二维空间时间分数阶色散方程的差分方法

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

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.

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.