Legendre spectral projection methods for Hammerstein integral equations with weakly singular kernel
type: Item
instance of: scholarly article
title: Legendre spectral projection methods for Hammerstein integral equations with weakly singular kernel
zbMATH Open document ID: 1406.65139
author: Bijaya Laxmi Panigrahi
published in: International Journal of Applied and Computational Mathematics
publication date: 2018-12-03T00:00:00Z
review text: This paper is devoted to the Fredholm-Hammerstein integral equation of the second kind \[ u(s)-\int\limits_{-1}^{1}k(s,t)\psi(t,u(t))dt=f(s), \; -1\leq s\leq 1, \tag{1} \] with respect to the unknown function \(u(s)\) defined in a Banach space \(\mathbb{X}=C[-1,1]\subset L^2[-1,1]\). The kernel \(k(s,t)\) is of weakly singular type and has the form \[ k(s,t)=m(s,t)g_{\alpha}(|s-t|), \] where \(m(s,t)\in C([-1,1]\times C[-1,1])\) and \[ g_{\alpha}(x)=\begin{cases} x^{\alpha-1} & \text{if } 0<\alpha<1 ;\\ \log x & \text{if } \alpha=1. \end{cases} \] The Legendre-Galerkin and Legendre collocation methods for solving (1) are studied. The convergence rates for both methods in both \(L^2\) and infinity-norm are evaluated. To improve the convergence rates, iterated Legendre-Galerkin and iterated Legendre collocation methods are considered. It is proved that iterated Legendre-Galerkin methods converge faster than Legendre-Galerkin methods in both \(L^2\) and infinity-norm. Numerical examples are presented to validate the theoretical estimate.
reviewed by: Alexander N. Tynda
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 76M22
zbMATH DE Number: 6987683
zbMATH Keywords: Hammerstein integral equations
zbMATH Keywords: weakly singular kernels
zbMATH Keywords: spectral method
zbMATH Keywords: Galerkin method
zbMATH Keywords: collocation method
zbMATH Keywords: Legendre polynomials.
MaRDI profile type: Publication
full work available at URL: s40819-018-0580-0
OpenAlex ID: W2900161647
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DOI: 10.1007/S40819-018-0580-0
Wikidata QID: Q128956189
description: scientific article; zbMATH DE number 6987683