Periodic Integral and Pseudodifferential Equations with Numerical Approximation

✍ Scribed by Jukka Saranen, Gennadi Vainikko

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 461
• Series: Springer Monographs in Mathematics
• Edition: Softcover reprint of hardcover 1st ed. 2002
• Category: Library

✦ Synopsis

An attractive book on the intersection of analysis and numerical analysis, deriving classical boundary integral equations arising from the potential theory and acoustics. This self-contained monograph can be used as a textbook by graduate/postgraduate students. It also contains a lot of carefully chosen exercises.

✦ Table of Contents

Preface
Contents
1. Preliminaries
1.1 Preliminaries from Operator Theory
1.2 Fredholm Theorems
1.3 The Fredholm Operators
1.4 The Regularizers
1.5 Krylov Subspace Methods
2. Single Layer and Double Layer Potentials
2.1 Classical Boundary Value Problems
2.2 Fundamental Solution
2.3 An Integral Representation of Functions
2.4 Jordan Ares and Curves
2.5 Boundary Potentials
3. Solution of Boundary Value Problems by Integral Equations
3.1 Integral Equations for Boundary Value Problems
3.2 Solution of the Laplace Equation by Integral Equations of the Second Kind
3.3 Solution of the HeImholtz Equation by Integral Equations of the Second Kind
4. Singular Integral Equations
4.1 Singular Integral Equations in Hölder Spaces
4.2 L2-Theory of Singular Integral Equations
5. Boundary Integral Operators in Periodic Sobolev Spaces
5.1 Distributions on the Real Line
5.2 Periodic Distributions
5.3 Periodic Sobolev Spaces
5.4 Finite Part of Hypersingular Integrals
5.5 Spectral Representation of a Convolution Integral Operator
5.6 Symm's Integral Operator
5.7 Hilbert Integral Operator
5.8 Cauchy Integral Operator on the Unit Circle
5.9 Cauchy Integral Operator on a Jordan Curve
5.10 Hypersingular Integral Operator
5.11 Biharmonic Problem
5.12 Operator Interpolation
5.13 Multiplication of Functions in H^λ
6. Periodic Integral Equations
6.1 Boundedness of Integral Operators Between Sobolev Spaces
6.2 Fredholmness of Integral Operators Between Sobolev Spaces
6.3 A Class of Periodic Integral Equations
6.4 Examples of Periodic Integral Equations
6.5 Analysis of the Modified Symm's Equations
6.6 A General Class of Periodic Integral Equations
6.7 Equations with Analytic Coefficient Functions
7. Periodic Pseudodifferential Operators
7.1 Prolongation of a Function Defined on Z
7.2 Two Definitions of PPDO and Their Equivalence
7.3 Boundedness of a PPDO
7.4 Asymptotic Expansion of the Symbol
7.5 Amplitudes
7.6 Asymptotic Expansion of Integral Operators
7.7 The Symbol of Dual and Adjoint Operators
7.8 The Symbol of the Composition of PPDOs
7.9 Pseudolocality
7.10 Elliptic PPDOs
7.11 Gårding's Inequality
7.12 Estimation of the Operator Norm
7.13 Classical PPDOs
7.14 Integral Operator Representation of Classical PPDOs
7.15 Functions κ^±_α(t)
8. Trigonometrie Interpolation
8.1 Subspace Tn
8.2 Orthogonal Projection
8.3 Interpolation Projection
8.4 Exponential Approximation Order
8.5 Two Dimensional Interpolation
9. Galerkin Method and Fast Solvers
9.1 Precondition of the Problem
9.2 Galerkin Method for the Preconditioned Problem
9.3 Matrix Representation of a PIO
9.4 A Full Discretization
9.5 Using Asymptotic Expansions
9.6 Stability Estimates
9.7 Regularization via Discretization
9.8 Standard Galerkin Method
10. Trigonometrie Collocation
10.1 Collocation Problem
10.2 Full Discretization
10.3 Modifications
10.4 Further Discrete Versions
10.5 Fast Solvers for Lippmann-Schwinger Equation
11. Integral Equations on an Open Arc
11.1 Equations on an Interval
11.2 Periodization
11.3 Even and Odd Operators
11.4 Analysis of the Periodic Problem
11.5 More About Convolution Operators on (-1,1)
11.6 Collocation Solution
11. 7 Fully Discrete Collocation Methods
12. Quadrature Methods
12.1 The Idea of a Quadrature Method
12.2 A Simple Quadrature Method
12.3 The ε-quadrature Method
12.4 A Modified Quadrature Method
12.5 Integral Equations of the Second Kind
12.6 Singular Integral Equations
12.7 Hypersingular Equations
12.8 Extensions by Localization
13. Spline Approximation Methods
13.1 Spline Spaces
13.2 Splines on Uniform Meshes
13.3 Approximation and Inverse Properties of Splines
13.4 Discrete Fourier Transform of Periodic Functions
13.5 Spline Collocation on Uniform Mesh
13.6 An Abstract Galerkin Method
13.7 The Spline Galerkin Method
13.8 Some Extensions of the Basic Methods
Bibliography
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Index
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