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

Abstract

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

Funding

National Natural Science Foundation of China(11561060)