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求解Riesz空间分数阶扩散方程的一种新的数值方法

杨晋平1, 李志强1, 闫玉斌2   

  1. 1. 吕梁学院 数学系, 吕梁 033001;
    2. 切斯特大学 数学系, 英国 CH1 4BJ
  • 收稿日期:2017-07-28 出版日期:2019-06-15 发布日期:2019-05-18
  • 基金资助:

    山西省自然科学基金(201801D121010)和吕梁学院校内基金(ZRXN201511)资助项目.

杨晋平, 李志强, 闫玉斌. 求解Riesz空间分数阶扩散方程的一种新的数值方法[J]. 计算数学, 2019, 41(2): 170-190.

Yang Jinping, Li Zhiqiang, Yan Yubin. A NEW NUMERICAL METHOD FOR SOLVING RIESZ SPACE-FRACTIONAL DIFFUSION EQUATION[J]. Mathematica Numerica Sinica, 2019, 41(2): 170-190.

A NEW NUMERICAL METHOD FOR SOLVING RIESZ SPACE-FRACTIONAL DIFFUSION EQUATION

Yang Jinping1, Li Zhiqiang1, Yan Yubin2   

  1. 1. Department of Mathematics, Luliang University, Lvliang 033001, China;
    2. Department of Mathematics, University of Chester, Chester CH1 4BJ, UK
  • Received:2017-07-28 Online:2019-06-15 Published:2019-05-18
本文利用Diethelm方法构造了一种逼近Riesz空间分数阶导数的O(△x3-α)格式,其中1 < α < 2,△x是空间步长.进一步对一阶时间导数采用Crank-Nicolson方法离散,得到了求解Riesz空间分数阶扩散方程的一种新的有限差分格式,并用矩阵方法证明了稳定性和收敛性,其误差估计为O(△t2+△x3-α),其中△t为时间步长.最后,数值算例验证了差分格式的正确性和有效性.
By using Diethelm's method, we construct an approximate scheme to the Riesz space fractional derivative with order O(△x3-α), where 1 < α < 2 and △x denotes the space step size. Further we discretize the time derivative with the Crank-Nicolson method and obtain a new finite difference method for solving Riesz space fractional diffusion equation. The stability and convergence are proved by the matrix method and the error estimate in the maximum norm is O(△t2 + △x3-α), where △t denotes the time step size. Finally, some numerical examples are given to illustrate their correctness and efficiency.
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[1] Chaves A S. A fractional diffusion equation to describe Lévy flights[J]. Phys. A, 1998, 239:13-16.

[2] Gorenflo R, Mainardi F. Feller fractional diffusion and Lévy stable motions[C]. Conference on Lévy Processes:Theory and Applications, 18-22 January 1999.

[3] Hanneken J W, Narahari Achar B N, Vaught D M, et al. A random walk simulation of fractional diffusion[J]. J. Mole. Liqu., 2004, 114(1-3):153-157.

[4] Klafter J, Shlesinger M F, Zumofen G. Beyond Brownian motion[J]. Phys. Today, 1996, 49:33-39.

[5] Molz F J, Fix G J, Lu S. A physics interpretation for fractional derivative in the Lévy diffusion[J]. Appl. Math. Lett., 2002,15(7):907-911.

[6] EI-Nabulsi R A. Fractional description of super and subdiffusion[J]. Phys. A, 2005, 340(5-6):361-368.

[7] Bagley R L, Calico R A. Fractional order state equations for the control of viscoelastic structures[J]. J. Guid. Contr. Dynam., 1991, 14:304-311.

[8] Koeller R C. Application of fractional calculus to the theory of viscoelasticity[J]. J. Appl. Mech., 1984, 51(2):299-307.

[9] Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena[J]. Chaos. Soliton. Fract., 1996, 7(9):1461-1477.

[10] Podlubny I. Fractional Differential Equations[M]. Academic Press, 1998.

[11] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Application of Fractional Differential Equations[M]. Elsevier, Amsterdam, 2006.

[12] 郭柏灵, 蒲学科, 黄凤辉. 分数阶偏微分方程及其数值解[M]. 北京:科学出版社, 2011.

