Discrete collocation for a first kind Cauchy singular integral equation with weakly singular solution
type: Item
instance of: scholarly article
title: Discrete collocation for a first kind Cauchy singular integral equation with weakly singular solution
zbMATH Open document ID: 0908.65130
author: Heinrich N. Mülthei
author: Claus Schneider
published in: Journal of Integral Equations and Applications
publication date: 1999-03-18T00:00:00Z
full work available at URL: cont9-4.htm
review text: A discrete collocation method for the numerical solution of the first kind Cauchy singular integral equation \[ {1\over 2\pi} \int^{2\pi}_0 u(s)\text{cot}{t- s\over 2} ds= f(t),\quad t\in[0, 2\pi] \] is investigated. Represented in the form \[ {1\over 2\pi} \int^{2\pi}_0 \{u(s)- u(t)\}\text{cot}{t- s\over 2} ds= f(t), \] and using the sequences of grids by the knots \[ x^n_0= 0< x^n_1<\cdots< x^n_n= 2\pi,\quad n\in\mathbb{N} \] and \[ t^n_i:= (1- \lambda^n_i) x^n_{i- 1}+ \lambda^n_i x^n_i,\quad 0<\lambda^n_i< 1,\quad i= 1,2,\dots, n \] the discrete collocation method \[ {1\over 2\pi} \sum^n_{j= 1} \omega^n_j\text{cot}{t^n_i- x^n_j\over 2} (u^n_j- (1- \lambda^n_i) u^n_{i- 1}- \lambda^n_i u^n_i)= f(t^n_i),\quad i= 1,2,\dots, n,\tag{\(*\)} \] where \(u^n_i= u_n(x^n_i)\), \(i= 0,1,\dots, n\), \(u^n_0= u^n_n\) is received. The choice of the collocation points \(t^n_i\), \(i= 1,\dots, n\), stability, convergence and exactness of the discrete collocation method \((*)\) are investigated.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E05
zbMATH DE Number: 1185928
zbMATH Keywords: weakly singular solution
zbMATH Keywords: discrete collocation method
zbMATH Keywords: Cauchy singular integral equation
zbMATH Keywords: stability
zbMATH Keywords: convergence
reviewed by: Ilia V. Boikov
MaRDI profile type: Publication
OpenAlex ID: W2092842283
cites work: Q3976442
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DOI: 10.1216/JIEA/1181076029
description: scientific article; zbMATH DE number 1185928