• 论文 • 上一篇    

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   

  1. 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 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.

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   

  1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105 China
  • Received:2019-04-18 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.
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.

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[1] Anh V V, Mcvinish R. Fractional differential equations driven by Lévy noise[J]. J. Appl. Math. Stoch. Anal., 2003, 16(2):97-119.
[2] Jumarie G. Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions[J]. Appl. Math. Model., 2008, 32(5):836-859.
[3] Sakthivel R, Revathi P, Anthoni S M. Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations[J]. Nonlinear Anal., 2012, 75(7):3339-3347.
[4] Sakthivel R, Revathi P, Ren Y. Existence of solutions for nonlinear fractional stochastic differential equations[J]. Nonlinear Analysis:Theory, Methods & Applications, 2013, 81:70-86.
[5] Benchaabane A, Sakthivel R. Sobolev-type fractional stochastic differential equations with nonLipschitz coefficients[J]. J. Comput. Appl. Math., 2017, 312:65-73.
[6] 黄鹤皋. 几类随机问题解的存在唯一性与Milstein方法的均方稳定性[D]. 湘潭大学硕士论文, 2013.
[7] 王文强, 孙晓莉. 一类随机分数阶微分方程隐式Euler方法的弱收敛性与弱稳定性[J]. 数值计算与计算机应用, 2014, 35(2):153-162.
[8] 王文强, 孙晓莉. 线性随机分数阶微分方程Euler方法的弱收敛性与弱稳定性[J]. 计算数学, 2014, 36(2):195-204.
[9] 鄢志平. 两类随机微分方程数值方法的强收敛性分析[D]. 湘潭大学硕士论文, 2015.
[10] Kamrani M. Numerical solution of stochastic fractional differential equations[J]. Numer. Algor., 2015, 68(1):81-93.
[11] Kamrani M. Convergence of Galerkin method for the solution of stochastic fractional integro differential equations[J]. Optik, 2016, 127(20):10049-10057.
[12] Dai X, Bu W, Xiao A. Well-posedness and EM approximation for nonlinear stochastic fractional integro-differential equations with weakly singular kernels[J]. J. Comput. Appl. Math., 2019, 356:377-390.
[13] 毛文亭, 王文强, 林伟贤. 一类带乘性噪声随机分数阶微分方程Euler方法的弱收敛性与弱稳定性[J]. 计算数学, 2016, 38(4):442-452.
[14] Mao X. Stochastic differential equations and applications[M]. UK:Horwood Publishing Limited, 1997.
[15] Wu Q. A new type of the Gronwall-Bellman inequality and its application to fractional stochastic differential equations[J]. Cogent Math., 2015.
[16] Podlubny I. Fractional differential equations[M]. San Diego:Academic Press, 1999.
[17] Kuo H H. Introduction to stochastic integration[M]. Springer Science and Business Media, 2006.
[18] 孙志忠, 高广花. 分数阶微分方程的有限差分方法[M]. 北京:科学出版社, 2015.
[19] Kloeden P E, Platen E. Numerical solution of stochastic differential equations[M]. Berlin:Springer-Verlag, 1992.
[20] 毛文亭, 张维, 王文强. 一类带乘性噪声随机分数阶微分方程数值方法的弱收敛性与弱稳定性[J]. 数值计算与计算机应用, 2018, 39(3):161-171.
[1] 毛文亭, 王文强, 林伟贤. 一类带乘性噪声随机分数阶微分方程Euler方法的弱收敛性与弱稳定性[J]. 计算数学, 2016, 38(4): 442-452.
[2] 王琦, 汪小明. 单物种人口模型指数隐式Euler方法的振动性[J]. 计算数学, 2015, 37(1): 57-66.
[3] 王文强, 孙晓莉. 线性随机分数阶微分方程Euler方法的弱收敛性与弱稳定性[J]. 计算数学, 2014, 36(2): 195-204.
[4] 范振成, 宋明辉. 非全局Lipschitz条件下随机延迟微分方程Euler方法的收敛性[J]. 计算数学, 2011, 33(4): 337-344.
[5] 王文强, 陈艳萍. 线性中立型随机延迟微分方程Euler方法的均方稳定性[J]. 计算数学, 2010, 32(2): 206-212.
[6] 张浩敏, 甘四清, 胡琳. 随机比例方程带线性插值的半隐式Euler方法的均方收敛性[J]. 计算数学, 2009, 31(4): 379-392.
[7] 范振成. 随机延迟微分方程的全隐式Euler方法[J]. 计算数学, 2009, 31(3): 287-298.
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