[13] 刘发旺, 庄平辉, 刘青霞. 分数阶偏微分方程数值方法及其应用[M]. 北京:科学出版社, 2015.

[14] 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.

[15] Li C P, Zeng F H. Numerical Methods for Fractional Calculus[M]. CRC Press, 2015.

[16] 林世敏, 许传炬. 分数阶微分方程的理论和数值方法研究[J]. 计算数学, 2016, 38:1-24.

[17] Meerschaert M M, Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations[J]. J. Comput. Appl. Math., 2004, 172(1):65-77.

[18] Meerschaert M M, Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations[J]. Appl. Numer. Math., 2006, 56(1):80-90.

[19] Ford N J, Kamal P, Yan Y. An algorithm for the numerical solution of two-sided space fractional partial differential equations[J]. Comput. Methods Appl. Math., 2015, 15:497-514.

[20] Celik C, Duman M. Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative[J]. J. Comput. Phys., 2012, 231(4):1743-1750.

[21] Liu F, Zhuang P, Burrage K. Numerical methods and analysis for a class of fractional advectiondispersion models[J]. Comput. Math. Appl., 2012, 64(10):2990-3007.

[22] Zhang H M, Liu F. Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term[J]. J. Appl. Math. Inform., 2008, 26(1-2):1-14.

[23] Yang Q, Liu F, Turner I. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives[J]. Appl. Math. Model., 2010, 34(1):200-218.

[24] Shen S, Liu F, Anh V, et al. A novel numerical approximation for the space fractional advectiondispersion equation[J]. IMA J. Appl. Math., 2014, 79(3):431-444.

[25] Ding H F, Li C P, Chen Y Q. High-order algorithms for Riesz derivative and their applications(I)[J]. Abst. Appl. Anal., 2014, Article ID 653797, 17 pages.

[26] Ding H F, Li C P, Chen Y Q. High-order algorithms for Riesz derivative and their applications(Ⅱ)[J]. J. Comput. phys., 2015, 293:218-237.

[27] Ding H F, Li C P. High-order algorithms for Riesz derivative and their applications(Ⅲ)[J]. Fract. Calc. Appl. Anal., 2016, 19:19-55.

[28] Zhao X, Sun Z Z, Hao Z P. A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation[J]. SIAM J. Sci. Comput., 2014, 36:A2865-A2866.

[29] Tian W Y, Zhou H, Deng W H. A class of second order difference approximation for solving space fractional diffusion equations[J]. Math. Comput., 2015, 84:1703-1727.

[30] Zhou H, Tian W Y, Deng W H. Quasi-compact finite difference schemes for space fractional diffusion equations[J]. J. Sci. Comput., 2013, 56:45-66.

[31] Chen M H, Deng W H. WSLD operators:A class of fourth order difference approximations for space Riemann-Liouville derivative[J]. Math., 2013, 16(2):516-540.

[32] Hao Z P, Sun Z Z, Cao W R. A fourth-order approximation of fractional derivatives with its applications[J]. J. Comput. Phys., 2015, 281:787-805.

[33] Diethelm K. An algorithm for the numerical solution of differential equations of fractional order[J]. Elect. Trans. Numer. Anal., 1997, 5:1-6.

[34] Yan Y, Pal K, Ford N J. Higher order numerical methods for solving fractional differential equations[J]. BIT Numer. Math., 2014, 54:555-584.

[35] Li Z Q, Yan Y, Ford N J. Error estimates of a high order numerical method for solving linear fractional differential equation[J]. Appl. Numer. Math., 2017, 114:201-220.

[36] Li Z Q, Liang Z Q, Yan Y. High-order numerical methods for solving time fractional partial differential equations[J]. J. Sci. Comput., 2017, 71:785-803.

[37] Diethelm K. The Analysis of Fractional Differential Equations[M]. New York:Springer, 2004.

[38] Diethelm K. Generalized compound quadrature formulae finite-part integral[J]. IMA J. Numer. Anal., 1997, 17:479-493.

[39] 张文生. 科学计算中的偏微分方程有限差分法[M]. 北京:高等教育出版社, 2006.

[40] Ding H F, Li C P. High-order numerical algorithms for Riesz derivatives via construction new generating functions[J]. J. Sci. Comput., 2017, 71:759-784.
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