Asymptotic solution of some singularly perturbed Fredholm integral equations
type: Item
instance of: scholarly article
title: Asymptotic solution of some singularly perturbed Fredholm integral equations
zbMATH Open document ID: 0676.45004
DOI: 10.1007/BF00945000
author: A. K. Gautesen
author: W. E. Olmstead
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1989-01-01T00:00:00Z
review text: The authors consider certain physical problems reducing to a singularly perturbed Fredholm integral equation of the type \(K_{\epsilon}u(x)+c(x)u(x)=f(x,\epsilon)\) with \[ K_{\epsilon}u(x)=1/\epsilon \int^{b}_{a}k((x- s)/\epsilon,x,s)u(s)ds,\quad a\leq x\leq b, \] where \(\epsilon >0\) is a small parameter. The kernel k(z,x,s) is continuous in x and s as well as in z except possibly at \(z=0\). The authors demonstrate a formal approach to construct the leading order boundary layer (at either or both of the end points \(x=a\), \(x=b)\) corrections to \(u_ 0(x)\) which is the solution of the reduced (as \(\epsilon\) \(\to 0)\) problem. They consider an asymptotic expansion of the form \(u(x)=u_ 0(x)V(\eta)W(\zeta),\) \(\eta =(x-a)/\epsilon,\) \(\zeta =(b-x)/\epsilon.\) Building the asymptotic expansion is reduced to solving two Wiener-Hopf problems for W and V. The authors consider concrete problems in heat transfer, diffraction theory, crack mechanics, Markov processes and low order eigenvalue problems.
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 80A20
Mathematics Subject Classification ID: 74R05
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 4107426
zbMATH Keywords: singularly perturbed Fredholm integral equation
zbMATH Keywords: small parameter
zbMATH Keywords: boundary layer
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: Wiener-Hopf problems
zbMATH Keywords: heat transfer
zbMATH Keywords: diffraction theory
zbMATH Keywords: crack mechanics
zbMATH Keywords: Markov processes
zbMATH Keywords: eigenvalue problems
reviewed by: Leonid Lebedev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4107426