Wavelet Based Approximation Schemes for Singular Integral Equations [1 ed.] 9780367199173

Table of contents :
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

Citation preview

Wavelet Based Approximation Schemes for Singular Integral Equations M M Panja Department of Mathematics Visva-Bharati, Santiniketan, India

B N Mandal Professor (retired), Physics and Applied Mathematics Unit Indian Statistical Institute, Kolkata, India

p, p,

A SCIENCE PUBLISHERS BOOK A SCIENCE PUBLISHERS BOOK

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2020 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20200414 International Standard Book Number-13: 978-0-3671-9917-3 (Hardback)

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Preface

The mathematical modelling of physical processes and the computational mathematics have complemented each other since the era of Newton. The underlying principle of numerical technique in computational mathematics was the interpolation based on the solid foundation, the Weierstrass approximation theorem. It is well known that here unknown functions are represented/approximated in polynomial basis whose coefficients are determined by the values of the unknown function prescribed at some points within its domain of definition. But around 1900, mathematician Runge observed that approximation based on interpolation scheme is unable to represent functions efficiently which are continuous and even differentiable within its domain of definition. On the other hand after the development of the formal theory of function spaces, another computational scheme known as Fourier approximation has been developed. Here the unknown functions are approximated/ represented by the linear combination of harmonics with coefficients involving integrals of the unknown function and the corresponding harmonics. This scheme is found to be well suited for approximating unknown solutions of differential and integral equations (arising in the mathematical analysis of physical processes) which are smooth enough within the domain of interest. Again, around 1900, J W Gibbs pointed out that approximation based on harmonics (trigonometric function, in particular) is unable to represent functions having finite discontinuities in their domain. Over and above, estimation of error in approximation of function in the numerical methods based on classical harmonics requires exhaustive mathematical analysis. It is thus desirable to search for a computational scheme which can effectively approximate functions which are smooth in most of the region but may have sharp variations within a narrow region, even may have finite/infinite discontinuities within the domain of interest as well as provide a posteriori error in a straightforward way. One of our objectives here is to present a computational scheme based on a novel mathematical structure, known as multiresolution analysis (MRA) of function space which may be regarded as the confluence of several existing computational schemes as well as a mathematical microscope. We will concentrate here on the L2-space or its subspace. The scheme of our presentation is as follows: • An overview of MRA of L2(R)/L2([a, b]) • Multiresolution approximation of functions and operators in L2(R)/L2([a, b]) • Wavelet based computational schemes for getting approximate solution of integral equations of second kind with singular kernels, in particular. In many fields of application of mathematics, progress is crucially dependent on the good flow of information between (i) theoretical mathematicians looking for applications, (ii) mathematicians working on applications in need of theory, and (iii) scientists and engineers applying mathematical models and methods. The intention of this book is to stimulate this flow of information. In the first chapter some mathematical prerequisites of singular integral equations have been presented. The underlying mathematical structure of wavelet bases as desired in this monograph have been described in chapter two. In chapter three, mathematical formulae and tricks for approximation of functions, representation of (differential and integral) operators have been described in somewhat details. The efficiency

iv < Wavelet Based Approximation Schemes for Singular Integral Equations

of the formulae derived here have been tested through their applications to the relevant test problems. The knowledge and techniques developed in the last two chapters have been applied in subsequent chapters for obtaining approximate solutions of Fredholm integral equation of second kind with a variety of singular kernels. In chapter four we have considered weakly singular kernel with both logarithmic and algebraic types. Chapter 5 deals with a Fredholm integral equation of second kind with a special type of kernel having singularity at a fixed point. Fredholm integral equation of second kind with Cauchy singular kernels in both bounded and unbounded domain have been studied in Chapter 6. The singular integral equation with hypersingular kernels have been discussed in Chapter 7. Some numerical data for several ingredients involved in the wavelet based numerical scheme for obtaining approximate solution of integral equations, with singular coefficients or kernels, in particular have been presented in the Appendices. The authors thank Dr. Swaraj Paul for providing some material of this book prepared jointly during the tenure of his Ph.D. MMP is thankful to Dr. Prakash Das, Debabrata, Sayan, and Mouzakkir for their participation in preparation of some results and help in typing of this monograph. He is deeply indebted to his family members, wife Manju, brother Amit, children Dibya, Rohini and Rivu in particular, who provided him with their continued encouragement, patience and support during the preparation of this book. The authors would highly appreciate any correspondence concerning constructive suggestion. Finally, the authors thank Mr. Vijay Primlani of Science Publishers (CRC Press) for his support and patience in the preparation of the monograph. M M Panja B N Mandal

Contents Preface 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

iii

1

1 2 3 4 6 7 9 10 11 13 13 14 15 20 22

23 23 23 24 25 27 27

28 33 36 40 40 43 47

vi < Wavelet Based Approximation Schemes for Singular Integral Equations

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

50

50 51 51 53 55 56 58 58 60 61 63 63 64 69 69 69 70 71 71 73 74 74 78 80 80 83 85 86 91 92 96 100 105 106 108 110 111 111 113

Contents < vii

3.4.7

3.5 3.6 3.7

3.8

3.9 3.10

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 Multiscale Representation of Differential Operators Representation of the Derivative of a Function in LMW Basis 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 Estimates of Local Hölder Indices 3.8.1 Basis in Daubechies family 3.8.2 Basis in Multiwavelet family Error Estimates in the Multiscale Approximation 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

114 114 115 116 117 118 121 121 125 125 125 126 126 126 128 134 135

135 137 137 137 138 143 143 143 144 147 148 150 150 152 152 157 162

163 163 165 169

viii < Wavelet Based Approximation Schemes for Singular Integral Equations

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' ∈ ⋀Vj I

6.2.1.2

6.2.1.3

6.2.1.4 6.2.1.5

k ∈ ⋀Vj IT , k' ∈ ⋀Vj LT k∈⋀ k∈⋀ k∈⋀

6.2.1.6 k ∈ ⋀

V LT j

V LT j

, k' ∈ ⋀

, k' ∈ ⋀

V RT j

176 177

V RT j

178

V RT j

178

, k' ∈ ⋀

V RT j

6.2.2 Evaluation of f 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 T j

172 174 175 176

V LT j

, k' ∈ ⋀

V IT j

171

179 179

180 180 184 184 185 188 188 189 190 193 193 198 199 200 201 201 202 208 210 215 220 221 222 230

Contents < ix

6.3.5

6.3.6

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 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 7.2 7.3 7.4

Finite Part Integrals Involving Hypersingular Functions Existing Methods Reduction to Cauchy Singular Integro-differential Equation 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

233 234 236 236 237 239 239 241 244

244 246 247 248 249 250 250 254

Appendices

258

References

269

Author Index

283

Subject Index

285

Chapter 1

Introduction 1.1

Singular Integral Equation

In mathematics, singular integrals and integral operators with singular kernels have a well-established theoretical basis (Muskhelishvili, 2013; Mikhlin, 2014). For example, the weakly singular (WS) integrals are considered as improper integrals, the singular integrals are considered in the sense of Cauchy principal values (CPV) and the hypersingular integrals integrals are considered in the sense of Hadamard finite parts (FP) (Muskhelishvili, 2013; Mikhlin, 2014; Kanwal, 1998). As for example, for a < x < b, b a

Man

N

B

and

ja

Pan

M

E→0

dal

x−E a

= lim

M

ln|x − y|dy

WS

ln|x − y|dy +

b x+E

ln|x − y |dy

= (b − x)ln(b − x) + (x − a)ln(x − a) − (b − a) b

WS a

1 dy |x − y|µ

=

lim

E1 →0,E2 →0

= b

CPV a

1 dy x−y

= lim

E→0

x−E1 1 |x−y |µ dy a

(x−a)1−µ 1−µ

+

x−E 1 x−y dy a

(b−x)1−µ 1−µ

+

b 1 dy x+E2 |x−y |µ

0= 0, k = 1, 2, · · · , n

(1.1.1.27)

dal

ja

M

M

Pan

and

B

N

Man

instead of condition (1.1.1.25) for the collocation method. The left side is the Fourier coefficient of rn associated with the element φk in the basis of Xn . If {φ1 , φ2 , ....., φn } are the leading members of an orthonormal family Φ ≡ {φ1 , φ2 , ....., φn , ....} that is complete in X, then (1.1.1.27) requires the leading terms to be zero in the Fourier expansion of rn with respect to Φ. n n ck! φk! (x), we apply (1.1.1.27) to (1.1.1.19) with To find the approximate solution un (x) = k! =1

xi = x. This yields the linear system n n

ck! {λ < φk! , φk > − < Kφk! , φk >} =< f, φk >, k = 1, 2, ...., n.

(1.1.1.28)

k! =1

This is Galerkin’s method for obtaining an approximate solution to (1.1.1.1). Now questions are: does the system of equations in (1.1.1.28) have any solution? If yes, is it unique? Does the resulting sequence of approximate solutions un converges to u in X ? Does the sequence converge in C(D)? Note also that, the above formulation contains double integrals < Kφk! , φi >. These must often be computed numerically. We return to a consideration of this later. To get answers to questions mentioned above it is convenient to write (1.1.1.28) in an abstract framework. We introduce here the projection operator Pn that maps X onto Xn . For general u ∈ X, define Pn u to be the solution of the following minimization problem u − Pn u = min u − v . v∈Xn

(1.1.1.29)

By assumption, Xn is finite dimensional. So, it can be shown that this problem has a solution; and by employing the fact that Xn is an inner product space, the solution can be shown to be unique. To obtain a better understanding of Pn , we give an explicit formula for Pn u. Introduce a new basis comprising elements {θ1 , θ2 , · · · , θn } for Xn by using Gram-Schmidt or any other method to

1.1. Singular Integral Equation

8

create an orthonormal basis from {φk , k = 1, · · · , n}. The elements {θk� , k � = 1, 2, · · · , n} are linear combinations of {φ1 , φ2 , ....., φn }, and moreover < θk , θk� >= δk k� ,

k, k � = 1, 2, · · · , n.

(1.1.1.30)

With this new basis, it is straightforward to show that Pn u =

n �

< u, θk > θk .

(1.1.1.31)

k=1

This shows immediately that Pn is a linear operator. With this formula, we can show the following results. 2 2 2 �u� = �Pn u� + �u − Pn u� , (1.1.1.32) 2

�Pn u� =

n �

2

|(u, θk )| ,

k=1

(Pn u, v) = (u, Pn v),

u, v ∈ X,

(1.1.1.33)

((Id − Pn )u, Pn v) = 0, u, v ∈ X.

(1.1.1.34)

Because of the last identity, the operator Pn u may be regarded as the orthogonal projection of u ∈ X onto Xn . Consequently, the operator Pn defined by (1.1.1.29) may be regarded as an orthogonal projection operator. The result (1.1.1.32) leads to

2

ja M

Pan

and

B

N

dal

(1.1.1.35)

M

Using (1.1.1.34), it can be established that

Man

�Pn � = 1.

2

2

�u − v� = �u − Pn u� + �Pn u − v� ,

v ∈ Xn .

(1.1.1.36)

This implies that Pn u is the unique solution to (1.1.1.29).

Using the fact that the elements {φk , k = 1, · · · , n} of the basis of Xn are independent,

Pn z = 0 if and only if (z, φi ) = 0,

i = 1, 2, · · · , n.

(1.1.1.37)

One can thus rewrite (1.1.1.27) as Pn rn = 0 or equivalently, Pn (λ − K)un = Pn f,

un ∈ Xn .

(1.1.1.38)

Note that this relation is similar to (1.1.1.26) appearing in the collocation method. There is a variant on Galerkin’s method, known as the Petrov-Galerkin method. Here, one chooses un ∈ Xn , but we require (rn , w) = 0,

∀ w ∈ Wn

where Wn is another finite dimensional subspace of dimension n. This method is not considered further in this monograph. It is an important method when looking at the numerical solution of boundary integral equations. Another variant to Galerkin’s method is to set it within a variational framework.

1.1. Singular Integral Equation

1.1.1.5

9

Quadratic spline collocation method

For n ∈ N, let us consider a grid Δn = {x0 , x1 , · · · , xn : 0 = x0 < x1 · · · < xn = 1}

(1.1.1.39)

(n)

on [0, 1] (a partition of the closed interval [0,1] with grid points xi ≡ xi , i = 0, 1, · · · , n). Definition 1.4. The grid Δn is said to be quasi-uniform if max (xi+1 − xi )

0≤i≤n−1

min (xi+1 − xi )

≤q

(1.1.1.40)

0≤i≤n−1

for some q ≥ 1 independent of n.

Definition 1.5. The partition is said to be a graded grid if

r

xi = 12 2i n x n2 +i = 1 − x n2 −i

i = 0, 1, · · · , n2 , i = 1, 2, · · · , n2 ,

(1.1.1.41)

where n ∈ 2N and r ≥ 1 a real number independent of the size of the number of nodes n + 1. Observation 1. The exponent r present in the definition characterizes the non-uniformity of the grid, e.g., the grid is uniform for the choice r = 1 which is densely clustered near the end points 0 and 1. dal

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Observation 2. The graded grid is not quasi-uniform for r > 1. Definition 1.6. The symbol S2,1 (Δn ) has been used as the collection S2,1 (Δn ) = {y(x) ∈ C 1 ([0, 1]) : y(x)|[xi ,xi+1 ] ∈ P2 , i = 0, 1, · · · , n − 1}

(1.1.1.42)

of quadratic splines with defect 1 on the grid Δn mentioned above. Here P2 is the collection of polynomials of degree not exceeding 2, C 1 ([0, 1]) is the set of all continuously differentiable functions y in [0, 1]. The explicit variable dependence of elements in S2,1 (Δn ) is ⎧ (x−xi−2 )2 ⎪ ⎪ ⎪ (xi −xi−2 )(xi−1 −xi−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (x−xi−2 )(xi −x) (xi+1 −x)(x−xi−1 ) ⎪ ⎨ (xi −xi−2 )(xi −xi−1 ) + (xi+1 −xi−1 )(xi −xi−1 ) B2,i (x) = ⎪ ⎪ (xi+1 −x)2 ⎪ ⎪ ⎪ (x −x ⎪ i+1 i−1 )(xi+1 −xi ) ⎪ ⎪ ⎪ ⎪ ⎩ 0

given by x ∈ [xi−2 , xi−1 ), x ∈ [xi−1 , xi ), (1.1.1.43) x ∈ [xi , xi+1 ), otherwise

for i = 0, 1, · · · , n and B2,n+1 (x) =

⎧ ⎪ ⎨

(x−xn−1 )2 (xn −xn−1 )2

⎪ ⎩ 0

x ∈ [xn−1 , xn ), (1.1.1.44) otherwise.

1.1. Singular Integral Equation

10

For given n ∈ N, an approximation un to the unknown u is defined as un (x) =

n +1 �

ci B2,i (x), x ∈ [0, 1]

(1.1.1.45)

i=0

where ci , i = 0, 1, · · · , n + 1 are constants to be determined. For getting approximate solution un (x) of equation � 1 u(x) − λ K(x, s)u(s) ds = f (x), x ∈ [0, 1] (1.1.1.46) 0

in the linear span of S2,1 (Δn ), one replaces u(x) by un (x) in the above. Then its evaluation at the nodes xi , i = 0, 1, · · · , n + 1 provides a system of linear simultaneous equations � 1 un (xi ) − λ K(xi , s)un (s) ds = f (xi ), i = 0, 1, · · · , n + 1 (1.1.1.47) 0

involving a (n + 2) × (n + 2) matrix. Here it is assumed that x−2 = x−1 = x0 , xn+2 = xn+1 = xn . Solution of this system of equations provide the unknown coefficients cn , whose substitution into (1.1.1.45) gives the approximation un (x) to the solution u(x). 1.1.1.6

Method based on product integration

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We consider here the numerical solution of Fredholm integral equations of the second kind with singular kernels, for which the associated integral operator K is still compact on C(D) into C(D). The main ideas presented here can be extended to higher dimensions, but it is more instructive to first present these ideas for integral equations of a single variable, b

� λ u(x) −

K(x, t) u(t)ds = f (x), a ≤ x ≤ b.

(1.1.1.48)

a

In this setting, kernel functions K(t, s) have an infinite singularity, and the most important examples are weakly singular kernels, viz., kernel ln|t − s| with logarithmic singularity, or the kernels of the form |t − s|γ−1 for some 0 < γ < 1 with singularities of algebraic nature and variants of them. We introduce the idea of product integration by considering a special case of (1.1.1.48) � λ u(x) −

b

L(x, t) ln|x − t| u(t)dt = f (x), a ≤ x ≤ b

(1.1.1.49)

a

with the kernel K(x, t) = L(x, t) ln|x − t|.

(1.1.1.50)

We assume that L(t, s) is a function smooth enough (that is, it is several times continuously differentiable), and initially we assume the unknown solution u(x) is also well-behaved. To solve (1.1.1.49), we define a method called the product integration (trapezoidal) rule. Let, n ≥ 1 be an integer (number of subdivisions of the domain [a, b]), h = b−a n (length of the (uniform) subintervals), and xj = a + jh, j = 0, 1, 2, ..., n (nodes). For general u ∈ C[a, b], define [L(x, t) u(t)]n =

1 [(xj − t)L(x, xj −1 )u(xj−1 ) + (t − xj−1 )L(x, xj )u(xj )], h

(1.1.1.51)

1.1. Singular Integral Equation

11

for xj −1 ≤ t ≤ xj , j = 1, 2, .., n and a ≤ x ≤ b. This is piecewise (linear) polynomial in t (in case of trapezoidal), and it interpolates L(x, t)u(t) at t = x0 , x1 , ..., xn , for all x ∈ [a, b]. Define a numerical approximation to the integral operator in (1.1.1.49) by b

[L(x, t)u(x)]n ln|x − t| dt,

Kn u(x) =

a ≤ x ≤ b.

(1.1.1.52)

a

This can also be written as Kn u(x) =

n n

ωj (x)L(x, xj )u(xj ), u ∈ C[a, b]

(1.1.1.53)

j=0

with weights ω0 (x) = ωj (x) =

1 h

xj

(t −xj −1 ) ln|x−t| dt+ xj −1

ωn (x) =

1 h

x1

1 h

(x1 − t) ln|x − t| dt,

(1.1.1.54)

x0

1 h

xj+1

(xj+1 −t) ln|x−t| dt, j = 1, 2, · · · , n − 1, (1.1.1.55) xj

xn

(t − xn−1 ) ln|x − t| dt. xn−1

To approximate the integral equation (1.1.1.49), we use (1.1.1.56)

dal

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ωj (x)L(x, xj )un (xj ) = f (x), Pan

j=0

M

n n

M

λ un (x) −

As with the other methods discussed earlier, this is equivalent to first solving the system of linear equations λ un (xi ) −

n n

ωj (xi )L(xi , xj )un (xj ) = f (xi ), i = 0, 1, 2, ..., n

(1.1.1.57)

j=0

followed by the use of the Nystr¨ om interpolation formula ⎤ ⎡ n n 1 ωj (x)L(x, xj )un (xj )⎦ , un (x) = ⎣f (x) + λ j=0

a ≤ x ≤ b.

(1.1.1.58)

With this method, we approximate those parts of the integrand in (1.1.1.49) that can be well­ aproximated by piecewise (linear) polynomial interpolation, and we integrate exactly the remaining more singular parts of the integrand.

