带Caputo导数的变分数阶随机微分方程的Euler-Maruyama方法

刘家惠; 邵林馨; 黄健飞

doi:10.21656/1000-0887.430250

带Caputo导数的变分数阶随机微分方程的Euler-Maruyama方法

doi: 10.21656/1000-0887.430250

扬州大学数学科学学院，江苏扬州 225002

基金项目:

江苏省自然科学基金项目 BK20201427

国家自然科学基金项目 11701502

国家自然科学基金项目 11871065

详细信息

作者简介:
刘家惠(1998—), 女, 硕士生(E-mail: 965440574@qq.com)

通讯作者:
黄健飞(1983—), 男, 副教授, 博士(通讯作者. E-mail: jfhuang@lsec.cc.ac.cn)

中图分类号: O211.5;O241.8
计量
- 文章访问数: 385
- HTML全文浏览量: 110
- PDF下载量: 59
- 被引次数: 0
出版历程
- 收稿日期: 2022-08-04
- 修回日期: 2022-11-29
- 刊出日期: 2023-06-01

An Euler-Maruyama Method for Variable Fractional Stochastic Differential Equations With Caputo Derivatives

School of Mathematical Sciences, Yangzhou University, Yangzhou, Jiangsu 225002, P.R.China

摘要

摘要: 该文构造了Euler-Maruyama(EM)方法求解一类带Caputo导数的变分数阶随机微分方程. 首先, 证明了该方程的适定性; 然后, 详细推导出对应的EM方法, 并对该方法进行了强收敛性的分析, 通过使用EM方法的连续形式证明了其强收敛阶为β-0.5, 其中β是Caputo导数的阶数，且满足0.5 < β < 1. 最后, 通过数值实验验证了理论分析结果的正确性.
- 变分数阶随机微分方程 /
- Caputo导数 /
- Euler-Maruyama方法 /
- 强收敛性
Abstract: A Euler-Maruyama (EM) method was constructed to solve a class of variable fractional stochastic differential equations with Caputo derivatives. Firstly, the well-posedness of the equation was proved. Then, the corresponding EM method was derived in detail, and the strong convergence of the method was analyzed. By means of the continuous form of the EM method, its strong convergence order was proved to be β-0.5, where β is the order of the Caputo derivative and 0.5 < β < 1. Numerical experiments verify the correctness of the theoretical results.
- variable fractional stochastic differential equation /
- Caputo derivative /
- Euler-Maruyama method /
- strong convergence

HTML全文

表 1 β=0.9时，EM方法的误差与收敛阶

Table 1. Errors and convergence orders of the EM method for β=0.9

h	α₁=0.2, α₂=0.6		α₁=0.6, α₂=0.2
h	error e_h	convergence order n_co	error e_h	convergence order n_co
1/32	0.180 197	-	0.180 329	-
1/64	0.135 958	0.406	0.135 998	0.407
1/128	0.101 704	0.419	0.101 716	0.419
1/256	0.076 984	0.402	0.076 986	0.402

下载: 导出CSV

表 2 β=0.8时，EM方法的误差与收敛阶

Table 2. Errors and convergence orders of the EM method for β=0.8

h	α₁=0.2, α₂=0.5		α₁=0.5, α₂=0.2
h	error e_h	convergence order n_co	error e_h	convergence order n_co
1/32	0.251 476	-	0.251 654	-
1/64	0.202 993	0.310	0.203 053	0.310
1/128	0.162 630	0.320	0.162 650	0.320
1/256	0.131 342	0.308	0.131 333	0.309

下载: 导出CSV

表 3 β=0.7时，EM方法的误差与收敛阶

Table 3. Errors and convergence orders of the EM method for β=0.7

h	α₁=0.1, α₂=0.5		α₁=0.5, α₂=0.1
h	error e_h	convergence order n_co	error e_h	convergence order n_co
1/32	0.343 325	-	0.343 776	-
1/64	0.296 312	0.212	0.296 487	0.214
1/128	0.254 176	0.221	0.254 244	0.222
1/256	0.218 904	0.216	0.223 625	0.185

下载: 导出CSV

参考文献(30)

