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)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0,1)$$\end{document}

. 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.,

{\text{L2-1}}_{\sigma }

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and L1-2 ones. The spatial fractional derivatives are discretized by the second order finite difference methods. When the

{\text{L2-1}}_{\sigma }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{L2-1}_{\sigma }$$\end{document}

discretization is used, the derived numerical schemes are unconditionally stable, with both theoretical and numerical convergence order

\mathcal{O}({\tau }^2+h^2)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}(\tau ^{2}+h^{2})$$\end{document}

for all

\alpha \in (0,1)

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, in which

\tau

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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)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}(\tau ^{2}+h^{2})$$\end{document}

for

\alpha \in (0,0.3738)

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, and numerically exhibit the stability for all

\alpha \in (0,1)

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with the numerical error being

\mathcal{O}({\tau }^{3-\alpha }+h^2)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {O}(\tau ^{3-\alpha }+h^2)$$\end{document}

. The illustrative examples displayed are in line with the theoretical analysis.

Keywords

Ultra-slow diffusion equation / Caputo-Hadamard derivative / Riesz derivative / Fractional Laplacian /

{\text{L2-1}}_{\sigma }

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formula')">

{\text{L2-1}}_{\sigma }

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formula / L1-2 formula / 35R11 / 65M06

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Funding

National Natural Science Foundation of China(12271339)

RIGHTS & PERMISSIONS

Shanghai University