Wavelet Based Approximation Schemes for Singular Integral Equations

Cover
Title Page
Copyright Page
Preface
Table of Contents
1. Introduction
1.1 Singular Integral Equation
1.1.1 Approximate solution of integral equations
1.1.1.1 The general scheme of approximation
1.1.1.2 Nyström method
1.1.1.3 Collocation method
1.1.1.4 Galerkin’s method
1.1.1.5 Quadratic spline collocation method
1.1.1.6 Method based on product integration
1.1.2 Kernel with weak (logarithmic and algebraic) singularity
1.1.3 Integral equations with Cauchy singular kernel
1.1.3.1 Method based on Legendre polynomials
1.1.3.2 Method based on Chebyshev polynomials
1.1.3.3 Method based on Jacobi polynomials
1.1.4 Integral equations with hypersingular kernel
2. Multiresolution Analysis of Function Spaces
2.1 Multiresolution Analysis of L2(R)
2.1.1 Multiresolution generator
2.1.2 Wavelets
2.1.3 Basis with compact support
2.1.4 Properties of elements in Daubechies family
2.1.5 Limitation of scale functions and wavelets in Daubechies family
2.2 Multiresolution Analysis of L2([a, b] ⊂ R)
2.2.1 Truncated scale functions and wavelets
2.2.2 Multiwavelets
2.2.3 Orthonormal (boundary) scale functions and wavelets
2.3 Others
2.3.1 Sinc function
2.3.2 Coiflet
2.3.3 Autocorrelation function
3. Approximations in Multiscale Basis
3.1 Multiscale Approximation of Functions
3.1.1 Approximation of f in the basis of Daubechies family
3.1.1.1 f ∈ L2(R)
3.1.1.2 Orthonormal basis for L2([a, b])
3.1.1.3 Truncated basis
3.1.2 Approximation of f ∈ L2([0, 1]) in multiwavelet basis
3.2 Sparse Approximation of Functions in Higher Dimensions
3.2.1 Basis for Ω ⊆ R2
3.2.1.1 Representation of f (x, y)
3.2.1.2 Homogeneous function K (λx, λy) = λμ K (x – y), μ ∈ R
3.2.1.3 Non-smooth function f (x, y) = |x + y|ν, ν ∈ R – {N ⋃ 0}
3.2.1.4 f (x, y) = ln|x ± y| involving logarithmic singularity
3.2.1.5 f ∈ Ω ⊂ R2
3.3 Moments
3.3.1 Scale functions and wavelets in R
3.3.2 Truncated scale functions and wavelets
3.3.3 Boundary scale functions and wavelets
3.4 Quadrature Rules
3.4.1 Daubechies family
3.4.1.1 Nodes, weights and quadrature rules
3.4.1.2 Formal orthogonal polynomials, nodes, weights of scale functions
3.4.1.3 Interior scale functions
3.4.1.4 Boundary scale functions (Φleft on R+, Φright on R–)
3.4.1.5 Truncated scale functions (ΦLT, ΦRT on [0, 2K – 1])
3.4.1.6 Formal orthogonal polynomials, nodes, weights of wavelets
3.4.1.7 Algorithm
3.4.1.8 Error estimates
3.4.1.9 Numerical illustrations
3.4.2 Quadrature rules for singular integrals
3.4.2.1 Integrals with logarithmic singularity
3.4.2.2 Quadrature rule for weakly (algebraic) singular integrals
3.4.2.3 Quadrature rule for Cauchy principal value integrals
3.4.2.4 Finite part integrals
3.4.2.5 Composite quadrature formula for integrals having Cauchy and weak singularity
3.4.2.6 Numerical examples
3.4.3 Logarithmic singular integrals
3.4.4 Cauchy principal value integrals
3.4.5 Hypersingular integrals
3.4.6 For multiwavelet family
3.4.7 Others
3.4.7.1 Sinc functions
3.4.7.2 Autocorrelation functions
3.4.7.3 Representation of function and operator in the basis generated by autocorrelation function
3.5 Multiscale Representation of Differential Operators
3.6 Representation of the Derivative of a Function in LMW Basis
3.7 Multiscale Representation of Integral Operators
3.7.1 Integral transform of scale function and wavelets
3.7.2 Regularization of singular operators in LMW basis
3.7.2.1 Principle of regularization
