Product integration methods for solving a system of nonlinear Volterra integral equations
type: Item
instance of: scholarly article
title: Product integration methods for solving a system of nonlinear Volterra integral equations
zbMATH Open document ID: 0858.65139
DOI: 10.1016/0377-0427(95)00041-0
author: Bartur Jumarhon
author: S. McKee
published in: Journal of Computational and Applied Mathematics
publication date: 1997-03-25T00:00:00Z
review text: In their earlier paper [J. Math. Anal. Appl. 190, No. 3, 806-820 (1995; Zbl 0822.35076)] the authors showed that the heat equation with nonlinear and nonlocal boundary conditions, \[ u_t(x,t)=u_{xx} (x,t)\;(0<x<1,\;t>0);\;u(x,0)=1; \] \[ u_x(0,t)=0;\;u_x(1,t)={Em\over 1+L} \biggl\{L \gamma (t)- \bigl(1-\gamma(t)\bigr) u(1,t) \biggr\}; \] \[ m\gamma (t)+ \int^1_0 u(x,t)dx = 1 \] (with given positive constants \(E\), \(L\), \(m)\) is equivalent to a system of two nonlinear Volterra integral equations of the second kind with weakly singular kernel. Here, they derive and analyze the product Euler methods and a product trapezoidal method for solving this system of integral equations numerically. The known singularity of the exact solution at \(t=0\) permits the use of singularity subtraction techniques and the derivation of (classical) optimal order convergence results. A numerical example illustrates these results.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 45G15
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 908417
zbMATH Keywords: heat equation
zbMATH Keywords: nonlinear and nonlocal boundary conditions
zbMATH Keywords: system
zbMATH Keywords: nonlinear Volterra integral equations of the second kind
zbMATH Keywords: weakly singular kernel
zbMATH Keywords: product Euler methods
zbMATH Keywords: product trapezoidal method
zbMATH Keywords: singularity subtraction techniques
zbMATH Keywords: convergence
zbMATH Keywords: numerical example
reviewed by: H. Brunner
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00041-0
OpenAlex ID: W2021840750
cites work: Nonpolynomial Spline Collocation for Volterra Equations with Weakly Singular Kernels
cites work: Q3750978
cites work: Product integration methods for second-kind Abel integral equations
cites work: Q3682852
cites work: Collocation methods for second-kind Volterra integral equations with weakly singular kernels
cites work: A nonlinear weakly singular Volterra integro-differential equation arising from a reaction-diffusion study in a small cell
cites work: On monotone Abel-Volterra integral equations on the half line
cites work: A mathematical model of a biosensor
cites work: On the heat equation with nonlinear and nonlocal boundary conditions
cites work: Q3681889
cites work: Runge-Kutta Theory for Volterra and Abel Integral Equations of the Second Kind
cites work: Smoothness of Solutions of Volterra Integral Equations with Weakly Singular Kernels
cites work: Volterra integral equations and a new Gronwall inequality (Part I: The linear case)
cites work: Volterra integral equations and a new Gronwall inequality (Part II: The nonlinear case)
cites work: Q5652137
cites work: Collocation Methods for Weakly Singular Second-kind Volterra Integral Equations with Non-smooth Solution
description: scientific article; zbMATH DE number 908417