Numerical simulation of the space fractional $ (3+1) $-dimensional Gray-Scott models with the Riesz fractional derivative

Dan-Dan Dai; Wei Zhang; Yu-Lan Wang; Dan-Dan Dai; Wei Zhang; Yu-Lan Wang

doi:10.3934/math.2022569

AIMS Mathematics

2022, Volume 7, Issue 6: 10234-10244. doi: 10.3934/math.2022569

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Research article

Numerical simulation of the space fractional $ (3+1) $-dimensional Gray-Scott models with the Riesz fractional derivative

Dan-Dan Dai ¹,
Wei Zhang ^{2
,
,},
Yu-Lan Wang ^{3
,
,}

1.
School of Physics and Electronic Information Engineering, Jining Normal University, Jining 012000, Inner Mongolia, China
2.
Institute of Economics and Management, Jining Normal University, Jining 012000, Inner Mongolia, China
3.
Department of Mathematics, Inner Mongolia University of Technology, Hohhot, 010051, China

Received: 20 January 2022 Revised: 12 March 2022 Accepted: 16 March 2022 Published: 22 March 2022
MSC : 65M30, 65D20

The reaction-diffusion process always behaves extremely magically, and any a differential model cannot reveal all of its mechanism. Here we show the patterns behavior can be described well by the fractional reaction-diffusion model (FRDM), which has unique properties that the integer model does not have. Numerical simulation is carried out to elucidate the attractive properties of the fractional (3+1)-dimensional Gray-Scott model, which is to model a chemical reaction with oscillation. The Fourier transform for spatial discretization and fourth-order Runge-Kutta method for time discretization are employed to illustrate the fractal reaction-diffusion process.
- numerical simulation,
- patterns,
- Fourier spectral method,
- Riesz fractional derivative
Citation: Dan-Dan Dai, Wei Zhang, Yu-Lan Wang. Numerical simulation of the space fractional $ (3+1) $-dimensional Gray-Scott models with the Riesz fractional derivative[J]. AIMS Mathematics, 2022, 7(6): 10234-10244. doi: 10.3934/math.2022569

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Abstract

The reaction-diffusion process always behaves extremely magically, and any a differential model cannot reveal all of its mechanism. Here we show the patterns behavior can be described well by the fractional reaction-diffusion model (FRDM), which has unique properties that the integer model does not have. Numerical simulation is carried out to elucidate the attractive properties of the fractional (3+1)-dimensional Gray-Scott model, which is to model a chemical reaction with oscillation. The Fourier transform for spatial discretization and fourth-order Runge-Kutta method for time discretization are employed to illustrate the fractal reaction-diffusion process.

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