Efficient numerical method for the Fitzhugh-Nagumo equations with Neumann boundary conditions

Volume 8, Issue 2, April 2023     |     PP. 41-53      |     PDF (395 K)    |     Pub. Date: February 26, 2023
DOI: 10.54647/physics140526    83 Downloads     2029 Views  

Author(s)

Hao Zhou, School of Digital Engineering, Zhejiang Dongfang Polytechnic,Wenzhou 323000, P. R. China
Xiang Liu, School of Digital Engineering, Zhejiang Dongfang Polytechnic,Wenzhou 323000, P. R. China
Weiguo Zhang, School of Digital Engineering, Zhejiang Dongfang Polytechnic,Wenzhou 323000, P. R. China
Yiwen Liao, School of Digital Engineering, Zhejiang Dongfang Polytechnic,Wenzhou 323000, P. R. China

Abstract
The objective of this work is to construct a new efficient numerical scheme to solve the Fitzhugh-Nagumo model. For the space discretization, Chebyshev spectral method proposed on Legendre orthogonal approximations on Gauss- Chebyshev- Lobatto points. A high-order Runge-Kutta algorithm was used in the time direction. The full-discrete scheme was expressed explicitly and was easy to be implemented with the Neumann boundary conditions. Numerical experiments are discussed to validate the accuracy and reliability of the proposed method.

Keywords
Chebyshev spectral method; FitzHugh-Nagumo equation; Neumann boundary condition

Cite this paper
Hao Zhou, Xiang Liu, Weiguo Zhang, Yiwen Liao, Efficient numerical method for the Fitzhugh-Nagumo equations with Neumann boundary conditions , SCIREA Journal of Physics. Volume 8, Issue 2, April 2023 | PP. 41-53. 10.54647/physics140526

References

[ 1 ] FitzHugh R, Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 1995, 17, 257–278 (1955). https://doi.org/10.1007/BF02477753
[ 2 ] Olmos D., Shizgal B.D., Pseudospectral method of solution of the Fitzhugh–Nagumo equation. Math. Comput. Simulat. 2009, 79, 2258–2278. https://doi.org/10.1016/j.matcom.2009.01.001
[ 3 ] Inan B., Ali K.K., Saha A., Ak T., Analytical and numerical solutions of the Fitzhugh–Nagumo equation and their multistability behavior. Numer. Meth. Part. D. E. 2020, 1–17. https://doi.org/10.1002/num.2251
[ 4 ] Moghaderi H., Dehghan M., Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Math. Method. Appl. Sci. 2017, 40, 1170 - 1200.
[ 5 ] Wang Y., Cai, L., Feng, X., Luo, X., Gao, H.. A ghost structure finite difference method for a fractional FitzHugh-Nagumo monodomain model on moving irregular domain. J. Comput. Phys. 2021, 428, 110081. https://doi.org/10.1016/j.jcp.2020.110081
[ 6 ] Ruiz-Ramirez J., Macias-Diaz J.E., A finite-difference scheme to approximate non-negative and bounded solutions of a FitzHugh Nagumo equation. Int. J. Comput. Math. 2011, 88, 3186–3201. http://dx.doi.org/10.1080/00207160.2011.579964
[ 7 ] Hariharan G., Kannan K., Haar wavelet method for solving FitzHugh–Nagumo equation, Int. J. Math. Comput. Sci. 2010, 4, 909–913.
[ 8 ] Al-Juaifri G. A., Harfash A. J., Finite element analysis of nonlinear reaction–diffusion system of fitzhugh–nagumo type with robin boundary conditions. Math. Comput. Simulat. 2023, 203, 486-517. https://doi.org/10.1016/j.matcom.2022.07.005
[ 9 ] Cai L., Sun Y., Jing F., Li Y., Shen X., Nie Y., A fully discrete implicit-explicit finite element method for solving the Fitzhugh-Nagumo model. J. Compu. Math. 2020, 38(3), 469. https://doi.org/10.4208/jcm.1901-m2017-0263
[ 10 ] Bueno-Orovio A., Kay D., Burrage K., Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer. Math. 2014, 54, 937-954.
[ 11 ] Zhang J., Lin S., Wang J.,. Stability and convergence analysis of Fourier pseudo-spectral method for FitzHugh-Nagumo model. Appl. Numer. Math. 2020, 157, 563-578. https://doi.org/10.1016/j.apnum.2020.07.009
[ 12 ] Li X., Han C., Wang Y. Novel patterns in fractional-in-space nonlinear coupled FitzHugh–Nagumo models with Riesz fractional derivative. Fractal Fract. 2022, 6(3), 136. https://doi.org/10.3390/fractalfract6030136
[ 13 ] Yasin M. W., Iqbal M. S., Ahmed N., et al, Numerical scheme and stability analysis of stochastic Fitzhugh–Nagumo model. Results Phys. 2022, 32, 105023. https://doi.org/10.1016/j.rinp.2021.105023
[ 14 ] Han, C., Wang, Y.L., Li, Z.Y. A high-precision numerical approach to solving space fractional Gray-Scott model. Appl. Math. Lett. 2022, 125, 107759. https://doi.org/10.1142/S0218348X21502467
[ 15 ] Tu H., Wang Y., Lan, Q., et al, A Chebyshev-Tau spectral method for normal modes of underwater sound propagation with a layered marine environment. J. Sound Vib. 2021. 492, 115784. https://doi.org/10.1016/j.jsv.2020.115784
[ 16 ] Hammad D A, El-Azab M S. Chebyshev–Chebyshev spectral collocation method for solving the generalized regularized long wave (GRLW) equation. Appl. Math. Comput. 2016, 285: 228-240. http://dx.doi.org/10.1016/j.amc.2016.03.033