PIMS/CSC Weekly Seminar: George C. Hsiao

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.