A Note on Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Diffusion Equation

Recently, Zhang and Ding developed a novel finite difference scheme for the time-Caputo and space-Riesz fractional diffusion equation with the convergence order O(tau(2-alpha) + h(2)) in Zhang and Ding (Commun. Appl. Math. Comput. 2(1): 57-72, 2020). Unfortunately, they only gave the stability and convergence results for alpha is an element of (0, 1) and beta is an element of [7/8 + (3)root 621+48 root 87/24 + 19/8(3)root 621+48 root 87, 2]. In this paper, using a new analysis method, we find that the original difference scheme is unconditionally stable and convergent with order O(tau(2-alpha) + h(2)) for all alpha is an element of (0, 1) and beta is an element of (1, 2]. Finally, some numerical examples are given to verify the correctness of the results.