广东工业大学学报 ›› 2019, Vol. 36 ›› Issue (03): 74-79.doi: 10.12052/gdutxb.180122

• 综合研究 • 上一篇    下一篇

带周期边界的时间分数阶扩散方程的差分格式

张会琴, 汪志波   

  1. 广东工业大学 应用数学学院, 广东 广州 510520
  • 收稿日期:2018-09-20 出版日期:2019-05-09 发布日期:2019-04-04
  • 通信作者: 汪志波(1987-),男,副教授,主要研究方向为微分方程数值解.E-mail:wzbmath@gdut.edu.cn E-mail:wzbmath@gdut.edu.cn
  • 作者简介:张会琴(1992-),女,硕士研究生,主要研究方向为微分方程数值解.
  • 基金资助:
    国家自然科学基金资助项目（11701103）；广东省自然科学基金资助项目（2017A030310538）；广东省青年拔尖人才项目（2017GC010379）；广东省高校特色创新项目（2017KTSCX062）；广东工业大学基金资助项目（220413131，220413550）

Finite Difference Schemes for Time Fractional Diffusion Equations with Periodic Boundary Conditions

Zhang Hui-qin, Wang Zhi-bo   

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China
  • Received:2018-09-20 Online:2019-05-09 Published:2019-04-04

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摘要： 鉴于分数阶方程的解析解实难求得，本文主要研究了带周期边界的时间分数阶扩散方程的有限差分方法，时间方向采用L2-1σ离散公式，空间方向采用二阶差分格式离散，数值格式整体可达到二阶精度.随后利用Fourier方法证明了有限差分格式的唯一可解性、稳定性和收敛性.最后用MATLAB语言对具体的模型进行了数值求解，数值实验能很好地印证理论结果.

关键词: 分数阶扩散方程, Fourier方法, 变系数, 稳定性, 收敛性

Abstract: Generally, the analytic solution for fractional differential equations is hard to obtain. The difference scheme of time fractional diffusion equations with periodic boundaries is mainly studied, adopting the L2-1σ formula in temporal direction, and the second order difference approximation in spatial direction. The proposed scheme can obtain second order accuracy globally. The unique solvability, stability and convergence of the proposed scheme are proved by the Fourier method. Finally, the specific fractional models are solved numerically by MATLAB language. Numerical experiments are carried out to support the theoretical results.

Key words: fractional diffusion equation, Fourier method, variable coefficients, stability, convergence

中图分类号: 

  • 35R11
[1] NIGMATULLIN R R. To the theoretical explanation of the "universal response"[J]. Physica Status Solidi, 1984, 123(2):739-745
[2] GORENFLO R, MAINARDI F, SCALAS E, et al. Fractional calculus and continuous-time finance Ⅲ:the diffusion limit[J]. Physica A Statistical Mechanics & Its Applications, 2000, 284(1-4):376-384
[3] 梅立泉, 李瑞, 李智. 三元期权定价问题的偏微分方程数值解[J]. 西安交通大学学报, 2006, 40(4):484-487 MEI L Q, LI R, LI Z. Numerical solutions of partial differential equations for trivariate option pricing[J]. Journal of Xi'an Jiaotong University, 2006, 40(4):484-487
[4] ZHANG H, LIU F, PHANIKUMAR M S, et al. A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model[J]. Computers & Mathematics with Applications, 2013, 66(5):693-701
[5] CAIN G, MEYER G H. Separation of variables for partial differential equations[M]. New York:Chapman and Hall, 2006.
[6] PODLUBNY I. Fractional differential equations[M]. San Diego:Academic Press, 1999.
[7] SUN Z, WU X. A fully discrete difference scheme for a diffusion-wave system[J]. Applied Numerical Mathematics, 2006, 56(2):193-209
[8] LIN Y, XU C. Finite difference/spectral approximations for the time-fractional diffusion equation[J]. Journal of Computational Physics, 2007, 225(2):1533-1552
[9] CUI M. Compact finite difference method for the fractional diffusion equation[J]. Journal of Comp-utational Physics, 2009, 228(20):7792-7804
[10] REN J, SUN Z, ZHAO X. Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions[J]. Journal of Computational Physics, 2013, 232(1):456-467
[11] WANG Z, VONG S. A compact difference scheme for a two dimensional nonlinear fractional Klein-Gordon equation in polar coordinates[J]. Computers & Mathematics with Applications, 2016, 71(12):2524-2540
[12] FENG Q, MENG F. Finite difference scheme with spatial fourth-order accuracy for a class of time fractional parabolic equations with variable coefficient[J]. Advances in Difference Equations, 2016, 2016(1):305
[13] YUSTE S B, ACEDO L. An explicit finite difference method and a new von neumann-type stability analysis for fractional diffusion equations[J]. SIAM Journal on Numerical Analysis, 2005, 42(5):1862-1874
[14] ZHUANG P, LIU F, ANH V, et al. New solution and analytical techniques of the implicit numerical method for the anomalous sub-diffusion equation[J]. SIAM Journal on Numerical Analysis, 2008, 46(2):1079-1095
[15] ALIKHANOV A A. A new difference scheme for the time fractional diffusion equation[J]. Journal of Computational Physics, 2015, 280(C):424-438
[16] VONG S, LYU P, CHEN X, et al. High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives[J]. Numerical Algorithms, 2016, 72(1):195-210
[17] ZHAO X, XU Q. Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient[J]. Applied Mathematical Modelling, 2014, 38(15-16):3848-3859
[18] VONG S, WANG Z. A high order compact finite difference scheme for time fractional Fokker-Planck equations[J]. Applied Mathematics Letters, 2015, 43:38-43
[19] CUI M. Compact exponential scheme for the time fractional convection-diffusion reaction equation with variable coefficients[J]. Journal of Computational Physics, 2015, 280(2):143-163
[20] 杜瑞连, 梁宗旗. 右侧Caputo分数阶导数的L2-1插值逼近[J]. 集美大学学报(自然科学版), 2017, 22(4):68-74 DU R L, LIANG Z Q. Interpolation approximation of the right side of the caputo fractional derivative[J]. Journal of Jimei University (Natural Science), 2017, 22(4):68-74
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