Wavelet Based Approximation Schemes for Singular Integral Equations

The Boundary Integral Equation Method (BIEM) is a powerful tool for solving mathematical problems in science and engineering defined by ordinary or partial differential equations with appropriate initial-boundary conditions. It transforms the problem into one for an integral/integro-differential operator, making it more tractable from an analytical or numerical perspective. Integral equations with singular kernels are important in various fields, and numerical simulations have become crucial in applied science and engineering. However, existing methods have limitations, such as difficulty in handling rapid changes in singularities. Research into developing new efficient schemes for approximate solutions/numerical simulations is ongoing, with numerical methods based on wavelet basis gaining popularity as they combine the advantages of different numerical schemes.

Format: Paperback / softback
Length: 290 pages
Publication date: 15 February 2022
Publisher: Taylor & Francis Ltd

Numerous mathematical issues 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 arguably the most effective. Its primary advantage lies in its ability to transform a problem from its formulation in terms of unbounded differential operator to one for an integral/integro-differential operator, which renders the problem tractable from an analytical or numerical perspective. Essentially, 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 are among the most important classes in the fields of elasticity, fluid mechanics, electromagnetics, and other domains in applied science and engineering. With the advancements in computer technology, numerical simulations have become indispensable 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 suffer from difficulties in problem solving when rapid changes appear in singularities. Even with the modern computing machines, the 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 is imperative.

Weight: 553g
Dimension: 254 x 178 (mm)
ISBN-13: 9780367565541