求解Riesz空间分数阶扩散方程的一种新的数值方法

A NEW NUMERICAL METHOD FOR SOLVING RIESZ SPACE-FRACTIONAL DIFFUSION EQUATION

本文利用Diethelm方法构造了一种逼近Riesz空间分数阶导数的O(△x^(3-α))格式,其中1 <α<2,△x是空间步长.进一步对一阶时间导数采用Crank-Nicolson方法离散,得到了求解Riesz空间分数阶扩散方程的一种新的有限差分格式,并用矩阵方法证明了稳定性和收敛性,其误差估计为O(△t^2+△x^(3-α)),其中△t为时间步长.最后,数值算例验证了差分格式的正确性和有效性.

By using Diethelm’s method,we construct an approximate scheme to the Riesz space fractional derivative with order O(△x3-α),where 1<a<2 and△x denotes the space step size.Further we discretize the time derivative with the Crank-Nicolson method and obtain a new finite difference method for solving Riesz space fractional diffusion equation.The stability and convergence are proved by the matrix method and the error estimate in the maximum norm is O(△t2+△x3-α),where At denotes the time step size.Finally,some numerical examples are given to illustrate their correctness and efficiency.