一类时间-空间Riesz分数阶扩散方程的数值方法

A Method for Solving Time-space Riesz Fractional Diffusion Equations

文章研究了一类带齐次Dirichlet边界条件的时间空间Riesz分数阶扩散方程的初边值问题,利用分数阶中心差分对空间方向进行离散,其误差估计为O(Δx^(2)),Δx是空间步长。在时间上,采用Diethelm方法离散导数,其误差估计为O(Δt2-α),其中Δt为时间步长。进一步得到了求解时间空间Riesz分数阶扩散方程的有限差分格式,并用最大范数法证明了稳定性和收敛性.最后,用实际数值算例验证了差分离散格式的有效性.

In this paper,the initial boundary value problem of a class of time-space Riesz fractional diffusion equation with homogeneous Dirichlet boundary conditions is studied.The space direction is discretized by fractional difference.The error is estimated as O(Δx^(2)),where Δx denotes the space step size.In terms of time,Diethelm method is adopted to discretize derivatives,and its error is estimated as O(Δx^(2)),where Δx denotes the time step size.We further obtain the finite difference scheme for solving the time-space Riesz fractional diffusion equation,and the stability and convergence are proved by the maximum norm method.Finally,the efficiency of the discrete difference scheme is verified by a numerical example.