An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation

Ömer Oruç

doi:10.33187/jmsm.1931851

An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation

Abstract

In the present study, we proposed an efficient numerical approach for Fitzhugh-Nagumo equation which is a nonlinear partial differential equation and has many applications in various branches of science. The suggested numerical method uses a meshless method called as enriched radial point interpolation method for discretization of space variables which forms a large system of ordinary differential equations (ODEs). In general, the acquired ODEs system is stiff therefore to obtain good results a small time step size should be used for time integration. In order to put relaxation on the time step size a splitting approach is utilized which divide the equation into non-stiff and stiff terms. Then a Rosenbrock method is devised to solve stiff term and a strong stability preserving Runge Kutta method is proposed to solve non-stiff term. To show efficiency of the suggested method some numerical tests are fulfilled and acquired results are compared with available numerical techniques in literature such as collocation, meshless and finite difference methods, and with exact solution. The comparisons reveal accuracy of the suggested method.

Keywords

Enriched radial point interpolation method, Meshless method, Rosenbrock method, Fitzhugh-Nagumo equation, Time splitting

References

A. M. Wazwaz, A. Gorguis, An analytic study of Fisher’s equation by using Adomian decomposition method, Appl. Math. Comput., 154(3) (2004), 609-620. https://doi.org/10.1016/S0096-3003(03)00738-0
A. H. Bhrawy, A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Appl. Math. Comput., 222 (2013), 255-264. https://doi.org/10.1016/j.amc.2013.07.056
R. Fitzhugh, Impulse and physiological states in models of nerve membrane, Biophys. J., 1(6) (1961), 445-466. https://doi.org/10.1016/S0006-3495(61)86902-6
J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE., 50(10) (1962), 2061-2070. https://doi.org/10.1109/JRPROC.1962.288235
M. E. Schonbek, Boundary value problems for the Fitzhugh-Nagumo equations, J. Differ. Equ., 30(1) (1978), 119-147. https://doi.org/10.1016/0022-0396(78)90027-X
G. Arioli, K. Hans, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation, Nonlinear Anal. Theory Methods Appl., 113 (2015), 51-70. https://doi.org/10.1016/j.na.2014.09.023
X. Ding, Limit cycle of Fitzhugh-Nagumo equation, J. Phys. Conf. Ser., 2669 (2023), Article ID 012002. https://doi.org/10.1088/1742-6596/2669/1/012002
D. E. Jackson, Existence and regularity for the Fitzhugh-Nagumo equations with inhomogeneous boundary conditions, Nonlinear Anal. Theory Methods Appl., 14(3) (1990), 201-216. https://doi.org/10.1016/0362-546X(90)90029-G
S. Cai, K. Nagatou, Y. Watanabe, A numerical verification method for a system of FitzHugh-Nagumo type, Numer. Funct. Anal. Optim., 33(10) (2012), 1195-1220. https://doi.org/10.1080/01630563.2012.677918
A. Czechowski, P. Zgliczynski, Existence of periodic solutions of the FitzHugh--Nagumo equations for an explicit range of the small parameter, SIAM J. Appl. Dyn. Syst., 15(3) (2016), 1615-1655. https://doi.org/10.1137/15M1007707

