## PIMS/CSC Weekly Seminar: George C. Hsiao

- Date: 03/16/2012
- Time: 14:30

Simon Fraser University

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.