3.7.2.2 Regularization of convolution operator in LMW basis
3.8 Estimates of Local Hölder Indices
3.8.1 Basis in Daubechies family
3.8.2 Basis in Multiwavelet family
3.9 Error Estimates in the Multiscale Approximation
3.10 Nonlinear/Best n-term Approximation
4. Multiscale Solution of Integral Equations with Weakly Singular Kernels
4.1 Existence and Uniqueness
4.2 Logarithmic Singular Kernel
4.2.1 Projection in multiscale basis
4.2.1.1 Basis in Daubechies family
4.2.1.2 LMW basis
4.3 Kernels with Algebraic Singularity
4.3.1 Existence and uniqueness
4.3.2 Approximation in multiwavelet basis
4.3.2.1 Scale functions
4.3.2.2 Scale functions and wavelets
4.3.2.3 Wavelets
4.3.2.4 Multiscale approximation (regularization) of integral operator KA in LMW basis
4.3.2.5 Reduction to algebraic equations
4.3.2.6 Multiscale approximation of solution
4.3.2.7 Error Estimates
4.3.3 Approximation in other basis
5. An Integral Equation with Fixed Singularity
5.1 Method Based on Scale Functions in Daubechies Family
5.1.1 Basic properties of Daubechies scale function and wavelets
5.1.2 Method of solution
5.1.3 Numerical results
6. Multiscale Solution of Cauchy Singular Integral Equations
6.1 Prerequisites
6.2 Basis Comprising Truncated Scale Functions in Daubechies Family
6.2.1 Evaluation of matrix elements
6.2.1.1 k , k' ∈ ⋀VIj
6.2.1.2 k ∈ ⋀VITj , k' ∈ ⋀VLTj
6.2.1.3 k ∈ ⋀VLTj , k' ∈ ⋀VLTj
6.2.1.4 k ∈ ⋀VLTj , k' ∈ ⋀VRTj
6.2.1.5 k ∈ ⋀VITj , k' ∈ ⋀VRTj
6.2.1.6 k ∈ ⋀VRTj , k' ∈ ⋀VRTj
6.2.2 Evaluation of fTj
6.2.3 Estimate of error
6.2.4 Illustrative examples
6.3 Multiwavelet Family
6.3.1 Equation with constant coefficients
6.3.1.1 Evaluation of integrals
6.3.1.2 Multiscale representation (regularization) of the operator KC in LMW basis
6.3.1.3 Multiscale approximation of solution
6.3.1.4 Estimation of error
6.3.1.5 Illustrative examples
6.3.2 Cauchy singular integral equation with variable coefficients
6.3.2.1 Evaluation of integrals involving function, Cauchy singular kernel and elements in LMW basis
6.3.2.2 Evaluation of the integrals involving product of a(x), scale functions and wavelets
6.3.2.3 Multiscale representation (regularization) of the operator ωKC in LMW basis
6.3.2.4 Multiscale approximation of solution
6.3.2.5 Estimate of Hölder exponent of u(x) at the boundaries
6.3.2.6 Estimation of error
6.3.2.7 Applications to problems in elasticity
6.3.3 Equation of first kind
6.3.3.1 Evaluation of integrals involving kernel with fixed singularity and elements in the LMW basis
6.3.3.2 Evaluation of integrals involving kernel with fixed singularity and weight factor
6.3.3.3 Multiscale representation (regularization) of the operator ωKF in LMW basis
6.3.3.4 Multiscale approximation of solution
6.3.3.5 Illustrative examples
6.3.4 Autocorrelation function family
6.3.5 In R
6.3.5.1 Transformation to the finite range of integration
6.3.5.2 Multiscale approximation of solution
6.3.5.3 Estimation of error
6.3.5.4 Illustrative examples
6.3.6 Other families
6.3.6.1 Hilbert transform
6.3.6.2 Integral equation of second kind
7. Multiscale Solution of Hypersingular Integral Equations of Second Kind
7.1 Finite Part Integrals Involving Hypersingular Functions
7.2 Existing Methods
7.3 Reduction to Cauchy Singular Integro-differential Equation
7.4 Method Based on LMW Basis
7.4.1 Multiscale approximation of the solution
7.4.2 Estimation of error
7.4.3 Illustrative examples
7.5 Other Families
Appendices
References
Author Index
Subject Index