Search

Abstract: This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the numerical solution of delay differential equations. The emphasis is on the nonlinear stability of multistep methods. It is shown that every A-stable linear multistep method with piecewise constant or linear interpolation can preserve the asymptotic stability of a class of nonlinear systems, which is an extension of the well known GP-stability result to the nonlinear case. As a comparison, it is also proved that strict stability at infinity is necessary to the asymptotic stability of one-leg methods for non-autonomous equations. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the numerical solution of delay integro-differential equations. The adaptation of linear multistep methods is considered. The emphasis is on the linear stability of numerical methods. It is shown that every A-stable, strongly 0-stable linear multistep method of Pouzet type can preserve the delay-independent stability of the underlying linear systems. In addition, some delay-dependent stability conditions for the stability of numerical methods are also given. [Copyright &y& Elsevier]

Investigates the stability of general linear methods for linear systems of neutral type. Demonstration that a general linear method with strict stability at infinity can preserve the asymptotic stability of the underlying system if an appropriate equi-stage interpolation is employed to approximate the delay argument.

This paper is devoted to a study of nonlinear stability of general linear methods for the numerical solution of delay differential equations in Hilbert spaces. New stability concepts are further introduced. The stability properties of (k, p, q)-algebraically stable general linear methods with piecewise constant or linear interpolation procedure are investigated. We also discuss stability of linear multistep methods viewed as a special subset of the class of general linear methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of delay differential equations(DDEs). We focus on the stability behaviour of Runge-Kutta methods for nonlinear DDEs. The new concepts of GR(l)-stability, GAR(l)-stability and weak GAR(l)stability are further introduced. We investigate these stability properties for (k, l)algebraically stable Runge-Kutta methods with a piecewise constant or linear interpolation procedure. [ABSTRACT FROM AUTHOR]

This paper is concerned with the delay dependent stability of the stochastic exponential Euler method for stochastic delay differential equations and stochastic delay partial differential equations. By using root locus technique, the necessary and sufficient condition of the numerical delay dependent stability of the method is derived for a class of stochastic delay differential equations and it is shown that the stochastic exponential Euler method can fully preserve the asymptotic mean square stability of the underlying system. Furthermore, we investigate the delay dependent stability of the semidiscrete and fully discrete systems for a linear stochastic delay partial differential equation. The necessary and sufficient condition for the delay dependent stability of the semidiscrete system based on the standard central difference scheme in space is given. Based on this condition, the delay dependent stability of the fully discrete system by using the stochastic exponential Euler method in time is studied. It is shown that the fully discrete scheme can inherit the delay dependent stability of the semidiscrete system completely. At last, some numerical experiments are given to validate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a projected Euler-Maruyama method for stochastic delay differential equations with variable delay under a global monotonicity condition. This condition admits some equations with highly nonlinear drift and diffusion coefficients. We appropriately generalize the idea of C-stability and B-consistency given by Beyn et al. (2016) to the case with delay. Moreover, the method is proved to be convergent with order one-half in a succinct way. Finally, some numerical examples are included to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a split-step theta (SST) method is introduced and analyzed for neutral stochastic differential delay equations (NSDDEs). It is proved that the SST method with θ ∈ [ 0 , 1 / 2 ] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize and the drift coefficient, but for θ ∈ ( 1 / 2 , 1 ] , the SST can reproduce the exponential mean square stability unconditionally. Then, based on the stability results of SST scheme, we examine the exponential mean square stability of the stochastic linear theta (SLT) approximation for NSDDEs and obtain the similar stability results to that of the SST method. Moreover, for sufficiently small stepsize, we show that the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately. These results show that different values of theta will drastically affect the exponential mean square stability of the two classes of theta approximations for NSDDEs. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the asymptotic behavior of the stochastic Lotka–Volterra model with multiple delays. By using certain matrix techniques, some results on nonexplosion, boundedness and pathwise estimation of the model are investigated. [Copyright &y& Elsevier]