A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations
type: Item
instance of: scholarly article
title: A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations
zbMATH Open document ID: 1317.65252
author: F. Talati
author: Seyyed Ahmad Hosseini
author: Sedaghat Shahmorad
published in: Numerical Algorithms
publication date: 2015-03-23T00:00:00Z
review text: This paper investigates a numerical method for solving two-dimensional nonlinear Volterra-Fredholm integral equations of the following form: \[ u(x,t)-\lambda_1\int_{0}^{t}\int_{0}^{x} k_1(x,t,y,z) u^r(y,z)dydz- \] \[ -\lambda_2\int_{0}^{T_2}\int_{0}^{T_1} k_2(x,t,y,z) u^s(y,z)dydz=f(x,t), \quad r,s\in \mathbb{Z}^{+}, \] for the unknown function \(u(x,t)\) given on some region \(\Omega:=[0,T_1]\times[0,T_2]\). A matrix based method is considered for this equation. The method is based on the operational tau method. Some theoretical results are given to simplify and reduce the computation costs. In addition, \(L^2\)-convergence is proved when the given data are sufficiently smooth. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 6418188
zbMATH Keywords: two-dimensional nonlinear Volterra-Fredholm integral equation
zbMATH Keywords: matrix based method
zbMATH Keywords: operational tau method
zbMATH Keywords: numerical examples
reviewed by: Alexander N. Tynda
MaRDI profile type: Publication
full work available at URL: s11075-014-9858-4
OpenAlex ID: W2017833503
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DOI: 10.1007/S11075-014-9858-4
description: scientific article; zbMATH DE number 6418188