1.1.2

Kernel with weak (logarithmic and algebraic) singularity

The conditions for existence, uniqueness and regularity of the solution of Eq. (1.1.1.1) and the estimate of error in its approximation have been presented in the following theorems: Assumption: It is assumed that the kernel K(x, s) is of the form K(x, s) = g(x, s)κ(x − s)

(1.1.2.1)

1.1. Singular Integral Equation

12

with g to be thrice continuously differentiable function on [0, 1] × [0, 1], κ is twice continuously differentiable function on [−1, 1] − {0} such that c , 0 0&&k2 ≥ Length[h] − 1&&class == ”T ”, nodewt[µRT [k1, n]]; k1 ≤ 0&&k2 ≥ 12 Length[h]&&class == ”B”, nodewt[µL [k2 − 1/2Length[h], n]], k1 ≤ −(1/2)Length[h]&&k2 ≥ 0&&class == ”B”, nodewt[µR [−k1 − 1/2Length[h] + 1, n]]]; ReP art = T able[Re[nw[[1, i]]], {i, 1, Length[nw[[1]]]}]; If [M in[ReP art] ≥ k1&&M ax[ReP art] ≤ k2, Return[nw], dal

ja

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and

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M essage[node :: ”OutsideSupport”, n]; n = n + 1; Goto[r1]]; ]

The output of the program is nodes (xi ) and pseudo- or quasi-weights (ωi ) for some n' ≥ n. In the evaluation of the four integrals mentioned above we have considered cases for which nodes and pseudo- or quasi-weights of the quadrature rule are all real or some of them are complex. As expected from the Theorem 3.7 in §3.4.1.3, appearance of complex nodes and weights in the quadrature formula do not pose any difficulty in the evaluation of the integrals. In § 3.4.1.8, it has been found that the error in the evaluation of integral by using Qn [f, ϕ] involves derivatives of order 2n of f (x). Here f (x) = sin x, cos 2(x − 2) + sin 3(x − 2), cos 2|x − 1|+sin 3|x − 1| f

(2n)

(x)

and cos 2|x − 2|+sin 3|x − 2|, which are all trigonometric functions. Hence, i(2n)! is bounded on [0, 5] for i = 1, 2; on [0, 3] − {1} for i = 3 and on [0, 5] − {2} for i = 4. Thus the value of each integral obtained by Qn [f, ϕI ] or composite formula Qn [f, ϕLT ] + Qn! [f, ϕRT ] converges smoothly to the exact value as n increases. Here, we regard the values obtained by using the quadrature rules Q24 [fi , ϕI = Dau3[0, 5]](i = 1, 2), Q20 [f3 , ϕLT ] + Q20 [f3 , ϕRT ](for ϕ = Dau2[0, 3]), and Q19 [f4 , ϕLT ] + Q20 [f4 , ϕRT ](for ϕ = Dau3[0, 5]) as the exact values of the integrals for the purpose of evaluation of the errors. In Tables 3.1 and 3.2, we have compared the absolute value of errors in the evaluation of �n 2K−1 I ϕ (x)fi (x)dx, i = 1, · · · , 4 by using the quadrature rule Qn [f, ϕI ] = i=1 ωiI f (xiI ) involv­ 0 ing n real or complex nodes and weights (RCNW) and other quadrature rules based on lifting trick

3.4. Quadrature Rules

88

(LTr) with c = 12 of Barinka et al. (Barinka et al., 2001) and shifting trick (STr) of Sweldens and Piessens (Sweldens and Piessens, 1994b). The numbers within parenthesis in this table indicate exponent of 10. The exact values of the integrals correct up to sixteen decimal places are given in the last row. From this table it appears that accuracy of the values of the integrals for smooth functions obtained by Gauss-type quadrature rule involving RCNW seems to be better than those obtained by the quadrature rule based on STr of Sweldens and Piessens. The accuracy appears to be uni­ form irrespective of nodes and weights of the quadrature rule being all real or some being complex. However, the results exhibit Runge-like phenomenon for integrals involving non-smooth function f3 and Dau2[0, 3], slow rate of convergence for non-smooth function f4 and the pseudo-weight function Dau3[0, 5]. It is interesting to note that the same trend in accuracy appears whenever the integrals have been evaluated by using the quadrature rule I

ϕ (x)f (x)dx ≈

Qcn [f,

I

ϕ ]=

n n

ωic f (xci )

−c

i=1

supp ϕI

n n

ωiχ f (xχi )

i=1

based on LTr of Barinka et al. (Barinka et al., 2001). However, it is important to note that for a given n, the above mentioned formula involves 2n distinct nodes ({xci , xχi , i = 1, 2, · · · , n)} and their corresponding weights ({ωic , ωiχ , i = 1, 2, · · · , n)}. Consequently, one may regard this formula as effectively a quadrature rule of 2n nodes with stability constant n �

dal

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M

M

|

and

B

σnc =

(|ωic |+c |ωiχ |)

i=1 n �

. (ωic − c ωiχ )|

i=1

Comparison of errors in Tables 3.1 and 3.2 exhibits the superiority of the proposed quadrature rule over the existing one for smooth functions. To evaluate the approximate values of integrals involving non-smooth functions, 2K−1 ϕ(x)fi (x)dx, i = 3, 4 (non-smooth at xcr = 1 for f3 (x) and xcr = 2 for f4 (x)) with a e.g., 0 rapidly convergent quadrature formula and to check the effectiveness of Gauss-Daubechies quadrara­ ture rule for integrals involving truncated scale functions, the above mentioned integrals have been x 2K −1 split into 0 cr ϕLT (x)fi (x)dx and xcr ϕRT (x)fi (x)dx. Now 2K−1

ϕ(x)fi (x)dx

≈

Qnn! [f, ϕLT , ϕRT ]

=

Qn [f, ϕLT ] + Qn! [f, ϕRT ]

0

=

n n i=1

ωiLT f (xiLT ) +

n! n

(3.4.1.10)

T ωiRT f (xR i ).

i=1

The number of nodes and weights used in the quadrature rules for ϕLT (x) and ϕRT (x) for given K are different due to the fact that equal number of nodes and weights may not always be available. For the purpose of calculating the functions fi (x) (i = 3, 4) at a complex node x, we use � x − xcr xcr < Re x, |x − xcr |= xcr − x xcr > Re x

3.4. Quadrature Rules

89

in the quadrature rule mentioned above. � 2K−1 The errors in the evaluation of 0 ϕ(x)fi (x)dx (i = 3, 4) by the present method and by the method given by Huybrechs and Vandewalle (Huybrechs and Vandewalle, 2005) are displayed in Table 3.3. From this table it is obvious that the present method is superior to the method of Huybrechs and Vandewalle (Huybrechs and Vandewalle, 2005). The stability constants for the quadrature rule Qn [f, ϕI ] developed here and for the rule Qnc [f, ϕI ] developed by Barinka et al. (Barinka et al., 2001) have been compared in Table 3.4. Regarding the choice of c in the quadrature rule Qcn , it may be noted that it is to be chosen such that the lifted function ϕc (x) = ϕ(x) + c χsupp ϕ should be non-negative for x ∈ supp ϕ. For scale functions in Daubechies family c ≥ 12 . We choose c = 21 as this choice leads to better accuracy as well as better stability constants which are evident from Tables 3.1, 3.2 and 3.4. From Table 3.4 it is found that the stability of the present method is better than that of the Barinka et al. (Barinka et al., 2001). Thus, Gauss-Daubechies quadrature rules having complex nodes and weights for integrals involv­ ing scale functions with variable signs regarded as pseudo- or quasi-weight functions and smooth or non-smooth functions can be treated as efficient and almost in the same footing as the classical Gauss quadrature rule for integrals restricted to positive semi-definite weight functions.

Table 3.1: Comparison of � 2K−1 I ϕ (x)fi (x)dx, i = 1, 2. 0

errors

evaluating

approximate

RCNW

dal N B and ja Pan M M

LTr

STr

of

j=1

j=0

RCNW

RCNW

j=1 LTr

RCNW

5

8.4(–8)

1.5(–6)

6.1(–4)

8.8(–11)

1.4(–3)

3.8(–2)

3.4(–6)

9/10ST r

3.3(–16)

3.9(–15)

9.8(–5)

≈ 10−29

2.2(–8)

1.1(–6)

3.5(–13)

20

≈ 10−47

≈ 10−45

4.3(–6)

≈ 10−59

≈ 10−29

≈ 10−26

≈ 10−42

.741 104 421 925 904 6

Table 3.2: Comparison of errors � 2K−1 I ϕ (x)f (x)dx, i = 3, 4 for j = 0. i 0 Nodes

integrals

ϕI = Dau3[0, 5] Man

j=0

value

f2 (x)

f1 (x) ϕI = Dau3[0, 5]

No. of Nodes n

in

in

–.644 487 735 893 018 1

evaluating

f3 (x) ϕI = Dau2[0, 3]

approximate

f4 (x) ϕI = Dau3[0, 5]

n

RCNW

LTr

n

RCNW

LTr

4

3.5(–2)

5.5(–3)

5

3.5(–2)

1.3(–1)

7

4.3(–2)

6.(–2)

11

1.9(–2)

9.4(–3)

13

1.5(–2)

1.3(–2)

21

4.5(–4)

1.1(–2)

1.38 501 797 074 570 12

–.604 713 724 795 161 5

value

of

integrals

3.4. Quadrature Rules

90

Table 3.3: Comparison of errors in evaluation of Qnn� [f, ϕLT , ϕRT ].

� 2K−1 0

f3 (x), ϕ = Dau2[0, 3]

ϕ(x)fi (x)dx,

i = 3, 4 by using

f4 (x), ϕ = Dau3[0, 5]

j=0

j=0

n

n’

RCNW

n=n’

HV

n

n’

RCNW

n=n’

HV

4

4

1.6(–5)

5

7.1(–2)

5

5

1.1(–6)

7

1.4(–2)

7

7

9.8(–11)

9

2.1(–4)

11

12

≈ 10−19

13

5.4 (–6)

13

13

≈ 10−24

17

4.3(–11)

21

21

≈ 10−37

25

9.6(–13)

Table 3.4: Stability constants σn and σnc for the quadrature rules Qn [f (x), ϕI ] and Qcn [f, ϕ] for c = and 1. ϕI n

Dau2[0, 3] 1 1 σn σn σn2

n

Dau3[0, 5] 1 1 σn σn σn2

4

1.1

4

7

5

1.3

6

11

7

1.2

4

7

9

1.2

6

11

13

1.2

4

7

20

1.3

6

11

1 2

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We now study the utility of Gauss-Daubechies quadrature rule Qn [f, ψ] =

n �

ωi f (xi ),

i=1

(xi , ωi are nodes and weights associated with pseudo-weight function ψ(x)) for evaluation of the integral I[f, ψ, −K + 1, K] involving wavelets. As mentioned earlier, this integral can also be evaluated with the help of the formula K � 1 · Qn [f, ψ, −K + 1, K] = √ gl Qn� [ f ( ), ϕl , −2K + 2, 2K]. 2 2 l=−K+1

The errors in Qn [f, ψ] and Qn [f, ψ, −K + 1, K] can be estimated by using the results in the previous section as � Pn (x)2 dx En [f, ψ] ≈ µ supp

and En [f, ψ, −K + 1, K] ≈ ||Pn ||2ϕ 2−2n

�

gl < f (2n) >ϕl .

l∈ Z

Here < f

(2n)

>ϕl is the average of f ϕl = l−K+1

f (2n) (x)ϕl (x)dx.

3.4. Quadrature Rules

91

dal

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The sum in the right side of En [f, ψ, −K + 1, K] is zero for f (x) = xs , s = 2n, · · · · · · , 2n + K − 1 in addition to s = 0, · · · · · · , 2n − 1 due to the vanishing property of the moment of ψ(x). Above mentioned two formulas for errors apparently suggest that for a given n, the quadrature rule Qn [f, ψ, −K + 1, K] will provide more accurate value of the integral I[f, ψ, −K + 1, K] than the quadrature rule Qn [f, ψ]. However, from a close observation of the results presented in the columns j = 0 and j = 1 of Table 3.1 and the two quadrature rules mentioned above, it is found that for given n, the number of arithmetic operations in Qn [f, ψ, −K + 1, K] is 2K times the arithmetic operations in Qn [f, ψ]. Thus, for a given computational time, while Qn [f, ψ, −K + 1, K] can accurately eval­ uate the integral of product of ψ(x) and a polynomial of degree 2n + K − 1, Qn [f, ψ] can evaluate the same for polynomials of degree up to 4Kn − 1. Therefore, for pre-assigned order of accuracy, computational cost for evaluating wavelet coefficients by using Gauss-Daubechies quadrature rule Qn [f, ψ] will be much less than the same for quadrature rule Qn [f, ψ, −K + 1, K]. Accurate computation of these types of integrals is of concern in numerical computations where wavelets are used for their multiresolution approximation. In that setting, one typically has to evaluate many such integrals. Many workers focus on quadrature rules for these integrals with equidistant nodes, since in that case function evaluation can be reused for other integrals resulting in a considerable reduction of overall computational cost. While studying compression techniques and optimal complexity estimates for boundary integral equations, Dahmen et al. (Dahmen et al., 2006) concluded that for stable Galerkin scheme with optimal order of convergence, such integrals have to be computed with full accuracy in the coarser resolution while the same on the finer scale is allowed to have less accuracy. The necessary accuracy can be achieved within the allowed expenses if one employ an exponentially convergent quadrature method. As mentioned above, Gauss-Daubechies quadrature rule is much stable and has higher rate of convergence than the quadrature rules involving equidistant nodes and for such integrals on higher resolution, just a few quadrature points are generally sufficient. The additional calculations due to non-equidistant nodes of Gauss-Daubechies quadrature rule is expected to be more than balanced by the reduction of number of nodes and higher rate of convergence in the quadrature rule. An in-depth discussion on this aspect of quadrature rule involving complex nodes and weights including relevant numerical data are available in the study of Panja and Mandal (Panja and Mandal, 2011; Panja and Mandal, 2015).

3.4.2

Quadrature rules for singular integrals

As mentioned earlier, evaluation of integrals of product of refinable functions having full or partial support within the domain of integration with a function f is an important step in the multiresolution approximation of the given function. The accuracy of the evaluated value of integrals by using some numerical technique depends on the smoothness of the integrand associated with it. Discontinuities of the integrand or any of its derivative usually disturbs the convergence of the method. This feature equally appears in case of evaluation of the integral of product of refinable function and singular functions. Although moment based quadrature rule enables us to evaluate the integral of product of refinable function and singular function in their regular domain, the same rule is unable to produce appropriate value of the same integral when the support of refinable function contains the singularity of the other function involved in the integrand. Perhaps the source of difficulty lies in inappropriateness of representing integrand by polynomials on the vicinity of the singularity.

3.4. Quadrature Rules

92

Fortunately, the refinement equation like (2.1.4.2) satisfied by the refinable function provides a way to get rid of this difficulty, particularly, for weakly (logarthmic or algebraic) singular functions. The method based on refinement equation is not equally successful when it is extended to treat integrals involving higher order singularities. Although some researchers (Kessler et al., 2003b; Li and Chen, 2007) developed methods for evaluation of Cauchy principal value integrals (CPVIs) with refinable function having full support within the domain of integration, such procedure is unable to evaluate the same integral when the support of refinable function not fully contained in the domain of integration or the singularities is of higher order. In the next section, we will follow the techniques followed by Panja and Mandal (Panja and Mandal, 2013b) to show that, with the aid of regularization principle, it is possible to evaluate integrals of product of singular functions and refinable functions with full as well as partial support within the domain of integration uniformly. To attain the goal, we first calculate the integrals involving product of refinable function and function with logarithmic singularity. 3.4.2.1

Integrals with logarithmic singularity

Interior scale functions (Kessler et al., 2003b) Here it is assumed that scale functions are in Daubechies family with suppφ = [0, 2K − 1]. In case of finite support [k, k +2K −1] ⊂ [a, b] of the refinable function ϕIk (x) within the domain of integration, the integral ∞

b

ln|x| ϕIk (x) dx =

ILk =

ln|x| ϕIk (x) dx

(3.4.2.1)

−∞

a dal

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becomes singular whenever a < k < 0 < k + 2K − 1 < b. The quadrature formula based on moment of the refinable function (Kessler et al., 2003b; Sweldens and Piessens, 1994b) cannot evaluate these singular integrals ILk s accurately.

Theorem 3.9. 2K singular integrals IL = {IL−2K+1 , · · · , IL0 } are the solution of the linear si­ multaneous equation (3.4.2.2) AL2K×2K IL2K×1 = b2K×1 where elements ap q and bp of matrices AL, b respectively are given by 1 ap q = δp q − √ hq−2 p , 2

−2K + 1 ≤ p, q ≤ 0

(3.4.2.3)

and, 1 bp = √ 2

−2K n r=−2k

1 hr−2p ILr + √ 2

2k+2K−1 n

hr−2p ILr − ln 2,

−2K + 1 ≤ p ≤ 0.

(3.4.2.4)

r=1

Proof : Substituting the expansion of ϕk (x) given in refinement equation (2.1.4.2) in the R.H.S. of (3.4.2.1) and using definition (3.4.2.1), one finds recurrence relation 1 ILk = √ 2

2k+2K−1 n r=2k

hr−2k ILr − ln 2, k ∈ Z.

(3.4.2.5)

3.4. Quadrature Rules

93

From a careful analysis of recurrence relation (3.4.2.5) it appears that whenever k > 0 or k < −2K + 1, formula (3.4.2.5) provides a relation in which integrals with lower k involve integrals with higher k’s and vice-versa. Integrals with higher k s (k ≥ 20) can be evaluated efficiently by using quadrature formula based on moment of refinable function. We may mention here that a 13−point quadrature rule yields result correct up to O(10−24 ) for polynomials of degree up to 25. Thus, one can evaluate ILk s accurately whenever (k > 0) or (k < −2K + 1) with appropriate combination of moment based quadrature rule (|k|≥ 20) and recurrence relation (3.4.2.5) for (0 < k < 20) and (−20 < k < −2K + 1). Therefore, judicious use of quadrature rule and recurrence relation into the relation (3.4.2.5) for k = −2K + 1, −2K + 2, · · · , 0 provides a system of linear equations AL IL = b. Here the matrix elements ap q in AL2K ×2K and bp in b2K×1 are given by 1 ap q = δp q − √ hq−2 p , 2

−2K + 1 ≤ p, q ≤ 0

and, 1 bp = √ 2

− 2K n r=−2k

1 hr−2p ILr + √ 2

2k+2 K−1 n

hr−2p ILr − ln 2,

−2K + 1 ≤ p ≤ 0.

r=1

dal ja M

M

Pan

and

B

N

Man

In formulae (3.4.2.3), (3.4.2.4) and (3.4.2.5), hl ∈ h = (h0 , · · · , h2K −1 ) and in (3.4.2.4), the summa­ tions take value 0 whenever their lower limit exceeds upper limit. For the case of Daubechies K = 3 refinable functions, numerical estimates of singular as well as regular integrals ILk , k = −8, · · · , 3 are given in Table 3.5.

Table 3.5: Numerical values of ILk , ILRT and ILkLT for few k close to singularity x = 0 in case of k scale function Dau3[0,5]. k

ILLT k

ILk

−8

1.971747367781395

−7

1.8218703338385005

−6

1.6455190058120355

−5

1.4312863770023681

ILRT k

−4

1.1573795241796710

−0.0012393149717146322

1.1586188391513856

−3

0.75046835527805196

−0.043273470823419856

0.79374182610147181

−2

0.31562430394302346

0.23612027918122357

0.079504024761799893

−1

−1.8364645639911718

−0.91037738301020185

−0.92608718098096999

0

−0.25845321316677250

1

0.59250182648710685

2

1.0344302834271699

3

1.3389980904432829

Corollary 3.10. For a dyadic point y ∈ (a, b − 2j y is an integer

2K−1 2j )

b

ln|x − y| ϕIj k (x) dx =

ILj k (y) = a

with an appropriate resolution j for which 1 j

22

(ILk − 2j y − j ln 2).

(3.4.2.6)

3.4. Quadrature Rules

94

Proof : The integral of the form (3.4.2.6) with logarithmic singularity at any dyadic point y ∈ (a, b − 2K−1 2j ) can be evaluated by using the formula � ∞ ILj k (y) = ln|x − y| ϕIj k (x) dx −∞ � ∞ j = 22 (ln|2j x|−j ln2) ϕI (2j x − k + 2j y) dx −∞ � ∞ − 2j (ln|t|−j ln2) ϕI (t − k + 2j y) dt = 2 −∞ �� ∞ � � ∞ − 2j I = 2 ln|t|ϕIk−2j y (t) dt − j ln2 ϕk−2 (t) dt jy −∞

− 2j

=

2

−∞

(ILk−2j y − j ln 2).

(by using definition 3.4.2.1)

(3.4.2.7)

Boundary scale functions In the following theorem and subsequent two corollaries, a different notation for the truncated scale functions have been used. The superscript RT used here is synonymous to LT of section 2.2.1 and vice versa. Theorem 3.10: The integrals involving logarithmic singularities at the edge of domain of integration and scaling functions ∈ ΛRT or LT viz., � b

RT ILp = ln|x − a| ϕRT

a − 2K + 2 ≤ p ≤ a − 1 (3.4.2.8) p (x) dx, Man

ILLT p

B

N

dal

a

or

ja M

M

Pan

and

b

�

ln|b − x| ϕLT p (x) dx b − 2K + 2 ≤ p ≤ b − 1

=

(3.4.2.9)

a

satisfy the system of linear simultaneous equation 1 (I − √ H RT 2

or LT

) ILRT

or LT

= bRT

or LT

.

