留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

二维空间时间分数阶色散方程的差分方法

张英晗 杨小远

张英晗, 杨小远. 二维空间时间分数阶色散方程的差分方法[J]. 北京航空航天大学学报, 2015, 41(12): 2296-2301. doi: 10.13700/j.bh.1001-5965.2014.0813
引用本文: 张英晗, 杨小远. 二维空间时间分数阶色散方程的差分方法[J]. 北京航空航天大学学报, 2015, 41(12): 2296-2301. doi: 10.13700/j.bh.1001-5965.2014.0813
ZHANG Yinghan, YANG Xiaoyuan. Difference methods for two-dimensional space-time fractional dispersion equation[J]. Journal of Beijing University of Aeronautics and Astronautics, 2015, 41(12): 2296-2301. doi: 10.13700/j.bh.1001-5965.2014.0813(in Chinese)
Citation: ZHANG Yinghan, YANG Xiaoyuan. Difference methods for two-dimensional space-time fractional dispersion equation[J]. Journal of Beijing University of Aeronautics and Astronautics, 2015, 41(12): 2296-2301. doi: 10.13700/j.bh.1001-5965.2014.0813(in Chinese)

二维空间时间分数阶色散方程的差分方法

doi: 10.13700/j.bh.1001-5965.2014.0813
基金项目: 国家自然科学基金(61271010);北京市自然科学基金(4152029);北京航空航天大学博士研究生创新基金
详细信息
    作者简介:

    张英晗(1986-),男,河北邢台人,博士研究生,zhangyinghan007@126.com

    通讯作者:

    杨小远(1964-),女,江苏淮安人,教授,xiaoyuanyang@vip.163.com,主要研究方向为分数阶随机偏微分方程.

  • 中图分类号: O241.82

Difference methods for two-dimensional space-time fractional dispersion equation

  • 摘要: 通过把标准的二维色散方程中的一阶时间导数替换成Caputo分数阶导数,两个二阶空间导数分别替换成Riemann-Liouville分数阶导数,得到二维空间时间分数阶色散方程.基于两个空间分数阶导数的转移Grünwald有限差分近似,分别构造了逼近二维空间时间分数阶色散方程的隐式差分格式和交替方向隐式差分格式.对两种差分格式分别进行了相容性、稳定性和收敛性分析.应用数学归纳法证明了两种隐式差分格式都是无条件稳定和收敛的并且得到了收敛阶.对两种隐式差分格式的收敛速度和计算复杂度进行了比较.基于以上所构造的差分格式,对精确解已知的一个空间时间分数阶色散方程进行了数值实验模拟,模拟结果验证了理论分析的正确性.

     

  • [1] Arrarás A,Portero L,Jorge J C.Convergence of fractional step mimetic finite difference discretizations for semilinear parabolic problems[J].Applied Numerical Mathematics,2010,60(4):473-485.
    [2] Meerschaert M,Tadjeran C.Finite difference approximations for fractional advection-dispersion flow equations[J].Journal of Computational and Applied Mathematics,2004,172(1):65-77.
    [3] Meerschaert M,Scheffler H,Tadjeran C.Finite difference methods for two-dimensional fractional dispersion equation[J].Journal of Computational Physics,2006,211(2):249-261.
    [4] Liu F,Anh V,Turner I.Numerical solution of the space fractional Fokker-Planck equation[J].Journal of Computational and Applied Mathematics,2004,166(1):209-219.
    [5] Tadjeran C,Meerschaert M,Scheffler H.A second-order accurate numerical approximation for the fractional diffusion equation[J].Journal of Computational Physics,2006,213(2):205-213.
    [6] Tadjeran C,Meerschaert M.A second-order accurate numerical method for the two-dimensional fractional diffusion equation[J].Journal of Computational Physics,2007,220(2):813-823.
    [7] Liu Q,Liu F,Turner I,et al.Approximation for the Levy-Feller advection-dispersion process by random walk and finite difference method[J].Journal of Computational Physics,2007,222(1):57-70.
    [8] Zhuang P,Liu F,Anh V,et al.Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term[J].SIAM Journal of Numerical Analysis,2009,47(3):1760-1781.
    [9] Liu F,Zhuang P,Anh V,et al.Stability and convergence of the difference methods for the space-time fractional advection- diffusion equation[J].Applied Mathematics and Computation,2007,191(1):12-20.
    [10] Podlubny I.Fractional differential equations[M].New York:Academic Press,1999:51-87.
    [11] Ervin J S,Roop J P.Variational solution of fractional advection dispersion equations on bounded domains in R d [J].Numerical Mathematics-Theory Methods and Applications,2007,23(2):256-281.
    [12] Richtmyer R D,Morton K W.Difference methods for initial-value problems[M].New York:Krieger Publishing Co.,1994:133-200.
    [13] Ghazizadeh H R,Maerefat M,Azimi A.Explicit and implicit finite difference schemes for fractional Cattaneo equation[J].Journal of Computational Physics,2010,229(19):7042-7057.
    [14] Gao G,Sun Z.A compact finite difference scheme for the fractional sub-diffusion equations[J].Journal of Computational Physics,2011,230(3):586-595.
    [15] Sousa E.Finite difference approximations for a fractional advection diffusion problem[J].Journal of Computational Physics,2009,228(11):4038-4054.
    [16] Li C,Zeng F.The finite difference methods for fractional ordinary differential equations[J].Numerical Functional Analysis and Optimization,2014,34(2):149-179.
    [17] Zhang Y,Liao Z,Lin H.Finite difference methods for the time fractional diffusion equation on non-uniform meshes[J].Journal of Computational Physics,2014,265(1):195-210.
    [18] Brunner H,Han H,Yin D.Artificial boundary conditions and finite difference approximations for a time-fractional diffusion wave equation on a two-dimensional unbounded spatial domain[J].Journal of Computational Physics,2014,276(3):541-562.
  • 加载中
计量
  • 文章访问数:  650
  • HTML全文浏览量:  2
  • PDF下载量:  517
  • 被引次数: 0
出版历程
  • 收稿日期:  2014-12-24
  • 修回日期:  2015-02-13
  • 刊出日期:  2015-12-20

目录

    /

    返回文章
    返回
    常见问答