S. Kutluay, A. Esen, A finite difference solution of the regularized long-wave equation, Math. Probl. Eng., 2006(1) (2006), Article ID 085743. https://doi.org/10.1155/MPE/2006/85743
A. Rouatbi, M. Labidi, K. Omrani, Conservative difference scheme of solitary wave solutions of the generalized regularized long-wave equation, Indian J. Pure Appl. Math., 51 (2020), 1317-1342. https://doi.org/10.1007/s13226-020-0468-7
A. Esen, S. Kutluay, Application of a lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput., 174(2) (2006), 833-845. https://doi.org/10.1016/j.amc.2005.05.032
N. M. Yağmurlu, Y. Uçar, İ. Ç. elikkaya, Operator splitting for numerical solutions of the RLW equation, J. Appl. Anal. Comput., 8(5) (2018), 1494-1510. http://dx.doi.org/10.11948/2018.1494
Ö. Oruç, Two meshless methods based on pseudo spectral delta-shaped basis functions and barycentric rational interpolation for numerical solution of modified Burgers equation, Int. J. Comput. Math., 98(3) (2021), 461-479. https://doi.org/10.1080/00207160.2020.1755432
Ö. Oruç, A meshless multiple-scale polynomial method for numerical solution of 3d convection-diffusion problems with variable coefficients, Eng. Comput., 36(4) (2020), 1215-1228. https://doi.org/10.1007/s00366-019-00758-5
S. Kutluay, N. M. Yağmurlu, A. S. Karakaş, A novel perspective for simulations of the modified equal-width wave equation by cubic Hermite B-spline collocation method, Wave Mot., 129 (2024), Article ID 103342. https://doi.org/10.1016/j.wavemoti.2024.103342
A. S. Karakas¸, M. Yağmurlu, A novel approach for simulations of EW equation by trigonometric collocation method, Fundam. Contempl. Math. Sci., 7(1) (2026), 70-90. https://doi.org/10.54974/fcmathsci.1706691
D. Olmos, B. D. Shizgal, Pseudospectral method of solution of the Fitzhugh-Nagumo equation, Math. Comput. Simul., 79(7) (2009), 2258-2278. https://doi.org/10.1016/j.matcom.2009.01.001
R. A. Van Gorder, K. Vajravelu, A variational formulation of the Nagumo reaction-diffusion equation and the Nagumo telegraph equation, Nonlinear Anal. Real World Appl., 11(4) (2010), 2957-2962. https://doi.org/10.1016/j.nonrwa.2009.10.016
R. Jiwari, R. K. Gupta, V. Kumar, Polynomial differential quadrature method for numerical solutions of the generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Ain Shams Eng. J., 5(4) (2014), 1343-1350. https://doi.org/10.1016/j.asej.2014.06.005
I. Ahmad, M. Ahsan, Z. Din, et al., An efficient local formulation for time-dependent PDEs, Math., 7(3) (2019), Article ID 216. https://doi.org/10.3390/math7030216
K. M. Agbavon, A. R. Appadu, Construction and analysis of some nonstandard finite difference methods for the FitzHugh-Nagumo equation, Numer. Methods Partial Differ. Equ., 36(5) (2020), 1145-1169. https://doi.org/10.1002/num.22468
S. Singh, Mixed-Type discontinuous Galerkin approach for solving the generalized FitzHugh-Nagumo reaction-diffusion model, Int. J. Appl. Comput. Math., 7 (2021), Article ID 207. https://doi.org/10.1007/s40819-021-01153-9
B. Inan, K.K. Ali, A. Saha, et al., Analytical and numerical solutions of the Fitzhugh-Nagumo equation and their multistability behavior, Numer. Methods Partial Differ. Equ., 37(1) (2021), 7-23. https://doi.org/10.1002/num.22516
J. Zhang, S. Lin, J. Wang, Stability and convergence analysis of Fourier pseudo-spectral method for FitzHugh-Nagumo model, Appl. Numer. Math., 157 (2020), 563-578. https://doi.org/10.1016/j.apnum.2020.07.009
S. Abbasbandy, E. Shivanian, K. H. AL-Jizani, On the analysis of a kind of nonlinear Sobolev equation through locally applied pseudo-spectral meshfree radial point interpolation, Numer. Methods Partial Differ. Equ., 37(1) (2020), 462-478. https://doi.org/10.1002/num.22536
W. Chang, C. S. Chen, X-Y. Liu, J. Li, Localized meshless methods based on polynomial basis functions for solving axisymmetric equations, Math. Comput. Simul., 177 (2020), 487-499. https://doi.org/10.1016/j.matcom.2020.05.006
Ö. Oruç, A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates, Appl. Math. Model., 122 (2023), 555-571. https://doi.org/10.1016/j.apm.2023.03.014
Z. Avazzadeh, O. Nikan, A. T. Nguyen, et al., A localized hybrid kernel meshless technique for solving the fractional Rayleigh-Stokes problem for an edge in a viscoelastic fluid, Eng. Anal. Bound. Elem., 146 (2023), 695-705. https://doi.org/10.1016/j.enganabound.2022.11.003
Ö. Oruç, A strong-form local meshless approach based on radial basis function-finite difference (RBF-FD) method for solving multi-dimensional coupled damped Schr¨odinger system appearing in Bose-Einstein condensates, Commun. Nonlinear Sci. Numer. Simul., 104 (2022), Article ID 106042. https://doi.org/10.1016/j.cnsns.2021.106042
G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, Vol. 6, World Scientific Publishers, Singapore, 2007. https://doi.org/10.1142/6437
R. Jiwari, A. Gerisch, A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems, Eng. Comput., 38(6) (2021), 2666-2691. https://doi.org/10.1108/EC-05-2020-0291
Y. T. Gu, An enriched radial point interpolation method based on weak-form and strong-form, Mech. Adv. Mater. Struct., 18(8) (2011), 578-584. https://doi.org/10.1080/15376494.2011.621832
Y. T. Gu, W. Wang, L. C. Zhang, et al., An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields, Eng. Fract. Mech., 78 (2011), 175-190. https://doi.org/10.1016/j.engfracmech.2010.10.014
H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2010. https://doi.org/10.1017/CBO9780511617539
H. P. Langtangen, S. Linge, Finite Difference Computing with PDEs: A Modern Software Approach, Springer, 2017. https://doi.org/10.1007/978-3-319-55456-3
G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5(3) (1968), 506-517. https://doi.org/10.1137/0705041
E. Hairer, G. Wanner, Solving Ordinary Differential Equations II, Springer, Berlin, Heidelberg, 1996. https://doi.org/10.1007/978-3-642-05221-7
J. G. Verwer, E. Spee, J. Blom, et al., A second-order Rosenbrock method applied to photochemical dispersion problems, SIAM J. Sci. Comput., 20(4) (1999), 1456-1480. https://doi.org/10.1137/S1064827597326651
K. Dekker, J. G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, Elsevier-North Holland, Amsterdam, 1984.
A. Tambue, I. Berre, J. M. Nordbotten, Efficient simulation of geothermal processes in heterogeneous porous media based on the exponential Rosenbrock–Euler and Rosenbrock-type methods, Adv. Water Resour., 53 (2013), 250-262. https://doi.org/10.1016/j.advwatres.2012.12.004
R. Djouad, B. Sportisse, N. Audiffren, Numerical simulation of aqueous- phase atmospheric models: use of a non-autonomous rosenbrock method, Atmos. Environ., 36(5) (2002), 873-879. https://doi.org/10.1016/S1352-2310(01)00497-6
D. A. Voss, A. Q. M. Khaliq, Parallel Rosenbrock methods for chemical systems, Comput. Chem., 25(1) (2001), 101-107. https://doi.org/10.1016/S0097-8485(00)00093-0
S. Yıldız, Numerical solution of semi-linear advection-diffusion-reaction equations by discontinuous Galerkin methods, Ph.D. Thesis, Middle East Technical University, 2016.
E. J. Kubatko, B. A. Yeager, D. I. Ketcheson, Optimal strong-stability-preserving Runge-Kutta time discretizations for discontinuous Galerkin methods, J Sci. Comput., 60 (2014), 313-344. https://doi.org/10.1007/s10915-013-9796-7
P. Virtanen, R. Gommers, E. T. E. Oliphant, et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods., 17 (2020), 261-272. https://doi.org/10.1038/s41592-019-0686-2
C. R. Harris, K. J. Millman, S. J. van der Walt, et al., Array programming with NumPy, Nature, 585 (2020), 357-362. https://doi.org/10.1038/s41586-020-2649-2
J. D. Hunter, Matplotlib: A 2D graphics environment, Comput. Sci. Eng., 9(3) (2007), 90-95. https://doi.org/10.1109/MCSE.2007.55