[1]	CHEN W, SUN H G, ZHANG X D, et al. Anomalous diffusion modeling by fractal and fractional derivatives[J]. Computers and Mathematics With Applications, 2010, 59(5): 1754-1758. doi: 10.1016/j.camwa.2009.08.020
[2]	SUN H G, CHEN W, CHEN Y Q. Variable-order fractional differential operators in anomalous diffusion modeling[J]. Physica A: Statistical Mechanics and Its Applications, 2009, 388(21): 4586-4592. doi: 10.1016/j.physa.2009.07.024
[3]	KILBAS A A, SRIVASTAVA H M, TRUJILLO J J. Theory and Applications of Fractional Differential Equations[M]. Amsterdam: Elsevier Science, 2006.
[4]	ROSSIKHIN Y A, SHITIKOVA M V. Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems[J]. Acta Mechanica, 1997, 120(1): 109-125.
[5]	AHMED E, ELGAZZAR A S. On fractional order differential equations model for nonlocal epidemics[J]. Physica A: Statistical Mechanics and Its Applications, 2007, 379(2): 607-614. doi: 10.1016/j.physa.2007.01.010
[6]	TAJADODI H, KHAN Z A, IRSHAD A, et al. Exact solutions of conformable fractional differential equations[J]. Results in Physics, 2021, 22(1): 103916.
[7]	KHODABIN M, MALEKNEJAD K, ASGARI M. Numerical solution of a stochastic population growth model in a closed system[J]. Advances in Difference Equations, 2013, 2013(1): 130. doi: 10.1186/1687-1847-2013-130
[8]	SHAH A, KHAN R A, KHAN A, et al. Investigation of a system of nonlinear fractional order hybrid differential equations under usual boundary conditions for existence of solution[J]. Mathematical Methods in the Applied Sciences, 2021, 44(2): 1628-1638. doi: 10.1002/mma.6865
[9]	朱帅润, 李绍红, 钟彩尹, 等. 时间分数阶的非饱和渗流数值分析及其应用[J]. 应用数学和力学, 2022, 43(9): 966-975. doi: 10.21656/1000-0887.420334 ZHU Shuairun, LI Shaohong, ZHONG Caiyin, et al. Numerical analysis of time fractional-order unsaturated flow and its application[J]. Applied Mathematics and Mechanics, 2022, 43(9): 966-975. (in Chinese) doi: 10.21656/1000-0887.420334
[10]	ALSHEHRI M H, DURAIHEM F Z, ALALYANI A, et al. A Caputo (discretization) fractional-order model of glucose-insulin interaction: numerical solution and comparisons with experimental data[J]. Journal of Taibah University for Science, 2021, 15(1): 26-36. doi: 10.1080/16583655.2021.1872197
[11]	LIU F W, ANH V, TURNER I. Numerical solution of the space fractional Fokker-Planck equation[J]. Journal of Computational and Applied Mathematics, 2004, 166(1): 209-219. doi: 10.1016/j.cam.2003.09.028
[12]	高兴华, 李宏, 刘洋. 非线性分数阶常微分方程的分段线性插值多项式方法[J]. 应用数学和力学, 2021, 42(5): 531-540. doi: 10.21656/1000-0887.410149 GAO Xinghua, LI Hong, LIU Yang. A piecewise linear interpolation polynomial method for nonlinear fractional ordinary differential equations[J]. Applied Mathematics and Mechanics, 2021, 42(5): 531-540. (in Chinese) doi: 10.21656/1000-0887.410149
[13]	GARRAPPA R. Numerical solution of fractional differential equations: a survey and a software tutorial[J]. Mathematics, 2018, 6(2): 16. doi: 10.3390/math6020016
[14]	JING Y Y, LI Z, XU L P. The averaging principle for stochastic fractional partial differential equations with fractional noises[J]. Journal of Partial Differential Equations, 2021, 34: 51-66. doi: 10.4208/jpde.v34.n1.4
[15]	GUO Z K, FU H B, WANG W Y. An averaging principle for Caputo fractional stochastic differential equations with compensated Poisson random measure[J]. Journal of Partial Differential Equations, 2021, 35: 1-10.
