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

LIU Jiahui; SHAO Linxin; HUANG Jianfei

doi:10.21656/1000-0887.430250

Volume 44 Issue 6

Jun. 2023

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2023 > 44(6): 731-743

LIU Jiahui, SHAO Linxin, HUANG Jianfei. An Euler-Maruyama Method for Variable Fractional Stochastic Differential Equations With Caputo Derivatives[J]. Applied Mathematics and Mechanics, 2023, 44(6): 731-743. doi: 10.21656/1000-0887.430250

Citation:

PDF( 460 KB)

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

doi: 10.21656/1000-0887.430250

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

Received Date: 2022-08-04
Rev Recd Date: 2022-11-29
Publish Date: 2023-06-01

Abstract

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

FullText(HTML)

References(30)

References

[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.