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Volume 9
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Article

A High-Precision Numerical Method for the Fractional-in-Time Complex Ginzburg–Landau Equation with Periodic Boundary Condition

by
Wei Zhang
1 and
Yulan Wang
2,*
1
Institute of Economics and Management, Jining Normal University, Ulanqab 012000, China
2
Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China
*
Author to whom correspondence should be addressed.
Fractal Fract. 2025, 9(11), 738; https://doi.org/10.3390/fractalfract9110738
Submission received: 24 September 2025 / Revised: 28 October 2025 / Accepted: 13 November 2025 / Published: 14 November 2025
Versions Notes

Abstract

This paper investigates the chaotic and pattern dynamics of the time-fractional Ginzburg–Landau equation. First, we propose a high-precision numerical method that combines finite difference schemes with an improved Grünwald–Letnikov fractional derivative approximation. Subsequently, the effectiveness of the proposed method is validated through systematic comparisons with classical numerical approaches. Finally, numerical simulations based on this method reveal rich dynamical phenomena in the fractional Ginzburg–Landau equation: the system exhibits complex behaviors including chaotic oscillations and novel two- and three-dimensional pattern structures. This study not only advances the theoretical development of numerical solutions for fractional GLE but also provides a reliable computational tool for deeper understanding of its complex dynamical mechanisms.
Keywords: a high-precision numerical method; Fractional Ginzburg–Landau equation; pattern formation; short-memory principle; periodic boundary condition a high-precision numerical method; Fractional Ginzburg–Landau equation; pattern formation; short-memory principle; periodic boundary condition

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MDPI and ACS Style

Zhang, W.; Wang, Y. A High-Precision Numerical Method for the Fractional-in-Time Complex Ginzburg–Landau Equation with Periodic Boundary Condition. Fractal Fract. 2025, 9, 738. https://doi.org/10.3390/fractalfract9110738

AMA Style

Zhang W, Wang Y. A High-Precision Numerical Method for the Fractional-in-Time Complex Ginzburg–Landau Equation with Periodic Boundary Condition. Fractal and Fractional. 2025; 9(11):738. https://doi.org/10.3390/fractalfract9110738

Chicago/Turabian Style

Zhang, Wei, and Yulan Wang. 2025. "A High-Precision Numerical Method for the Fractional-in-Time Complex Ginzburg–Landau Equation with Periodic Boundary Condition" Fractal and Fractional 9, no. 11: 738. https://doi.org/10.3390/fractalfract9110738

APA Style

Zhang, W., & Wang, Y. (2025). A High-Precision Numerical Method for the Fractional-in-Time Complex Ginzburg–Landau Equation with Periodic Boundary Condition. Fractal and Fractional, 9(11), 738. https://doi.org/10.3390/fractalfract9110738

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