(3.4.2.10)

Proof : We first consider the integrals (3.4.2.8). Assuming |b − a|> 2K − 1 and since a − 2K + 2 ≤ p ≤ a − 1, substituting x − a = u and changing limit of integration appropriately one can rewrite the integral in (3.4.2.8) into the form � ∞ ILpRT = ln|x| ϕRT p� = −2K + 2, · · · , −1. � p� (x) dx, 0

Next use of the two-scale relation (2.1.4.2) into the R.H.S. of the above equation gives, ILpRT = �

∞

� 0

√ ln|x|{ 2

−1 � q = −2K+2

T HpR� Tq ϕR q (2x) +

� √ 2K−3 2 HpR� Tq I ϕlI (2x)} dx l=0

� ∞ � ∞ −1 � 1 RT HpRT { ln|y| ϕ (y) dy − ln2 ϕRT =√ � q q q (y) dy } 2 q = −2K+2 0 0 � ∞ 2K −3 1 � RT I Hp� l { +√ ln|y| ϕIl (y) dy − ln2}. 2 l=0 −∞

3.4. Quadrature Rules

95

Using the definitions (3.4.2.1) and (3.4.2.8) in the last formula one gets a system of linear equations RT RT , IL−2K+3 , · · · · · · , ILRT for ILRT = {IL−2K+2 −1 } as 1 (I − √ H RT ) ILRT = bRT . 2

(3.4.2.11)

RT The elements bRT in the above equation are given by p; in the inhomogeneous term b

ln 2 bRT p; = − √ 2

−1 K

2K−1

HpRT ; q |q|

q=−2K+2

2K−3 1 K RT I ϕ(y) dy + √ Hp; l {ILl − ln 2}. 2 l=0

(3.4.2.12)

LT LT Following the same procedure, equations for ILLT = {ILLT −2K+2 , IL−2K+3 , · · · · · · , IL−1 } can be found as 1 (I − √ H LT ) ILLT = bLT (3.4.2.13) 2 LT with the elements bLT as p; in the inhomogeneous term b −2K+1 K 1 ln 2 I √ = HpLT bLT ; ; l {ILl − ln 2} − √ p 2 l=−4K+4 2

−1 K

|q|

HpLT ; q

ϕ(y) dy.

(3.4.2.14)

0

q=−2K+2

dal

ja

M

M

Pan

and

B

N

Man

Numerical values of ILRT and ILLT can be evaluated easily by solving the Eqs. (3.4.2.11) and (3.4.2.13), are presented in Table 3.5 for K = 3. Their accuracy can be checked through the verification of the condition LT ILRT p + ILp = ILp . Corollary 3.11. Whenever 2J a and 2J b are integers for some J ∈ Z and 2j (a or b) − 2K + 2 ≤ p ≤ 2j (a or b) − 1, the integrals b

ILRT jp

or LT

ln|x − a or b| ϕRT jp

=

or LT

(x) dx

(3.4.2.15)

a

at the j th (j ≥ J) resolution are related to the same at 0th resolution by the relation or LT ILRT = j p;

1 j

22

{ILRT p;

or LT

or LT }. − j ln2 < x0 >ϕRT ;

(3.4.2.16)

p

Here −2K + 2 ≤ pi = p − 2j (a or b) ≤ −1. Proof: We first consider the integral containing singularity at the left edge of domain of inte­ gration b

ILRT jp =

ln|x − a| ϕRT j;p (x) dx,

2j a − 2K + 2 ≤ p ≤ −1

a b

j

{ln|2j x − 2j a|−j ln2} ϕRT (2j x − p) dx

= 22 a

=

1 2

j 2

2j b

{ln|y − 2j a|−j ln2} ϕRT (y − p) dy. 2j a

3.4. Quadrature Rules

96

Substituting y − 2j a = z and p − 2j a = p� , one gets � ∞ � ∞ 1 RT ILRT = { lnz ϕ (z)dz − j ln2 ϕRT � � j jp p p� (z) dz } 22 0 0 1 RT − j ln2 < x0 >ϕRT }. j {ILp� p� 22 The relation for the other end can be established similarly. =

3.4.2.2

Quadrature rule for weakly (algebraic) singular integrals

Weakly singular integrals appear in diverse fields of mathematical sciences. Since most of them cannot be evaluated analytically, several numerical methods have been developed for evaluation of their approximate numerical values. Observing the success in evaluation integrals with logarithmic singular function and scale functions in truncated (Meyer) basis (Goswami and Chan, 2011) we present here a quadrature rule developed in (Panja and Mandal, 2012) for numerical evaluation of weakly singular integrals with singularities at the edges a and b. Integrals of the form

�b

ϕj k (x) dx, a (x−a)µ

0 (β −α) to assure enough interior scale function within [0, 2j (b − a) − (β − α)]. Using the two-scale relation for ϕk� (x) in (3.4.2.19), a recurrence relation for ωkL� [µ] can be found as 1

ωkL� [µ] = 2µ− 2

β �

L hl ω2k � +l (µ).

(3.4.2.20)

l=α

From this relation it is obvious that the determination of ωkL� [µ] for particular k � > 0, involves nu­ merical values ωlL [µ] for l > k � . These quantities, usually called asymptotic values, can be evaluated following the fact that within the support of ϕl (x), l >> 1, the factor x1µ behaves like a regular function. Therefore, one may evaluate ωkL� [µ]’s , l >> 1 but within [0, 2j (b − a) − (β − α)] by either of the results obtained by using one-point quadrature rule ωlL [µ] ≈

1 , (l + �x�)µ

3.4. Quadrature Rules

97

or by using the series ωlL [µ] ≈

rM ax (µ)r F ull M om 1 � I (r). µ l r = 0 lr 0

Here I0F ull M om (r) = µr in section 3.3.1. Once the asymptotic values are known, ωkL� [µ]’s for other positive values of k � can be easily evaluated with the help of the formula (3.4.2.20). The values of ωkL� [µ]’s for −β + 1 ≤ k � ≤ −α, are determined by solving a system of linear simultaneous equations generated with the help of (3.4.2.20) whose solution for Daubechies-3 scale function for µ = 12 are presented in Table 3.6. Furthermore, for 2j (b − a) − β + 1 ≤ k � ≤ 2j (b − a) − α − 1, the scale function has the partial support within the domain of integration [0, 2j (b − a) ]. However, due to the regular behaviour of x1µ within the partial support of ϕk� (x) one may estimate ωkL� [µ] by using either by one point quadrature rule ωkL� [µ] ≈ wkR�

1 (k � +

< x >[0,2j (b−a)−k� ] )µ

,

or by summing the series ωkL� [µ] ≈

rmax 1 � (−1)r (µ)r R Ij � (r) � µ (k ) r=0 r! (k � )r 2 (b−a)−k

dal

ja

M

M

Pan

and

B

N

Man

r where IkR (r) = µRT . Therefore, k ⎧ ⎪ ⎪ ⎪ ⎪ 0 for k � ≤ −β, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Solution of system of Eqs. formed by (3.4.2.20) for − β + 1 ≤ k � ≤ −α, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Formula (3.4.2.20) for − α < k � < 2(β − α), ⎪ ⎪ ⎪ ⎨ 1 ωkL� [µ] = for 2(β − α) ≤ k � ≤ (b − a)2j µ ⎪ ⎪ ([k� ,k� +2K −1] ) ⎪ ⎪ −(β − α), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ �rmax (−1)r (µ)r R ⎪ ⎪ ⎪ (k1� )µ r=0 for (b − a)2j − (β − α + 1) ⎪ r!(k� )r I2j (b−a)−k� (r) ⎪ ⎪ ⎪ ≤ k � ≤ (b − a)2j − 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 for k � ≥ (b − a)2j . (3.4.2.21) Here (µ)r is the Pochhammer symbol.

Integrals of the form

�b

ϕj k (x) a (b−x)ν

dx, 0 < µ < 1

Following a similar method with appropriate modification, this integral can be estimated by using the formula � 2j b � b ϕj k (x) ϕk (x) R (ν− 12 )j ωjk [ν, b] = dx, = 2 dx, 0 < ν < 1 (3.4.2.22) ν j b − x)ν (b − x) (2 j a 2 a

3.4. Quadrature Rules

98

R can be found as where the two-scale relation for ωjk

ωjRk [ν, b]

ν − 12

=2

β �

R hl ωj+1 2k+l [ν, b].

(3.4.2.23)

l=α R The expressions for ωjk [ν, b] for the admissible values of k are summarised as

⎧ ⎪ ⎪ ⎪ ⎪ for k ≤ a 2j − β + α, ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ �rmax ⎪ (ν)r ⎪ 1 j L j ⎪ (2j (b− ⎪ r=0 r!(2j (b−a))r Ik−2j a (r) for a 2 − (2K − 2) ≤ k ≤ a 2 , a))ν ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ for |b 2j − k|≥ 2(β − α) and ⎪ ⎪ ⎨ (2j b−[k,k+2K −1] )ν a 2j ≤ k ≤ b 2j − β + α − 1, ωjRk [ν, b] = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ for 2j a ≤ k ≤ b 2j − β + α − 1 Formula (3.4.2.23) ⎪ ⎪ ⎪ ⎪ and |2j b − k|≤ 2(β − α), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Solution of system of for 2j b − (β − α) ≤ k ≤ 2j b − 1, ⎪ ⎪ ⎪ ⎪ Eqs. formed by (3.4.2.23) ⎪ ⎪ ⎩ 0 for k ≥ 2j b.

(3.4.2.24)

dal

ja

M

M

Pan

and

B

N

Man

of section 3.3.2. Here IkL (r) = µRT k The main formulae for evaluating inner product of scale functions and weakly singular functions at left or right edges are formulae (3.4.2.21) and (3.4.2.24) supported by their asymptotic values whose errors may be made as small as possible. The numerical values of ωjLk or ωjRk without prefactors for the scale function Dau3[0,5] are presented in Table 3.6.

1 1 L R Table 3.6: Numerical values of ωk−2 j a [ 2 ] and ωk−2j b [ 2 ].

k − 2j a

W ω0k

k − 2j b

W ω0k

−4

0.002418025890650

−5

0.4888247649751728

−3

0.074507561327850

−4

0.5607046201248957

−2

−0.383612087110153

−3

0.6826054160639504

−1

1.438658438411280

−2

0.9542008763684104

0

1.171967541211238

−1

1.6438144798009484

Quadrature formula for integrals

�b

F (x) dx, a (x−a)µ (b−x)ν

0 < µ, ν < 1

We are now well equipped to develop quadrature formula for numerical evaluation of above integral in terms of raw image in the truncated basis with elements of compact support. We first split the

3.4. Quadrature Rules

99

above integral into b

I[F ; µ, ν]

= a

F (x) ν µ dx (b − x) (x − a)

a+b 2

= a

fb (x) µ dx + (x − a)

with

(3.4.2.25)

b a+b 2

fa (x) ν dx, (b − x)

(3.4.2.26)

fb (x) =

F (x) ν (b − x)

(3.4.2.27)

fa (x) =

F (x) µ. (x − a)

(3.4.2.28)

and,

� � Substituting expansion (3.1.1.17) for the regular function fb (x), fa (x) within their domain a, a+b 2 � a+b � and 2 , b respectively, and then using the values of the integrals whenever they appear, the es­ timate for the weakly singular integral in (3.4.2.25) can be found as b j ( a+ 2 )2 −1

n

I [F ; µ, ν ] =

fb, j l ωjLl [µ, a] +

l = 2j a−β+1

j b2 −1 n

r =(

fa, j r ωjRr [ν, b],

(3.4.2.29)

a+b j 2 )2

dal

ja

M

M

Pan

and

B

N

Man

where the raw images fb, j,l ’s and fa, j,r ’s for fb (x) and fa (x) are determined by using formulas � � 1 k+ < x > fj,k = j f , (3.4.2.30) 2j 22 L fj,k = fj,k =

2j a−α− n 1

L j j ¯L (N LL )−1 jl ), k ∈ {2 a−β +1, · · · , 2 a−α−1} (3.4.2.31) k−2j a,l−2j a ωjl [µ, a]f (x

l=2j a−β+1

and fj,k =

R fj,k

=

2j b−α− n 1

1 R

j j (N RR )− ¯R jl ), k ∈ {2 b − β + 1, · · · , 2 b − α − 1} (3.4.2.32) k−2j b,l−2j b ωjl [ν, b]f (x

l=2j b−β+1

respectively. This formula can be written in terms of values of the function fa (x) and fb (x) at different nodes by reversing the summation over l or r and k as

I[F ; µ, ν]

=

2j a−α− n 1

a+b j 2 2 −β

ΩL j l (µ)

f (x ¯jLl )

l = 2j a−β+1 j b−β 2n

+ r =(

a+b j 2 ) 2 −β+1

+

n

ωjLl [µ, a]ωjI l f (¯ xIj l )

l =2j a−α

ωjRr [ν,

b]ωjI r f (¯ xIj r )

2j b−α− n 1

+ r

=2j b−β+1

ΩR ¯R j r (ν)f (x j r ).

(3.4.2.33)

3.4. Quadrature Rules

100

Table 3.7: Relative error for the two weakly singular integrals. Relative error for

�1 0

ex √ x

dx

Relative error for

�1 −1

√ |x|

1−x2

dx

j

Method adopted here

Hashish et al.

j

Method adopted here

n

Method based on n-point formula

7 9

5.9(−7) 3.4(−7)

3.2(−2) 1.6(−2)

4 5

3.4(−5) 8.4(−6)

20 40

1.3(−2) 4.5(−3)

11

1.5(−7)

8.1(−3)

7

4.8(−7)

80

1.6(−3)

R The quantities ΩL j l (µ) and Ωj r (ν) are given by

ΩL j l (µ)

2j an −α−1

=

1 L (N LL )− k−2j a, l−2j a ωj k [µ, a],

(3.4.2.34)

R (N RR )−1 k−2j b, r −2j b ωj k [ν, b].

(3.4.2.35)

k =2j a−β+1

ΩR j r (ν)

2j b−α−1 n

= k

=2j b−β+1

dal ja M

M

Pan

and

B

N

Man

To check the efficiency of our formula (3.4.2.29) or (3.4.2.33) for evaluation of numerical values 1 ex 1 |x| of weakly singular integrals a comparison of results for the integrals 0 √ dx and −1 √1−x dx 2 x have been presented in Table 3.7. The Table 3.7 shows that the method adopted here is superior to the methods adopted by Hashish et al. (Hashish et al., 2009) and Jung et al. (Jung and Kwon, 1998) for evaluating the weakly singular integrals. 3.4.2.3

Quadrature rule for Cauchy principal value integrals

Numerical evaluation of Cauchy principal value (CPV) integrals within a finite domain by using scale function is a major issue when wavelet analysis is invoked to boundary integral approach for boundary value problems. Encouraged by the successful application of scale function with compact support based raw image dependent quadrature formula for evaluating regular or weakly singular integrals within a finite interval, we now develop here quadrature rule for CPV integrals b

I C [f, t] = − a

f (x) dx x−t

f (x) ∈ L2 [a, b],

(3.4.2.36)

with singularity t within the interval (a, b). The underlying idea behind the construction of formula is the application of formula 2j b−α− n 1 f (x) ≈ fj,k ϕj,k (x)χ[a,b] (3.4.2.37) 2j a−β+1 1 in the integral of (3.4.2.36) and then evaluation of the integrals involving product x−t and ϕj k (x) within the interval [a, b]. So, the prime objective of numerical estimate of Cauchy singular integral is the evaluation of the integral b

ωjCk [t] = − a

or IT or RT

ϕLT (x) j k

dx; 2j a − β + 1 ≤ k ≤ 2j b − α − 1. x−t

(3.4.2.38)

3.4. Quadrature Rules

101

Table 3.8: Numerical values of ωkC� for ϕ in Dau3[0,5]. k

C ω0k

k

C ω0k

−5

−0.23891481914902063

0

1.514314419700848

−4

−0.30768588635390487

1

0.5580063741076433

−3

−0.30259404864095946

2

0.35636164539588144

−2

−1.7516332044823586

3

0.2623962192581948

−1

−0.17177891031342468

4

0.20775723506648994

5

0.1719821939613779

j

Using relation ϕj k (x) = 2 2 ϕ(2j x − k) followed by transformation of variables, (3.4.2.38) can be recast into the form j ωjCk [t] = 2 2 ω Ck [2j t], (3.4.2.39) where ω Ck [2j t] is defined as � 2j b LT ϕk ω Ck [2j t] = − 2j

a

or IT or RT

x − 2j t

(x)

dx.

(3.4.2.40)

dal

ja

M

M

Pan

and

B

N

Man

Evaluation of integrals in (3.4.2.40) whenever the point t is dyadic and the point of singularity or RT (x), has been discussed in details 2j t falls beyond the supports of truncated scale function ϕLT k in a series of works by Kessler et al. (Kessler et al., 2003a). So, skipping the details of the procedure of evaluation of such integrals, we mention just the formulae which will be used here. The values of ω Ck [2j t], for k ∈ {2j t − {2K − 1}, ....2j t} presented in Table 3.8, was calculated by Kessler et al. (Kessler et al., 2003a) by extending the limit of the integral in (3.4.2.40) to (−∞, ∞) using the properties of ϕ k (x). The evaluation of ω Ck [2j t] for other values of k are carried out with the help of recurrence relation ω Ck [2j t] = ω Ck� = k−2j t [0] =

√ � 2 hl ω2Ck� +l (0)

(3.4.2.41)

in conjunction to the asymptotic value of ω Ck� [0] given by ωkC� [0] ≈

k�

1 + �x�

for k >> 1.

Numerical values of ω Ck [2j t], whenever 2j a − β + 1 ≤ k ≤ 2j a − α − 1 and 2j b − β + 1 ≤ k ≤ 2j b − α − 1 are performed by summing the series ω Ck [2j t]

r� r M ax 1 (−1) ≈ �x�[2j a−k,β] or [α,2j b−k] k − 2j t r=0 (k − 2j t)r

(3.4.2.42)

for 2j a − β + 1 ≤ k ≤ 2j a − α − 1 or 2j b − β + 1 ≤ k ≤ 2j b − α − 1. Using the expansion (3.4.2.37) for f (x) ∈ L2 [a, b] in combination with the formulae (3.4.2.40)­ (3.4.2.42), the integral of (3.4.2.36) can be written as � b j f (x) − dx ≈ 2 2 x − t a

−α−1 b2j� k

= a2j −β+1

fj k ω Ck (2j t)

(3.4.2.43)

3.4. Quadrature Rules

102

where fjk ’s are raw images of the function f (x) in the basis ϕjk (x) determined by using formulae (3.4.2.30)-(3.4.2.32). In the notation ΩCL j l (t)

=

2j a−α− � 1

C j (N LL )−1 k−2j a, l−2j a ωj k (2 t)

(3.4.2.44)

k=2j a−β+1

and R ΩC j r (t)

=

2j b� −α−1

C j (N RR )−1 k−2j b, r−2j b ωj k (2 t),

(3.4.2.45)

k=2j b−β+1

the quadrature rule in (3.4.2.43) can be recast into Q[f ; t] =

2j� aα−1

ΩCL j l (t)

ωjLl

f (¯ xL j l)

j b−β 2�

+

l =2j a−β+1

ωjCi (2j t)ωjI i f (x ¯Ij i )

i =2j a−α j

−α−1 2 b�

+

R ΩjCR ¯R r (t)ωj r f (x j r ).

