13 pages
English

Numerical stability and oscillation of the Runge-Kutta methods for equation x ′ ( t ) = a x ( t ) + a 0 x ( M [ t + N M ] )

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This paper deals with the numerical properties of Runge-Kutta methods for the alternately of retarded and advanced equation x ′ ( t ) = a x ( t ) + a 0 x ( M [ t + N M ] ) . Necessary and sufficient conditions for the stability and oscillation of the numerical solution are given. The conditions that the Runge-Kutta methods preserve the stability and oscillations of the analytic solutions are obtained. Some numerical experiments are illustrated.

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Publié le 01 janvier 2012
Nombre de lectures 4
Langue English

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Song and Liu Advances in Difference Equations 2012, 2012 :146 http://www.advancesindifferenceequations.com/content/2012/1/146 R E S E A R C H Open Access Numerical stability and oscillation of the Runge-Kutta methods for equation x ( t ) = ax ( t ) + a x ( M [ t + MN ]) Minghui Song * and MZ Liu * Correspondence: songmh@lsec.cc.ac.cn Abstract Department of Mathematics, Harbin Institute of Technology, Harbin, This paper deals with the numerical properties of Runge-Kutt a a 0 x m(e M t[h t o + N d]s)forthe 150001, P.R. China alternately of retarded and advanced equation x ( t ) = ax ( t ) + M . Necessary and sufficient conditions for the stability and oscillation of the numerical solution are given. The conditions that the Runge-Kutta methods preserve the stability and oscillations of the analytic solutions are obtained. Some numerical experiments are illustrated. Keywords: stability; oscillation; differential equation; Runge-Kutta method 1 Introduction This paper deals with the numerical solutions of the alternately of retarded and advanced equation with piecewise continuous arguments (EPCA) x ( t ) = f x ( t ), x M t + MN  , (.) where [ · ] is the greatest integer function, and M and N are positive integers such that N < M . Differential equations of this form have simulated considerable interest and have been studied by Wiener and Aftabizadeh [ ]. Cooke and Wiener [] studied the special cases of this form with M = , N =  and M = , N = . In these equations the argument derivation T ( t ) = t M [ t + MN ] are piecewise linear periodic function with periodic M . Also, T ( t ) is negative for Mn N t < Mn and positive for Mn < t < M ( n + ) – N ( n is inte-ger). Therefore, Eq. (.) is of advanced type on [ Mn N , Mn ) and of retarded type on ( Mn , M ( n + ) – N ). EPCA describe hybrid dynamics systems, combine properties of both differential and difference equations, and have applications in certain biomedical models in the work of Busenberg and Cooke []. For these equations of mixed type, the change of sign in the argu-ment derivation leads not only to interesting periodic properties but also to complications in the asymptotic and oscillatory behavior of solutions. Oscillatory, stability and periodic properties of the linear EPCA of the form with alternately of retarded and advanced have been investigated in []. There exist some papers concerning the stability of the numerical solutions of delay dif-ferential equations with piecewise continuous arguments, such as [ , , ]. Also, there © 2012 Song and Liu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.