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Novel patterns in a class of fractional reaction–diffusion models with the Riesz fractional derivative

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  • Che, Han
  • Wang, Yu-Lan
  • Li, Zhi-Yuan

Abstract

Pattern is a non-uniform macro structure with some regularity in space or time, which is common in nature. In this manuscript, we introduce the Fourier transform for spatial discretization and Runge–Kutta method for time discretization to solve a class of fractional reaction–diffusion models such as Allen–Cahn model, FitzHugh–Nagumo model and Gray–Scott model with space derivatives described by the fractional Laplacian. Numerical experiments show that compared with semi-implicit Fourier spectral method, present method has higher precision and low computational complexity. The patterns of 2D FitzHugh–Nagumo model with standard diffusion obtained in this manuscript are in line with the numerical simulations and theoretical analysis that made by other academics. Then, we discuss the limit case of fractional order: the process of pattern formations of the fractional order tends to corresponding integer-order reaction–diffusion model when super diffusion tends to standard diffusion. Finally, some long time diffusion behaviors of 3D FitzHugh–Nagumo model and 3D Gray–Scott model are observed.

Suggested Citation

  • Che, Han & Wang, Yu-Lan & Li, Zhi-Yuan, 2022. "Novel patterns in a class of fractional reaction–diffusion models with the Riesz fractional derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 149-163.
  • Handle: RePEc:eee:matcom:v:202:y:2022:i:c:p:149-163
    DOI: 10.1016/j.matcom.2022.05.037
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    References listed on IDEAS

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    1. Zhiyuan Li & Meichun Wang & Yulan Wang & Jing Pang, 2020. "Using Reproducing Kernel for Solving a Class of Fractional Order Integral Differential Equations," Advances in Mathematical Physics, Hindawi, vol. 2020, pages 1-12, March.
    2. Jia, Yunfeng, 2020. "Bifurcation and pattern formation of a tumor–immune model with time-delay and diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 92-108.
    3. Gangwei Wang, 2021. "SYMMETRY ANALYSIS, ANALYTICAL SOLUTIONS AND CONSERVATION LAWS OF A GENERALIZED KdV–BURGERS–KURAMOTO EQUATION AND ITS FRACTIONAL VERSION," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(04), pages 1-11, June.
    4. Meerschaert, Mark M. & Mortensen, Jeff & Wheatcraft, Stephen W., 2006. "Fractional vector calculus for fractional advection–dispersion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 181-190.
    5. Tang, Xiaosong, 2022. "Periodic solutions and spatial patterns induced by mixed delays in a diffusive spruce budworm model with Holling II predation function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 420-429.
    6. Jhinga, Aman & Daftardar-Gejji, Varsha, 2018. "A new finite-difference predictor-corrector method for fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 418-432.
    7. Che Han & Yu-Lan Wang & Zhi-Yuan Li, 2021. "NUMERICAL SOLUTIONS OF SPACE FRACTIONAL VARIABLE-COEFFICIENT KdV–MODIFIED KdV EQUATION BY FOURIER SPECTRAL METHOD," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(08), pages 1-19, December.
    8. Wang, Junjie & Xiao, Aiguo, 2019. "Conservative Fourier spectral method and numerical investigation of space fractional Klein–Gordon–Schrödinger equations," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 348-365.
    9. Hans Engler, 2010. "On the Speed of Spread for Fractional Reaction-Diffusion Equations," International Journal of Differential Equations, Hindawi, vol. 2010, pages 1-16, November.
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