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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 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]

The Euler–Maruyama method is presented for linear stochastic Volterra integral equations. Then the strong convergence property is analyzed for convolution kernels and general kernels, respectively. It is well known that for stochastic ordinary differential equations, the strong convergence order of the Euler–Maruyama method is 1 2 . However, the strong superconvergence order of 1 is obtained for linear stochastic Volterra integral equations with convolution kernels if the kernel K 2 of the diffusion term satisfies K 2 ( 0 ) = 0 ; and this strong superconvergence property is inherited by linear stochastic Volterra integral equations with general kernels if the kernel K 2 of the diffusion term satisfies K 2 ( t , t ) = 0 . The theoretical results are illustrated by extensive numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we concern with the theoretical and numerical analysis of the generalized stochastic Volterra integro-differential equations (SVIDEs). The existence, uniqueness, boundedness and Hölder continuity of the analytic solutions for generalized SVIDEs are investigated. The Euler–Maruyama method for generalized SVIDEs is presented. The boundedness of the numerical solution is proved, and the strong convergence order is obtained. The theoretical results are illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]