Exploring fractional dynamics in the fractional-in-time Gierer-Meinhardt reaction-diffusion model with periodic boundary condition

Xinzhi Wang; Wei Zhang; Xinzhi Wang; Wei Zhang

doi:10.3934/nhm.2026003

Networks and Heterogeneous Media

2026, Volume 21, Issue 1: 55-69. doi: 10.3934/nhm.2026003

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Research article

Exploring fractional dynamics in the fractional-in-time Gierer-Meinhardt reaction-diffusion model with periodic boundary condition

Xinzhi Wang ¹,
Wei Zhang ^{2
,
,}

1.
School of Physics and Electronic Information Engineering, Jining Normal University, Ulanqab, Inner Mongolia 012000, China
2.
Institute of Economics and Management, Jining Normal University, Ulanqab, Inner Mongolia 012000, China

Received: 19 August 2025 Revised: 19 December 2025 Accepted: 30 December 2025 Published: 16 January 2026

This paper presents a new numerical method for simulating the dynamic behavior of the fractional-in-time Gierer-Meinhardt reaction-diffusion model with periodic boundary conditions. A recursive algorithm for binomial coefficients is introduced, avoiding numerical instabilities associated with Gamma functions. High-precision polynomial expansions and the short-memory principle are employed to enhance efficiency and accuracy. Numerical simulations reveal diverse pattern formation.
- fractional-in-time Gierer-Meinhardt model,
- pattern formation,
- numerical method,
- short-memory principle
Citation: Xinzhi Wang, Wei Zhang. Exploring fractional dynamics in the fractional-in-time Gierer-Meinhardt reaction-diffusion model with periodic boundary condition[J]. Networks and Heterogeneous Media, 2026, 21(1): 55-69. doi: 10.3934/nhm.2026003

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Abstract

This paper presents a new numerical method for simulating the dynamic behavior of the fractional-in-time Gierer-Meinhardt reaction-diffusion model with periodic boundary conditions. A recursive algorithm for binomial coefficients is introduced, avoiding numerical instabilities associated with Gamma functions. High-precision polynomial expansions and the short-memory principle are employed to enhance efficiency and accuracy. Numerical simulations reveal diverse pattern formation.

References

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