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ASYMPTOTIC STABILITY AND NUMERICAL ANALYSIS FOR SYSTEMS OF GENERALIZED NEUTRAL DELAY DIFFERENTIAL EQUATIONS
广义中立型系统的渐近稳定性及数值分析

Cong Yuhao Yang Biao Kuang Jiaoxun,
丛玉豪
,杨彪,匡蛟勋

计算数学 , 2001,
Abstract: This paper deals with the stability analysis of implicit Runge-Kutta methods for the numerical solutions of systems of generalized neutral delay differential equations. The stability behaviour of implicit Runge-Kutta methods is analysed for the solution of the generalized system of linear neutral test equations. After an establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, we show that a implicit Runge-Kutta method is NGPG-stable if and only if it is A-stable under some Lagrange interpolation condition.
Almost Surely Asymptotic Stability of Numerical Solutions for Neutral Stochastic Delay Differential Equations  [PDF]
Zhanhua Yu,Mingzhu Liu
Discrete Dynamics in Nature and Society , 2011, DOI: 10.1155/2011/217672
Abstract: We investigate the almost surely asymptotic stability of Euler-type methods for neutral stochastic delay differential equations (NSDDEs) using the discrete semimartingale convergence theorem. It is shown that the Euler method and the backward Euler method can reproduce the almost surely asymptotic stability of exact solutions to NSDDEs under additional conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results. 1. Introduction The neutral stochastic delay differential equation (NSDDE) has attracted much more attention, and much work (see [1–4]) has been done. For example, Mao [2] studied the existence and uniqueness, moment and pathwise estimates, and the exponential stability of the solution to the NSDDE. Moreover, Mao et al. [4] studied the almost surely asymptotic stability of the NEDDE with Markovian switching: Since most NSDDEs cannot be solved explicitly, numerical solutions have become an important issue in the study of NSDDEs. Convergence analysis of numerical methods for NSDDEs can be found in [5–7]. On the other hand, stability theory of numerical solutions is one of the fundamental research topics in the numerical analysis. For stochastic differential equations (SDEs) as well as stochastic delay differential equations (SDDEs), moment stability and asymptotic stability of numerical solutions have received much more attention (e.g., [8–13] for moment stability and [12–14] for asymptotic stability). Recently, Wang and Chen [15] studied the mean-square stability of the semi-implicit Euler method for NSDDEs. We aim in this paper to study the almost surely asymptotic stability of Euler-type methods for NSDDEs using the discrete semimartingale convergence theorem. The discrete semimartingale convergence theorem (cf. [16, 17]) plays an important role in the almost surely asymptotic stability analysis of numerical solutions to SDEs and SDDEs [17–19]. Using the discrete semimartingale convergence theorem, we show that Euler-type methods for NSDDEs can preserve the almost surely asymptotic stability of exact solutions under additional conditions. In Section 2, we introduce some necessary notations and state the discrete semimartingale convergence theorem as a lemma. In Section 3, we study the almost surely asymptotic stability of exact solutions to NSDDEs. Section 4 gives the almost surely asymptotic stability of the Euler method. In Section 5, we discuss the almost surely asymptotic stability of the backward Euler method. Numerical experiments are presented in Section 6. 2. Preliminaries Throughout this
On a class of third order neutral delay differential equations with piecewise constant argument  [cached]
Garyfalos Papaschinopoulos
International Journal of Mathematics and Mathematical Sciences , 1994, DOI: 10.1155/s0161171294000153
Abstract: In this paper we study existence, uniqueness and asymptotic stability of the solutions of a class of third order neutral delay differential equations with piecewise constant argument.
Asymptotic properties of the spectrum of neutral delay differential equations  [PDF]
Y. N. Kyrychko,K. B. Blyuss,P. Hoevel,E. Schoell
Physics , 2012, DOI: 10.1080/14689360902893285
Abstract: Spectral properties and transition to instability in neutral delay differential equations are investigated in the limit of large delay. An approximation of the upper boundary of stability is found and compared to an analytically derived exact stability boundary. The approximate and exact stability borders agree quite well for the large time delay, and the inclusion of a time-delayed velocity feedback improves this agreement for small delays. Theoretical results are complemented by a numerically computed spectrum of the corresponding characteristic equations.
Stability Properties of Neutral Delay Integro-Differential Equation
Ali Fuat Yeni?erio?lu
Sel?uk Journal of Applied Mathematics , 2009,
