A quadrature method for the hypersingular integral equation on an interval
type: Item
instance of: scholarly article
title: A quadrature method for the hypersingular integral equation on an interval
zbMATH Open document ID: 0846.65081
author: Kathrin Bühring
published in: Journal of Integral Equations and Applications
publication date: 1996-10-03T00:00:00Z
review text: The paper is concerned with quadrature methods for the approximate solution of the hypersingular integral equation \[ p.f.\;\int^1_0 {u(\tau)\over |\tau- t|^2} d\tau= f(t),\quad 0\leq t\leq 1.\tag{1} \] Before discretization the parametrization \[ \gamma(s)= v(s)^\alpha/(v(s)^\alpha+ v(1- s)^\alpha),\quad 0\leq s\leq 1,\;\alpha> 3/2 \] with a certain cubic polynomial \(v\) is applied to (1), leading to the transformed equation \[ p.f.\;\int^1_0 {\gamma'(s) \gamma'(\sigma)\over |\gamma(\sigma)- \gamma(s)|^2} u(\gamma(s)) d\sigma= f(\gamma(s)) \gamma'(s),\quad 0\leq s\leq 1,\tag{2} \] with a regular solution. Then (2) is discretized by using a simple quadrature rule on a uniform mesh, and results on stability and error estimates in the norm of the Sobolev space \(H^{1/2}\) are proved. Moreover, corresponding results are obtained for a quadrature method based on the cosine transformation \[ \gamma(s)= (1- \cos\pi s)/2,\qquad 0\leq s\leq 1. \]
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E05
zbMATH DE Number: 851873
zbMATH Keywords: quadrature methods
zbMATH Keywords: hypersingular integral equation
zbMATH Keywords: stability
zbMATH Keywords: error estimates
zbMATH Keywords: cosine transformation
reviewed by: Johannes Elschner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075880
OpenAlex ID: W2067512324
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DOI: 10.1216/JIEA/1181075880
description: scientific article; zbMATH DE number 851873