A Trigonometric Approach to Time Fractional FitzHugh-Nagumo Model on Nerve Pulse Propagation

Year 2022, , 135 - 145, 09.09.2022

Berat Karaagac

https://doi.org/10.36753/mathenot.1025072

Cited By: 1

Abstract

The aim of this paper is to put on display the numerical solutions and dynamics
of time fractional Fitzhugh-Nagumo model, which is an important nonlinear
reaction-diffusion equation. For this purpose, finite element method based
on trigonometric cubic B-splines are used to obtain numerical solutions of the model.
In this model, the derivative which is fractional order is taken in terms of Caputo.
Thus, time dicretization is made using $L1$ algorithm for Caputo
derivative and space discretization is made using trigonometric cubic B-
spline basis. Also, the non-linear term in the model is linearized by the
Rubin Graves type linearization. The error norms are calculated for measuring the
accuracy of the finite element method. The comparison of numerical and
exact solutions are exhibited via tables and graphics.

Keywords

Time fractional Fitzhugh-Nagumo model, finite element method, collocation

References

• [1] Freed, A. D., Diethelm K.: Fractional calculus in biomechanics: a 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad. Biomechanics and modeling in mechanobiology. 5(4), 203-215 (2006).
• [2] Ullah, N., Khan W., Wang S.: High performance direct torque control of electrical aerodynamics load simulator using fractional calculus. Acta Polytechnica Hungarica 11 (10), 59-78 (2014).
• [3] Baleanu, D., Atangana, A., Oukouomi Noutchie,S. C., Kurulay, M., Bildik, N.,Kilicman, A.: Fractional calculus: models and numerical methods. World Scientific, 3, 2012.
• [4] Changpin, L., Zeng, F.: Numerical methods for fractional calculus. Chapman and Hall/CRC, 2019.
• [5] Oruç, Ö., Esen, A., Bulut F. : Highly accurate numerical scheme based on polynomial scaling functions for equal width equation. Wave Motion: 102760,(2021).
• [6] Günerhan, H., Çelik, E.: Analytical and approximate solutions of fractional partial differential-algebraic equations. Applied Mathematics and Nonlinear Sciences 5 (1): 109-120 (2020).
• [7] Mahdy,AmrM.S.:Numericalsolutionsforsolvingmodeltime-fractionalFokker–Planckequation.NumericalMethods for Partial Differential Equations 37 (2), 1120-1135 (2021).
• [8] Dokuyucu, M. A., Baleanu,D., Çelik E. : Analysis of Keller-Segel model with Atangana-Baleanu fractional derivative, Filomat 32 (16): 5633-5643 (2018).
• [9] Owolabi,K.M.,Hammouch,Z.:SpatiotemporalpatternsintheBelousov–ZhabotinskiireactionsystemswithAtangana– Baleanu fractional order derivative. Physica A: Statistical Mechanics and its Applications 523, 1072-1090(2019).
• [10] Onal, M., Esen, A. : A Crank-Nicolson approximation for the time fractional Burgers equation, Applied Mathematics and Nonlinear Sciences 5 (2), 177-184(2020).
• [11] Oruç,Ö., Esen, A., Bulut, F.: A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers’ equation. Discrete & Continuous Dynamical Systems-S 12 (3), 533(2019).
• [12] Owolabi, Kolade M., Atangana A. : Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative. Chaos, Solitons & Fractals 105,111-119(2017).
• [13] Oruç,Ö., Esen, A., Bulut, F.: A Haar wavelet approximation for two-dimensional time fractional reaction–subdiffusion equation, Engineering with Computers 35 (1), 75-86(2019).
• [14] Owolabi, K. M.: Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order. Communications in Nonlinear Science and Numerical Simulation 44, 304-317(2017).
• [15] FitzHugh R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophysics, 17,257- 278(1955)
• [16] Nagumo J., Arimoto S., Yoshizawa S: An active pulse transmission line simulating nerve axon. Proc. IRE. 50, 2061– 2070 (1962).
• [17] Xu, B., Binczak, S., Jacquir, S., Pont, O., Yahia, H.: Parameters analysis of FitzHugh-Nagumo model for a reliable simulation. Annu Int Conf IEEE Eng Med Biol Soc. 2014;2014:4334-7. doi: 10.1109/EMBC.2014.6944583. PMID: 25570951.
• [18] Alphen ,F.I. M: Model order reductionon Fitz Hugh-Nagumo model. BSthesis. University of Twente,2 020.
• [19] Singh, S. : Mixed-Type Discontinuous Galerkin Approach for Solving the Generalized FitzHugh–Nagumo Reaction– Diffusion Model. International Journal of Applied and Computational Mathematics 7 (5), 1-16 (2021).
• [20] Yongheng, W., 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. Journal of Computational Physics 428, 110081 (2021).
• [21] Hemami, M., Parand,K., Rad J. A.: An efficient meshfree machine learning approach to simulate the generalized Fitzhugh-Nagumo equation inspired by neuroscience. In: The 51th Annual Iranian Mathematics Conference, 16–19 February, University of Kashan .1-4,(2021)
• [22] Jiwari,R.,Gupta,R.K.,KumarV.:Polynomialdifferentialquadraturemethodfornumericalsolutionsofthegeneralized Fitzhugh–Nagumo equation with time-dependent coefficients. Ain Shams Engineering Journal 5 (4), 1343-1350(2014).
• [23] Liu,F.,Turner,I.,Yang,Q.,Burrage,K.:AnumericalmethodforthefractionalFitzhugh–Nagumomonodomainmodel. Anziam Journal 54, C608-C629(2012).
• [24] Kumar, D., Singh,J., Baleanu, D.:A new numerical algorithm for fractional Fitzhugh–Nagumo equation arising in transmission of nerve impulses. Nonlinear Dynamics 91 (1), 307-317(2018).
• [25] Injrou, S., Mahmoud, S., Kassem, A.:A Stable Numerical Scheme for Fitzhugh-Nagumo Time-Fractional Partial Differential Equation. Tishreen University Journal-Basic Sciences Series 41 (4), (2019).
• [26] Chapwanya, M., Jejeniwa, O. A., Appadu, A. R., Lubuma, J. M.-S.: An explicit nonstandard finite difference scheme for the FitzHugh–Nagumo equations. International Journal of Computer Mathematics 96 (10), 1993-2009(2019).
• [27] Jiwari,R.,Gupta,R.K.,Kumar,V.:Polynomialdifferentialquadraturemethodfornumericalsolutionsofthegeneralized Fitzhugh–Nagumo equation with time-dependent coefficients. Ain Shams Engineering Journal 5 (4), 1343-1350(2014). ̇
• [28] ErsoyHepson,O.,Dag ̆,I,SakaB.,AyB.:TheCubicB-splineLeastSquaresFiniteElementMethodfortheNumerical Solutions of Regularized Long Wave Equation. International Journal of Computer Mathematics, 1-12 (2021). https://doi.org/10.1080/00207160.2021.1940979
• [29] Yuste, Santos B., Acedo, L. An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM Journal on Numerical Analysis 42 (5), 1862-1874(2005).
• [30] Tchier,F.,IncM.,KorpinarZ.S.:Solutionsofthetimefractionalreaction–diffusionequationswithresidualpowerseries method. Advances in Mechanical Engineering 8 (10) : 1687814016670867(2016).