Pacific Institute for the Mathematical Sciences - PIMS

Applications of Integral Equation Methods to a Class of Fundamental Problems in Mechanics and Mathematical Physics

Abstract

The monograph Singular Integral Equations by N.I.
Muskhelishvili was published originally in Russian in 1946 and was
revised and trans­lated into English in 1958. In this monograph, the
solution of the Dirichlet problem is expressed in terms of the potential
of a simple layer, whichleads to a Fredholm integral equation of the
first kind. This new approach introduced by Muskhelishvili in 1946 for
solving boundary value problems by using integral equations of the first
kind has made significant contribution 30years later to the development
of variational methods for boundary integralequations and their
nu­merical discretizations. The later is known as the boundary element
method and has become one of the most popular numerical schemes in
nowadays.This lecture discusses mathematical foundation of
boundary inte­gral equations of the first kind and its applications to a
class of fun­damental problems in elasticity, fluid mechanics and other
branches of mathematical physics. Applications are drawn from various
disciplines including topics such as singular perturbation theory for
viscous flow past an obstacle, coupling procedure and domain
decomposition for linear and nonlinear interface problems in non
homogeneous medium. The presentation of these topicsindicates also the
chronological or­der of the development of the Muskhelishvili’s method
concerning first kind integral equations and its generalizations. This is
an expository lecture meant for general audience.

Location: TASC-2

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