A polynomial collocation method for Cauchy singular integral equations over the interval
type: Item
instance of: scholarly article
title: A polynomial collocation method for Cauchy singular integral equations over the interval
zbMATH Open document ID: 1024.65131
author: Peter Junghanns
author: Andreas Rathsfeld
publication date: 2003-11-06T00:00:00Z
full work available at URL: https://eudml.org/doc/123017
full work available at URL: pp79-126.html
review text: The authors consider a polynomial collocation method for the numerical solution of a singular integral equation over the interval. More precisely, the operator of the integral equation is supposed to be of the form \(R:=aI + \mu^{-1}bS\mu I + K\), where \(S\) is the Cauchy integral operator, \(a\) and \(b\) are piecewise continuous coefficients, \(K\) is a regular integral operator, and \(\mu\) is a Jacobi weight. The collocation method is applied to the equation \(Ru=f\), where the collocation points are the Chebyshev nodes of the second kind and where the trial space is the space of polynomials multiplied by another Jacobi weight. For the stability and convergence of this collocation in weighted \(L^2\) spaces, necessary and sufficient conditions are derived. To show stability and convergence, a general technique due to \textit{S. Roch} [Lokale Theorie des Reduktionsverfahrens für singuläre Integraloperatoren mit Carlemanschen Verschiebungen Diss. (1988; Zbl 0727.47038)] and \textit{S. Prössdorf} and \textit{B. Silbermann} [Numerical analysis for integral and related operator equations. Basel: Birkhäuser (1991; Zbl 0763.65103)] is applied. The authors introduce a special Banach algebra \(\mathcal F\) of sequences of discretized operators such that the stability of a sequence is equivalent to the invertibility of four limit operators and to the invertibility of a corresponding coset in a suitable quotient algebra \(\mathcal F/\mathcal J\). In particular, the collocation sequence \(\{A_n\}\) is shown to be an element of the algebra \(\mathcal F\). To show the invertibility of the corresponding element in the quotient algebra, a subalgebra \(\mathcal A/\mathcal J\) of this quotient algebra and a subalgebra inthe center of \(\mathcal A/\mathcal J\) are introduced. Using the Allan-Douglas local principle [cf. \textit{R. G. Douglas}, Banach algebra techniques in operator theory. New York: Academic Press (1972; Zbl 0247.47001)] the invertibility is reduced to localized p…
reviewed by: Nikolay Yakovlevich Tikhonenko
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E05
zbMATH DE Number: 1893813
zbMATH Keywords: Cauchy singular integral equation
zbMATH Keywords: polynomial collocation method
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: invertibility in a Banach algebra
zbMATH Keywords: Allan-Douglas local principle
zbMATH Keywords: local invertibility
zbMATH Keywords: quotient algebra
MaRDI profile type: Publication
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published in: ETNA - Electronic Transactions on Numerical Analysis
description: scientific article; zbMATH DE number 1893813