二维调和分数阶扩散方程的数值模拟

Numerical simulation of two-dimensional tempered fractional diffusion equation

讨论一个二维调和分数阶扩散方程,其中的调和分数阶导数是分数阶导数的推广,可模拟粒子在早期的超扩散向后期的次扩散的渐进行为.采用隐式交替方向法(ADI)和Crank-Nicolson(C-N)格式建立方程的数值离散格式,并采用外推法得到差分格式的二阶精度,运用矩阵分析的方法给出稳定性和收敛性的证明,同时给出一个数值例子说明所建立的数值离散格式的有效性.

This paper discusses a two-dimensional tempered fractional diffusion equation,in which the tempered fractional derivative is the extension of fractional derivative.The scheme can model the transition from super-diffusion early time to diffusive late-time behavior.We apply the alternating directions implicit approach and the Crank-Nicolson(C-N)algorithan to establish our numerical discretization scheme.Furthermore,we obtain the second-order accurate difference method by a Richardson extrapolation.The stability and the convergence of the numerical scheme are proven via the technique of matrix analysis.A numerical example is given to demonstrate the efficiency of the designed schemes.