Two kinds of numerical algorithms for ultra-slow diffusion equations
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In this article, two kinds of numerical algorithms are derived for the ultra-slow (or superslow) diffusion equation in one and two space dimensions, where the ultra-slow diffusion is characterized by the Caputo-Hadamard fractional derivative of order α∈ (0,1). To describe the 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., L2-1_σ and L1-2 methods. The spatial fractional derivatives are discretized by the 2-nd order finite difference methods. When L2-1_σ discretization is used, the derived numerical scheme is unconditionally stable with error estimate 𝒪(τ^2+h^2) for all α∈ (0, 1), in which τ and h are temporal and spatial stepsizes, respectively. When L1-2 discretization is used, the derived numerical scheme is stable with error estimate 𝒪(τ^3-α+h^2) for α∈ (0, 0.3738). The illustrative examples displayed are in line with the theoretical analysis.
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