Linear Integral Equations

2., Softcover Nachdruck of the original 2. Auflage 1999
ISBN: 978-1-4612-6817-8
Verlag: Springer

Buch, Englisch, Band 82, 367 Seiten, PB, Format (B × H): 155 mm x 235 mm, Gewicht: 587 g

Reihe: Applied Mathematical Sciences

The result of the author's fascination with the mathematical beauty of integral equations, this book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter.

Zielgruppe

Research

Autoren/Hrsg.

Kress, Raimer

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Integralrechnungen- und -gleichungen

Inhaltsverzeichnis

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 and Kernels.- 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.