Numerical oscillation and non-oscillation analysis of the mixed type impulsive differential equation with piecewise constant arguments

Abstract

The purpose of this paper is to study oscillation and non-oscillation of Runge–Kutta methods for linear mixed type impulsive differential equations with piecewise constant arguments. The conditions for oscillation and non-oscillation of numerical solutions are obtained. Also conditions under which Runge–Kutta methods can preserve the oscillation and non-oscillation of linear mixed type impulsive differential equations with piecewise constant arguments are obtained. Moreover, the interpolation function of numerical solutions is introduced and the properties of the interpolation function are discussed. It turns out that the zeros of the interpolation function converge to ones of the analytic solution with the same order of accuracy as that of the corresponding Runge–Kutta method. To confirm the theoretical results, the numerical examples are given.

Keywords:

• Oscillation
• numerical solution
• Runge–Kutta methods
• impulsive delay differential equations
• piecewise constant arguments

Mathematics Subject Classifications:

• 65L06
• 65L05

Acknowledgments

The authors are grateful to the handling editor and anonymous referees for their careful reading and valuable comments which have improved the presentation and contents of this paper significantly.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Nature Science Foundation of China [grant number 12001143].