Numerical Approximations of Impulsive Delay Differential Equations

The article deals with numerical approximations of impulsive delay differential equations with a non-fixed time of impulses. The right-hand side of the approximation is assumed to be Lipschitz with respect to the norm of the measurable functions, which allows us to estimate the distance between functions with different times of jumps. Illustrative examples are provided.     

A Newton-Picard Collocation Method for Periodic Solutions of Delay Differential Equations

This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We exploit the equivalence of the linearized collocation system and the discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using a variant of the Newton-Picard method [Int. J. Bifurcation Chaos, 7 (1997), pp. 2547–2560]. This method combines a direct method in the low-dimensional subspace of the weakly stable and unstable modes with an iterative solver in the high-dimensional orthogonal complement. As a side effect, we also obtain good estimates for the dominant Floquet multipliers. We have implemented the method in the DDE-BIFTOOL environment to test our algorithm. AMS subject classification (2000) 65J15, 65P30, 65Q05     

A Stable Numerical Approach for Implicit Non-Linear Neutral Delay Differential Equations

In this paper we consider implicit non-linear neutral delay differential equations to derive efficient numerical schemes with good stability properties. The basic idea is to reformulate the original problem eliminating the dependence on the derivative of the solution in the past values. Our hypothesis on the original equation allow us to study the boundedness and asymptotic stability of the true and numerical solutions by the theory of stability with respect to the forcing term.     

一类时滞微分方程解的振动准则

本文建立了一类时滞微分方程一切解振动的必要充分条件     

广义时滞微分方程的渐近稳定性和数值分析

考虑了广义时滞微分方程的初值问题，分析了用线性多步法求解一类广义滞后型微分系统数值解的稳定性，在一定的Lagrange插值条件下，给出并证明了求解广义滞后型微分系统的线性多步法数值稳定的充分必要条件。     

Numerical Solutions of Stochastic Differential Delay Equations with Jumps

Abstract

In this article, we investigate the strong convergence of the Euler–Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R.     

具变系数时滞微分方程解的振动性

研究了时滞微分方程解的振动性，其中P（t）、τ（t）非负连续．我们证明了：如果对充分大的，且，则方程（*）每一解振动．该结论改进和推广了许多已知结果．     

一阶中立型线性微分系统解的振动性

本文给出一类一阶中立型线性微分系统解振动的充分条件 ,改进和推广了文 [1 ]的工作 .     

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

Abstract

In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 _p -consistency, numerical 𝕃 _p -stability and 𝕃 _p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.     

二阶非线性中立型时滞微分方程的振动准则

文章考虑二阶非线性中立型微分方程a(t)x(t)+∑li=1ci(t)x(t-τi(t))″+∑mi=1pi(t)fi(x(t-δi(t)))-∑ni=1qi(t)gi(x(t-σi(t)))=0的振动性,获得了该方程所有解振动的充分条件,推广了有关文献的结果.     

一类带滞量的微分方程解的表达式的讨论

本讨论了方程ａｎｘ＾（ｎ）（ｔ）＋…＋ａ０ｘ（ｔ）＋ｂｘ（ｔ－τ）＝ｔ＾ｋ的解的一些表达式。     

Multiple Solutions to Stochastic Differential Delay Equations and a Related Comparison Theorem

In this article, we discuss the existence of multiple solutions to a one-dimensional stochastic differential delay equation with continuous drift coefficients and derive a related comparison theorem.     

脉冲中立型时滞微分方程解的振动性

本文讨论一阶脉冲中立型时滞微分方程［ｙ（ｔ）＋Ｐｙ（ｔ－σ）］′＋Ｑ（ｔ）ｙ（ｔ－σ）＝０，ｔ０，ｔ≠ｔｋ，ｋ＝１，２，…，ｙ（ｔ＋ｋ）－ｙ（ｔ－ｋ）＝ｂｋｙ（ｔｋ），ｋ＝１，２，…，｛（Ｅ）这里τ，σ，Ｐ均为常数，τ＞０，σ＞０，Ｑ（ｔ）∈Ｃ（［０，∞），Ｒ＋），ｂｋ＞－１，ｋ＝１，２，…．分三种情况，Ｐ－１；－１＜Ｐ＜０；Ｐ＞０给出了方程（Ｅ）所有解振动的充分条件．     

稳定性切换在一阶时滞微分方程中的应用

介绍了时滞动力系统中的对零解稳定性讨论的稳定性切换法,并应用此方法对时滞动力系统中的三个一阶时滞微分方程基本定理给予证明.同时表明了在局部稳定性分析中,该方法有着更大的优势.     

一类二阶时滞微分方程在控制系统中的应用

基于时滞微分方程稳定性理论,针对一类状态向量中含有时滞的控制系统,研究其时滞相关稳定性问题.通过构造Lyapunov函数,获得了系统稳定、分岔的时滞相关充分条件,所得结论推广和改进了某些已有文献的相应结果.且借助实例及仿真结果解释所得结果的有效性.     

脉冲时滞微分方程解的整体存在唯一性、振动性与非振动性

本文讨论脉冲时滞微分方程X’(t)=f(t,x(t-T_1(t)),…,x(t-T_n(t))),x(t_k)-x(t_k~-)=I_k(x(t_k~- )).获得了方程(E) 解的一个整体存在唯一性定理.当(E)是线性方程时,给出了由时滞微分方程解的振动性或非振动性刻划出相应的脉冲时滞微分方程的同样性质的一般性脉冲条件.     

二阶时滞微分方程奇异半正边值问题

该文致力于讨论二阶时滞微分方程奇异半正边值问题正解的存在性,非线性项f(t,y)在y=0处具有奇性.      

具无限时滞中立型泛函微分方程解的稳定性与有界性

以相空间为基础，研究了具有无限时滞中立型泛函微分方程解的稳定性和有界性，建立了方程解为一致稳定，一致渐近稳定的充要性判据；证明了当方程右端泛函满足Lipschitz条件时，解的一致渐近稳定性蕴涵了有界解的存在性，推广了文献［4-6］中已有的相关结果.     

Legendre-Gauss Spectral Collocation Method for Second Order Nonlinear Delay Differential Equations

In this paper, we present and analyze a single interval Legendre-Gauss spectral collocation method for solving the second order nonlinear delay differential equations with variable delays. We also propose a novel algorithm for the single interval scheme and apply it to the multiple interval scheme for more efficient implementation. Numerical examples are provided to illustrate the high accuracy of the proposed methods.