On the stability of integral equations of the first kind with logarithmic kernels
type: Item
instance of: scholarly article
title: On the stability of integral equations of the first kind with logarithmic kernels
zbMATH Open document ID: 0606.65089
DOI: 10.1007/BF00280433
author: George C. Hsiao
published in: Archive for Rational Mechanics and Analysis
publication date: 1986-01-01T00:00:00Z
review text: Among other ones, the author discusses the problem \[ - \int_{\Gamma}\sigma (y)\log | x-y| dS_ y+\omega =f(x),\quad x\in \Gamma,\quad \int_{\Gamma}\sigma (y)dS_ y=b \] where \(\Gamma\) is a smooth, simple, closed curve on the plane, f is a given function and by is a given constant (both not known exactly, \(| b_{\epsilon}-b| <\epsilon\), \(\| f_{\epsilon}-f\|_{L^ 2(\Gamma)}\leq \epsilon)\). The function \(\sigma\) and the constant \(\omega\) are to be determined. The problem is discretized by means of Galerkin finite elements. The author derives error estimates on the norms of \(H^ t(\Gamma)\times {\mathbb{R}}\), -\(\leq t\leq m\). The results are consistent with those obtained by the Tikhonov regularization procedure.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E10
zbMATH DE Number: 3980434
zbMATH Keywords: first kind
zbMATH Keywords: logarithmic kernels
zbMATH Keywords: weakly singular
zbMATH Keywords: Galerkin finite elements
zbMATH Keywords: error estimates
zbMATH Keywords: Tikhonov regularization
reviewed by: G. Vainikko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3980434