[16]	HIGHAM D J. Stochastic ordinary differential equations in applied and computational mathematics[J]. IMA Journal of Applied Mathematics, 2011, 76(3): 449-474. doi: 10.1093/imamat/hxr016
[17]	MAO X R. The truncated Euler-Maruyama method for stochastic differential equations[J]. Journal of Computational and Applied Mathematics, 2015, 290: 370-384. doi: 10.1016/j.cam.2015.06.002
[18]	钱思颖, 张静娜, 黄健飞. 带有弱奇性核的多项分数阶非线性随机微分方程的改进Euler-Maruyama格式[J]. 应用数学和力学, 2021, 42(11): 1203-1212. doi: 10.21656/1000-0887.420067 QIAN Siying, ZHANG Jingna, HUANG Jianfei. A modified Euler-Maruyama scheme for multi-term fractional nonlinear stochastic differential equations with weakly singular kernels[J]. Applied Mathematics and Mechanics, 2021, 42(11): 1203-1212. (in Chinese) doi: 10.21656/1000-0887.420067
[19]	WANG H T, ZHENG X C. Wellposedness and regularity of the variable-order time-fractional diffusion equations[J]. Journal of Mathematical Analysis and Applications, 2019, 475(2): 1778-1802. doi: 10.1016/j.jmaa.2019.03.052
[20]	YANG Z W, ZHENG X C, ZHANG Z Q, et al. Strong convergence of Euler-Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noise[J]. Chaos, Solitons and Fractals, 2021, 142: 110392. doi: 10.1016/j.chaos.2020.110392
[21]	薛益民, 戴振祥, 刘洁. 一类Riemann-Liouville型分数阶微分方程正解的存在性[J]. 华南师范大学学报(自然科学版), 2019, 51(2): 105-109. https://www.cnki.com.cn/Article/CJFDTOTAL-HNSF201902017.htm XUE Yimin, DAI Zhenxiang, LIU Jie. On the existence of positive solutions to a type of Riemann-Liouville fractional differential equations[J]. Journal of South China Normal University (Natural Science Edition), 2019, 51(2): 105-109. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-HNSF201902017.htm
[22]	TUAN N H, MOHAMMADI H, REZAPOUR S. A mathematical model for COVID-19 transmission by using the Caputo fractional derivative[J]. Chaos, Solitons and Fractals, 2020, 140: 110107. doi: 10.1016/j.chaos.2020.110107
[23]	张敬凯, 徐家发, 柏仕坤. 一类Caputo型分数阶微分方程边值问题多正解的存在性[J]. 重庆师范大学学报(自然科学版), 2022, 39(4): 87-91. https://www.cnki.com.cn/Article/CJFDTOTAL-CQSF202204012.htm ZHANG Jingkai, XU Jiafa, BAI Shikun. Existence of multiple positive solutions for a class of Caputo type fractional differential equations boundary value problems[J]. Journal of Chongqing Normal University (Natural Science), 2022, 39(4): 87-91. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-CQSF202204012.htm
[24]	DELAVARI H, BALEANU D, SADATI J. Stability analysis of Caputo fractional-order nonlinear systems revisited[J]. Nonlinear Dynamics, 2012, 67(4): 2433-2439.
[25]	SON D T, HUONG P T, KLOOEDEN P E, et al. Asymptotic separation between solutions of Caputo fractional stochastic differential equations[J]. Stochastic Analysis and Applications, 2018, 36(4): 654-664.
[26]	SONJA C, MARTIN H, ARNULF J. Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions[J]. IMA Journal of Numerical Analysis, 2021, 41(1): 493-548.
[27]	CONT R, FOURNIE D. A functional extension of the Ito formula[J]. Comptes Rendus Mathematique, 2010, 348(1): 57-61.
[28]	CONG N, SON D, TUAN H. On fractional Lyapunov exponent for solutions of linear fractional differential equations[J]. Fractional Calculus and Applied Analysis, 2014, 17(2): 285-306.
[29]	SAMKO S G, KILBAS A A, MARICHEV O I. Fractional Integrals and Derivatives: Theory and Applications[M]. New York: Gordon and Breach, 1993.
[30]	HUANG J F, TANG Y F, VAZQUEZ L. Convergence analysis of a block-by-block method for fractional differential equations[J]. Numerical Mathematics: Theory, Methods and Applications, 2012, 5: 229-241.