一类变延迟中立型微分方程梯形方法的渐近估计

数学物理学报 ›› 2019, Vol. 39 ›› Issue (3): 560-569.

一类变延迟中立型微分方程梯形方法的渐近估计

张根根^1,*(),王晚生³,肖爱国²

¹ 广西师范大学数学与统计学院广西桂林 541004
² 湘潭大学数学与计算科学学院湖南湘潭 411105
³ 上海师范大学数理学院上海 200234

收稿日期:2018-02-01 出版日期:2019-06-26 发布日期:2019-06-27
通讯作者: 张根根 E-mail:zhanggen036@163.com
基金资助:
国家自然科学基金(11701110)

Asymptotic Estimation of the Trapezoidal Method for a Class of Neutral Differential Equation with Variable Delay

Gengen Zhang^1,*(),Wansheng Wang³,Aiguo Xiao²

¹ School of Mathematics and Statistics, Guangxi Normal University, Guangxi Guilin 541004
² School of Mathematics and Computational Science, Xiangtan University, Hunan Xiangtan 411105
³ Department of Mathematics, Shanghai Normal University, Shanghai 200234

Received:2018-02-01 Online:2019-06-26 Published:2019-06-27
Contact: Gengen Zhang E-mail:zhanggen036@163.com
Supported by:
the NSFC(11701110)

摘要/Abstract

摘要：

该文研究了一类变延迟中立型微分方程梯形方法的稳定性，并借助于一个泛函不等式得到了数值解的渐近估计.此渐近估计对数值解的性态不仅比数值渐近稳定性描述得更加精确，而且能给出非稳定情形数值解的上界估计式.

关键词: 中立型延迟微分方程, 梯形方法, 渐近估计, 渐近稳定性

Abstract:

In this paper, we investigate the stability of the trapezoidal method for a class of neutral differential equation with variable delay and obtain the asymptotic estimation of numerical solution with the aid of a functional inequality. The asymptotic estimation is more accurate than asymptotic stability in describing the behaviours of the numerical solution, and gives the upper bound estimates of the numerical solution for the nonstable case.

Key words: Neutral delay differential equation, Trapezoidal method, Asymptotic estimation, Asymptotic stability

中图分类号:

O241.8

张根根,王晚生,肖爱国. 一类变延迟中立型微分方程梯形方法的渐近估计[J]. 数学物理学报, 2019, 39(3): 560-569.

Gengen Zhang,Wansheng Wang,Aiguo Xiao. Asymptotic Estimation of the Trapezoidal Method for a Class of Neutral Differential Equation with Variable Delay[J]. Acta mathematica scientia,Series A, 2019, 39(3): 560-569.

图/表 1

参考文献 13

1	Bellen A , Zennaro M . Numerical Methods for Delay Differential Equations. New York: Oxford University Press, 2003
2	Čermák J , Jansky J . On the asymptotics of the trapezoidal rule for the pantograph equation. Math Comput, 2009, 78 (268): 2107- 2126 doi: 10.1090/S0025-5718-09-02245-5
3	Gugllelmi N , Zennaro M . Stability of one-leg methods for the variable coefficient pantograph equation on the quasi-geometric. IMA J Numer Anal, 2003, 23 (3): 421- 438
4	Heard M L . A family of solutions of the IVP for the equation $x'(t)=ax(λt), λ > 1$. Aequations Math, 1973, 9 (3): 273- 280
5	Huang C , Vandewalle S . An analysis of delay dependent stablity for ordinary and partial differential equations with fixed and distributed delays. SIAM J Sci Comput, 2004, 25 (5): 1608- 1632 doi: 10.1137/S1064827502409717
6	Iserles A . On the generalized pantograph functional-differential equation. Europ J Appl Math, 1993, 4 (1): 1- 38
7	Kuang J , Cong Y . Stablity of Numerical Methods for Delay Differential Equations. Beijing: Science Press, 2005
8	Lehniger H , Liu Y . The functional-differential equation $y'(t)=Ay(t) + By(λt) + Cy'(qt) + f(t)$. Europ J Appl Math, 1998, 9 (1): 81- 91
9	Liu Y . Regular solutions of the Shabat equation. J Diff Equ, 1999, 154 (1): 1- 41 doi: 10.1006/jdeq.1998.3541
10	Vandewalle S . Discretized stablity and error growth of the non-autonomous pantograph equation. SIAM J Numer Anal, 2005, 42 (5): 2020- 2042
11	Wang W , Li S . Stability analysis of θ-methods for nonlinear neutral functional differential equations. SIAM J Sci Comput, 2008, 30 (4): 2181- 2205 doi: 10.1137/060654116
12	Wang W , Li S , Su K . Nonlinear stability of Runge-Kutta methods for neutral delay differential equations. J Comput Appl Math, 2008, 214 (1): 175- 185
13	Zhang G , Xiao A , Wang W . The asymptotic behaviour of the θ-methods with constant stepsize for the generalized pantograph equation. Int J Comput Math, 2016, 93 (9): 1484- 1504 doi: 10.1080/00207160.2015.1061124

[1]	崔静,梁秋菊,毕娜娜. 分数布朗运动驱动的带脉冲的中立性随机泛函微分方程的渐近稳定性[J]. 数学物理学报, 2019, 39(3): 570-581.
[2]	王慧灵, 高建芳. 一类带有多项延迟的非线性中立型延迟微分方程解析解的振动性分析[J]. 数学物理学报, 2018, 38(4): 740-749.
[3]	王晚生, 钟鹏, 赵新阳. 非线性中立型变延迟微分方程的长时间稳定性[J]. 数学物理学报, 2018, 38(1): 96-109.
[4]	罗日才, 许弘雷, 王五生. 一类新的变时滞中立型神经网络的全局渐近稳定性条件[J]. 数学物理学报, 2015, 35(3): 634-640.
[5]	吴事良, 刘三阳. 反应扩散系统双稳波前解的全局渐近稳定性[J]. 数学物理学报, 2010, 30(2): 440-448.
[6]	陈福来;文贤章. 脉冲泛函微分方程的渐近性态[J]. 数学物理学报, 2006, 26(2): 287-296.
[7]	黄枝姣, 张诚坚. 数值求解NDDEs系统的单支方法的非线性稳定性[J]. 数学物理学报, 2002, 22(3): 421-426.
[8]	李小平, 蒋建初. 中立型时滞微分方程的渐近稳定性[J]. 数学物理学报, 2002, 22(2): 163-170.
[9]	滕志东. 一类无穷时滞周期Lotka-Volterra型系统的正周期解[J]. 数学物理学报, 2001, 21(1): 94-101.
[10]	滕志东陈兰荪. 具有时滞的周期Lotka-Volterra型系统的全局渐近稳定性[J]. 数学物理学报, 2000, 20(3): 296-303.
[11]	沈根成高国柱李力锋. 具有扰动和无界时滞的一维泛函微分方程的渐近稳定性[J]. 数学物理学报, 1999, 19(5): 493-500.
[12]	罗跃虎, 冯德兴. 关于C₀半群的渐近稳定性[J]. 数学物理学报, 1997, 17(4): 396-402.