Abstract
This paper is concerned with the numerical properties of Runge-Kutta methods for the alternately of retarded and advanced equation ˙(t )= ax(t )+ a0x(2[ t+1 2 ]). The stability region of Runge-Kutta methods is determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained. A necessary and sufficient condition for the oscillation of the numerical solution is given. And it is proved that the Runge-Kutta methods preserve the oscillations of the analytic solutions. Some numerical experiments are illustrated.
Highlights
1 Introduction This paper deals with the numerical solution of the alternately of retarded and advanced equation with piecewise continuous arguments (EPCA)
There are some papers concerning the stability of numerical solutions of delay differential equations with piecewise continuous arguments, such as [ – ]
We investigate the numerical properties, including the stability and oscillation, of Runge-Kutta methods of delay differential equations with piecewise continuous arguments
Summary
This paper deals with the numerical solution of the alternately of retarded and advanced equation with piecewise continuous arguments (EPCA). Where [·] is the greatest integer function Differential equations of this form have stimulated considerable interest and have been studied by Cooker and Wiener [ ], Jayasree and Deo [ ], Wiener and Aftabizadeh [ ]. There are some papers concerning the stability of numerical solutions of delay differential equations with piecewise continuous arguments, such as [ – ]. ). In this paper, we investigate the numerical properties, including the stability and oscillation, of Runge-Kutta methods of delay differential equations with piecewise continuous arguments. The following theorems give existence and uniqueness of solutions and provide necessary and sufficient conditions for the asymptotic stability and the oscillation of all solutions of ) is called non-oscillatory if all nontrivial solutions of Eq Theorem . [ ] A necessary and sufficient condition for all solutions of Eq ( . ) to be oscillatory is either aea ea – or a a ea –
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