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2013 Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations
Haiyan Yuan, Cheng Song
Abstr. Appl. Anal. 2013(SI01): 1-13 (2013). DOI: 10.1155/2013/679075

Abstract

This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of ( k , l ) -algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a ( k , l ) -algebraically stable two-step Runge-Kutta method with 0 < k < 1 is proved. For the convergence, the concepts of D -convergence, diagonally stable, and generalized stage order are firstly introduced; then it is proved by some theorems that if a two-step Runge-Kutta method is algebraically stable and diagonally stable and its generalized stage order is p , then the method with compound quadrature formula is D -convergent of order at least min { p , ν } , where ν depends on the compound quadrature formula.

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Haiyan Yuan. Cheng Song. "Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations." Abstr. Appl. Anal. 2013 (SI01) 1 - 13, 2013. https://doi.org/10.1155/2013/679075

Information

Published: 2013
First available in Project Euclid: 26 February 2014

zbMATH: 1275.65097
MathSciNet: MR3055973
Digital Object Identifier: 10.1155/2013/679075

Rights: Copyright © 2013 Hindawi

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Vol.2013 • No. SI01 • 2013
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