(3.4.2.46)

r=2j b−β+1

dal

ja

M

M

Pan

and

B

N

Man

To verify the efficiency of the formulae derived here we have computed approximate value of � 1 sin−1 x −−1 x dx at several resolution j and the values of Legendre function of second kind Q3 (x) � 1 3 (t) from its integral representation − 12 −−1 Pt−x dt for several values of x at fixed resolution j = 5 by using (3.4.2.43) or (3.4.2.46) for ϕ in Dau3[0,5]. The relative error of the approximate values are presented in Tables 3.9 and 3.10 and seem to be reliable to apply for the evaluation of approximate values of other CPV integrals. �1 Table 3.9: Relative error in I C = −−1

sin−1 x x

dx in different j.

j

3

5

7

IC

7.2 × 10−5

1.5 × 10−5

2.1 × 10−6

Table 3.10: Relative error in evaluation of Q3 (x) from its integral representation by using (3.4.2.43) at resolution j = 5. x

− 34

− 12

− 14

0

1 4

1 2

3 4

Rel. Error

2.7(−5)

3.3(−5)

1.7(−6)

9.9(−6)

3.0(−5)

1.1(−4)

4.9(−5)

An alternative scheme for evaluation of these integrals have been suggested as the following. Numerical evaluation of CPVIs of product of refinable functions having full or partial support within the domain of integration is a major issue when wavelet Galerkin approximation is involved to boundary integral approach in boundary value problem (BVP). In their works Kessler et al. (Kessler et al., 2003a; Kessler et al., 2003b) and Li and Chen (Li and Chen, 2007) used a method leading to solving a system of linear equations for the singular integrals with the help of the refine­ ment equation (2.1.4.2). However, unlike the case of logarithmic singular integrals, the system of

3.4. Quadrature Rules

103

Table 3.11: Numerical values of ICk and IHk for few k’s close to singularity x = 0 in case of K = 3. k

ICk

IHk

−8

−0.13919879017032227

0.019296500171771583

−7

−0.16173181086431092

0.027877462010513194

−6

−0.19321532248743284

0.021786335123957215

−5

−0.23373864975756627

0.15120203563534345

−4

−0.36846021161004507

−0.26016497428329957

−3

−0.18102687560643768

0.96488831919994888

−2

−1.7346754045340097

0.45940638247675755

−1

−0.40669622999391133

−2.4789308000964914

0

1.7144459926562902

−0.39629986541816989

1

0.52107246148948010

1.0904423098492243

2

0.34361952157538996

−0.036342657640222082

3

0.26408765866206860

0.073127241955033627

linear equations for the integral � ∞ � b I ϕk (x) ϕ (x) I Ck = − k dx, a < −β + α + 1 < β − α − 1 < b dx = − x x a −∞

(3.4.2.47)

dal

ja

M

M

Pan

and

B

N

Man

becomes redundant. As a consequence of redundancy of the system of equations for ICk s, −β + α + 1 ≤ k ≤ β − α − 1 with null space of dimension 1, presence of an arbitrary constant in their solutions is inevitable. Kessler et al. (Kessler et al., 2003a) suggested the conditions � a dx − = 0, (3.4.2.48a) −a x � ϕn (x) = 1 (3.4.2.48b) n

may be used simultaneously to fix such arbitrary constant and found the numerical values of IC−4 , IC−3 , IC−2 , IC−1 for scale function in Daubechies family with K = 3 as presented in Table 3.11. Instead we propose here an alternative procedure based on regularization of the singular integral (Martin and Rizzo, 1989; Lifanov et al., 2004) � b � b f (x) − dx = f (a) ln|t − a|−f (b) ln|t − b|+ f � (x) ln|x − t| dx (3.4.2.49) a t−x a for evaluation of ICk s by using values of logarithmic singular integrals. Theorem 3.12. The integrals ICk in (3.4.2.47) for Daubechies-K scale function satisfy the relation ICk = −

β−1 �

rs(1) {ILk+s − ILk−s },

(3.4.2.50)

s=α+1 (1)

where ILk is defined in (3.4.2.1). The symbol rs mentioned above is known as connection coefficient involved in the formula connecting the scale function and its first order derivative. A discussion on the connection coefficient has been presented in somewhat details in a subsequent section of this chapter.

3.4. Quadrature Rules

104

Proof. Due to finite support (α + k, β + k) ⊂ [a, b] of ϕk (x), first two terms in the right hand side of (3.4.2.49) vanish for f (x) = ϕk (x). In addition choosing t = 0 in (3.4.2.49) and recalling the definition (3.4.2.47) one gets ∞

ln|t| ϕ'k (t) dt.

ICk = −

(3.4.2.51)

−∞ (1)

Further use of definition (3.5.0.3) and antisymmetric property rs (3.4.2.1) in the R.H.S. gives I Ck = −

−n α−1

(1) rl

ILk+s = −

β −α− n1

(1)

rl

(1)

= −r−s for ϕ and the definition

{ILk+s − ILk−s }.

(3.4.2.52)

s=1

s=−β+1

It is important to note that one does not need to bring any additional condition to determine CPVIs as it is essential in traditional approach developed by Kessler et al. (Kessler et al., 2003a). Over and above all the integrals whether the support of scale function contains a singularity or not, can be evaluated by summing a few terms, once the values of integrals involving logarithmic term are known. Interestingly, numerical values of ICk ’s obtained by using the formula (3.4.2.50) are different from those obtained by Kessler et al. in (Kessler et al., 2003a). In spite of these differences we will see that values of ICk ’s obtained by using formula (3.4.2.50) based on regularization principle yield quite accurate approximate value for variety of CPVIs. ∞

j ϕjk (x) dx = 2 2 ICk x

−

(3.4.2.53)

ja M

Pan

and

B

N

Man

−∞

dal

Corollary 3.13.

M

Proof. If we define

∞

ICjk = − −∞ j 2

ϕjk (x) dx, x

j

then using the definition ϕjk (x) = 2 ϕ(2 x − k) into the R.H.S of the above equation one gets, ∞

j

ICjk = 2 2 − −∞

ϕ(2j x − k) dx. x

Transforming the variable x → u = 2j x and imparting the definition (3.4.2.47) one gets, j

ICjk = 2 2 ICk . Corollary 3.14. For any dyadic y ∈ Q, i.e., for some J ∈ Z, 2J y ∈ Z ∞

ICjk (y) = − −∞

j ϕjk (x) dx = 2 2 ICk−2j y . x−y

Proof. By definition, j

∞

ICjk (y) = 2 2 − −∞

(3.4.2.54)

ϕ(2j x − k) dx. x−y

Multiplying numerator and denominator of the integrand by 2j changing the variable x → u = 2j x one gets, ∞ j ϕ(u − k) du. ICjk (y) = 2 2 − j −∞ u − 2 y

3.4. Quadrature Rules

105

Since 2J y ∈ Z for some J ∈ Z, 2j y ∈ Q ∀j ≥ J. Thus, further change of integration variable u → v = u − 2j y leads to � ∞ j j ϕ(v − k − 2j y) ICjk (y) = 2 2 − dv = 2 2 ICk−2j y ∀j ≥ J. v −∞ 3.4.2.4

Finite part integrals

Finite part integrals which are often called hypersingular integrals (De Klerk, 2005) play an im­ portant role in continuum mechanics, particularly, in the area of fracture mechanics (Chan et al., 2003a) and linearized theory of water waves (Martin et al., 1997). We denote such integral by � b f (t) IH[f ](x) = = dt, a < x < b. (3.4.2.55) (t − x)2 a This integral is defined in the sense of � b � x−� � b f (t) f (t) f (t) = dt := lim { dt + dx 2 2 �→ 0 (t − x) (t − x)2 (t − x) a a x+� f (x − �) + f (x + �) }. (3.4.2.56) � Moreover, one can evaluate the integral (3.4.2.55) by using the formula based on regularization, for f ∈ C 1,α � b � b � f (t) f (a) f (b) f (t) = dt := − − +− dt, a < x < b. (3.4.2.57) 2 x−a b−x a (t − x) a (t − x) ja M

M

Pan

and

B

N

Man

Theorem 3.15. The integrals

dal

−

� b ϕk (x) IHk = = dx, a < k + α < k + β < b x2 a are related to the Cauchy principal value integrals in (3.4.2.47) by the formulae IHk =

β− α−1 �

rs(1) {ICk+s − ICk−s }.

(3.4.2.58)

(3.4.2.59)

s=1

Proof: For f (x) = ϕk (x) and x = 0 in (3.4.2.57), f (x) is zero outside [k + α, k + β]. Thus, whenever a < k + α, b > k + β the domain of the above integral can be regarded as R so that f (a) = f (b) = 0 in (3.4.2.57) for any finite k. Therefore, for finite k, x = 0 and a and b satisfying inequalities mentioned above, integral in (3.4.2.58) can be recast into � ∞ � ∞ � � b ϕk (x) ϕk (x) ϕ k (x) = IHk = = dx = dx = − dx. 2 2 x x x a −∞ −∞ The CPV in the R.H.S. can be evaluated by using definition of ϕ � in (3.5.0.3) and results provided in the previous subsection to get IHk =

β− α−1 �

rs(1) {ICk+s − ICk−s }.

s=1

The numerical values of finite part integrals IHk , k = −2K + 2, · · · , −1 for Dau3 refinable function are presented in Table 3.11.

3.4. Quadrature Rules

106

Corollary 3.16. For any dyadic y ∈ Q ∞

IHjk (y) = = −∞

1 ϕjk (x) dx = 2j (2− 2 ) IHk−2j y . 2 (x − y)

Proof: By definition,

∞

j

IHjk (y) = 2 2 = −∞

(3.4.2.60)

ϕ(2j x − k) dx. (x − y)2

Multiplying numerator and denominator of the integrand by 22j and then changing the variable x → u = 2j x one gets, ∞ 1 ϕ(u − k) IHjk (y) = 2(2− 2 )j = du. (u − 2j y)2 −∞ By assumption, 2J y ∈ Z for some J ∈ Z, 2j y ∈ Q ∀j ≥ J. Thus, further change of variable u → v = u − 2j y leads to 1

∞

IHjk (y) = 2(2− 2 )j = −∞

1 ϕ(v − k − 2j y ) dv = 2(2− 2 )j IHk−2j y ∀j ≥ J. v2

The main ingredient of this subsection is getting simple algebraic rules relating integrals involving Daubechies scale function and functions with singularities with pole of order two. In this approach any additional condition is not necessary for determination of IHk s, k = α − β + 1, · · · , −1.

dal ja M

M

Pan

and

B

N

Man

3.4.2.5 Composite quadrature formula for integrals having Cauchy and weak singu­ larity During the last few decades, the numerical evaluation of a combination of weakly singular and Cauchy singular integrals became one of the important problems in numerical analysis and computational mathematics. For example, it is well known that the singular integral equation of first kind with Cauchy kernel 1 f (t) − 1 < x < 1 − dt = g(x) (3.4.2.61) −1 t − x where the integral is in the sense of CPV, has four kinds of solutions: √ 1 A0 1 − x2 g(t) 1 dt, , 1. f (x) = √ + √ 2 2 t−x 1−x π 1 − x −1 � � 1 + t g(t) 1 1−x 1 2. f (x) = dt, π2 1 + x −1 1 − t t − x � � 1 1+x 1 1 − t g(t) 3. f (x) = dt 2 π 1 − x −1 1 + t t − x 1 1 � 1 g(t) 2 √ 4. f (x) = 1 − x dt 2 π2 t −x 1−t −1

(3.4.2.62a) (3.4.2.62b) (3.4.2.62c) (3.4.2.62d)

subject to the condition that 1 −1

g(t) √ dt = 0. 1 − t2

(3.4.2.63)

3.4. Quadrature Rules

107

Here A0 is an arbitrary constant. From the outward appearance of the integrals in (3.4.2.62a)­ (3.4.2.63) it appears that although the integrals involved in (3.4.2.62a) and (3.4.2.63) can be eval­ uated numerically by using the scale function based raw image dependent quadrature formula (3.4.2.43) and (3.4.2.33) respectively, integrals involved in other solutions (3.4.2.62b)-(3.4.2.62d) remain intractable due to presence of multiple singularities of different types within the limit of integration. It is thus desirable to develop quadrature rule that may be called composite quadrature rule which can estimate singular integral with multiple singularities with the same order of accuracy as was achieved in case of single singular cases. So, we consider the integral b

I[µ, ν; x] = − a

1 g(t) dt (t − a)µ (b − t)ν t − x

− a < x < b.

(3.4.2.64)

Assuming α = 0, β = 2K − 1 for the support of scale function ϕ we divide the range Λ = {2j a − (2K − 2), · · · · · · , 2j b − 1} of raw images for regular part of the integrand into three parts ΛL = {2j a − (2K − 2), · · · · · · , 2j (c − δ) − 1}, ΛC = {2j (c − δ), · · · · · · , 2j (c + δ) − K} and, ΛR = {2j (c + δ) − K + 1, · · · · · · , 2j b − 1} with a suitable choice of δ > 0 and treat gb (t, x)

=

gc (t, x)

=

ga (t, x)

=

g(t) , (b − t)ν (t − x) g(t) , (b − t)ν (t − a)µ g(t) (t − a)ν (t − x)

(3.4.2.65a) (3.4.2.65b) (3.4.2.65c)

dal

ja

M

M

Pan

and

B

N

Man

as the regular functions within the support of scale functions spanned by the respective index set ΛL , ΛC and ΛR . Then using the quadrature formulae for weakly and Cauchy singular integrals (3.4.2.29) and (3.4.2.43) with the raw images for gb , gc , ga , the composite quadrature formula for the integral in (3.4.2.64) can be found as (c−δ)2j −1

n

I[µ, ν; x] ≡ Q[µ, ν; x] = k

+

(c+δ)2j −K

gb; j k ωjWk L

= a2j −(2K−2) j b2 −1 n

n

+ k

ga;j k ωjWk R .

gc; j k ω Ck−2j x

= (c−δ)2j

(3.4.2.66)

k =( c+δ) 2j −K+1

We now compute the integral appearing in the fourth kind solution (3.4.2.62d) of Cauchy singular integral equation of first kind (3.4.2.61) for g(t) = tn , n = 0, 1, ...4 and for x = ± 34 , ± 21 , ± 14 , 0. n 1 During evaluation of the integral −1 √1−tt2 (x−t) dt at x = ± 14 , 0 we have partitioned the domain of integration into [−1, −δ] ∪ [−δ, δ] ∪ [δ, 1] with δ = 21 at the resolution j = 5. But for the evaluation of integrals for x = ± 12 or ± 34 one needs to adjust both δ and the resolution j to 14 , 81 and 6, 7 4K−2 respectively so that the condition j ≥ for each component of the partition upper limit−lower limit [−1, −δ], [−δ, δ], and [δ, 1] is satisfied. The approximate numerical values of this integral evaluated by our quadrature rule have been compared with the numerical values obtained from the exact ex­ pressions

3.4. Quadrature Rules

108

Table 3.12: Relative error. δ = 18 j=7

δ = 14 j=6

δ = 12 j=5

δ = 12 j=5

δ = 12 j=5

δ = 14 j=6

δ = 18 j=7

− 34

− 12

− 14

0

1 4

1 2

3 4

0

4.0 × 10−4

2.5 × 10−4

4.3 × 10−5

4.6 × 10−5

4.9 × 10−5

2.6 × 10−4

4.5 × 10−4

1

3.0 ×

10−4

1.2 ×

10−4

1.3 ×

10−6

3.1 ×

10−6

1.2 ×

10−5

1.4 ×

10−4

3.6 × 10−4

2.2 ×

10−4

6.2 ×

10−5

2.7 ×

10−6

3.2 ×

10−6

5.9 ×

10−6

8.1 ×

10−5

2.9 × 10−4

1.7 ×

10−4

3.7 ×

10−5

4.3 ×

10−6

3.7 ×

10−6

8.2 ×

10−6

5.2 ×

10−5

2.5 × 10−4

1.4 ×

10−4

3.1 ×

10−5

4.7 ×

10−5

5.2 ×

10−5

5.7 ×

10−5

4.3 ×

10−5

2.2 × 10−4

x n

2 3 4

⎧ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ π ⎨

if n = 0,

if n = 1, � 1 tn dt = − √ πx if n = 2, ⎪ 1 − t2 (x − t) −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ if n = 3, π (x2 + 12 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ πx(x2 + 12 ) if n = 4.

(3.4.2.67)

dal

ja

M

M

Pan

and

B

N

Man

The absolute errors of our approximate values are presented in Table 3.12 and found to be O(10−5 ) at the minimum resolution j. 3.4.2.6

Numerical examples

In order to establish the merit of our approach for evaluation of integrals of product of Daubechies scale function and functions with varied singularities (logarithmic, algebraic, poles of order one and two) it is desirable to verify the efficiency of quadrature rules related to such integrals when applied to evaluate approximate values of variety of singular integrals appearing in physical problems. In our exercise it is assumed that Supp ϕ is [0, 2K − 1] in case of interior scale function having K vanishing moments of their wavelets. It is important to note the following observations. �b • The evaluation of integration a f (x)dx based on the approximation PVj

0

[f ](x) = [I]

j0 a2� −1

ϕjLT (x) + f˜jLT 0 p 0 p

p=a2j0 −2K+2

b2j0� −2K+1

f˜j0 l ϕj0 l (x)

l=a2j0

+

j0 b2� −1

f˜jRT ϕjRT (x) 0 q 0 q

(3.4.2.68)

q=b2j0 −2K+2

generated by ϕ ∈ BVJT (x) of section (3.1.1.3) of the regular integrand f (x) produces exact value for f (x) = xn , n = 0, 1, 2 for Dau3 refinable function despite the loss of orthonormality of truncated V LT V RT }, {ϕRT }. refinable functions {ϕLT j0 k (x), k ∈ Λj0 j0 k (x), k ∈ Λj0

3.4. Quadrature Rules

109

b a

• In spite of producing exact values of integrals of projection

f (x)dx by

b a

fj0 (x)dx for f (x) = xn , n = 0, 1, 2, the

a2j − 1

PWj≥j

0

[I ]

b2j −2K+1 LT d˜LT j p ψj p (x) +

[f ](x) =

d˜j l ψj l (x)

p=a2j −2K+2

l=a2j b2j − 1 RT d˜RT j q ψj q (x)

+

(3.4.2.69)

q=b2j −2K+2

into the detail spaces BWJT (x), j ≥ j0 restricted within finite interval I do not vanish identically as expected. The source of disagreement is perhaps hidden in the determination of d˜LT and d˜RT by solving system of ill-conditioned equations NψRT d˜RT = dRT

and NψLT d˜LT = dLT

(3.4.2.70)

involving matrices NψRT and NψLT having high condition numbers. • The quadrature rule Qj [f, s](x) for evaluation of the integral I[f, s](x) =

f (t)s(t, x)dt

(3.4.2.71)

I

involving regular function f (t) and singular function s(t, x) which is singular at t = x, can be formed in two ways, viz., b2j −2K+1 B and Panja

p=a2j −2K+2

b2j − 1

w ˜jRqT (x) fjRT q (3.4.2.72)

wj l (x) fj l +

N

Man

+

M

=

M

w ˜jLTp (x) fjLT p

dal

a2j − 1

Qsj [f, s](x)

l=a2j

q=b2j −2K+2

obtained by using the projection a2j − 1

b2j −2K+1

s˜LT j0 p (x)

PVj [s](t, x) =

ϕjLTp (t)

p=a2j −2K+2

+

b2j − 1 RT s˜jRT q (x) ϕj q (t)

sj l (x) ϕj l (t) + l=a2j

q=b2j −2K+2

of

singular function s(t, x) into f (t)s(t, x)dt and then evaluating I LT or I or RT (t)dt with the help of quadrature rule for regular function or, f (t)ϕ jk I a2j − 1

Qjf [f, s](x)

b2j −2K+1

wjLTp (x) f˜jLT p

=

+

p=a2j −2K+2

(3.4.2.73) the integral

b2j − 1

˜RT

wjRT q (x) fj q (3.4.2.74)

wj l (x) fj l + l=a2j

q=b2j −2K+2

derived by substituting the projection a2j − 1

b2j −2K+1 LT f˜jLT p ϕj p (t) +

PVj [f ](t) = p=a2j −2K+2

b2j − 1 RT f˜jRT q ϕj q (t)

fj l ϕj l (t) + l=a2j

(3.4.2.75)

q=b2j −2K+2

of regular function f (t) into the integral I[f, s](x). or I or RT (t) appearing both the formula are evaluated by using quadraThe integrals I s(t, x)ϕLT jk ture rules for regular integral in case the singularities are within I. But such integrals are evaluated

3.4. Quadrature Rules

110

by solving the appropriate algebraic equations whenever singularity appear at one or both ends of the interval I. The coefficients with the symbol ˜ having superscript RT or LT are obtained by using the formula (3.4.2.76) (f˜ or w ˜) RT or LT = (N LL or RR )−1 (f or w) RT or LT where the matrices N LL or RR are given by (2.2.1.9) and (2.2.1.10) respectively. Here we adopt the formula Qsj [f, s](x) in (3.4.2.72) for evaluating the integrals involving s(t, x) = ln|t − x|,

1 1 and t−x (t − x)2

(3.4.2.77)

for x ∈ (−1, 1).

