Numerical Analysis and Stability of Higher-Order Schemes for Non-linear Generalized Time-Fractional Diffusion Equations

Abstract

This paper presents the development of a stable, higher-order numerical scheme for solving non-linear generalized time-fractional diffusion equations (GTFDEs). The proposed difference scheme achieves a convergence order of

3-\alpha

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$3-\alpha $$\end{document}

,

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

. Spatial discretization is performed using a second-order finite difference operator, and the non-linear term is approximated via higher-order Taylor series expansion. A rigorous stability and convergence analysis is carried out in the

L_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_{2}$$\end{document}

norm. In addition, the numerical stability of the scheme is examined under random noisy perturbations. Numerical experiments on three test problems demonstrate the robustness and efficiency of the proposed method.

Keywords

Fractional derivative with generalized memory kernel / Weight function / Non-linear / Generalized L2 scheme / Convergence and stability / 65M06 / 65M12 / 65M50 / 35R11

References

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