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Difference methods for two-dimensional space-time fractional dispersion equation
ZHANG Yinghan , YANG Xiaoyuan
School of Mathematics and Systems Science, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract: The two-dimensional space-time fractional dispersion equation is obtained from the standard two-dimensional dispersion equation by replacing the first order time derivative by the Caputo fractional derivative, and the two second order space derivatives by the Riemann-Liouville fractional derivatives, respectively. Base on the shifted Grünwald finite difference approximation for the two space fractional derivatives, an implicit difference method and a practical alternate direction implicit difference method were proposed to approximate the fractional dispersion equation. The consistency, stability, and convergence of the two implicit difference methods were analyzed. By using mathematical induction method, it was proven that the two implicit difference methods were all unconditionally stable and convergent and the order of convergence were obtained. The convergence speed and computational complexity of the two implicit difference methods were compared. A numerical simulation for a space-time fractional dispersion equation with known exact solution was also presented, and correctness of the theoretical analysis was verified by the numerical results.
Key words: fractional derivatives     fractional dispersion equation     finite difference method     stability     convergence
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1 问题描述

2 隐式差分格式

tn=nτ,n=0,1,…,N;xi=ihx,i=0,1,…,Nx;yj=jhy,j=0,1,…,Ny,其中$\tau =\frac{T}{N}$为时间步长,hx=$\frac{{{L}_{x}}}{{{N}_{x}}}$和hy=$\frac{{{L}_{y}}}{{{N}_{y}}}$为空间步长.令ui,jn为差分方法的数值解.引入记号Δtu(xi,yj,tn)=u(xi,yj,tn+1)－u(xi,yj,tn),按如下方式对时间分数阶导数进行离散:

L1εi,jn+1=L2εi,jn

En=[ε1,1n,ε2,1n,…,εNx－1,1n,ε1,2n,ε2,2n,…,εNx－1,Ny－1n]T

3 交替方向隐式差分格式

1) 首先对每个固定的yj求解x方向,产生中间解ui,j*,n+1:

2) 对每个固定的xi求解y方向:

4 数值实验

u(x,0,t)=u(0,y,t)=0
u(1,y,t)=(t2+1)y2
u(x,1,t)=(t2+1)x2

u(x,y,t)=(t2+1)x2y2

 时间步长 空间步长 最大误差 误差率 0.100 000 0 0.100 000 0 0.014 269 5 0.025 000 0 0.050 000 0 0.008 392 7 1.700 227 6 0.006 250 0 0.025 000 0 0.004 567 3 1.837 562 7 0.001 562 5 0.012 500 0 0.002 380 1 1.918 953 0
5 结 论

1) 隐式差分格式是以O(Δt)+O(Δx)+O(Δy)无条件收敛的.

2) 交替方向隐式差分格式的收敛阶为O(Δtα)+O(Δx)+O(Δy).

3) 虽然交替方向隐式差分格式在时间方向上的收敛精度低于隐式差分格式,但在计算复杂度上考虑,交替方向隐式差分格式优于隐式差分格式.

4) 数值模拟结果与理论分析是一致的.

#### 文章信息

ZHANG Yinghan, YANG Xiaoyuan

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

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(12): 2296-2301.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0813