3.4.3

Logarithmic singular integrals

We consider the integral �

1

f (t) ln|x − t| dt

I[f, s](x) =

(3.4.3.1)

−1

with s(t, x) = ln|x − t| and f (t) = tn , whose exact values for positive integral exponent n is given by (Carley, 2007) x

1 2m+1

�

±1

(1 − xm+1 ) ln|1 − x|+{(−1)m + xm+1 } × �m 2 x2k ln|1 + x|] − 2m+1 k=0 2m−2k+1

2 2m+1 (ln2

−

�m

1 q=0 2q+1 )

ja

M

M

Pan

and

B

N

Man

2m

�= ±1

dal

n

2m + 1

1 2m+2 [(1

− xm+1 ) ln|1 − x|+{(−1)m + xm+1 }× �m 2 x2k+1 ln|1 + x|] − 2m+2 k=0 2m−2k+1

±1 m+1

�m

1 q=0 2q+1

where m = 0, 1, 2, · · ·. The integral I[f, s](x) for n = 0, 3, 6 and x = {±1, ±.75, ±.5, ±.25, 0} ∈ I are evaluated by the present method at several resolutions and their absolute errors are presented in Fig. 3.8. In evaluating integrals at the interior singular points, we have calculated sRT or LT involved in quadrature formula Qsj [f, s](x) in (3.4.2.72) by regarding s(t, x) as the regular function adjacent to the boundaries. In this case, sRT and LT have been calculated by using the quadrature rule for integrals involving regular function and Daubechies refinable function with partial supports. Since the singular function s(t, x) possesses integrable singularities at the terminal points x = ±1, elements of sRT are calculated by solving Eq. (3.4.2.11) and sLT by using quadrature rule for x = −1. In case of x = 1, sLT have been calculated by solving Eq. (3.4.2.13) and sRT by using quadrature rules. Whenever s(t, x) is singular at both ends x = ±1, solutions of Eqs. (3.4.2.11) and (3.4.2.13) are appropriate for the values of sRT and LT respectively involved in Qsj [f, s](±1).

3.4. Quadrature Rules

3.4.4

111

Cauchy principal value integrals

In order to check the correctness of values of ICk ’s in (3.4.2.47) obtained by using regularization principle, we first verify condition (3.4.2.48a) used by Kessler et al. (Kessler et al., 2003b) to make their system equations for singular ICk ’s consistent. In such comparison Cauchy principal value integral � M 1 IC = − dt −M t in the L.H.S. of (3.4.2.48a) have been calculated with help of the quadrature rule (3.4.2.72). Com­ parison of the above integral with (3.4.2.71) leads to f (x) = 1 and s(t, x) = 1t with singularity at x = 0. Values of IC have been calculated for several choices of M and found to be, as expected, approaches to zero as M gradually increases, e.g., IC ≡ 10−18 for M = 500. The values of CPV integral � 1 f (t) ICP V [f ](x) = − dt (3.4.4.1) −1 t − x 1

with f (x) = (1 − x2 )m− 2 Tn (x) or Un (x), where Tn (x) and Un (x) are the Chebyshev polynomials of first and second kind respectively, for (m, n) = (1, 3),(1, 4), (1, 5) are evaluated by the quadrature rule (3.4.2.72) and comparison of their absolute errors are presented in Figs. 3.9a & 3.9b. In order to compute the absolute error we have calculated exact value of the above integrals by using the formulae (Chan et al., 2003a)

M

M

ja

Pan

and

B

N

dal

Man

� � � 1 1 m−1 2� Tn (t)(1 − t2 )m− 2 1 2m − 1 dt = π(−1)m+1 ( )2m−1 − (−1)k Tn+1−2m+2k (x) k t−x 2 −1 k=0

and � � � 1 1 m−2 2� Un (t)(1 − t2 )m− 2 1 2m − 2 − dt = π(−1)m ( )2m−2 (−1)k Tn+3−2m+2k (x). k t−x 2 −1 k=0

3.4.5

Hypersingular integrals

As in the case of CPVI, we have to first check (3.17a) like consistency condition � M 1 2 = dt = − 2 t M −M

(3.4.5.1)

for finite part integrals by evaluating the integrals in L.H.S. with the help of (3.4.2.72). Comparison of approximate value of the singular integral obtained by using quadrature rule derived here with the exact finite part values given in R.H.S reveals that approximate values of L.H.S. converges as good as in the case of CPVI to value in the R.H.S. which suggest that numerical values of IHk s obtained by using regularization principle are equally reliable as in the case of ICk s. Observing this agreement we have applied proposed quadrature rule (3.4.2.72) to get approximate value of finite part integrals � 1 f (t) IH[f ](x) = = dt (3.4.5.2) 2 −1 (t − x)

3.4. Quadrature Rules

112

LogHEAbs L -7 0 5 -8 -1

3 5

-0.5

6 5

1

0.5 -10

6 7

-11

6 9

-12

R1 Figure 3.8: Log10 (Eabs ) for the integrals −1 tn ln|t − x|dt for x = ± 4i , i = 0, 1, · · · , 4. Numbers in the first column besides the figure correspond the exponent n of the integrand, whereas the numbers in the second column indicate the resolution j of the quadrature rule.

Log10 HEAbs L

Log10 HEAbs L

-5

-5

3 6

-6

4 6

-7

5 6

-8

5 7

4 6

-7

5 6

-8

5 9 -0.75-0.5-0.25 -10

3 6

-6

5 7

-0.75-0.5-0.25 -10

0.25 0.5 0.75

0.25 0.5 0.75

5 9

-11

dt for x = ± 8i , i = 0, 1, · · · , 7.

dal

√ 1−t2 Tn (t) or Un (t) t−x

ja M

M

Pan

and

B

N

Man

R1 Figure 3.9: Log10 (Eabs ) for the integrals −−1

Log10 HEAbs L

Log10 HEAbs L

-2

-2 -3 -4 -5 -6 -7 -8

3 6

-3

4 6

-4

5 6

-5

5 7

-6

5 9

-7 -0.75-0.5-0.25

0.25 0.5 0.75

-0.75-0.5-0.25

R1 Figure 3.10: Log10 (Eabs ) for the integrals =−1

1

(1−t2 )m− 2 Tn (t) (t−x)2

or

Un (t)

3 6 4 6 5 6 5 7 5 9 0.25 0.5 0.75

dt for x = ± 8i , i = 0, 1, · · · , 7.

1

with f (x) = (1 − x2 )m− 2 Tn (x) or Un (x), for (m, n) = (1, 3),(1, 4), (1, 5) in case of Tn (x) and (m, n) = (2, 3),(2, 4), (2, 5) in case of Un (x) and comparison of their absolute errors are presented in Figs. 3.10a and 3.10b. In order to compute the absolute error we have calculated the exact value of the above integrals by using the formulae (Chan et al., 2003a) Z 1 1 Tn (t)(1 − t2 )m− 2 = dt (t − x)2 −1 m+1

= π (−1)

  2m−1 1 2m−1 X 2m − 1 k ( ) (−1) (n + 1 − 2m + 2k)Un−2m+2k (x) k 2 k=0

3.4. Quadrature Rules

and

1

= −1

113

1

Un (t)(1 − t2 )m− 2 dt (t − x)2

� � 2m −2 n 1 2m − 2 = π(−1)m ( )2m−2 (−1)k (n + 3 − 2m + 2k)Un+2−2m+2k (x). k 2 k=0

3.4.6

For multiwavelet family

Here we evaluate the integrals of product of the functions and elements of LMW basis as 1

f (x)φi (x) dx,

(3.4.6.1)

f (x)ψ i (x) dx.

(3.4.6.2)

0

or

1

0

Now by applying Gauss-Legendre quadrature rule, we get (Hildebrand, 1987) 1 i

f (x)φ (x) dx =

−1 n n

wm f (xm )φi (xm ),

(3.4.6.3)

m=0

0

dal

ja

M

M

Pan

and

B

N

Man

where xm denotes the nodes and wm denotes the weights of Guass-Legendre quadrature rule. x0 , x1 , ..., xn−1 are the roots of Legendre polynomial Pn (2x − 1), and the weight functions wm ’s are given by 1 wm := . (3.4.6.4) ' n Pn (2xm − 1) Pn−1 (2xm − 1) Now we compute the values of the coefficients cJ,k and dj,k of (3.1.2.8) and (3.1.2.9) respectively k+1 2J

cJi ,k

1

f (x)φiJ,k (x)

=

dx =

k 2J

1 2

J 2

f(

x+k i ) φ (x) dx, 2J

(3.4.6.5)

f(

x+k ) ψ i (x) dx. 2j

(3.4.6.6)

0

and k+1 2J

dij,k

1 i f (x)ψj,k (x)

= k 2J

dx =

1 2

j 2

0

Using the relation (2.2.2.4) and after some algebraic simplifications, we obtain ⎛ ⎞ 1 1 K −1 n x + 2k x + 2k + 1 l 1 ⎝ (0) (1) dij,k = gi l f ( j+1 ) φl (x) dx + gi l f( ) φ (x) dx⎠ . j 2 2j+1 21+ 2 l=0 0

0

(3.4.6.7)

3.4. Quadrature Rules

114

Now, by using the quadrature formula given in Eq. (3.4.6.3), it follows that n−1 1 �

ciJ,k =

2

J 2

wm f (

m=0

xm + k i ) φ (xm ), 2J

(3.4.6.8)

and dij,k

�K −1 �

1

= 2

1+ 2j

l=0

(1) l

+gi

n−1 �

wm

m=0

3.4.7

Others

3.4.7.1

Sinc functions

n−1 �

xm + 2k l ) φ (xm ) 2j+1 m=0 � xm + 2k + 1 l f( ) φ (xm ) . 2j+1

(0) l

gi

wm f (

(3.4.6.9)

Definition 3.17. (Lund and Bowers, 1992) Let f be a function defined on R and let 0 < h ∈ R. Then the series � C(f, h)(x) = f (kh)S(k, h)(x) (3.4.7.1) k∈Z

when it is convergent is known as cardinal function or cardinal function representation of f .

=

C(f, h)(x)

=

�∞

f (p) (x)

=

dp dxp C(f, h)(x)

=

1 hp

dal

f (x)

Man

Corollary 3.18. (i) For f ∈ B(h), ja

M

M

Pan

and

B

N

k=−∞

f (kh)S(k, h)(x) (3.4.7.2)

�∞

k=−∞

��∞

p � l=−∞ f (lh)δl−k S(k, h)(x).

(ii) For f ∈ L2 (R) − B(h), f (x)

=

� � sinc (x) f (kh)S(k, h)(x) + χ h E h Nkh ( ) N ( ) k=−∞

�∞

2

kh

2

(3.4.7.3) f (p) (x)

=

�∞

k=−∞

���

∞ l=−∞

� dp sinc f (lh)δlp−k S (k, h)(x) + χNkh ( h ) dx . h (x) pE N ( ) �

2

kh

2

Definition 3.19. (Lund and Bowers, 1992) Let f be a function defined on R and M, N ∈ N, 0 < h ∈ R. Then CM,N (f, h)(x) ≡

N � k=−M

N � x f (kh)sinc( − k) = f (kh)S(k, h)(x) h

(3.4.7.4)

k=−M

is known as truncated cardinal series or truncated cardinal function representation of f . The single exponential transformation (2.3.1.12a) maps x ∈ R into t ∈ [a, b]. Then sinc approx­ imation of f ∈ L2 ([a, b]) is given by f (t) = f (φSE (x)) ≈

N � k=−M

f (φSE (kh))S(k, h)(φ−1 SE (t)), t ∈ [a, b].

(3.4.7.5)

3.4. Quadrature Rules

115

The lower- and upper limits, M and N respectively of the sum will depend on the space of functions that contains f . This representation can be used to get the sinc approximation of definite and indefinite integral of functions f ∈ L2 ([a, b]) as b

�

∞

� f (τ ) dτ

�

=

f (φSE (x))φSE (x)dx −∞

a

≈

N �

h

�

f (φSE (kh))φSE (kh)),

(3.4.7.6a)

k=−M t

�

� f (τ ) dτ

φ−1 SE (t)

=

�

f (φSE (x))φSE (x)dx −∞

a

N �

≈

�

1 f (φSE (kh))φSE (kh))J(k, h)(φ− S E (t))

(3.4.7.6b)

k=−M

Here J(k, h)(x) is the function defined in (2.3.1.10). Approximations in two formulae mentioned above are known as single exponential sinc quadrature and the single exponential sinc indefinite integration, respectively. Similarly, the double exponential sinc approximation, quadrature and indefinite integration of f ∈ L2 ([a, b]) corresponding to the double exponential transformation (2.3.1.13a) are given by (Sugihara and Matsuo, 2004; Okayama et al., 2013) f (φDE (x)) M

Panja

and

B

N

dal

=

Man

f (t)

≈

M

N �

f (φDE (kh))S(k, h)(φ−1 DE (t)), t ∈ [a, b],

(3.4.7.7a)

k=−M b

�

∞

� f (τ ) dτ

=

�

f (φDE (x))φDE (x)dx −∞

a

≈

h

N �

�

f (φDE (kh))φDE (kh)),

(3.4.7.7b)

k=−M

�

t

� f (τ ) dτ

φ−1 DE (t)

=

�

f (φDE (x))φDE (x)dx −∞

a

≈

N �

�

f (φDE (kh))φDE (kh))J(k, h)(φ−1 DE (t)).

(3.4.7.7c)

k=−M

3.4.7.2

Autocorrelation functions

Both Φ(x) and Ψ(x) at different scales for autocorrelation family are defined as Φj,k (x) = Φ(2j x − k), j

Ψj,k (x) = Ψ(2j x − k) j

(3.4.7.8)

Unlike functions φj,k (x) = 2 2 φ(2j x − k) and ψj,k (x) = 2 2 ψ(2j x − k) in Daubechies or Coiflet families, functions Φj,k (x), k ∈ Z and Ψj � ,k (x), k ∈ Z, j � ≥ j are not orthogonal, instead they are independent for different k.

3.4. Quadrature Rules

116

3.4.7.3 Representation of function and operator in the basis generated by autocorre­ lation function Approximation of f ∈ L2 (R) Any smooth function f ∈ L2 (R) can be approximated by f (x) � fJ (x) =

n k

f(

k ) ΦJ,k (x). 2J

(3.4.7.9)

� k Let f (x) = Pn (x), a polynomial of degree n. The approximation fJ (x) = k Pn ( 2J )ΦJk (x) of f (x) = Pn (x) is exact for n ≤ 2K − 1 in the basis {ΦJ,k (x), k ∈ Z} having 2K − 1 vanishing moment starting from x, J ∈ N ∪ 0. This relation is valid in spite of Pn (x) is not an element in L2 (R). For f ∈ L2 (R), the dyadic scaling function and wavelet transforms, Pj f (x) and Qj f (x) respec­ tively of f are defined as (Pj f )(x)

=

(Qj f )(x)

=

1 2j 1 2j

y−x )dy 2j R y−x f (y) Ψ( j )dy. 2 R f (y ) Φ(

(3.4.7.10) (3.4.7.11)

For f ∈ C r (R) ∩ L2 (R) with r ≥ 2K, f (x) + O(

(Qj f )(x)

=

O(

f (2K) (x) < ξ 2K >Φ ), 22Kj (2K)!

(3.4.7.12)

dal

=

Man

(Pj f )(x)

(3.4.7.13)

< ξ 2K >Ψ < ξ 2K >Φ � , K ∈ N (2K)! (2K)!

(3.4.7.14)

ja

M

M

Pan

and

B

N

f (2K) (x) < ξ 2K >Ψ ). 22Kj (2K)!

It is important to observe that

as is evident from the following table: K Φ (2K)! Ψ (2K)!

K Φ (2K)! Ψ (2K)!

1

2

3

4

5

8.33(–2)

–1.25 (–2)

2.48(–2)

–5.36(–4)

1.2(–4)

–6.25(–2)

–1.17 (–2)

2.44(–2)

–5.34(–4)

1.2(–4)

6

7

8

9

10

–2.75(–5)

6.39(–6)

-1.49(–6)

3.54(–7)

–8.4(–8)

2.75(–5)

6.39(–6)

1.49(–6)

–3.54(–7)

–8.4(–8)

Thus, for f ∈ C r (R) ∩ L2 (R) with r ≥ 2K, (Pj f )(x) = f (x) + (Qj f )(x).

(3.4.7.15)

Unlike wavelet basis, detail information (Qj f )( 2kj ) of the function f around 2kj can be extracted from the difference of average values (Pj f )( 2kj ) − (Pj+1 f )( 2kj ) without recourse to the exercise of pyramid algorithm.

3.5. MSR Diff. Op.

3.5

117

Multiscale Representation of Differential Operators

The wavelet bases with compact support are useful in the approximation of solution of ordinary/ partial differential equations or integro-differential equations by using Galerkin technique or collo­ cation method. One of the important steps in this approach is the representation of differential operators in the wavelet basis. To derive boundary conditions adapted representation of differential operators, one desires numerical values of some connection coefficients which plays an important role to approximate the derivatives of smooth or non-smooth functions. We have used the notations (Beylkin, 1992) 1

rjκ11,k,κ12 ;j2 ,k2 (p)

=

,κ2 αjκ11,k (p) 1 ;j2 ,k2

=

βjκ11,k,κ12;j2 ,k2 (p)

=

γjκ11,k,κ12;j2 ,k2 (p)

=

0

1 0

φκj11,k1 (x)

dp κ2 φ (x)dx, dxp j2 ,k2

(3.5.0.1a)

φjκ11,k1 (x)

d p κ2 ψ (x)dx, dxp j2 ,k2

(3.5.0.1b)

ψjκ11,k1 (x)

dp κ2 φ (x)dx, dxp j2 ,k2

(3.5.0.1c)

ψjκ11,k1 (x)

dp κ2 ψ (x)dx dxp j2 ,k2

(3.5.0.1d)

1 0 1 0

ja M

M

Pan

and

B

N

Man

Connection coefficients

dal

whenever necessary. Here, each of κ1 , κ2 represents the symbols lef t, LT, I, RT, right as the case may be. In case of (interior) scale function ϕ in Daubechies family, some properties of derivatives available in (Beylkin, 1992; Lin et al., 2005) are summarized in the present context as follows:

If ϕ(p) (x) denotes the pth derivative of the refinable function ϕ(x) with K vanishing moments of their wavelets, then differentiation of both sides of formula (2.1.4.2)(j = 0) leads to √ ϕ(p) (·) = 2p 2 h · Φ(p) (2·) p < K. (3.5.0.2) Since support of ϕ(p) (·) is contained in the support of ϕ(·), clearly ϕ(p) (·) can be expanded as ϕ(p) (·) =

β n

(p)

rl

ϕ(· − l).

(3.5.0.3)

l=α (p)

The coefficients rl in (3.5.0.3) are known as connection coefficients, and are determined by using the following theorems (Beylkin, 1992)

Theorem 3.20. If the integral (p)

rl (p)

exists, then the coefficients rl

∞

ϕ(p) (x) ϕ(x − l)dx

=

(3.5.0.4)

−∞

satisfy the system of linear equations (p)

rl

= 2p

β n β n

(p)

hl1 hl2 r2l+l1 −l2

l1 =α l2 =α

(3.5.0.5)

3.6. Rep. of Der. in LMW

118

Table 3.13: Connections (Beylkin, 1992) for Daubechies scale function Dau3[0,5]. l (1) rl

0/5

1/4

2/3

0/0

1 − 272 /− 2920 365

53 16 /− 1095 365

subject to the condition �

(p)

lp rl

= (−1)p p! .

