非线性随机分数阶延迟积分微分方程Euler-Maruyama方法的强收敛性

doi:10.12286/jssx.j2021-0830

计算数学 ›› 2023, Vol. 45 ›› Issue (1): 57-73.DOI: 10.12286/jssx.j2021-0830

非线性随机分数阶延迟积分微分方程Euler-Maruyama方法的强收敛性

王琳, 许珊珊, 王文强

湘潭大学科学工程计算与数值仿真湖南省重点实验室, 湘潭 411105

收稿日期:2021-07-01 出版日期:2023-02-14 发布日期:2023-02-13
通讯作者: 王文强, Email: wwq@xtu.edu.cn
基金资助:
国家自然科学基金(12071403)和湖南省教育厅重点项目(18A049)资助.

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王琳, 许珊珊, 王文强. 非线性随机分数阶延迟积分微分方程Euler-Maruyama方法的强收敛性[J]. 计算数学, 2023, 45(1): 57-73.

Wang Lin, Xu Shanshan, Wang Wenqiang. STRONG CONVERGENCE OF THE ELUER-MARUYAMA METHOD FOR STOCHASTIC FRACTIONAL DELAY INTEGRO-DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2023, 45(1): 57-73.

STRONG CONVERGENCE OF THE ELUER-MARUYAMA METHOD FOR STOCHASTIC FRACTIONAL DELAY INTEGRO-DIFFERENTIAL EQUATIONS

Wang Lin, Xu Shanshan, Wang Wenqiang

Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China

Received:2021-07-01 Online:2023-02-14 Published:2023-02-13
Supported by:
The project was supported by the National Key Research and Development Program of China (2019YFC1905301);National Natural Science Foundation of China (22078115,21776108,21690083,22008078).

本文研究了一类新的模型问题: 非线性随机分数阶延迟积分微分方程. 当方程中的漂移项和扩散项满足全局 Lipschitz 条件和线性增长条件时, 基于压缩映射原理给出了该方程解存在唯一的充分条件. 由于理论求解的困难, 构造了一种数值方法(Euler-Maruyama 方法), 并证得强收敛阶为α-1/2,α∈ (2/1, 1]. 最后通过数值试验, 验证了这一理论结果.

关键词： 随机分数阶延迟积分微分方程, Eluer-Maruyama方法, 强收敛性, 解的存在唯一性, Caputo分数微分算子

This paper studies a new type of model problem: nonlinear stochastic fractional delay integro-differential equations. Under the assumptions that the drift and diffusion terms in the equation satisfy the global Lipschitz condition and linear growth condition, the existence and uniqueness of the solution is proved by using contraction mapping principle. Due to the difficulty of theoretical solution, a numerical method (Euler-Maruyama method) is constructed and the order of strong convergence is proved to be α-1/2, α∈(1/2, 1].Finally, the theoretical results are verified by numerical experiments.

Key words: Stochastic fractional delay integro-differential equation, Eluer-Maruyama method, Strong convergence, Existence and uniqueness of solution, Caputo fractional differential operator

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非线性随机分数阶延迟积分微分方程Euler-Maruyama方法的强收敛性

STRONG CONVERGENCE OF THE ELUER-MARUYAMA METHOD FOR STOCHASTIC FRACTIONAL DELAY INTEGRO-DIFFERENTIAL EQUATIONS

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