﻿ 分数阶导数双边空间微分方程的显式差分解法

# 分数阶导数双边空间微分方程的显式差分解法Finite Difference Methods for Space-Time Fractional Two-Sided Space Partial Differential Equations

Abstract:
Fractional order differential equations are generalizations of classical differential equations. Now, they are widely used in the fields of physics, information; finance and others. In this paper, an explicit finite difference method for space-time fractional two-sided space partial differential equations is established. Be- sides, the stability and convergence order are analyzed.

[1] R. Metzler, J. Klafter. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics, 2004, A37: 161-208.

[2] Z. Deng, V. P. Singh and L. Bengtsson. Numerical solution of fractional advection-dispersion equation. Journal of Hydraulic Engineering, 2004, 130(5): 422-431.

[3] R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto. Fractional calculus and continuous-time finance III: The diffusion limit. In: Kolhmann, S. Tang, Eds., Mathematical Finance, Basel: Birkhauser Verlag, 2001: 171-180.

[4] 苏丽娟, 王文洽. 双边分数阶对流–扩散方程的一种有限差分解法[J]. 山东大学学报(理学版), 2009, 10: 29-32.

[5] F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage. Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Applied Mathematics and Computation, 2007, 191(1): 12-21.

[6] 周激流, 蒲亦非, 廖科. 分数阶微积分原理及其在现代信号分析与处理中的应用[M]. 北京: 科学出版社, 2010.

[7] R. Hilfer. Application of fractional calculus in physics. Singapore, New Jersey, London and Hong Kong: World Scientific Publication Company, 2000.

[8] A. A. Kibas, H. M. Srivastava and J. J. Trujillo. Theory and application of fractional differential equations. Amsterdam: Elservier, 2006.

[9] M. M. Meerschaert, C. Tadjeran. Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computational & Applied Mathematics, 2004, 172(1): 65-77.

[10] M. M. Meerschaert, C. Tadjeran. Finite difference approximations for two-sided space-fractional partial differential equations. Applied Nu- merical Mathematics, 2006, 56(1): 80-90.

[11] V. K. Tuan, R. Gorenflo. Extrapolation to the limit for numerical fractional differentiation. Zeitschrift für Angewandte Mathematik und Me- chanik, 1995, 75: 646-648.

[12] M. M. Meerschaert, H. P. Scheffler and C. Tadjeran. Finite difference method for two dimensional fractional dispersion equations. Journal of Computational Physics, 2006, 211: 249-261.

[13] 夏源, 吴吉春. 分数阶对流—弥散方程的数值求解[J]. 南京大学学报(自然科学版), 2007, 43(4): 44-446.

[14] F. Liu, V. Ahn and I. Turner. Numerical solution of the space fractional Fokker-Planck equation. Journal of Computational & Applied Mathematics, 2004, 166(1): 209-219.

[15] 周璐莹, 吴吉春, 夏源. 二维分数阶对流–弥散方程的数值解[J]. 高校地质学报, 2009, 15(4): 569-575.

[16] Y. Zhang. A finite difference method for fractional partial differential equation. Applied Mathematics and Computation, 2009, 215(2): 524- 529.

[17] M. M. Meerschaert, D. A. Benson, H. P. Scheffler and B. Baeumer. Stochastic solution of space-time fractional diffusion equations. Physical Review, 2002, E65: 1103-1106.

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