On stability of approximation methods for the Muskhelishvili equation
type: Item
instance of: scholarly article
title: On stability of approximation methods for the Muskhelishvili equation
zbMATH Open document ID: 1016.65108
DOI: 10.1016/S0377-0427(02)00393-X
published in: Journal of Computational and Applied Mathematics
publication date: 2003-02-23T00:00:00Z
review text: Let \(D\) be a domain in the complex plane \(\mathbb C\) bounded by a simple closed curve \(\Gamma\) with a finite number of corner points. The authors consider some numerical methods to solve the so-called Muskhelishvili integral equations on \(\Gamma\) \[ R\varphi(t)\equiv-k\overline{\varphi(t)}-\frac{k}{2\pi i}\int_\Gamma\overline{\varphi(\tau)}d \log\frac{\overline\tau-\overline t}{\tau-t}-\frac{1}{2\pi i}\int_\Gamma\varphi(\tau)d \frac{\overline\tau-\overline t}{\tau-t}=f_0(t). \] Such equations often arise in applications, especially in plane elasticity theory. In the considered case, the integral operator \(R\) is noncompact, which generates difficulties with both Fredholm properties of \(R\) and stability of the approximation methods. In this paper, the Fredholm properties of the operator \(R\) and their relations to the invertibility of \(R\) are investigated. Necessary and sufficient conditions for the stability of approximation methods based on piecewise constant splines, of the quadrature and Galerkin methods are established. The study is based on the local principle, originally proposed by G. R. Allan to investigate complex structures. The original proof of that principle essentially uses the theory of analytic functions. The authors provide another version directly applicable in real algebras.
reviewed by: Nikolay Yakovlevich Tikhonenko
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 74B05
zbMATH DE Number: 1872836
zbMATH Keywords: plane elasticity theory
zbMATH Keywords: Muskhelishvili integral equation
zbMATH Keywords: approximate solution
zbMATH Keywords: Fredholm properties
zbMATH Keywords: invertibility
zbMATH Keywords: curves with corner points
zbMATH Keywords: piecewise constant splines
zbMATH Keywords: quadrature method
zbMATH Keywords: Galerkin method
zbMATH Keywords: stability
zbMATH Keywords: Galerkin projection operator
author: Victor D. Didenko
author: B. Silbermann
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1872836