关键词: Caputo derivative, Riesz derivative, Time-Caputo and space-Riesz fractional difusion equation

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

Key words: Caputo derivative, Riesz derivative, Time-Caputo and space-Riesz fractional difusion equation