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Stability Analysis of Runge-Kutta Methods for Nonlinear Neutral Volterra Delay-Integro-Differential Equations

Published online by Cambridge University Press:  28 May 2015

Wansheng Wang  and
Dongfang Li
Wansheng Wang*
Affiliation:
School of Mathematics and Computational Sciences, Changsha University of Science & Technology, Yuntang Campus, Changsha 410114, China
Dongfang Li*
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
*
Corresponding author.Email address:w.s.wang@163.com
Corresponding author.Email address:lidongfang1983@gmail.com
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Abstract

This paper is concerned with the numerical stability of implicit Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations with constant delay. Using a Halanay inequality generalized by Liz and Trofimchuk, we give two sufficient conditions for the stability of the true solution to this class of equations. Runge-Kutta methods with compound quadrature rule are considered. Nonlinear stability conditions for the proposed methods are derived. As an illustration of the application of these investigations, the asymptotic stability of the presented methods for Volterra delay-integro-differential equations are proved under some weaker conditions than those in the literature. An extension of the stability results to such equations with weakly singular kernel is also discussed.

Keywords

65L0565L0665L2034K40Neutral differential equationsVolterra delay-integro-differential equationsRunge-Kutta methodsstability
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 4 , November 2011 , pp. 537 - 561
Copyright
Copyright © Global Science Press Limited 2011

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