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Volume 4, Issue 4
Stability Analysis of Runge-Kutta Methods for Nonlinear Neutral Volterra Delay-Integro-Differential Equations

Wansheng Wang & Dongfang Li

Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 537-561.

Published online: 2011-04

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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 is 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.

  • AMS Subject Headings

65L05, 65L06, 65L20, 34K40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-4-537, author = {}, title = {Stability Analysis of Runge-Kutta Methods for Nonlinear Neutral Volterra Delay-Integro-Differential Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2011}, volume = {4}, number = {4}, pages = {537--561}, 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 is 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.m1041}, url = {http://global-sci.org/intro/article_detail/nmtma/5982.html} }
TY - JOUR T1 - Stability Analysis of Runge-Kutta Methods for Nonlinear Neutral Volterra Delay-Integro-Differential Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 537 EP - 561 PY - 2011 DA - 2011/04 SN - 4 DO - http://doi.org/10.4208/nmtma.2011.m1041 UR - https://global-sci.org/intro/article_detail/nmtma/5982.html KW - Neutral differential equations, Volterra delay-integro-differential equations, RungeKutta methods, stability. AB -

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 is 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.

Wansheng Wang & Dongfang Li. (2020). Stability Analysis of Runge-Kutta Methods for Nonlinear Neutral Volterra Delay-Integro-Differential Equations. Numerical Mathematics: Theory, Methods and Applications. 4 (4). 537-561. doi:10.4208/nmtma.2011.m1041
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