Author(s)

WenYe Jiang, QiZhi Zhang, Yu Li^*

Affiliation(s)

Department of Mathematics, College of Science, Northeast Forestry University, Harbin 150040, Heilongjiang, China.

Corresponding Author

Yu Li

ABSTRACT

Mathematical modeling of the spatial fractional-order FitzHugh-Nagumo equations provides a critical framework for describing the propagation of electrical potentials in heterogeneous cardiac tissues, a problem that continues to attract significant research attention. Due to the geometrically irregular cross-sections of cardiac tissue, numerical solutions on conventional regular or approximately irregular domains remain limited, and analytical solutions are generally unavailable. In this study, we employ a high-order finite difference method in space coupled with a fourth-order Runge-Kutta scheme in time to solve this nonlinear fractional-order system. Numerical experiments are conducted to validate the high-order convergence and computational stability of the proposed numerical approach.

KEYWORDS

Riesz fractional derivative; FitzHugh-Nagumo model; Finite difference method; The fourth-order explicit Runge-Kutta method

CITE THIS PAPER

WenYe Jiang, QiZhi Zhang, Yu Li. Mathematical modeling and high-order finite difference methods for spatial fractional FitzHugh-Nagumo equations. Eurasia Journal of Science and Technology. 2025, 7(6): 44-50. DOI: https://doi.org/10.61784/ejst3125.

REFERENCES

[1] Han C, Wang Y L, Li Z Y. Novel patterns in a class of fractional reaction–diffusion models with the Riesz fractional derivative. Mathematics and Computers in Simulation, 2022, 202, 149–163.

[2] Zhang L, Sun H W. Numerical solution for multi-dimensional Riesz fractional nonlinear reaction–diffusion equation by exponential Runge–Kutta method. Journal of Applied Mathematics and Computing, 2019, 60 (1-2): 357-378.

[3] Zeng F, Liu F, Li C P, et al. A Crank--Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM Journal on Numerical Analysis, 2014, 52(6): 2599-2622.

[4] Li X, Han C, Wang Y. Novel Patterns in Fractional-in-Space Nonlinear Coupled FitzHugh-Nagumo Models with Riesz Fractional Derivative. Fractal and Fractional, 2022, 6(3), 136. 

[5] Prakash A, Kaur H. A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses. Nonlinear Engineering, 2019, 8(1):719-727.

[6] Wang Y H, Feng X L, Li H. Ghost structure finite difference method for fractional FitzHugh-Nagumo monodomain model on moving irregular domains. Journal of Computational Physics, 2021, 424: 110862.

[7] Kumar P, Erturk V S. A fractional-order improved FitzHugh-Nagumo neuron model. Chinese Physics B, 2025, 34(1): 018704.

[8] Lee H G. A second-order operator splitting Fourier spectral method for fractional-in-space reaction–diffusion equations. Journal of Computational and Applied Mathematics, 2018, 333: 395-403.

[9] Huang Y Y, Qu W, Lei S L. On τ-preconditioner for a novel fourth-order difference scheme of two-dimensional Riesz space-fractional diffusion equations. Computers & Mathematics with Applications, 2023, 145: 124-140.

[10] Han C, Wang Y L, Li Z Y. A High-Precision Numerical Approach to Solving Space Fractional Gray-Scott Model. Applied Mathematics Letters, 2022, 125: 107759.

[11] Chin P W M. The effect on the solution of the FitzHugh-Nagumo equation by the external parameter using the Galerkin method. Journal of Applied Analysis & Computation, 2023, 13(4): 1983-2005.

[12] Han C, Wang Y L, Li Z Y. Numerical solutions of space fractional variable-coefficient KdV–modified KdV equation by Fourier spectral method. Fractals, 2021, 29(08): 2150246.

[13] Zhang J J, Chen Y M, Wang X M. High-precision compact difference method for the FitzHugh-Nagumo equation. Journal of China West Normal University (Natural Science Edition), 2021, 42(2): 126–132.

[14] Luo Y, Song L Y. Solve the generalized Fitzhugh-Nagumo equation with time-varying coefficients based on the Chebyshev spectral method. Journal of Jiamusi University (Natural Science Edition), 2025, 43(3): 166–170.