The approximate solution of a class of Fredholm integral equations with a weakly singular kernel
type: Item
instance of: scholarly article
title: The approximate solution of a class of Fredholm integral equations with a weakly singular kernel
zbMATH Open document ID: 1205.65331
author: Sumit K. Garg
published in: Journal of Computational and Applied Mathematics
publication date: 2010-11-30T00:00:00Z
review text: The authors consider weakly singular integral equations and singular integral equations as \[ a(x) \varphi(x) + b(x)\int\limits^1_{-1} \frac{\varphi(y)}{(y-x)^\alpha}dy = f(x), \, |x| < 1, \, 0< \alpha \leq 1, \tag{1} \] where \(\varphi(x)\) is the unknown function. A method for finding the numerical solution of equation (1) is presented. The Taylor series is used to remove the singularity and Legendre polynomials are used as a basis. The Legendre function of the second kind is used to remove the singularity in Cauchy type integral equations. The integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the Cauchy principal value sense without using numerical quadratures. Four examples are given to show the accuracy of the method.
reviewed by: Ilia V. Boikov
Mathematics Subject Classification ID: 65R20
zbMATH DE Number: 5821389
zbMATH Keywords: Cauchy kernel
zbMATH Keywords: weakly singular
zbMATH Keywords: Taylor series
zbMATH Keywords: Galerkin method
zbMATH Keywords: Legendre functions
zbMATH Keywords: numerical examples
zbMATH Keywords: weakly singular integral equations
zbMATH Keywords: singular integral equations
MaRDI profile type: Publication
full work available at URL: j.cam.2010.07.025
OpenAlex ID: W2071934340
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DOI: 10.1016/J.CAM.2010.07.025
description: scientific article; zbMATH DE number 5821389