(3.5.0.6)

l

Theorem 3.21. If K ≥ p+1 2 , then equations (3.5.0.5), (3.5.0.6) have unique solution with a finite (p) (p) number of non-zero coefficients rl , viz., rl �= 0 for −β + 1 ≤ l ≤ −α − 1 such that for every even p � � (p) p (p) (p) (p) l2p¯rl = 0, p¯ = 0, 1, 2, · · · · · · , − 1, (iii) rl = 0 (3.5.0.7) (i) rl = r−l , (ii) 2 l

l

and for every odd p (p)

(i) rl

(p)

= −r−l and, (ii)

�

(p)

l2p¯−1 rl

l

= 0,

p¯ = 1, 2, · · · · · · ,

p−1 . 2

(3.5.0.8)

(1)

dal ja M

M

Pan

and

B

N

Man

We present values of rl for Daubechies K = 3 scale functions in Table 3.13 for their use in the subsequent part of this study. The formulae for evaluation of other cases (boundary elements in truncated or orthogonal classes) are available in (Panja et al., 2016). Values of derivatives at dyadic points Since the exact rule of correspondence between the independent and dependent variables involved in the basis of Daubechies family are not known, except at dyadic points in their support, determination of values of their derivatives is not straightforward. Instead, one has to exploit the formula � 1 �β (p) 2p+ 2 k=α hk ϕk (2x) x ∈ supp ϕ (p) ϕ (x) = (3.5.0.9) 0 otherwise, to determine the values of ϕ(p) (x) at any dyadic point whenever its value at the integers in suppϕ is known.

3.6 Representation of the Derivative of a Function in LMW Basis MSR of the derivative of a function in LMW basis is now given. Due to the finite discontinuity of the elements of LMW basis in their domain, representation of the derivative is defined in the weak � d� denote the derivative operator. In order to construct the representation of D sense. Let D ≡ dx

3.6. Rep. of Der. in LMW

119

in the LMW basis, we consider the evaluation of integrals involving product of elements in the basis and their images under D. Now we can write (Paul et al., 2016c) ⎛ ⎞ j2 K−1 J−1 −1 n n 2n ⎝ρDl ,l φl2 (x) + βDl2 l1 (j2 , k2 )ψjl22 ,k2 (x)⎠ , Dφl1 (x) = 2 1 j2 =0 k2 =0

l2 =0

� � Dψjl11 ,k1 (x)

=

K−1 n

⎛ ⎝αDl

2 l1

(j1 , k1 )φl2 (x) +

j2 J−1 −1 n 2n

⎞ γDl2 l1 (j2 , k2 ; j1 , k1 )ψjl22 ,k2 (x)⎠ ,

j2 =0 k2 =0

l2 =0

where 1

ρDl1 l2

=

αDl1 l2 (j2 , k2 )

=

βDl1 l2 (j1 , k1 )

=

γDl1 l2 (j1 , k1 ; j2 , k2 )

=

d φl2 (x) dx, dx 0 1 � d � l2 φl1 (x) ψj2 ,k2 (x) dx, dx 0 φl1 (x)

(3.6.0.1)

1

d φl2 (x) dx, dx 0 1 � d � l2 ψj2 ,k2 (x) dx. ψjl11 ,k1 (x) dx 0 ψjl11 ,k1 (x)

Since the explicit forms of the elements in LMW basis are known, the integrals in RHS of (3.6.0.1) can be evaluated by splitting the domain of integration at the points of discontinuity of the wavelets.

⎛

DJM S

⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

ρD β D (0) β D (1) . . . β D (J − 1)

dal

Man

(J−1) Ψ)�

of D in the basis (Φ0 ,

N

D(Φ0 ,

(J−1) Ψ)

can be written

ja M

Pan

and

B

(J−1) Ψ),

M

Thus the MSR � (Φ0 , in the form

α D (0) γ D (0, 0) γ D (1, 0) . . . γ D (J − 1, 0)

α D (1) γ D (0, 1) γ D (1, 1) . . . γ D (J − 1, 1)

...... ...... ...... ...... ...... ...... ......

α D (J − 1) γ D (0, J − 1) γ D (1, J − 1) . . . γ D (J − 1, J − 1)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

,

(2J K)×(2J K)

(3.6.0.2) where � � ρDl1 l2 K×K , � � α D (j) = αDl1 l2 (j, k) K×2j K , � � β D (j) = βDl1 l2 (j, k) 2j K×K , � � γ D (j1 , j2 ) = γDl1 l2 (j1 , k1 ; j2 , k2 ) 2j1 K×2j2 K . ρD

=

(3.6.0.3)

Thus, D(Φ0 ,

(J−1) Ψ)

≡ =

(Φ0 , (Φ0 ,

(J−1) Ψ)

� (Φ0 ,

(J−1) Ψ)

DJM S .

(J−1) Ψ),

D(Φ0 ,

(J−1) Ψ)�

(3.6.0.4)

3.6. Rep. of Der. in LMW

120

We now show that DJM S is a nilpotent matrix of order K. From (3.6.0.4) we find DK (Φ0 ,

(J−1) Ψ)

=

(Φ0 ,

DJM S

(J−1) Ψ)

K

.

(3.6.0.5)

Since elements in the LMW basis consist of piecewise continuous polynomials of degree at most K K − 1, all the elements of DK (Φ0 , (J−1) Ψ) are zero so that DJM S ≡ 0. Explicit value of DJM S for K = 3, J = 0 is √ ⎛ ⎞ 0 2 3 � MS� √0 D0 K=3 = ⎝ 0 0 2 15 ⎠ . 0 0 0 Clearly, here � MS� D0 K=3 while for K = 4

2

D0M S

= 0,

√ 2 7 √0 2 35 0

⎛ ⎜ � MS� D0 K=4 = ⎜ ⎝

3

� MS� D0 K=3

= 0 and

K=3

0

⎞ ⎟ ⎟ ⎠.

Moreover, � MS� D0 K=4 while for K = 5

3

⎛

dal

K=4

ja M

Pan

and

B

N

Man

D0M S

= 0, ⎞

0 √ 6 3 0 √ 6 7 0

M

⎜ ⎜ � MS� D0 K=4 = ⎜ ⎜ ⎝

4

� MS� D0 K=4

= 0 and

0

⎟ ⎟ ⎟. ⎟ ⎠

Again � MS� D0 K=5

4

� MS� D0 K=5

= 0 and

5

= 0.

Explicit value of DJM S for K = 4, J = 1 is ⎛

�

D1M S

� K=4

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎝ 0 0

�

D0M S

�

18

K=4

6 0

0

0

0

0

0

0 0

0 0

0 0

0 �

10 5 17

0 �

6

0 �

105 17

0 � 5 −6 119 0 − √18

119

�

80

0 �

3 7

2

0 �

15 7

14 5 119

0 � 15 −10 17 0

105 17

0 �

3

= 0 and

� MS� D1 K=4

4

5 17

0 � 5 32 51 0 142 −√

51

with the property � MS� D1 K=4

6

= 0.

�

15 7

0 � −10 37 0 √128 119

0 � 3 −16 17 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

3.7. MSR of Int. Op.

3.7

121

Multiscale Representation of Integral Operators

3.7.1

Integral transform of scale function and wavelets

We consider here the integral operator K involving the kernel K(x, t) in the form b

K[f ](x) =

K(x, t)f (t)dt.

(3.7.1.1)

a

Then

b

b

K(x, t)ϕj k (t)dt, K[ψj k ](x) =

K[ϕj k ](x) = a

K(x, t)ψj k (t)dt. a

j

j

Use of the the definition ϕj k (x) = 2 2 ϕ(2j x − k), ψj k (x) = 2 2 ψ(2j x − k) followed by the transformation of variable τ = 2j t − k leads to K[ϕj k ](x) =

1 2

j 2

2j b 2j a

τ +k K(x, j )ϕ(τ )dτ, 2

K[ψj k ](x) =

1 2

j 2

2j b

K(x, 2j a

τ +k )ψ(τ )dτ. 2j

Logarithmic singular kernel K(x) = ln|x|

The integral transform of functions involving logarithmic singular kernel is defined as

b

KL [f ](x)

ln|x − t| f (t)dt.

=

(3.7.1.2)

ja M

M

Pan

and

B

N

Man

For f = ϕj,k (x) and supp ϕj,k (x) ∈ [a, b],

dal

a

∞

KL [ϕj,k ](x)

ln|x − t| ϕj,k (t)dt.

=

(3.7.1.3)

−∞

A change in variable 2j t − k = s and some algebraic rearrangements provide KL [ϕj,k ](x)

=

� 1 � KL [ϕ](2j x − k) − j ln2 j 2

(3.7.1.4)

where ∞

KL [ϕ](x) = KL [ϕ0,0 ](x)

ln|x − t| ϕ(t)dt.

=

(3.7.1.5)

−∞ ∞ ϕ(x)dx = 1 has −∞ lef t supp ϕj,k (x) ∈ [a, b],

During calculation, the property For f =

t ϕlef j,k (x)

∈

t Φlef j

and

been used.

∞ t KL [ϕlef j,k ](x)

= a

j

t ln|x − t|ϕlef j,k (t − a)dt ∞

t j j ln|x − a − s|ϕlef k (2 s)ds (using 2 s = t) 0 � ∞ � 1 lef t j {ln | 2 (x − a) − t | −j ln2}ϕ (t)dt , j k 22 0 � 1 � lef t lef t 0 j K [ϕ ](2 (x − a)) − j ln2 µ . L j k k 22

= 22 = t KL [ϕlef j,k ](x) =

(3.7.1.6)

3.7. MSR of Int. Op.

122

right On the other end, f = ϕright and supp ϕright j,k (x) ∈ Φj j,k (x) ∈ [a, b], b

KL [ϕright j,k ](x)

= −∞

ln|x − t|ϕright

j,k (t)dt 0

j

=

22 −∞

0

�

� {ln | 2j (x − b) + s | −j ln2}ϕk� (s)dt ∞ � ∞ {ln | 2j (b − x) − s | −j ln2}ϕk� (s)dt ,

j

2 2� 1

= KL [ϕright j,k ](x)

1

−

=

ln|x − t|ϕ�k (2j (b − t))ds (using 2j (b − t) = s)

j

22 0 � 1 � KL [ϕ�k ](2j (b − x)) − j ln2 µ�k 0 . j 22

=

(3.7.1.7)

Homogeneous kernel K(λx) = λµ K(x) In case of the kernel K(x, t) = K(x − t) and K(λx) = λµ K(x), the integral transforms of the elements in the multiscale basis becomes K[ϕj k ](x)

=

K[ψj k ](x)

2j b

j(µ− 12 )

2

1

2j(µ− 2 )

=

K(2j x − k, τ )ϕ(τ )dτ,

(3.7.1.8)

K(2j x − k, τ )ψ(τ )dτ.

(3.7.1.9)

2j a 2j b 2j a

dal M

M

∞

1

1

K[ϕj k ](x) = 2j(µ− 2 ) 1

ja Pan

and

B

N

Man

Moreover, for ϕj k (x) ∈ φIj , ψj k (x) ∈ ψ Ij the above integral transforms reduces to K(2j x − k, τ )ϕ(τ )dτ = 2j(µ− 2 ) K[ϕ](2j x − k) −∞ ∞

K[ψj k ](x) = 2j(µ− 2 )

1

K(2j x − k, τ )ψ(τ )dτ = 2j(µ− 2 ) K[ψ](2j x − k).

(3.7.1.10) (3.7.1.11)

−∞

In case of boundary scale functions and wavelets these expressions become t K[ϕlef j0 ,k ](x) = t K[ψjlef ](x) 0 ,k

K[ϕright j0 ,k ](x) K[ψjright ](x) 0 ,k

=

∞

1

t K(2j (x − a), τ )ϕlef k (τ )dτ,

2j(µ− 2 ) 0 ∞

j(µ− 12 )

K(2j (x − a), τ )ψklef t (τ )dτ,

2

0

∞

1

=

2j(µ− 2 )

=

j(µ− 12 )

(3.7.1.12)

K(2j (b − x), τ )ϕk� (τ )dτ,

0 ∞

2

0

K(2j (b − x), τ )ψk� (τ )dτ.

(3.7.1.13)

Formulae for obtaining K[Φlef t ](x) and K[Φright ](x) It is observed that all the formulae mentioned for variety of kernels depend on the integrals involved in K[Φlef t ](x) or K[Φright ](x). So, it is desirable to provide either recurrence relations or formulae involving those integrals at two neighbouring scales as the case may be for those integrals.

3.7. MSR of Int. Op.

123

Definition 3.22. We use the symbols K[Φlef t ](x) and K[Φright ](x) to represent images of Φlef t (x) and Φright (x) respectively under the integral operator involving kernel K(x, t) as ∞

K[Φlef t ](x) =

Φlef t (t) K(x, t) dt,

(3.7.1.14a)

Φright (t) K(x, t) dt.

(3.7.1.14b)

0 0

K[Φright ](x) = −∞

Lemma 3.23. K[Φ

lef t

](x)

=

� 1 √ H lef t 2

�

∞

!

t x, 2

K 0

� !

!

Φlef t (t )dt �

� � ! ! t LI ! Φ (t )dt , K x, +H 2 −∞ � � !� 0 ! ! 1 t right √ H Φright (t )dt K x, 2 2 −∞ � !� � ∞ ! t RI RI ! Φ (t )dt . K x, +H 2 −∞ ∞

LI

K[Φright ](x)

=

(3.7.1.15a)

(3.7.1.15b)

dal

ja

M

M

Pan

and

B

N

Man

Proof: We first consider the case left. Using the two scale relation for Φlef t (t) into the definition (3.7.1.14a) of K[Φlef t ](x), one gets: ∞� √ � 2 H lef t Φlef t (2t) + H LI ΦLI (2t) K(x, t)dt K[Φlef t ](x) = 0 � � ∞ ∞ √ lef t lef t LI LI = 2 H Φ (2t)K(x, t)dt + H Φ (2t)K(x, t)dt . −∞

0 '

Performing a change t = 2t in variable gives � 1 lef t K[Φ ](x) = √ H lef t 2

�

∞ lef t

Φ

!

(t ) K

0 ∞

+H LI

!

ΦLI (t ) K −∞

!

t x, 2 �

� !

dt !

t x, 2

�

� !

dt

.

Hence, result (3.7.1.15a) is proved. Following similar steps for the case right, result (3.7.1.15b) can be proved. Definition 3.24. For K(x, t) = K(|x − t|), we introduce here the following definitions. ∞

KH [Φlef t ](x)

=

Φlef t (t) K(|x − t|)dt

(3.7.1.16a)

ΦLI (t) K(|x − t|)dt

(3.7.1.16b)

Φright (t) K(|x − t|)dt

(3.7.1.17a)

ΦRI (t) K(|x − t|)dt.

(3.7.1.17b)

0 ∞

KH [ΦLI ](x)

= −∞ 0

KH [Φright ](x)

= −∞ ∞

KH [ΦRI ](x)

= −∞

3.7. MSR of Int. Op.

124

Theorem 3.25. For K(x, t) = K(|x − t|) and K(λx) = λµ K(x)(homogeneity property), KH [Φlef t ](x) =

1

1

KH [Φright ](x) =

H lef t KH [Φlef t ](2x) + H LI KH [ΦLI ](2x)

(3.7.1.18a)

H right KH [Φright ](2x) + H RI KH [ΦRI ](2x) .

(3.7.1.18b)

1 2µ+ 2

1 2µ+ 2

Proof : To establish this result (3.7.1.18a), we use the homogeneity property and the transfor­ l mation of variables x = 2x, in (3.7.1.15a) to get KH [Φlef t ](x) = ∞ l l l l 1 lef t Φlef t (t )K(x , t )dt + H LI 1 {H µ+ 2 2 0

∞

l

l

l

l

ΦLI (t )K(x , t )dt }. −∞

Using the definitions (3.7.1.16a) and (3.7.1.16b), the above expression becomes KH [Φlef t ](x)

= =

1 1 2µ+ 2

l

l

{H lef t KH [Φlef t ](x ) + H LI KH [ΦLI ](x )}

1 1 2µ+ 2

H lef t KH [Φlef t ](2x) + H LI KH [ΦLI ](2x) .

Thus, result (3.7.1.18a) is proved. Following similar steps, the result (3.7.1.18b) can be proved with the help of definitions (3.7.1.17a) and (3.7.1.17b). Definition 3.26. For K(x, t) = ln|x − t|, we give the following definitions: dal N B and ja Pan M M

=

Man

∞

KL [Φlef t ](x)

Φlef t (t) ln|x − t|dt

(3.7.1.19a)

ΦLI (t) ln|x − t|dt.

(3.7.1.19b)

Φright (t) ln|x − t|dt

(3.7.1.20a)

ΦRI (t) ln|x − t|dt.

(3.7.1.20b)

0 ∞

KL [ΦLI ](x)

= −∞

0

KL [Φright ](x)

= −∞ ∞

KL [ΦRI ](x)

= −∞

Theorem 3.27. For K(x, t) = ln|x − t| and K(λx, λt) = K(x, t) + ln|λ|, KL [Φlef t ](x)

=

KL [Φright ](x) =

1 √ {H lef t KL [Φlef t ](2x) + H LI KL [ΦLI ](2x) 2 −ln2(H lef t µlef t 0 + H LI I(β−2α−k+1)×1 )} 1 √ {H right KL [Φright ](2x) + H RI KL [ΦRI ](2x) 2 −ln2(H right µright 0 + H RI I(β−2α−k+1)×1 )}.

Here, Im×n denotes an m × n matrix with all entries equal to 1.

(3.7.1.21a)

(3.7.1.21b)

3.7. MSR of Int. Op.

125

Proof: To prove the result in (3.7.1.21a), we substitute K(x, t) = ln|x − t| and K(λx, λt) = K(x, t) + ln(|λ|) in (3.7.1.15a) to get � � ∞ � � � 1 KL [Φlef t ](x) = √ H lef t (ln|2x − t |−ln2) Φlef t (t )dt 2 0 � � ∞ � � � + H LI (ln|2x − t |−ln2) ΦLI (t )dt . −∞

Use of definitions (3.7.1.19a),(3.7.1.19b),(3.3.3.1a) into the right hand side of above relation followed by appropriate rearrangement gives 1 � KL [Φlef t ](x) = √ H lef t KL [Φlef t ](2x) + H LI KL [ΦLI ](2x) 2 � − ln2 (H lef t µlef t 0 + H LI I(β −2α−k+1)×1 ) . Hence, result (3.7.1.21a) is proved. The result in (3.7.1.21b) can be obtained by following the similar steps in conjunction with the definitions (3.7.1.19b),(3.7.1.20b), (3.3.3.1a).

3.7.2

Regularization of singular operators in LMW basis

Here we present a multiresolution approach of regularization of singular integral operators � 1 K[f ](x) = K(x − y)f (y)dy, (3.7.2.1) 0

dal

ja

M

M

Pan

and

B

N

Man

with convolution kernels K(x − y) of homogeneous type ( K(λ(x − y)) = λα K(x − y), α ∈ R). We limit our discussion on numerical procedure based on LMW for their construction with a view to their applications to SIEs. 3.7.2.1

Principle of regularization

Let T be an operator as mentioned above. Provided unique solution ρ to a system of linear equations 2α+2ρ = A ρ + b

(3.7.2.2)

exists for some matrix A and matrix b , the regularized kernel (not singular in case of α < 0) K0 (x, y) =

K −1 �

ρl01 l2 φl1 (x)φl2 (y)

(3.7.2.3)

l1 ,l2 =0

with coefficients ρ can be obtained through the steps presented below. Then the operator T0 : V0 → V0 , with the kernel K0 (x, y) ∈ V0 × V0 , is defined as the multiresolution regularization (MRR) � of the operator T on wavelet basis φl (x), l ∈ ∧, an appropriate index set}. 3.7.2.2

Regularization of convolution operator in LMW basis

The main ingredient in the process multiresolution representation (MRR) of singular integral oper­ ator is the evaluation of the integral � 1 � � ρln1 l2 = φl1 (x − n) T φl2 (x)dx (3.7.2.4) 0

3.8. H¨ older Exponent

126

on scale j = 0 and n ∈ Z. Using the two scale relation (2.2.2.3) for LMW with K vanishing moments, this integral can be recast into the form ρln1 l2

= 2−α−2

K −1 K −1 � � �

� � (1) (1) (0) (1) (0) (0) 1 k2 hl1 k1 hl2 k2 ρk2n−1 + hl1 k1 hl2 k2 + hl1 k1 hl2 k2 ρk2n1 k2

k1 =0 k2 =0 (1) (0) 1 k2 +hl1 k1 hl2 k2 ρk2n+1

�

.

