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Full Description
Up till now there has been no text on integral equations adequately combining theory, applications and numerical methods: this book sets out to cover each of these areas with the same weighting. The first part provides the basic Riesz-Fredholm theory for equations of the second kind with compact operators in dual systems, including all functional analytic concepts necessary. The second part then illustrates the classical applications of integral equation methods to boundary value problems for the Laplace equation and the heat equation as one of the main historical sources for the development of integral equations, and also introduces Cauchy type singular integral equations. The third part describes the fundamental ideas for the numerical solutions of integral equations. In the final part, ill-posed integral equations of the first kind and their regularization are studied in a Hilbert space setting. So that the book is accessible not only to mathematicians but also to physicists and egineers, it is kept as self-contained as possible, requiring only a solid foundation in differential and integral calculus and, for some parts in complex function theory.
Some background in functional analysis will be helpful, but the basic concepts needed are reviewed or developed within the book: the reader must of course be willing to accept the modern functional analytic language for describing the theory and the numerical solution of integral equations.
Contents
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 Nystrom's Method.- 12.3 Nystrom'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.
