Solving a parabolic PDE with nonlocal boundary conditions using the sinc method
type: Item
instance of: scholarly article
title: Solving a parabolic PDE with nonlocal boundary conditions using the sinc method
zbMATH Open document ID: 1264.65170
author: R. Zolfaghari
author: Abdollah Shidfar
published in: Numerical Algorithms
publication date: 2013-03-27T00:00:00Z
review text: The authors use the sinc method to determine the solution \(u\) of the heat equation \(u_t-u_{xx}=q(x,t)\) for \(x\in(0,1)\) and \(t\in(0,T]\) with initial condition \(u(x,0)=f(x)\). They consider two problems for \(0<t\leq T\). Problem I: \(u(0,t)=\int_0^1 k_1(x,t)u(x,t)\,dx+g_1(t)\) and \(u(1,t)=\int_0^1 k_2(x,t)u(x,t)\,dx+g_2(t)\). Problem II: \(\alpha_1(t)u_x(0,t)+\beta(t)u(0,t)=g_1(t)\) and \(\alpha_2(t)u_x(1,t)+\gamma(t)u(1,t)=g_2(t)\). Problem I appears in the study of the quasi-static flexure of a thermoelastic bar. Problem II can be regarded as a heat transfer process where \(u\) represents the temperature distribution. In Section 2, both problems are transformed into systems of Volterra integral equations. In the Sections 3 and 4, convolution integrals are approximated by the sinc collocation method. Convergence analysis is discussed in Section 5. In Section 6, the performance of the algorithm in solving parabolic partial differential equations (PDEs) with nonlocal boundary conditions and the accuracy and efficiency of the method are illustrated with two examples.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45F05
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 6148996
zbMATH Keywords: parabolic equation
zbMATH Keywords: sinc function
zbMATH Keywords: collocation method
zbMATH Keywords: nonlocal boundary conditions
zbMATH Keywords: numerical examples
zbMATH Keywords: heat equation
zbMATH Keywords: systems of Volterra integral equations
zbMATH Keywords: convolution integrals
zbMATH Keywords: sinc collocation method
zbMATH Keywords: convergence
zbMATH Keywords: algorithm
MaRDI profile type: Publication
full work available at URL: s11075-012-9595-5
OpenAlex ID: W2042403004
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DOI: 10.1007/S11075-012-9595-5
description: scientific article; zbMATH DE number 6148996