(3.7.2.5)

The two-scale difference equation (3.7.2.5) takes the form (3.7.2.2) for unknowns ρ0

1×K 2

= {ρl01 l2 , l1 , l2 ∈ {0, ..., K − 1}}

(0) (0) (1) (1) with the matrix element Al1 l2 ;k1 k2 = hl1 k1 hl2 k2 + hl1 k1 hl2 k2 in the matrix A and elements bl1 l2 = �K−1 (0) (1) k1 k2 (1) (0) k1 k2 ) in the vector b . The elements ρln1 l2 , n = −1, 1, l1 , l2 ∈ k1 ,k2 =0 (hl1 k1 hl2 k2 ρ−1 + hl1 k1 hl2 k2 ρ1 {0, ..., K − 1} again follow Eq. (3.7.2.5). We skip the details of their evaluation here. It will be discussed in somewhat details at the time of their applications to integral equations of different types of singular kernels.

3.8 3.8.1

Indices ¨ Estimates of Local Holder Basis in Daubechies family

dal

ja

M

M

Pan

and

B

N

Man

Theorem 3.28. If r is the Holder ¨ index of interior scale function ϕ and wavelet ψ (i.e., ϕ, ψ ∈ C r ), then the collection in Theorem 2.2 is an unconditional basis for C s ([a, b]) for s < r. A bounded function f is in C s ([a, b]) if and only if �

� � � t > |, |< f, ψj k > |, k ∈ ΛjW I , |< f, Ψjright > | ≤ |< f, Ψlef j

C 1 2j(s+ 2 )

.

(3.8.1.1)

� � Here the constant C is independent of the resolution j and the location k ∈ 2j a, · · · , 2j b − 1 and I is the index set representing collection of indices for interior wavelets at resolution j whose ΛW j coefficients are not negligible.

3.8.2

Basis in Multiwavelet family

� � of [0, 1]. The We find the behaviour of u(x) at any point in the dyadic interval Ij,k = 2kj , k+1 2j estimate of the H¨older exponent of a function whose wavelet coefficients dijk are known, is given below (Paul et al., 2016a). Theorem 3.29. If νj,k is the Ho¨lder exponent ν of the solution u(x) in Ij,k , then the estimate of νj,k in terms of the wavelet coefficient blj,k is ⎛ 1 ⎜ νj,k ≈ − + log2 ⎝ 2

sup

dlj,k

⎞

l∈{0,1,···,K−1} ⎟ ⎠. sup dlj+1,2k l∈{0,1,···,K−1}

3.8. H¨ older Exponent

127

Proof. Assume that � �ν k u(x) ≈ constant x − j for x ∈ Ij,k . 2

(3.8.2.1)

If the value of ν is an integer, then u(x) is well behaved in Ij,k , but if ν is otherwise then u(x) is non-smooth in Ij,k . Thus the behaviour at each point of [0, 1] can be found from the wavelet coefficient in the following way. The wavelet coefficients of u(x) in [0, 1] are given by 1

blj,k

l u(x) ψj,k (x)dx

= 0

k+1 2j

=

k 2j

≈

l u(x) ψj,k (x)dx k+1 2j

j

constant 2 2

� x−

k 2j

= =

k 2j

�ν

ψ l (2j x − k)dx

1

constant

tν ψ l (t)dt

1 2j(ν+ 2 )

0

c1

where c1 is a constant. 1 , 2j(ν+ 2 )

(3.8.2.2)

Similarly, 1

blj+1,2k

l (x)dx u(x) ψj+1,2k

= 0

dal

ja

Pan

and

B

N

Man

u(x) ψjl +1,2k (x)dx M

M

=

k+ 1 2 2j k 2j

≈

constant 2

k+ 1 2 2j

j+1 2

k 2j

= =

1

constant 1 2(j+1)(ν+ 2 )

c1 1 . (j+1)(ν+ 2) 2

� �ν k x− j ψ l (2j x − 2k)dx 2

tν ψ l (t)dt 0

(3.8.2.3)

Thus, blj,k l bj+1,2k

1

≈ 2ν+ 2 ,

so that νj,k

1 ≈ − + log2 2

�

blj,k blj+1,2k

� ,

(3.8.2.4)

where νj,k stands for the H¨ older exponent ν, as we were considering the behaviour of u(x) in Ij,k . It may be noted that for a given j and k, l takes up the values l = 0, 1, . . ., K − 1. Thus we choose an estimate for νj,k in Ij,k as ⎛ ⎞ sup dlj,k 1 ⎜ l∈{0,1,···,K−1} ⎟ νj,k ≈ − + log2 ⎝ (3.8.2.5) ⎠. 2 sup dlj+1,2k l∈{0,1,···,K−1}

3.9. Error Estimation

128

S This result may be used in estimating error in MSA uM of u(x) as discussed in the following section. J

3.9

Error Estimates in the Multiscale Approximation

Approximations in the basis of Daubechies family For f ∈ L2 (R, in the approximation (3.1.1.3) f (x) � fJ (x)

=

cV W Rj0 J · BV W Rj0 J (x)

=

(BV Rj0 , BW Rj , j = j0 , · · · , J − 1)

(3.9.0.1)

·(cVRj0 , dWRj , j = j0 , · · · , J − 1)T , the error EJ (x) can be written as EJ (x) = (BW Rj , j = J, · · ·) · (dWRj , j = J, · · ·)T .

(3.9.0.2)

Then the L2 -error in the approximation is � EJ (x) �2

= =

< EJ (x), EJ (x) > ∞ n n |dj ! ,k |2 j ! =J k∈ΛRW j ∞ � n

C

� maxk∈ΛRW! |dj ! ,k |2 . j

M

M

ja Pan

and

B

N

dal

j ! =J Man

≤ For f ∈ L2 (R), there exists ν so that

maxk∈ΛRW! |dj ! k | j

maxk∈ΛRW! |dj ! −1,k | j −1

∼

1 , ν > 0. 2ν

(3.9.0.3)

Then � EJ (x) �2

≤

C

∞ � � n maxk∈ΛRW! |dj ! ,k |2 j

j ! =J

= = =

� � 1 1 1 2 W max |d | 1 + + + · · · k∈ΛRJ−1 J−1,k 22ν 22ν 24ν � � 1 1 C 2ν maxk∈ΛRW |dJ−1,k |2 . J−1 2 1 − 212ν 1 C 2ν maxk∈ΛRW |dJ−1,k |2 . J−1 2 −1 C

(3.9.0.4)

!

We redefine the exponent ν so that 22ν − 1 = 22ν . Then an estimate for a posteriori error can be found as � EJ (x) �

∼

C

! 1 W maxk∈ΛRW |dJ−1,k |= e−ν ln2 maxk∈ΛRJ−1 |dJ−1,k | J−1 2ν !

(3.9.0.5)

3.9. Error Estimation

129

Approximations in multiwavelet Basis In order to estimate the error in the MSA fJM S (x) of f (x), we use the fact that the multiscale expansion of f (x) is (Paul et al., 2016a) ⎡ ⎤ j K−1 ∞ 2� −1 � � l ⎣al0,0 φl0,0 (x) + f (x) = blj,k ψj,k (x)⎦ , (3.9.0.6) j=0 k=0

l=0

which can be recast into the form f (x) =

fJM S (x)

j K−1 ∞ 2� −1 ��

+

l l bj,k ψj,k (x),

(3.9.0.7)

l=0 j=J k=0

where fJM S (x) is given by (3.1.2.2). Hence if we write f = fJM S + δf,

(3.9.0.8)

where δf is the error in the MSA at resolution J, then δf =

j K−1 ∞ 2� −1 ��

l blj,k ψj,k (x).

(3.9.0.9)

l=0 j=J k=0

⎡

dal

l Using orthonormality property of ψj,k (x), we find

M

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and

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j K−1 ∞ 2� −1 ��

⎤ 12

|blj,k |2 ⎦ .

||δf ||L2 [0,1] = ⎣

(3.9.0.10)

l=0 j=J k=0

If f ∈ C ν [0, 1], then the RHS of (4.3.2.34) is always bounded (cf. (Alpert, 1993)) and a bound is given by 1 2 sup |u[ν] (x)|, 2Jν 22ν ν! x∈[0,1]

(3.9.0.11)

where [ν] is the integer part of ν. Thus ||δf ||L2 [0,1] ≤ A 2−Jν = A e−(ν ln2)J ,

(3.9.0.12)

sup |f [ν] (x)| x∈[0,1] 22ν−1 ν!

where A = so that as J increases, the error decreases exponentially. The regularity of a function f (x) can be measured in different ways. If f ∈ C n but f not in C n+1 , then its global H¨older exponent is given by µ = n + ν, where ν is the supremum of the α ∈ [0, 1) such that φ(p) ∈ C α , or equivalently � � log|f (n) (x + t) − f (n) (x)| ν = inf lim inf . log|t| |t|→0 x t

3.9. Error Estimation

130

Sinc approximations The sinc function approximates (in the rectangular domain Nkh ( h2 ) around the point (kh, 0) on the real line with side h × d, 0 < h, d ∈ R) any smooth function f ∈ L2 (R) by a one point interpolation formula f (x) = f (kh) sinc(

x − kh sinc sinc ) + EN (x) h (x) = f (kh) S(k, h)(x) + E Nkh ( h kh ( 2 ) 2) h

with the error Esinc (x) =

sin( πx h ) 2πi

� along the rectangle

f (z) dz. (z − x)sin( πz h )

(3.9.0.13)

(3.9.0.14)

Clearly the approximation of f (x) by the formula (3.9.0.13) provide the exact value at x = kh and the error in its approximation at any point in the h2 neighbourhood of kh can estimated from the formula given in (3.9.0.14). Definition 3.30. In the notation DS ≡ {z ∈ C : z = x + iy, |y|< d} representing the infinite strip of width 2d, 0 < d ∈ R containing the real line R, B(DS ) is the collection of functions f analytic in DS with � d

|f (x + iy|dy = O(|x|α ), |x|→ ∞, 0 ≤ α < 1

(3.9.0.15)

�� � p1 �∞ p limy→d− |f (x + iy| dx −∞ �� � p1 � ∞ + −∞ |f (x − iy|p dx < ∞.

(3.9.0.16)

and

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and

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N p (f, DS ) ≡

dal

−d

Theorem 3.31. (Lund and Bowers, 1992; Stenger, 2012; Stenger, 2016, Th.2.13,pp. 35) If f ∈ B(DS )(p = 1 or 2) and h > 0, the error εCard (x) = f (x) − C(f, h)(x) =

sin( πx h ) I(f, h)(x) 2πi

(3.9.0.17)

with � I(f, h)(x)

∞

≡ −∞

�

f (t − i d− ) � � (t − x − i d)sin πh (t − i d− )

f (t + i d− ) � � − (t − x + i d)sin πh (t + i d− )

� dt.

(3.9.0.18)

Moreover, ||f − C(f, h)||∞ ≤ O(e−

πd h

).

(3.9.0.19)

Theorem 3.32. For f ∈ B(DS )(p = 1 or 2) and there exists positive constants α, β and C so that � −α|x| e , x ∈ (−∞, 0) |f (x)|≤ C (3.9.0.20) e−βx , x ∈ [0, −∞). Choose

α N = geatest integer less than | M + 1| β

(3.9.0.21)

3.9. Error Estimation

131

and

�

πd 2πd ≤ . αM ln2

h= Then

(3.9.0.22)

√ √ ||f − CM,N (f, h)||∞ ≤ K1 M e− πdαM .

(3.9.0.23)

where K1 is a constant depending on f, p and d. It is assumed that [a, b] is contained in D, a simply-connected domain in C, 0 < K, α, β ∈ R. We use the symbols Lα,β K (D)

=

� � f : f is analytic, |f (z)|≤ K|(z − a)α (b − z)β | ,

� � �� � � z − a �� z ∈ Z : ��arg < d , b−z � ⎧ � ⎫ ⎡ ⎤� � � � � � � ��2 � ⎨ ⎬ � � 1 z−a 1 z−a ⎦� < d . = z ∈ Z : ��arg ⎣ ln + 1+ ln � ⎩ ⎭ π b−z π b−z � � �

DdS E

DdDE

=

Theorem 3.33. (Lund and Bowers, 1992; Stenger, 2012; Stenger, 2016, Th 4.2.5) For f ∈ � α,β πd LK (DdSE ), d ∈ (0, π), µ = min{α, β}, n ∈ N, h = µn , lower- and upper limits � N=

� αn β � µ=α , n µ=β

(3.9.0.24)

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and

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M=

n µ=α , βn �α� µ=β dal

�

� � N � � n √ −√πdµn � � 1 sup �f (t) − f (φSE (kh)S(k, h)(φ− (t))) ≤ C ne � SE � t∈(a,b) �

(3.9.0.25)

k=−M

with the constant (Okayama et al., 2013)[Th. 2.3] � � � 2K(b − a)α+β 2 µ √ C= + . d −2 πdµ α+β µ π d πd(1 − e )cos (2)

(3.9.0.26)

α,β Theorem 3.34. (Stenger, � 2012, Th 4.2.6) For (z − a)(b − z)f ∈ LK (DdSE ), d ∈ (0, π), µ = 2πd min{α, β}, n ∈ N, h = µn . The lower- and upper limits M, N are defined as the previous theorem. Then the error in single exponential sinc quadrature is � � N � � b n √ ! � � f (t)dt − h f (φSE (kh)φSE (kh))� ≤ CQ e− 2πdµn (3.9.0.27) � � � a k=−M

with the constant (Okayama et al., 2013)[Th. 2.3] � � 2K(b − a)α+β−1 2 √ CQ = +1 . µ (1 − e− 2πdµ )cosα+β ( d2 )

(3.9.0.28)

3.9. Error Estimation

132

Theorem 3.35. (Okayama et al., 2013, Th. 2.3) For (z − a)(b − z)f ∈ Lα,β K (DdSE ), d ∈ (0, π), µ = y πd min{α, β}, n ∈ N, h = µn . The lower- and upper limits M, N are defined as the previous theorem. Then the error in single exponential sinc indefinite integral is

N

t

�

!

− f (φSE (kh)φSE (kh))J(k, h)(φ−1 SE (t)) ≤ CI e

f (τ )dτ −

sup t∈(a,b)

a

√ πdµn

(3.9.0.29)

k=−M

with the constant CI =

2K(b − a)α+β−1 µ

1

d(1 −

πd + 1.1 . µ

√ e−2 πdµ )cosα+β ( d2 )

(3.9.0.30)

The convergence rate of double exponential sinc approximation, quadrature and indefinite inte­ grals appear to be faster as is evident from the following three theorems. π Theorem 3.36. (Okayama et al., 2013, Th. 2.9) For f ∈ Lα,β K (DdDE ), d ∈ (0, 2 ), µ = min{α, β}, ν = 2dn ln( µ ) max{α, β}, νe , lower- and upper limits 2d ≤ n ∈ N, h = n

M=

n µ=α , n − lln(α/β)/hJ µ = β

sup f (t) −

t∈(a,b)

N k=−M

n − lln(β/α)/hJ µ = α , n µ = β

N=

(3.9.0.31)

f (φDE (kh))S(k, h)(φ−1 DE (t)) (3.9.0.32) 2

dal

ja

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and

B

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cosd cosα+β ( π2 sind) (1−e−πµe ) + µe M

π

M

≤

2K(b−a)α+β πdµ

π 2ν

πdn

−

e ln(2dn/µ) .

π Theorem 3.37. (Okayama et al., 2013, Th. 2.11) For (z −a)(b−z)f ∈ Lα,β K (DdDE ), d ∈ (0, 2 ), µ = 4dn ln( µ ) min{α, β}, ν = max{α, β}, νe . The lower- and upper limits M, N are the 4d ≤ n ∈ N, h = n same as mentioned above. Then an estimate of error in double exponential sinc quadrature is given by ! b N f (t)dt − h k=−M f (φDE (kh))φDE (kh) a

≤

2K(b−a)α+β−1 µ

2

π

cos d cosα+β ( π2 sin d) (1−e− 2 µe )

+ µe

π 2ν

−

(3.9.0.33)

2πdn

e ln(4dn/µ) .

α,β Theorem 3.38. (Okayama et al., 2013, Th. 2.11) For (z −a)(b−z)f ∈ LK (DdDE ), d ∈ (0, π2 ), µ = 2dn ln( µ ) min{α, β}, ν = max{α, β}, νe . The lower- and upper limits M, N are the 2d ≤ n ∈ N, h = n same as in the previous theorem. Then an estimate of error in double exponential sinc indefinite integral is given by

sup t∈(a,b)

t a

f (τ )dτ −

N k=−M

!

−1 f (φDE (kh))φDE (kh)J(k, h)(φDE (t))

(3.9.0.34) ≤

2K(b−a)α+β−1 dµ

1

π

cos d cosα+β ( π2 sin d) (1−e− 2 µe )

+ µe

π 2 (α+β)

πdn

ln(2dn/µ) e− ln(2dn/µ) n

3.9. Error Estimation

133

Approximations in Coiflet basis Theorem 3.39. (Resnikoff and Raymond Jr, 2012, Th.9.3,p.206, Cor.9.4,p.208) For an orthogonal Coifman wavelet system of degree 2K, K ∈ N, � f (x) − (PVJ f )(x) �L2 ≤

C 22KJ

� f (x) − SJ (f )(x) �L2 ≤

,

(3.9.0.35)

C

(3.9.0.36)

22KJ

where the constant C depends on f ∈ C 2K (R) and the low-pass filter h. Here we have used the symbol 1 � k SJ (f )(x) = J f ( J )ϕj, k(x) (3.9.0.37) 2 2 2 k∈Z for the wavelet sampling approximation of the function f at resolution J. Theorem 3.40. Assume that the low- and high pass filters are of finite length and satisfy vanishing moment conditions (for scale function) � � h2k = h2k+1 = 1, (3.9.0.38) k∈Z

� k∈Z

p

(2k) h2k =

�

k∈Z

p

(2k + 1) h2k+1 = 0, p = 1, 2, · · · , 2K.

(3.9.0.39)

k∈Z

If, in addition, the scale function ϕ(x) ∈ C n (R), 0 ≤ n ≤ 2K, then C 22K−nJ where the constant C depends only on f and the low-pass filter h.

dal

(3.9.0.40)

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� f (x) − (PVJ f )(x) �H n ≤

Definition 3.41. The wavelet sampling approximation function of the function f : Rm → R in

C02K (Rm ) is defined as �k k � � km 1 1 2 SJ (f )(x1 , x2 , · · · , xm ) = jm k1 ,k2 ,···,km ∈Z f 2J , 2J · · · , 2J 2 2 (3.9.0.41) ×ϕ1J,k1 (x1 )ϕ2J,k2 (x2 ) · · · ϕm J,km (xm ). Here ϕi , i = 1, 2, · · · , m are scale functions with low-pass filter hi . Theorem 3.42. Assuming that low-pass filters hi , i = 1, 2, · · · , m are of finite length and each of them satisfy vanishing moment conditions mentioned in the previous theorem up to degree 2K. If ϕi (xi ), i = 1, 2, · · · , m ∈ L2 (R), then L2 −error in wavelet sampling approximation for any function f (x1 , x2 , · · · , xm ) ∈ C02K (Rm ) is � f (x1 , x2 , · · · , xm ) − SJ (f )(x1 , x2 , · · · , xm ) �L2 ≤

C . 22KJ

(3.9.0.42)

In addition, if ϕi (xi ), i = 1, 2, · · · , m ∈ C n (R), 0 ≤ n ≤ 2K, then H n −error in SJ (f )(x1 , x2 , · · · , xm ) for any function f (x1 , x2 , · · · , xm ) ∈ C02K (Rm ) is � f (x1 , x2 , · · · , xm ) − SJ (f )(x1 , x2 , · · · , xm ) �H n ≤

C 2(2K−n)J

.

In both cases the constant C depends on f and the low-pass filters hi , i = 1, 2, · · · , m.

(3.9.0.43)

3.10. Nonlinear/Best n-term Approximation

134

Autocorrelation function Theorem 3.43. (Urban, 2009) Let ϕ ∈ C r (R) be a compactly supported, refinable, interpolating scaling function which is exact of order d, i.e., � xm = ckm ϕ(x − k), 0 ≤ m ≤ d, x ∈ R, (3.9.0.44) k∈Z

with appropriate coefficients cm k ∈ R, where the convergence of the sum is to be understood locally. Then the following characterization for the interpolatory wavelets ψ(·) = ϕ(2 · −1): for 12 < σ < min{r, d}, 0 < p, q < ∞

�

�

djk ψjk

⎛ � �q ⎞ q1 � � σ (R) ∼ ⎝ �B2,q |djk |2 ⎠ . 2jσ j

jk

(3.9.0.45)

k

σ Here Bp,q is the Besov-space of smoothness order σ in the Lp (R) and fine tuning parameter q. A comprehensive discussion on this space of functions can be found in Appendix B of Urban cited above.

