Numerical Algorithms for Ultra-slow Diffusion Equations

Abstract

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 $\alpha \in (0,1)$. 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., $\textrm{L2-1}_{\sigma }$ and L1-2 ones. The spatial fractional derivatives are discretized by the second order finite difference methods. When the $\textrm{L2-1}_{\sigma }$ discretization is used, the derived numerical schemes are unconditionally stable, with both theoretical and numerical convergence order $\mathcal {O}(\tau ^{2}+h^{2})$ for all $\alpha \in (0, 1)$, in which $\tau$ 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 $\mathcal {O}(\tau ^{2}+h^{2})$ for $\alpha \in (0, 0.373\,8)$, and numerically exhibit the stability for all $\alpha \in (0, 1)$ with the numerical error being $\mathcal {O}(\tau ^{3-\alpha }+h^2)$. The illustrative examples displayed are in line with the theoretical analysis.

Funding

National Natural Science Foundation of China(12271339)