Continuity.- 1.2 Open and Closed Sets.- 1.3 Completeness.- 1.4 Compactness.- 1.5 Scalar Products.- 1.6 Best Approximation.- 2. Bounded and Compact Operators.- 2.1 Bounded Operators.- 2.2 Integral Operators.- 2.3 Neumann Series.- 2.4 Compact Operators.- 3. The Riesz Theory.- 3.1 Riesz Theory for Compact Operators.- 3.2 Spectral Theory for Compact Operators.- 3.3 Volterra Integral Equations.- 4. Dual Systems and Fredholm Theory.- 4.1 Dual Systems Via Bilinear Forms.- 4.2 Dual Systems Via Sesquilinear Forms.- 4.3 Positive Dual Systems.- 4.4 The Fredholm Alternative.- 4.5 Boundary Value Problems.- 5. Regularization in Dual Systems.- 5.1 Regularizers.- 5.2 Normal Solvability.- 5.3 Index.- 6. Potential Theory.- 6.1 Harmonic Functions.- 6.2 Boundary Value Problems: Uniqueness.- 6.3 Surface Potentials.- 6.4 Boundary Value Problems: Existence.- 6.5 Supplements.- 7. Singular Integral Equations.- 7.1 Holder Continuity.- 7.2 The Cauchy Integral Operator.- 7.3 The Riemann Problem.- 7.4 Singular Integral Equations with Cauchy Kernel.- 7.5 Cauchy Integral and Logarithmic Potential.- 7.6 Supplements.- 8. Sobolev Spaces.- 8.1 Fourier Expansion.- 8.2 The Sobolev Space Hp[0, 2?].- 8.3 The Sobolev Space Hp[?].- 8.4 Weak Solutions to Boundary Value Problems.- 9. The Heat Equation.- 9.1 Initial Boundary Value Problem: Uniqueness.- 9.2 Heat Potentials.- 9.3 Initial Boundary Value Problem: Existence.- 10. Operator Approximations.- 10.1 Approximations Based on Norm Convergence.- 10.2 Uniform Boundedness Principle.- 10.3 Collectively Compact Operators.- 10.4 Approximations Based on Pointwise Convergence.- 10.5 Successive Approximations.- 11. Degenerate Kernel Approximation.- 11.1 Finite Dimensional Operators.- 11.2 Degenerate Kernels Via Interpolation.-11.3 Degenerate Kernels Via Expansions.- 12. Quadrature Methods.- 12.1 Numerical Integration.- 12.2 Nyström's Method.- 12.3 Nyström's Method for Weakly Singular Kernels.- 13. Projection Methods.- 13.1 The Projection Method.- 13.2 The Collocation Method.- 13.3 The Galerkin Method.- 14. Iterative Solution and Stability.- 14.1 The Method of Residual Correction.- 14.2 Multi-Grid Methods.- 14.3 Stability of Linear Systems.- 15. Equations of the First Kind.- 15.1 Ill-Posed Problems.- 15.2 Regularization of Ill-Posed Problems.- 15.3 Compact Self Adjoint Operators.- 15.4 Singular Value Decomposition.- 15.5 Regularization Schemes.- 16. Tikhonov Regularization.- 16.1 The Tikhonov Functional.- 16.2 Weak Convergence.- 16.3 Quasi-Solutions.- 16.4 Minimum Norm Solutions.- 16.5 Classical Tikhonov Regularization.- 17. Regularization by Discretization.- 17.1 Projection Methods for Ill-Posed Equations.- 17.2 The Moment Method.- 17.3 Hilbert Spaces with Reproducing Kernel.- 17.4 Moment Collocation.- 18. Inverse Scattering Theory.- 18.1 Ill-Posed Integral Equations in Potential Theory.- 18.2 An Inverse Acoustic Scattering Problem.- 18.3 Numerical Methods in Inverse Scattering.
