Journals
Resources
About Us
Open Access

A Second-Order Alikhanov Type Implicit Scheme for the Time-Space Fractional Ginzburg-Landau Equation

A Second-Order Alikhanov Type Implicit Scheme for the Time-Space Fractional Ginzburg-Landau Equation

Year:    2024

Author:    Yufang Gao, Shengxiang Chang, Changna Lu

Advances in Applied Mathematics and Mechanics, Vol. 16 (2024), Iss. 6 : pp. 1451–1473

Abstract

In this paper, we consider an implicit method for solving the nonlinear time-space fractional Ginzburg-Landau equation. The scheme is based on the $L2-1_σ$ formula to approximate the Caputo fractional derivative and the weighted and shifted Grünwald difference method to approximate the Riesz space fractional derivative. In order to overcome the non-local property of Riesz space fractional derivatives and the historical dependence brought by Caputo time fractional derivatives, this paper introduces the fractional Sobolev norm and the fractional Sobolev inequality. It is proved in detail that the difference scheme is stable and uniquely solvable by the discrete energy method. In particular, the difference scheme is unconditionally stable when $\gamma≤0,$ where $\gamma$ is a coefficient of the equation. Moreover, the scheme is shown to be convergent in $l^2_h$ norm at the optimal order of $\mathcal{O}(\tau^2+h^2)$ with time step $\tau$ and mesh size $h.$ Finally, we provide a linearized iterative algorithm, and the numerical results are presented to verify the accuracy and efficiency of the proposed scheme.

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2022-0097

Advances in Applied Mathematics and Mechanics, Vol. 16 (2024), Iss. 6 : pp. 1451–1473

Published online:    2024-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    23

Keywords:    Time-space fractional Ginzburg-Landau equation Caputo fractional derivative Riesz fractional derivative $L2−1_σ$ formula convergence.

Author Details

Yufang Gao

Shengxiang Chang

Changna Lu