Author | Kress, Rainer. author |
---|---|
Title | Linear Integral Equations [electronic resource] / by Rainer Kress |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1989 |
Connect to | http://dx.doi.org/10.1007/978-3-642-97146-4 |
Descript | XI, 299 p. online resource |
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 Nystrรถmโs Method -- 12.3 Nystrรถmโs 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