Numerical Algorithms for Ultra-slow Diffusion Equations

doi:10.1007/s42967-024-00380-3

Abstract

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.

Key words: Ultra-slow diffusion equation, Caputo-Hadamard derivative, Riesz derivative, Fractional Laplacian, $ \textrm{L2-1}_{\sigma } $ formula, L1-2 formula

CLC Number:

Min Cai, Changpin Li, Yu Wang. Numerical Algorithms for Ultra-slow Diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(6): 2339-2384.

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