Details

Primary Language

English

Subjects

Partial Differential Equations, Applied Mathematics (Other)

Journal Section

Research Article

Authors

Ömer Oruç ^*
0000-0002-6655-3543
Türkiye

Early Pub Date

July 24, 2026

Publication Date

September 29, 2026

Submission Date

April 16, 2026

Acceptance Date

July 20, 2026

Published in Issue

Year 2026 Volume: 9 Number: 3

DOI

https://doi.org/10.33187/jmsm.1931851

IZ

https://izlik.org/JA59TY24RJ

APA

Oruç, Ö. (2026). An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation. Journal of Mathematical Sciences and Modelling, 9(3), 152-161. https://doi.org/10.33187/jmsm.1931851

AMA

1.Oruç Ö. An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation. Journal of Mathematical Sciences and Modelling. 2026;9(3):152-161. doi:10.33187/jmsm.1931851

Chicago

Oruç, Ömer. 2026. “An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation”. Journal of Mathematical Sciences and Modelling 9 (3): 152-61. https://doi.org/10.33187/jmsm.1931851.

EndNote

Oruç Ö (September 1, 2026) An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation. Journal of Mathematical Sciences and Modelling 9 3 152–161.

IEEE

[1]Ö. Oruç, “An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation”, Journal of Mathematical Sciences and Modelling, vol. 9, no. 3, pp. 152–161, Sept. 2026, doi: 10.33187/jmsm.1931851.

ISNAD

Oruç, Ömer. “An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation”. Journal of Mathematical Sciences and Modelling 9/3 (September 1, 2026): 152-161. https://doi.org/10.33187/jmsm.1931851.

JAMA

1.Oruç Ö. An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation. Journal of Mathematical Sciences and Modelling. 2026;9:152–161.

MLA

Oruç, Ömer. “An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation”. Journal of Mathematical Sciences and Modelling, vol. 9, no. 3, Sept. 2026, pp. 152-61, doi:10.33187/jmsm.1931851.

Vancouver

1.Ömer Oruç. An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation. Journal of Mathematical Sciences and Modelling. 2026 Sep. 1;9(3):152-61. doi:10.33187/jmsm.1931851

29237 Journal of Mathematical Sciences and Modelling 29238

The published articles in JMSM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

download?token=eyJhdXRoX3JvbGVzIjpbXSwiZW5kcG9pbnQiOiJqb3VybmFsIiwib3JpZ2luYWxuYW1lIjoibG9nby5wbmciLCJwYXRoIjoiNDU3Yi9kNTk0L2RiNWMvNmFiYTgyM2Q4NGUwODkuNjY5MzgxNTIucG5nIiwiZXhwIjoxNzkwNjExNTMzLCJub25jZSI6ImJlM2RjNTdiYTRmYmZhODZiYzcwNWEzZDAwZjk3OGZhIn0.n9YOtmz9GKxZ76z7deSN6NJT4P16Drf7DEE0YG-51e0

Technical Support by PiCODE Software

https://www.picodesoftware.com

An Enriched Radial Point Interpolation - Rosenbrock Method for Numerical Simulation of Fitzhugh-Nagumo Equation

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Early Pub Date

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

Cite