Novel Patterns in Fractional-in-Space Nonlinear Coupled FitzHugh–Nagumo Models with Riesz Fractional Derivative

Abstract

In this paper, the Fourier spectral method is used to solve the fractional-in-space nonlinear coupled FitzHugh–Nagumo model.Numerical simulation is carried out to elucidate the diffusion behavior of patterns for the fractional 2D and 3D FitzHugh–Nagumo model. The results of numerical experiments are consistent with the theoretical results of other scholars, which verifies the accuracy of the method. We show that stable spatio-temporal patterns can be sustained for a long time; these patterns are different from any previously obtained in numerical studies. Here, we show that behavior patterns can be described well by the fractional FitzHugh–Nagumo and Gray–Scott models, which have unique properties that integer models do not have. Results show that the Fourier spectral method has strong competitiveness, reliability, and solving ability for solving 2D and 3D fractional-in-space nonlinear reaction-diffusion models.

Ginzburg–Landau equation with fractional Laplacian on a upper- right quarter plane

“…Theoretical analysis of Cauchy problem for the fractional Ginzburg-Landau equation has also well developed. Among them, [10,22,30,39,40,43]. Due to the nonlocality of fractional operators, the study of a boundary value problem even in the case of a half-line is difficult.…”

Numerical simulation of fractal wave propagation of a multi-dimensional nonlinear fractional-in-space Schrödinger equation

“…[31], which showed Fourier spectral method has high precision. Besides, we used the Fourier spectral method to investigate a (2+1)-dimensional fractional FitzHugh-Nagumo Models [32] and a fractional Gray-Scott model [33], and achieved better results. Some scholars used Fourier spectral method to solve modified Swift-Hohenberg equation [34], space fractional nonlinear Schrödinger equation [35] and so on [36].…”

Numerical Simulation of the Fractional-Order Lorenz Chaotic Systems with燙aputo Fractional Derivative

“…In [4][5][6], the authors gave a dynamic analysis of a fractional-order Lorenz chaotic system. Although some numerical and analytical methods of the FDEs have been announced, such as spectral method [7][8][9][10][11], reproducing kernel method [12][13][14][15][16][17][18][19], homotopy perturbation method [20][21][22][23], high-precision numerical approach [24][25][26][27], and so on numerical and analytical methods [28][29][30][31][32][33][34][35][36]. These researchers all say their own approach can accurately simulate chaotic systems.…”