非线性随机延迟微分方程Heun方法的数值稳定性

doi:10.12286/jssx.2011.1.69

计算数学 ›› 2011, Vol. 33 ›› Issue (1): 69-76.DOI: 10.12286/jssx.2011.1.69

非线性随机延迟微分方程Heun方法的数值稳定性

王文强^1,2, 陈艳萍³

1. 湘潭大学数学与计算科学学院, 湖南湘潭 411105;
2. 湘潭大学土木工程与力学学院, 湖南湘潭 411105;
3. 华南师范大学数学科学学院, 广州 510631

收稿日期:2009-11-03 出版日期:2011-02-15 发布日期:2011-03-08
基金资助:
广东省高等学校珠江学者计划、国家自然科学基金(10871207)、973项目(2005CB321703)、教育部高校博士点基金(20094301110001)、湖南省自科基金(09JJ3002)和湘潭大学博士后科学基金资助项目.

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王文强, 陈艳萍. 非线性随机延迟微分方程Heun方法的数值稳定性[J]. 计算数学, 2011, 33(1): 69-76.

Wang Wenqiang, Chen Yanping. NUMERICAL STABILITY OF HEUN METHODS FOR NONLINEAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2011, 33(1): 69-76.

NUMERICAL STABILITY OF HEUN METHODS FOR NONLINEAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

Wang Wenqiang^1,2, Chen Yanping³

1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China;
2. Civil Engineering & Machanics College, Xiangtan University, Xiangtan 411105, Hunan, China;
3. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received:2009-11-03 Online:2011-02-15 Published:2011-03-08

本文讨论一般非线性随机延迟微分方程Heun方法的数值稳定性,证明了如果问题本身满足零解是均方指数稳定和均方渐近稳定的充分条件,则当方程的漂移项进一步满足一定的条件时,Heun方法是MS-稳定的, 带线性插值的Heun方法是均方指数稳定的和GMS-稳定的理论结果. 文末的数值试验进一步验证了所得的相关结论.

关键词： 随机延迟微分方程, Heun方法, 插值, 均方指数稳定, MS-稳定, GMS-稳定

In this paper, the authors investigated the numerical stability of Heun methods for nonlinear stochastic delay differential equations. When the analytical solution satisfies the conditions of mean-square stability, and if the drift term satisfy some restrictions, then the Heun methods with linear interpolation procedure is exponential mean-square stable and GMS-stable, the Heun methods is mean-square stable(MS-stable). Moreover, these results are also verified by some numerical examples.

Key words: stochastic delay differential equations, Heun methods, interpolation, exponential mean-square stability, MS-stability, GMS-stability

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摘要

非线性随机延迟微分方程Heun方法的数值稳定性

NUMERICAL STABILITY OF HEUN METHODS FOR NONLINEAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

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