Wavelet Based Approximation Schemes for Singular Integral Equations  book cover
1st Edition

Wavelet Based Approximation Schemes for Singular Integral Equations

ISBN 9780367199173
Published September 25, 2020 by CRC Press
300 Pages 23 Color & 29 B/W Illustrations

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Book Description

Many mathematical problems in science and engineering are defined by ordinary or partial differential equations with appropriate initial-boundary conditions. Among the various methods, boundary integral equation method (BIEM) is probably the most effective. It’s main advantage is that it changes a problem from its formulation in terms of unbounded differential operator to one for an integral/integro-differential operator, which makes the problem tractable from the analytical or numerical point of view. Basically, the review/study of the problem is shifted to a boundary (a relatively smaller domain), where it gives rise to integral equations defined over a suitable function space. Integral equations with singular kernels areamong the most important classes in the fields of elasticity, fluid mechanics, electromagnetics and other domains in applied science and engineering. With the advancesin computer technology, numerical simulations have become important tools in science and engineering. Several methods have been developed in numerical analysis for equations in mathematical models of applied sciences.

Widely used methods include: Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM) and Galerkin Method (GM). Unfortunately, none of these are versatile. Each has merits and limitations. For example, the widely used FDM and FEM suffers from difficulties in problem solving when rapid changes appear in singularities. Even with the modern computing machines, analysis of shock-wave or crack propagations in three dimensional solids by the existing classical numerical schemes is challenging (computational time/memory requirements). Therefore, with the availability of faster computing machines, research into the development of new efficient schemes for approximate solutions/numerical simulations is an ongoing parallel activity. Numerical methods based on wavelet basis (multiresolution analysis) may be regarded as a confluence of widely used numerical schemes based on Finite Difference Method, Finite Element Method, Galerkin Method, etc. The objective of this monograph is to deal with numerical techniques to obtain (multiscale) approximate solutions in wavelet basis of different types of integral equations with kernels involving varieties of singularities appearing in the field of elasticity, fluid mechanics, electromagnetics and many other domains in applied science and engineering.

Table of Contents


Singular integral equation

MRA of Function Spaces

Multiresolution analysis of L2(R)

Multiresolution analysis of L2([a, b] ⊂ R)


Approximations in Multiscale Basis

Multiscale approximation of functions

Sparse approximation of functions in higher dimensions


Quadrature rules

Multiscale representation of differential operators

Representation of the derivative of a function in LMW basis

Multiscale representation of integral operators

Estimates of local Holder indices

Error estimates in the multiscale approximation

Nonlinear/Best n-term approximation

Weakly Singular Kernels

Existence and uniqueness

Logarithmic singular kernel

Kernels with algebraic singularity

An Integral Equation with Fixed Singularity

Method based on scale functions in Daubechies family

Cauchy Singular Kernels


Basis comprising truncated scale functions in Daubechies family

Multiwavelet family

Hypersingular Kernels

Finite part integrals involving hypersingular functions

Existing methods

Reduction to Cauchy singular integro-differential equation

Method based on LMW basis

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M M Panja has a MSc in Applied Mathematics (1987) from Calcutta University, India, and a PhD (1993) from Visva-Bharati University, India. He investigated the origin of (hidden) geometric phase on quantum mechanical problems and initiated studies on Lie group theoretic approach of differential equations during his postdoctoral research. His investigations (2007) on approximation theory based on multiresolution analysis, has been published several international journals. His current research interests are (i) multiscale approximation based on wavelets, and (ii) similarity (exact) solution of mathematical models involving differential and integral operators.

B N Mandal has a MSc in Applied Mathematics (1966) and a PhD (1973) from Calcutta University, India. He was a postdoctoral Commonwealth Fellow at Manchester University, 1973-75. He was faculty at Calcutta University, 1970-89 and later at Indian Statistical Institute (ISI), Kolkata, 1989-2005. He was a NASI Senior Scientist, 2009-14 in ISI. His research work encompasses several areas of applied mathematics including water waves, integral transforms, integral equations, inventory problems, wavelets etc. He has published a number of works with reputable publishers. He has supervised PhD theses of more than 20 candidates and has more than 275 research publications.