非线性随机分数阶微分方程Euler方法的弱收敛性

doi:10.12286/jssx.j2019-0587

计算数学 ›› 2021, Vol. 43 ›› Issue (1): 87-109.DOI: 10.12286/jssx.j2019-0587

非线性随机分数阶微分方程Euler方法的弱收敛性

朱梦姣, 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:2019-04-18 出版日期:2021-02-15 发布日期:2021-02-04
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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朱梦姣, 王文强. 非线性随机分数阶微分方程Euler方法的弱收敛性[J]. 计算数学, 2021, 43(1): 87-109.

Zhu Mengjiao, Wang Wenqiang. THE WEAK CONVERGENCE OF EULER METHOD FOR NONLINEAR STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2021, 43(1): 87-109.

THE WEAK CONVERGENCE OF EULER METHOD FOR NONLINEAR STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS

Zhu Mengjiao, Wang Wenqiang

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

Received:2019-04-18 Online:2021-02-15 Published:2021-02-04

论文首先证明了非线性随机分数阶微分方程解的存在唯一性, 然后构造了数值求解该方程的Euler 方法, 并证明了当方程满足一定约束条件时, 该方法是弱收敛的. 特别地, 当分数阶α=0时, 该方程退化为非线性随机微分方程, 所获结论与现有文献中的相关结论是一致的; 当α ≠ 0, 且初值条件为齐次时, 所获结论可视为现有文献中线性随机分数阶微分方程情形的推广和改进. 随后, 文末的数值试验验证了所获理论结果的正确性.

关键词： 随机分数阶微分方程, 解的存在唯一性, Euler方法, 弱收敛性, Caputo导数

This paper is concerned with the existence and uniqueness of solutions for nonlinear stochastic fractional differential equations and the weak convergence of Euler method constructed for solving the equations when they satisfy certain constraints. Especially, when fractional order α = 0, the equations are degenerated to nonlinear stochastic differential equations, and the conclusions obtained from this paper are consisted with the relevant results; when α ≠ 0 and the initial condition is homogeneous, the conclusions can be regarded as the generalization and improvement of linear stochastic fractional differential equations in the existing literature. Finally, numerical examples illustrate the effectiveness of the theoretical results.

Key words: Nonlinear stochastic fractional differential equations, Existence and uniqueness of solutions, Euler method, Weak convergence, Caputo derivative

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摘要

非线性随机分数阶微分方程Euler方法的弱收敛性

THE WEAK CONVERGENCE OF EULER METHOD FOR NONLINEAR STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS

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