3.10

Nonlinear/Best n-term Approximation

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Wavelet theory provides simple and powerful decompositions of the target function into a series of building blocks. It is natural, then, to approximate the target function by selecting terms of this series. If we take partial sums of this series we are approximating again from linear spaces. It was easy to establish that this form of linear approximation offered little, if any, advantage over the already well established spline methods. However, it is also possible to let the selection of terms to be chosen from the wavelet series depend on the target function / and keep control only over the number of terms to be used. This is a form of nonlinear approximation which is called n-term approximation. Most function norms can be described in terms of wavelet coefficients. Using these descriptions not only simplifies the characterization of functions with a specified approximation order but also makes transparent strategies for achieving good or best n-term approximations. Indeed, it is enough to retain the n terms in the wavelet expansion of the target function that are largest relative to the norm measuring the error of approximation. Viewed in another way, it is enough to threshold the properly normalized wavelet coefficients. This leads to approximation strategies based on what is called wavelet shrinkage by Donoho and Johnstone (Donoho et al., 1994). Wavelet shrinkage is used by these two authors and others to solve several extremal problems in statistical estimation, such as the recovery of the target function in the presence of noise. Because of the simplicity in describing n-term wavelet approximation, it is natural to try to incorporate a good choice of basis into the approximation problem. This leads to a double stage nonlinear approximation problem where the target function is used both to choose a good (or best) basis from a given library of bases and then to choose the best n-term approximation relative to the good basis. This is a form of highly nonlinear approximation. Underlying principles and implementation strategies on this topic have been discussed by Donoho et al. (Donoho et al., 1994) and DeVore (DeVore, 1998; DeVore, 2009).

Chapter 4

Multiscale Solution of Integral Equations with Weakly Singular Kernels

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Integral equations with weakly singular kernels of the logarithmic and algebraic types appear in the mathematical formulation of many physical processes, e.g., potential problems with Dirichlet boundary conditions, the description of hydrodynamic interaction between elements in a polymer chain in solution, mathematical analysis of radiative equilibrium and transport problems, etc. Many researchers have studied for numerical methods to solve Fredholm integral equations with weakly singular kernels (Mandal and Nelakanti, 2019). The Galerkin, collocation, Petrov-Galerkin, Nystr¨om methods, piecewise polynomial approximation are commonly used approximation methods for find­ ing numerical solution to such equations. In this chapter our attempt is to get the approximate solution of singular integral equations of second kind with logarithmic and algebraic singularities by using wavelet basis.

4.1

Existence and Uniqueness

Before proceeding to the scheme based on wavelet we state here the existence and uniqueness of the solution of singular integral equation of second kind of Fredholm type with weakly (logarithmic and algebraic) singular kernel (Vainikko and Pedas, 1981) � u(x) −

1

K(|x − t|) u(t) dt = f (x), 0 ≤ x ≤ 1.

(4.1.0.1)

0

Theorem 4.1. Let γk0 xα0 +k

f ∈ C m ([0, b]), K(x) ∈ C m−1 [(0, b)] m ≥ 1,

(4.1.0.2)

≤ K (k) (x) for x ∈ (0, x0 ], x0 ≤ 1, k = 0, 1, 2, · · · , m − 1,

(4.1.0.3)

K (k) (x) ≤

γk , α+k x

for x ∈ (0, 1], k = 0, 1, 2, · · · , m − 1

(4.1.0.4)

4.1. Existence and Uniqueness

136

with 0 < α < 1, 2α < α0 < α, γk , γk0 , x0 are positive constants.

If the homogeneous equation corresponding to Eq. (4.1.0.1) has only the trivial solution in C[a, b],

then

i) Eq. (4.1.0.1) has unique solution u |u(k) (x)|

∈

C[0, 1] ∩ C m [0, 1], �

≤ ηk

1 xα+k−1

+

1 (1 − x)α+k−1

(4.1.0.5a) � , x ∈ (0, 1)

(4.1.0.5b)

provided conditions (4.1.0.2), (4.1.0.3) on the input function and kernel hold,

ii) if the conditions (4.1.0.2)–(4.1.0.4) hold, then

u(k) (x) = u(0)K (k−1) (x) − u(1)K (k−1) (1 − x) + vk (x), k = 1, 2, · · · , m,

(4.1.0.6)

where vk ∈ C m−k (0, b),

lim

x→0+

vk (x) vk (x) = 0 and lim = 0. x→1− K (k−1) (1 − x) K (k−1) (x)

(4.1.0.7)

Here ηk , k = 0, 1, · · · , m are positive constants.

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Remark 1. If the condition (4.1.0.3) holds only for some k = k0 , then the relation in (4.1.0.5) holds for k = k0 + 1. Remark 2. For K(x) ∈ C([0, 1]) whose some derivatives have singularity at the point zero, the property of the solution can be guessed from the equality in (4.1.0.5b). Many problems of physical interest can be reduced to the problem of solving the integral equa­ tion (4.1.0.1) or some of its variants, viz., equation with variable coefficients with a kernel of the form K(x) = lnx + K0 (x) = KL (x) + K0 (x)

(4.1.0.8)

or, 1 µ (x) + K0 (x), 0 < µ < 1 (4.1.0.9) + K0 (x) = KA xµ or their combinations, where K0 ∈ C m−1 ([0, 1]) is a smooth function without singularities. Here µ (x)) to indicate weak we have used the subscripts “L” and “A” in the singular parts KL (x) and KA singularity of logarithmic and algebraic nature respectively. K(x) =

Observation. (Vainikko and Pedas, 1981) The integral operator 1

� K[u](x) =

K(|x − t|) u(t) dt

(4.1.0.10)

0

where K(x) satisfies (4.1.0.4) and is a compact (completely continuous) linear operator in Banach spaces C([0, 1]) and Lp ((0, 1)), 1 ≤ p < ∞. The operator K maps the space Lp into Lq with p . p < q < 1−(1−α)p

4.2. Logarithmic Singular Kernel

4.2

137

Logarithmic Singular Kernel

Integral equation with logarithmic singular kernel follows from the representation of harmonic func­ tion by single layer potential or by the direct boundary integral equation method for plane Dirichlet boundary value problems (Jaswon, 1977; Estrada and Kanwal, 1989; Orav-Puurand, 2013). We consider here the integral equations 1

a(x) u(x) − λ b(x)

ln|x − t| u(t)dt =

f (x), x ∈ [0, 1],

(4.2.0.1a)

{a(x, t) ln|x − t| + b(x, t)} u(t)dt =

f (x), 0 ≤ x ≤ 1.

(4.2.0.1b)

0 1

u(x) − 0

4.2.1

Projection in multiscale basis

4.2.1.1

Basis in Daubechies family

Let us choose the multiscale approximation of u ∈ L2 ([0, 1]) as ⎧ in VJortho , ⎨ BVJortho (x) · cVJortho u(x) ≈ uJ (x) = ⎩ −1 BV Wjortho (x) · cV Wjortho in Vjortho ∪Jj=j Wjortho 0 0 ,J 0 ,J 0

(4.2.1.1)

dal ja M

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and

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in the basis of Daubechies family described in section 3.1.1.2. Here the symbol J plays the role of parameter h of the sequence of finite dimensional subspaces of L2 ([0, 1]). Use of these approximate solutions into Eq. (4.2.0.1a) provides 1

� a(x) BVJortho (x) − λ b(x)

� ln|x − t| BVJortho (t) dt · cVJortho = f (x),

0

or, � (x) − λ b(x) a(x) BV Wjortho 0 ,J

1 0

(4.2.1.2a) � ln|x − t| BV Wjortho (t) dt · cV Wjortho = f (x). 0 ,J 0 ,J (4.2.1.2b)

2j

Using the symbol KLJ [f ](x) = 0 ln|x − t|f (t) dt corresponding to the operator defined in (4.1.0.10) for K(x, t) = ln|x − t| , these two equations can be written in compact form as a(x) BVJortho (x) − λ b(x)KLJ [BVJortho ](x) · cVJortho = f (x) (4.2.1.3a) or,

(x) − λ b(x)KLJ [BV Wjortho ](x) · cV Wjortho = f (x). (4.2.1.3b) a(x) BV Wjortho 0 ,J 0 ,J 0 ,J

To reduce these equations to linear simultaneous equations for the sets of unknown coefficients , one may use the principles of either collocation or the Galerkin method. In case cVJortho or cV Wjortho 0 ,J � � (= ΛVJ L ∪ ΛJV I ∪ ΛJV R or in of collocation method we use nodes in xVJ = xJ,k = 2kJ , k ∈ Λortho J � � � � J −1 k k W ortho xV Wj0 ,J = xj0 ,J,k = 2kj0 , k ∈ ΛVj0 ortho ∪j= respectively. Evaluation j0 2j + 2j+1 , k ∈ Λj of Eqs. in (4.2.1.3a,b) at the respective collocation points mentioned above provide the system of

4.2. Logarithmic Singular Kernel

138

algebraic equations ALVJ · cVJortho = f VJortho

(4.2.1.4a)

= f V Wjortho . ALV Wj0 J · cV Wjortho 0 ,J 0 ,J

(4.2.1.4b)

or,

(ALV Wj0 J ) have been used to represent Here the symbols f VJortho (ALVJ ) and f V Wjortho 0 ,J vectors (matrices) obtained by evaluation of f (x) (vector a(x) BVJortho (x) − λ b(x)KLJ [BVJortho ] tho (x) − λ b(x)KLJ [BV Wjor ](x)) at x ∈ xVJ and at x ∈ xV Wj0 ,J respectively. /a(x) BV Wjortho 0 ,J 0 ,J However, this scheme cannot be implemented conveniently in Eq. (4.2.0.1b) due to non avail­ �1 �1 ability of integral transforms 0 b(x, t)ln|x − t| BVJortho (t) dt or 0 b(x, t)ln|x − t| BV Wjortho (t) dt of 0 ,J elements in the bases present in the approximation (4.2.1.1). It is hoped that such difficulty can be overcome with the aid of sparse representation discussed in subsection 3.2.1.1 for f (x, y) = ln(|x−y|) into the integration in Eq. (4.2.0.1b). 4.2.1.2

LMW basis

Let us choose the multiscale approximation (3.1.2.14) for u ∈ L2 ([0, 1]) u(x) ≈ uJ (x) = (Φj0 (x), Ψj0 (x), Ψj0 +1 (x), · · · ΨJ −1 (x)).(aj0 , bj0 , bj0 +1 , · · · bJ −1 )T .

(4.2.1.5)

Use of (4.2.1.5) into (4.2.0.1a) leads to dal ja M

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N

Man

a(x) [(Φj0 (x), Ψj0 (x), Ψj0 +1 (x), · · · ΨJ −1 (x)) −λ b(x)(KL [Φj0 ](x), KL [Ψj0 ](x), KL [Ψj0 +1 ](x), · · · KL [ΨJ −1 ](x))]

(4.2.1.6)

T

.(aj0 , bj0 , bj0 +1 , · · · bJ −1 ) = f (x). The expressions for elements in KL [Φj0 ](x) can be obtained by using the result ( for m = 0, 1, · · · , K− 1, K being the number of scale functions in the LMW basis at resolution 0) m

1

�

KL [x ](x) = 0

ln|x − t|tm dt (4.2.1.7) � � ⎧ m � 1 xi ⎪ ⎪ ln|1 − x|−2xm+1 arccoth(1 − 2x) − x < 0 or x > 1, ⎪ 1+m 1+m−i ⎪ ⎪ i=0 ⎪ 1 ⎨ − (1+m)2 x = 0, � � = (4.2.1.8) . m � i ⎪ 1 x x 1+m ⎪ x ln | | +ln |1 − x | ) − 0 < x < 1, ⎪ 1−x 1+m−i ⎪ ⎪ 1+m i=0

⎪ ⎩ H1+m

x = 1, − 1+m

into the relation KL [φi ](x) =

i � l=0

cil

�

1

ln|x − t|tl dt

(4.2.1.9)

0

where cil is the coefficient of xl in ϕi (x). Here Hm represent the mth harmonic number (Olver et al., 2010). The x−dependence of elements in KL [Φj0 ](x) can then be used in the definition (2.2.2.4) to obtain x−dependence of elements in KL [Ψj ](x), j = j0 , · · · , J − 1. To recast this equation into an

4.2. Logarithmic Singular Kernel

139

algebraic equation we evaluate the relation (4.2.1.6) at the nodes { 2kj0 , k ∈ {0, 1, · · · , 2j0 − 1} ∪J−1 j=j0 1 { 2j+1 + 2kj , k ∈ {0, 1, · · · , 2j − 1}} and get the system of linear simultaneous equations AL j0 J · Cj0 J = fj0 J .

(4.2.1.10)

for unknown coefficients Cj0 J = (aj0 , bj0 , bj0 +1 , · · · bJ−1 ). Use of solution to this system of equations into (4.2.1.5) provides the approximate solution of Eq. (4.2.1.5) in LMW basis. Coiflet basis Here we consider the integral equation � u(x) + λ

1

ln|x − t| u(t) dt = f (x).

(4.2.1.11)

0

For the unknown solution u(x), we use the approximation u(x) � uj (x)( = (PVj u)(x)) � BVjT (x) · cVjT .

(4.2.1.12)

Using (4.2.1.12) in (4.2.1.11) one gets �

BVjT (x)

1

� +λ

� ln | x − t | BVjT (t) dt · cVjT = f (x).

(4.2.1.13)

0

We now denote 1

�

Use of (4.2.1.14) in (4.2.1.13) leads to � � BVjT (x) + λ KL [BVjT ](x) · cVjT = f (x).

(4.2.1.15)

M

Panja

and

B

N

dal

(4.2.1.14)

Man

ln | x − t | BVjT (t) dt. M

KL [BVjT ](x) =

0

Using the definitions of BVjT (x) and cVjT given in (3.1.1.1a) one gets the Eq. (4.2.1.15) in explicit form � 1 � �� φjk (x) + λ ln | x − t | φjk (t) dt cjk = f (x) (4.2.1.16) k∈Λj

0

where Λj = {0, 1, ..., 2j }. Substitution of 2j x = ξ and 2j t = t� reduces (4.2.1.16) to � 2j � j � �� λ 2 λ � � � φk (t� ) dt� cjk φk (ξ) + j ln | ξ − t | φk (t ) dt − j j ln2 2 0 2 0

k∈Λj

=

1 j

22

f ( 2ξj ).

(4.2.1.17)

Evaluation of both sides of (4.2.1.17) at the nodes in ξVj = {k, k ∈ Λj } provides a system of linear simultaneous equations (4.2.1.18) ALj cVjT = fj .

4.2. Logarithmic Singular Kernel

140

Here ALj is the stiffness matrix whose elements are given by � �� j � 2j 2 λ � � � � � φk (ξk ) + j ln | ξk − t | φk (t ) dt − j ln2 φk (t ) dt , k ∈ Λj , ξk ∈ ξVj , 2 0 0

(4.2.1.19)

and the inhomogeneous term (vector) is fj =

1 j

� f

22

ξk 2j

� , ξk ∈ ξVj .

(4.2.1.20)

It is important to mention here that unlike classical approximation schemes values of singular integrals 2j

� KLj [φ ](ξ) =

�

2j −k�

ln | ξ − k � − t | φ(t) dt,

ln | ξ − t | φ (t) dt =

k�

k�

−k �

0

for (ξ−k � ) ∈ suppφ have been determined without using any quadrature rule. Instead their numerical values at integers are obtained as a solution of equation 4K−1 1 � KLj [φ](x) = √ hl {KLj [φ](2x − l) − ln2}. 2 l=−2K

Their values for K = 1 are presented in the following table: 0

1

2

3

0.693636

0.043631

-1.912173

0.023524

0.698321

1.099638

M N B and ja Pan

al

-1 and

-2

M

KLj [φ](ξ −

k� )

M

ξ − k�

Values of KLj [φ](ξ − k � ) for other (ξ − k � ) can be obtained by simultaneous use of recurrence relation mentioned above and their asymptotic values KLj [φ](ξ − k � ) = ln | ξ − k � | ,

| ξ − k � |� 4K.

Example 4.2. These results have been used to find elements of ALj in (4.2.1.19) and applied to get approximate solution of the following sample problem. � 1 u(x) + λ ln | x − t | u(t) dt = f (x) (4.2.1.21) 0

with λ = − 15 and ⎧ λ ⎪ ⎨− 4 , f (x) = λ2 x2 ln(x) + λ2 (1 − x2 )ln(1 − x) + (1 − λ2 )x − λ4 , ⎪ ⎩ 1 − 3λ 4 ,

x = 0, 0 < x < 1, x = 1.

(4.2.1.22)

The exact solution to this equation can be found as u(x) = x. The approximate solution is found to be reasonably accurate even for K = 1. The absolute error in the approximate solution gradually decreases with increase in the resolution as it appears from the following figure.

4.2. Logarithmic Singular Kernel

141

0.2

0.4

0.6

0.8

1.0

-1

-2

j=4 -3

j=5 j=6

-4

-5

-6

Figure 4.1: Absolute error in approximate solution (log10 scale) in Coiflet basis for J = 4, 5, 6. Autocorrelation scale function For the unknown solution u(x) of 1

�

ln|x − t| u(t) dt = f (x)

u(x) + λ

(4.2.1.23)

0

we have projected u(x) into the space of linear span of BVjT (x) by

dal

(4.2.1.24)

ja M

M

Pan

and

B

N

Man

u(x) � uj (x)( = (PVj u)(x)) � BVjT (x) · cVjT

whose elements are autocorrelation functions having full and partial supports. Using (4.2.1.24) in (4.2.1.23) one gets �

BVjT (x)

1

� +λ

� ln | x − t | BVjT (t) dt · cVjT = f (x).

(4.2.1.25)

0

We now denote KL [BVjT ](x) =

�

1

ln | x − t | BVjT (t) dt.

(4.2.1.26)

Use of (4.2.1.26) in (4.2.1.25) leads to � � BVjT (x) + λ KL [BVjT ](x) · cVjT = f (x).

(4.2.1.27)

0

Using the definitions of BVjT (x) and cVjT given in (3.1.1.1a) one gets the Eq. (4.2.1.27) in explicit form � 1 � �� φjk (x) + λ ln | x − t | φjk (t) dt cjk = f (x) (4.2.1.28) k∈Λj

where Λj = {0, 1, ..., 2j }.

0

4.2. Logarithmic Singular Kernel

142

If the basis is chosen from autocorrelation function family, the equation (4.2.1.28) with substi­ tutions 2j x = ξ and 2j t = t� takes the form � � � j � 2j � �� λ 2 λ ξ � � � ln | ξ − t | φk (t ) dt − j j ln2 φk (t� ) dt� cjk = f φk (ξ) + j 2 2j 2 0 0 k∈Λj (4.2.1.29) A system of linear equations can be obtained as given in (4.2.1.18) where the stiffness matrix is given by (4.2.1.19) and the inhomogeneous term will be � � ξk fj = f , ξk ∈ ξVj . 2j The integral 2j

�

�

ln | ξ − t� | φk (t� ) dt� =

2j −k

ln | ξ − k − t� | φ(t� ) dt� −k

0

(for φ with K = 1) can be evaluated explicitly by the use of the analytical expressions of the integrals presented the following table: Sl. No.

Integral

I

�0

II

�1

III

�0

IV

�1

−1 0

Value

ln(x − t)(1 + t)dt

1 4

�

−3 − 2x + 2x(2 + x)ln(1 +

ln(t − x) (1 − t) dt

1 4

ln(t − x)(1 + t) dt

1 4

Condition 1 ) x

� + 2 ln(1 + x)

{−3 + 2x + 4x(x − 2) arccoth(1 − 2x)

x>0 x1

+2 ln(x − 1)} V

�x

VI

�0

VII

�x

VIII

�1

−1 x 0 x

ln(x − t)(1 + t) dt

ln(t − x)(1 + t) dt

1 x 4

1 (1 + x)2 {−3 + 2 ln(1 + x)} �4 � � � 4 + x 3 + ln( x12 ) − 4ln(−x)

− 14 x {4 − 3x + 2(x − 2)ln(x)}

ln(x − t)(1 − t) dt

1 (x 4

ln(t − x)(1 − t) dt

−

1)2

{−3 + 2 ln(1 − x)}

−1 < x < 0 −1 < x < 0 0