Volume 42 Issue 6
Jun.  2021
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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. doi: 10.21656/1000-0887.410337

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

doi: 10.21656/1000-0887.410337
  • Received Date: 2020-10-29
  • Rev Recd Date: 2020-11-21
  • The fractional Langevin equation is of great scientific significance and engineering application value. Based on the classical block-by-block method, the numerical solution of a class of fractional Langevin equations with Caputo derivatives was obtained. Through introduction of the quadratic Lagrange basis function interpolation, the block-by-block convergent nonlinear equations were constructed, and the numerical solution of the Langevin equation was obtained by coupling in each block. Under the condition of 0<α<1, the stochastic Taylor expansion was used to prove that the block-by-block method is (3+α)-order convergent. Numerical experiments show that, the block-by-block method is stable and convergent under different values of α and time step h,and overcomes the existing methods’ disadvantages of slow speed and poor accuracy for solving fractional Langevin equations.
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  • 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))
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