一类分数阶导数微分方程的隐式差分解法
An Implicit Finite Difference Scheme for Space-Time Fractional Partial Differential Equation
DOI: 10.12677/ORF.2013.32002, PDF, HTML, 下载: 3,374  浏览: 11,123  国家自然科学基金支持
作者: 张 阳*：南开大学数学科学学院，天津；王瑞怡：河北工业大学理学院，天津
关键词: 分数阶导数；隐格式；稳定性；收敛性；误差估计；Fractional Derivative; Implicit Methods; Stability; Convergence; Error Estimate
摘要: 分数阶导数微分方程作为通常微分方程的推广，被广泛地应用于工程，物理，信息处理，金融，水文等领域。本文给出了数值求解一类时间空间分数阶导数的双边空间微分方程的一种隐式差分格式，并对其稳定性和收敛性进行了理论分析，证明了格式的无条件稳定性并给出了收敛阶估计。
Abstract: Fractional order differential equations are generalizations of classical differential equations. They are widely used in the fields of diffusive transport, finance, nonlinear dynamics, signal processing and others. In this paper, an implicit finite difference method for a class of initial-boundary value space-time fractional two-sided space partial differential equations with variable coefficients on a finite domain is established. The stability and convergence order are analyzed for the resulted implicit scheme. With mathematical induction skills, the scheme is proved to be unconditionally stable and convergent.
文章引用：张阳, 王瑞怡. 一类分数阶导数微分方程的隐式差分解法[J]. 运筹与模糊学, 2013, 3(2): 7-14. http://dx.doi.org/10.12677/ORF.2013.32002

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] Y. Zhang. A finite difference method for fractional partial differential equation. Applied Mathematics and Computation, 2009, 215(2): 524- 529.
[16] 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.
[17] 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, 1-7.
[18] 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.

为你推荐