A Fast Averaged L1 Finite Difference Method for Time Fractional Mobile/Immobile Diffusion Equation with Weakly Singular Solution

Abstract

In this paper, the time fractional mobile/immobile diffusion equation with the weak singular solution at the initial time is studied. The averaged L1 finite difference scheme is established for the equation. The stability of the numerical scheme is analyzed by the Fourier analysis method. The convergence order of the scheme is

O({\tau }^2|ln\tau |+h^2)

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

, where

\tau

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

and h are the sizes of the time and space steps, respectively. In addition, due to the historical dependence of the time fractional derivative, we establish a fast method based on the exponential-sum-approximation, effectively reducing computation and storage. Furthermore, we provide an error estimate of the fast algorithm. Finally, a numerical experiment verifies the effectiveness of the algorithm.

Keywords

Time fractional mobile/immobile diffusion equation / Averaged L1 scheme / Exponential-sum-approximation / Weak singularity

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Funding

Natural Science Foundation of Shandong Province(ZR2022QA038)

Doctoral Research Foundation of Liaocheng University(318052155)

RIGHTS & PERMISSIONS

Shanghai University

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