## 留言板

 引用本文: 张嫚, 曹艳华, 杨晓忠. 一类分数阶Langevin方程block-by-block算法的数值分析[J]. 应用数学和力学, 2021, 42(6): 562-574.
ZHANG Man, CAO Yanhua, YANG Xiaozhong. Numerical Analysis of a Class of Fractional Langevin Equations With the Block-by-Block Method[J]. Applied Mathematics and Mechanics, 2021, 42(6): 562-574. doi: 10.21656/1000-0887.410337
 Citation: ZHANG Man, CAO Yanhua, YANG Xiaozhong. Numerical Analysis of a Class of Fractional Langevin Equations With the Block-by-Block Method[J]. Applied Mathematics and Mechanics, 2021, 42(6): 562-574.

## 一类分数阶Langevin方程block-by-block算法的数值分析

##### doi: 10.21656/1000-0887.410337

###### 通讯作者: 杨晓忠(1965—)，男，教授，博士生导师(通讯作者. E-mail: yxiaozh@ncepu.edu.cn).
• 中图分类号: O211.63

## Numerical Analysis of a Class of Fractional Langevin Equations With the Block-by-Block Method

• 摘要: 分数阶Langevin方程有重要的科学意义和工程应用价值，基于经典block-by-block算法，求解了一类含有Caputo导数的分数阶Langevin方程的数值解.Block-by-block算法通过引入二次Lagrange基函数插值，构造出逐块收敛的非线性方程组，通过在每一块耦合求得分数阶Langevin方程的数值解.在0<α<1条件下，应用随机Taylor展开证明block-by-block算法是3+α阶收敛的，数值试验表明在不同α和时间步长h取值下，block-by-block算法具有稳定性和收敛性，克服了现有方法求解分数阶Langevin方程速度慢精度低的缺点，表明block-by-block算法求解分数阶Langevin方程是高效的.
•  包景东. 反常动力学导论[M]. 北京: 科学出版社, 2012.(BAO Jingdong. Introduction to Anomalous Statistical Dynamics[M]. Beijing: Science Press, 2012.(in Chinese)) [2]COFFCY W T, KALMYKOV Y P, WALDRON J T. The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering[M]. Beijing: World Scientific Press, 2004. [3]EAB C H, LIM S C. Fractional generalized Langevin equation approach to single-file diffusion[J]. Physica A: Statal Mechanics and Its Applications,2010,389: 2510-2521. [4]KOSINSKI R A, GRABOWSKI A. Langevin equations for modeling evacuation processes[J]. Acta Physica Polonica B: Proceedings,2010,3(2): 365-376. [5]ROSSIKHIN Y A, SHITIKOVA M V. Analysis of the viscoelastic rod dynamics via models involving fractional derivatives or operators of two different orders[J]. Shock and Vibration Digest,2004,36(1): 3-26. [6]GUO B L, PU X K, HUANG F H. Fractional Partial Differential Equations and Their Numerical Solutions[M]. Beijing: Science Press, 2015. [7]黄凤辉, 郭柏灵. 一类时间分数阶偏微分方程的解[J]. 应用数学和力学, 2010,31(7): 781-790.(HUANG Fenghui, GUO Boling. General solution for a class of time fractional partial differential equation[J]. Applied Mathematics and Mechanics,2010,31(7): 781-790.(in Chinese)) [8]UCHAIKIN V V. Fractional Derivatives for Physicists and Engineers, Vol Ⅱ: Applications[M]. Beijing: Higher Education Press, 2013. [9]SABATIER J, AGRAWAL O P, TENREIRO M J A. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering[M]. Beijing: World Publishing Corporation, 2014. [10]BHRAWY A H, ALGHAMDI M A. A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals[J]. Boundary Value Problems,2012,2012(1): 62. DOI: 10.1186/1687-2770-2012-62. [11]GUO P, LI C P, ZENG F H. Numerical simulation of the fractional Langevin equation[J]. Thermal Science,2012,16(2): 357-363. [12]孙春艳, 徐伟. 随机分数阶微分方程初值问题基于模拟方程法的数值求解[J]. 应用数学和力学, 2014,35(10): 1092-1099.(SUN Chunyan, XU Wei. An analog equation method-based numerical scheme for initial value problem of stochastic fractional differential equations[J]. Applied Mathematics and Mechanics,2014,35(10): 1092-1099.(in Chinese)) [13]KATANI R, SHAHMORD S. Block by block method for the systems of nonlinear Volterra integral equations[J]. Applied Mathematical Modelling,2010, 34(2): 400-406. [14]HUANG J F, TANG Y F, VZQUEZ L. Convergence analysis of a block-by-block method for fractional differential equations[J]. Numerical Mathematics-Theory Methods and Applications,2012, 5(2): 229-241. [15]CAO J Y, XU C J. A high order schema for the numerical solution of the fractional ordinary differential equations[J]. Journal of Computational Physics,2013,238(1): 154-168. [16]ESMAEILI S. A piecewise nonpolynomial collocation method for fractional differential equations[J]. Journal of Computational and Nonlinear Dynamics,2017, 12(5): 051020. [17]包景东. 经典和量子耗散系统的随机模拟方法[M]. 北京: 科学出版社, 2009.(BAO Jingdong. Stochastic Simulation Methods for Classical and Quantum Dissipative Systems[M]. Beijing: Science Press, 2009.(in Chinese)) [18]蒋锋. 随机系统数值方法的动力学分析及应用[M]. 北京: 科学出版社, 2016.(JIANG Feng. Dynamic Analysis and Application for Numerical Methods of Stochastic Systems[M]. Beijing: Science Press, 2016.(in Chinese)) [19]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.(SUN Zhizhong, GAO Guanghua. Finite Difference Methods for Fractional Differential Equations[M]. Beijing: Science Press, 2015.(in Chinese)) [20]DIETHELM K, FORD N J. Analysis of fractional differential equations[J]. Journal of Mathematical Analysis and Applications,2002,265: 229-248. [21]KUMAR P, AGRAWAL O P. An approximate method for numerical solution of fractional differential equations[J]. Signal Processing,2006,86(10): 2602-2610. [22]Lü T, HUANG Y. A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind[J]. Journal of Mathmatical Analysis and Applications,2003,282(10): 56-62. [23]SERGIO P, WEST B J. Fractional Langevin model of memory in financial markets[J]. Physical Review E,2002,66(2): 046118. [24]MAINARDI F, GORENFLO R. On Mittag-Leffler-type functions in fractional evolution processes[J]. Journal of Computational and Applied Mathematics,2000,118(1/2): 283-299. [25]蒲林娟, 杨晓忠, 孙淑珍. 一类分数阶Langevin方程预估校正算法的数值分析[J]. 数学物理学报, 2020,40A(4): 1018-1028.(PU Linjuan, YANG Xiaozhong, SUN Shuzhen. Numerical analysis of a class of fractional Langevin equation by predictor-corrector method[J]. Acta Mathematica Scientia,2020,40A(4): 1018-1028.(in Chinese))
##### 计量
• 文章访问数:  764
• HTML全文浏览量:  159
• PDF下载量:  81
• 被引次数: 0
##### 出版历程
• 收稿日期:  2020-10-29
• 修回日期:  2020-11-21

/

• 分享
• 用微信扫码二维码

分享至好友和朋友圈