Author | Kress, Rainer. author |
---|---|
Title | Linear Integral Equations [electronic resource] / by Rainer Kress |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1999 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0559-3 |
Descript | XIV, 367 p. online resource |
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 Hรถlder 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 Nystrรถmโs Method -- 12.3 Weakly Singular Kernels -- 12.4 Nystrรถmโs 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