运筹与模糊学  >> Vol. 2 No. 1 (February 2012)

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

DOI: 10.12677/orf.2012.21001, PDF, HTML, 下载: 3,020  浏览: 10,265  国家自然科学基金支持

作者: 张阳*, 李宁平, 陈璐:南开大学数学科学学院

关键词: 分数阶导数显格式稳定性收敛性误差估计
Fractional Derivative; Explicit Methods; Stability; Convergence; Error Estimates

摘要: 分数阶微分方程作为广义的微分方程,被广泛地应用于物理,信息处理,金融等领域。本文给出了数值求解时间空间分数阶导数的双边空间微分方程的一种显式差分格式,并对其稳定性和收敛性进行了理论分析。
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.

文章引用: 张阳, 李宁平, 陈璐. 分数阶导数双边空间微分方程的显式差分解法[J]. 运筹与模糊学, 2012, 2(1): 1-7. http://dx.doi.org/10.12677/orf.2012.21001

参考文献

[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.