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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. [ABSTRACT FROM AUTHOR]

The paper is mainly concerned with the oscillations of numerical solutions by the linear θ − methods for a kind of linear delay differential equations with positive and negative coefficients. Some conditions under which the numerical solutions are oscillatory are obtained and it is proved that oscillations of the analytical solutions are preserved by the numerical solutions under mild conditions. Numerical experiments are given to demonstrate our results. [ABSTRACT FROM AUTHOR]

The paper deals with the oscillation and non-oscillation of the Runge–Kutta methods for a differential equation with piecewise continuous arguments of mixed type. The conditions of the oscillation and non-oscillation for the Runge–Kutta method are obtained. It is proved that oscillation of the analytic solution is not preserved by the Runge–Kutta method under any conditions. The conditions under which the non-oscillation of analytic solutions is preserved by the Runge–Kutta method are obtained. Some numerical experiments are given. [ABSTRACT FROM AUTHOR]

The paper deals with the oscillation analysis of numerical solution in the ?-methods for differential equations with piecewise constant arguments of advanced type. The conditions of the oscillation for the ?-method are obtained. It is proved that the oscillation of the analytic solution is preserved by the ?- method. Some numerical experiments are given. [ABSTRACT FROM AUTHOR]

This article is concerned with the preservation of oscillations for differential equations with piecewise constant arguments of advanced type. By using the Runge-Kutta method, new oscillation conditions for numerical solution are established. We prove that oscillations of the analytic solution are preserved by the numerical solution in the Runge-Kutta method under some conditions. Some experiments are given. [ABSTRACT FROM PUBLISHER]

This paper is concerned with oscillation and non-oscillation of numerical solutions for nonlinear delay differential equation of hematopoiesis with unimodal production rate. By investigating the roots of characteristic equation, some conditions under which the numerical solution is oscillatory and non-oscillatory are obtained. The properties of non-oscillatory numerical solutions are investigated. To verify our results, we give numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with oscillations of numerical solutions for the nonlinear delay differential equation of population dynamics. The equation proposed by Mackey and Glass for a ”dynamic disease” involves respiratory disorders and its solution resembles the envelope of lung ventilation for pathological breathing, called Cheyne-Stokes respiration. Some conditions under which the numerical solution is oscillatory are obtained. The properties of non-oscillatory numerical solutions are investigated. To verify our results, we give numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with the oscillation analysis of numerical solution in the θ-methods for equation x′(t)+ ax(t)+ a 1 x([t −1])=0. The conditions of the oscillation for the θ-method are obtained. It is proved that the oscillation of the analytic solution is preserved by the θ-method. It turns out that the zeros of the linear interpolation function of the numerical solution can converge to the zeros of the analytic solution with the order of accuracy 1 and 2 . Some numerical experiments are given. [Copyright &y& Elsevier]