Numerical Algorithms for Ultra-slow Diffusion Equations

In this article, numerical algorithms are derived for ultra-slow (or superslow) diffusion equations in one and two space dimensions, where the ultra-slow diffusion is characterized by the Caputo-Hadamard fractional derivative of order \(\alpha \in (0,1)\) $\alpha \in (0,1)$ . To describe the non-locality in spatial interaction, the Riesz fractional derivative and the fractional Laplacian are used in one and two space dimensions, respectively. The Caputo-Hadamard derivative is discretized by two typical approximate formulae, i.e., \(\textrm{L2-1}_{\sigma }\) ${\text{L2-1}}_{\sigma }$ and L1-2 ones. The spatial fractional derivatives are discretized by the second order finite difference methods. When the \(\textrm{L2-1}_{\sigma }\) ${\text{L2-1}}_{\sigma }$ discretization is used, the derived numerical schemes are unconditionally stable, with both theoretical and numerical convergence order \(\mathcal {O}(\tau ^{2}+h^{2})\) $\mathcal{O}({\tau }^2+h^2)$ for all \(\alpha \in (0, 1)\) $\alpha \in (0,1)$ , in which \(\tau\) $\tau$ and h are temporal and spatial stepsizes, respectively. When the L1-2 discretization is used, the derived numerical schemes are proved to be stable with the error estimate \(\mathcal {O}(\tau ^{2}+h^{2})\) $\mathcal{O}({\tau }^2+h^2)$ for \(\alpha \in (0, 0.373\,8)\) $\alpha \in (0,0.3738)$ , and numerically exhibit the stability for all \(\alpha \in (0, 1)\) $\alpha \in (0,1)$ with the numerical error being \(\mathcal {O}(\tau ^{3-\alpha }+h^2)\) $\mathcal{O}({\tau }^{3-\alpha }+h^2)$ . The illustrative examples displayed are in line with the theoretical analysis.