Abstract: Some new stability results are given for a neutral delay integro-differential equation with constant delays. The stability of the trivial solution is described by the use of an appropriate real root of an equation, which is in a sense the corresponding characteristic equation. A basis theorem on the behavior of solutions of neutral delay integro-differential equations is established. As a consequence of this theorem, a stability criterion is obtained.
A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential Equations  [cached]
Wang Wansheng
Journal of Inequalities and Applications , 2010,
Abstract: This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs) and nonlinear neutral delay integrodifferential equations (NDIDEs) are obtained.
A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential Equations  [cached]
Wansheng Wang
Journal of Inequalities and Applications , 2010, DOI: 10.1155/2010/475019
Abstract: This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs) and nonlinear neutral delay integrodifferential equations (NDIDEs) are obtained.
Asymptotic properties, nonoscillation, and stability for scalar first order linear autonomous neutral delay differential equations  [cached]
Christos G. Philos,Ioannis K. Purnaras
Electronic Journal of Differential Equations , 2004,
Abstract: We study scalar first order linear autonomous neutral delay differential equations with distributed type delays. This article presents some new results on the asymptotic behavior, the nonoscillation and the stability. These results are obtained via a real root (with an appropriate property) of the characteristic equation. Applications to the special cases such as (non-neutral) delay differential equations are also presented.
The Asymptotic Behavior for Second-Order Neutral Stochastic Partial Differential Equations with Infinite Delay  [PDF]
Huabin Chen
Discrete Dynamics in Nature and Society , 2011, DOI: 10.1155/2011/584510
Abstract: By establishing two Lemmas, the exponential stability and the asymptotical stability for mild solution to the second-order neutral stochastic partial differential equations with infinite delay are obtained, respectively. Our results can generalize and improve some existing ones. Finally, an illustrative example is given to show the effectiveness of the obtained results. 1. Introduction The neutral stochastic differential equations can play an important role in describing many sophisticated dynamical systems in physical, biological, medical, chemical engineering, aero-elasticity, and social sciences [1–3], and the qualitative dynamics such as the existence and uniqueness, stability, and controllability for first-order neutral stochastic partial differential equations with delays have been extensively studied by many authors; see, for example, the existence and uniqueness for neutral stochastic partial differential equations under the non-Lipschitz conditions was investigated by using the successive approximation [4–6]; in [7], Caraballo et al. have considered the exponential stability of neutral stochastic delay partial differential equations by the Lyapunov functional approach; in [8], Dauer and Mahmudov have analyzed the existence of mild solutions to semilinear neutral evolution equations with nonlocal conditions by using the fractional power of operators and Krasnoselski-Schaefer-type fixed point theorem; in [9], Hu and Ren have established the existence results for impulsive neutral stochastic functional integrodifferential equations with infinite delays by means of the Krasnoselskii-Schaefer-type fixed point theorem; some sufficient conditions ensuring the controllability for neutral stochastic functional differential inclusions with infinite delay in the abstract space with the help of the Leray-Schauder nonlinear alterative have been given by Balasubramaniam and Muthukumar in [10]; Luo and Taniguchi, in [11], have studied the asymptotic stability for neutral stochastic partial differential equations with infinite delay by using the fixed point theorem. Very recently, in [12], the author has discussed the exponential stability for mild solution to neutral stochastic partial differential equations with delays by establishing an integral inequality. Although there are many valuable results about neutral stochastic partial differential equations, they are mainly concerned with the first-order case. In many cases, it is advantageous to treat the second-order stochastic differential equations directly rather than to convert them to first-order systems.
MEAN-SQUARE STABILITY OF EULER METHOD FOR LINEAR NEUTRAL STOCHASTIC DELAY DIFFERENTIAL EQUATIONS
线性中立型随机延迟微分方程Euler方法的均方稳定性

Wang Wenqiang,Chen Yanping,
王文强
,陈艳萍

计算数学 , 2010,
Abstract: So far there are not many results on the numerical stability of linear neutral stochastic delay differential equations. The main aim of this paper is to establish new results on the numerical stability. It is proved that the Euler method is mean-square stable under suitable condition. Moreover, the result is also verified by a numerical example.
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