Abstract
The purpose of this paper is to study oscillation of Runge-Kutta methods for linear advanced impulsive differential equations with piecewise constant arguments. We obtain conditions of oscillation and nonoscillation for Runge-Kutta methods. Moreover, we prove that the oscillation of the exact solution is preserved by the θ-methods. It turns out that the zeros of the piecewise linear interpolation functions of the numerical solution converge to the zeros of the exact solution. We give some numerical examples to confirm the theoretical results.
Highlights
In the past three decades, the theory of differential equations with piecewise constant arguments has been intensively studied
Oscillation of discontinuous solutions of differential equations with piecewise constant arguments has been proposed by Wiener as an open problem
We study oscillation of Runge-Kutta methods for the following equation:
Summary
In the past three decades, the theory of differential equations with piecewise constant arguments has been intensively studied. Oscillation of discontinuous solutions of differential equations with piecewise constant arguments has been proposed by Wiener as an open problem [ ], p. Oscillation of advanced impulsive differential equations with piecewise constant arguments has been studied in [ ]. In [ ], asymptotical stability of Runge-Kutta methods for the advanced linear impulsive differential equation with piecewise constant arguments was studied. We study oscillation of Runge-Kutta methods for the following equation:. In Section , the results about oscillation of the exact solutions of In Section , the conditions of oscillation and nonoscillation for θ -methods are obtained. It is proved that the oscillation of the exact solution is preserved by the θ -methods. All solutions of ( . ) are oscillatory if and only if b
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