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.
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ISBN-13
978-1-4612-0559-3 (9781461205593)
DOI
10.1007/978-1-4612-0559-3
Schweitzer Klassifikation
1 Normed Spaces.- 1.1 Convergence and Continuity.- 1.2 Completeness.- 1.3 Compactness.- 1.4 Scalar Products.- 1.5 Best Approximation.- Problems.- 2 Bounded and Compact Operators.- 2.1 Bounded Operators.- 2.2 Integral Operators.- 2.3 Neumann Series.- 2.4 Compact Operators.- Problems.- 3 Riesz Theory.- 3.1 Riesz Theory for Compact Operators.- 3.2 Spectral Theory for Compact Operators.- 3.3 Volterra Integral Equations.- Problems.- 4 Dual Systems and Fredholm Alternative.- 4.1 Dual Systems via Bilinear Forms.- 4.2 Dual Systems via Sesquilinear Forms.- 4.3 The Fredholm Alternative.- 4.4 Boundary Value Problems.- Problems.- 5 Regularization in Dual Systems.- 5.1 Regularizers.- 5.2 Normal Solvability.- 5.3 Index.- Problems.- 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 Nonsmooth Boundaries.- Problems.- 7 Singular Integral Equations.- 7.1 Hölder Continuity.- 7.2 The Cauchy Integral Operator.- 7.3 The Riemann Problem.- 7.4 Integral Equations with Cauchy Kernel.- 7.5 Cauchy Integral and Logarithmic Potential.- 7.6 Logarithmic Single-Layer Potential on an Arc.- Problems.- 8 Sobolev Spaces.- 8.1 The Sobolev Space Hp[0, 2?].- 8.2 The Sobolev Space Hp(?).- 8.3 Weak Solutions to Boundary Value Problems.- Problems.- 9 The Heat Equation.- 9.1 Initial Boundary Value Problem: Uniqueness.- 9.2 Heat Potentials.- 9.3 Initial Boundary Value Problem: Existence.- Problems.- 10 Operator Approximations.- 10.1 Approximations via Norm Convergence.- 10.2 Uniform Boundedness Principle.- 10.3 Collectively Compact Operators.- 10.4 Approximations via Pointwise Convergence.- 10.5 Successive Approximations.- Problems.- 11 Degenerate Kernel Approximation.- 11.1 Degenerate Operators andKernels.- 11.2 Interpolation.- 11.3 Trigonometric Interpolation.- 11.4 Degenerate Kernels via Interpolation.- 11.5 Degenerate Kernels via Expansions.- Problems.- 12 Quadrature Methods.- 12.1 Numerical Integration.- 12.2 Nyström's Method.- 12.3 Weakly Singular Kernels.- 12.4 Nyström's Method in Sobolev Spaces.- Problems.- 13 Projection Methods.- 13.1 The Projection Method.- 13.2 Projection Methods for Equations of the Second Kind.- 13.3 The Collocation Method.- 13.4 Collocation Methods for Equations of the First Kind.- 13.5 The Galerkin Method.- Problems.- 14 Iterative Solution and Stability.- 14.1 Stability of Linear Systems.- 14.2 Two-Grid Methods.- 14.3 Multigrid Methods.- 14.4 Fast Matrix-Vector Multiplication.- Problems.- 15 Equations of the First Kind.- 15.1 Ill-Posed Problems.- 15.2 Regularization of 1ll-Posed Problems.- 15.3 Compact Self-Adjoint Operators.- 15.4 Singular Value Decomposition.- 15.5 Regularization Schemes.- Problems.- 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.- Problems.- 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.- Problems.- 18 Inverse Boundary Value Problems.- 18.1 Ill-Posed Equations in Potential Theory.- 18.2 An Inverse Problem in Potential Theory.- 18.3 Approximate Solution via Potentials.- 18.4 Differentiability with Respect to the Boundary.- Problems.- References.
