THE EFFECT ON THE SOLUTION OF THE FITZHUGH-NAGUMO EQUATION BY THE EXTERNAL PARAMETER $\alpha$ USING THE GALERKIN METHOD

Pius W. M. Chin

doi:10.11948/20220316

2023 Volume 13 Issue 4

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Pius W. M. Chin. THE EFFECT ON THE SOLUTION OF THE FITZHUGH-NAGUMO EQUATION BY THE EXTERNAL PARAMETER $\alpha$ USING THE GALERKIN METHOD[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1983-2005. doi: 10.11948/20220316

Citation:

Pius W. M. Chin. THE EFFECT ON THE SOLUTION OF THE FITZHUGH-NAGUMO EQUATION BY THE EXTERNAL PARAMETER $\alpha$ USING THE GALERKIN METHOD[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1983-2005. doi: 10.11948/20220316

THE EFFECT ON THE SOLUTION OF THE FITZHUGH-NAGUMO EQUATION BY THE EXTERNAL PARAMETER $\alpha$ USING THE GALERKIN METHOD

Pius W. M. Chin^1, ,

1.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa 0204, Ga-rankuwa, Pretoria, South Africa

Corresponding author: Pius W. M. Chin, Email: pius.chin@smu.ac.za
Fund Project: The research contained in this article has been supported by Sefako Makgatho Health Sciences University, Medunsa 0204, Ga-rankuwa, Pretoria, South Africa

Abstract

Abstract

The Fitzhugh-Naguno equation is one of the most popular and attractive equation in real life. This equation is applicable in many different areas of physics, biology, population genetics and applied sciences to mention a few. In this paper, we design and analyze a coupled scheme consisting of the non-standard finite difference and the Galerkin methods in both time and space variables respectively. We show analytically by the use of the Galerkin method and the compactness theorem that the solution of this equation exists uniquely in appropriate spaces with the parameter $\alpha$ that determines the main dynamics of the equation, under controlled. We further show numerically that the above scheme is stable and converge optimally in specified norms with its numerical solution replicating the qualitative properties of the exact solution. We finally present numerical experiments with the help of an example and a careful choice of $\alpha$ to validate the theoretical results.
- Fitzhugh-Nagumo equation /
- non-standard finite difference /
- Galerkin and compactness methods /
- stability and optimal rate of convergence
MSC: 35J47, 35J70, 65N15, 65N30

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References

[1]	S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with the homotophy analysis method, Applied Mathematical Modeling, 2008, 32, 2706-2714. doi: 10.1016/j.apm.2007.09.019 CrossRef Google Scholar
[2]	A. R. Adams, Sobolev Space, Academic Press, New York, 1975. Google Scholar
[3]	A. A. Aderogha and M. Chapwanya, An explicit nonstandard finite difference scheme for the Allen-Cahn equation, Journal of Difference Equations and Applications, 2015, 21(10), 875-886. doi: 10.1080/10236198.2015.1055737 CrossRef Google Scholar
[4]	J. G. Alford, Bifurcation structure of rotating wave solutions of the Fitzhugh-Nagumo equations, Communications in Nonlinear Science and Numerical Simulation, 2009, 14, 3282-3291. doi: 10.1016/j.cnsns.2009.01.011 CrossRef Google Scholar
[5]	R. Anguelov and J. M. S. Lubuma, Contributions to the mathematics of the nonstandard finite difference method and applications, Numerical Methods for Partial Differential Equations, 2001, 17(5), 518-543. doi: 10.1002/num.1025 CrossRef Google Scholar
[6]	R. Anguelov and J. M. S. Lubuma, Nonstandard finite difference method by nonlocal approximation, Mathematics and Computers in Simulation, 2003, 61(3), 465-475. Google Scholar
[7]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, 1978, 30, 33-76. doi: 10.1016/0001-8708(78)90130-5 CrossRef Google Scholar
[8]	J. P. Aubin, Un théoréme de compacité, C. R. Acad. Sc. Paris, 1963, 256, 5012-5014. Google Scholar
[9]	P. W. M. Chin, Optimal Rate of Convergence for a Nonstandard Finite Difference Galerkin Method Applied to Wave Equation Problems, Journal of Applied Mathematics, 2013. Google Scholar
[10]	P. W. M. Chin, J. K. Djoko and J. M. S. Lubuma, Reliable numerical schemes for a linear diffusion equation on a nonsmooth domain, Applied Mathematics Letters, 2010, 23(5), 544-548. doi: 10.1016/j.aml.2010.01.008 CrossRef Google Scholar
[11]	P. W. M. Chin, The Galerkin reliable scheme for the numerical analysis of the Burgers'-Fisher equation, Progress in Computational Fluid Dynamics, 2021, 21(4), 234-247. doi: 10.1504/PCFD.2021.116530 CrossRef Google Scholar
[12]	P. W. M. Chin, The study of the numerical treatment of the Real Ginzburg-Landau equation using the Galerkin method, Numerical Functional Analysis and Optimization, 2021, 42(10), 1154-1177. DOI: 10.1080/016305632021.1948863. CrossRef Google Scholar
[13]	P. G. Ciarlet, The finite element method for elliptic problems, Elsevier, 1978. Google Scholar
[14]	L. C. Evan, Partial Differential Equations. Graduate, Studies in Mathematics, American Mathematical Society, Rhode Island, 1998, 19. Google Scholar
[15]	R. Fitzhugh, Impulse and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1961, 1(6), 445-466. doi: 10.1016/S0006-3495(61)86902-6 CrossRef Google Scholar
[16]	A. Hodgkin and A. Huxley, A quanlitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 1952, 117(4), 500. doi: 10.1113/jphysiol.1952.sp004764 CrossRef Google Scholar
[17]	S. Hussain, N. Malluwawadu and P. Zhu, A weak Galerkin finite element method for the second order elliptic problem with mixed boundary conditions, Journal of Applied Analysis and Computation, 2018, 8(5), 1452-1463. DOI: 10.11948/2018.1452. CrossRef Google Scholar
[18]	D. E. Jackson, Existence and regularity of the Fitzhugh-Nagumo equations, Nonlinear Analysis: Real World Applications, 2002, 3, 515-541. doi: 10.1016/S1468-1218(01)00046-3 CrossRef Google Scholar
[19]	D. E. Jackson, Error estimates for the semi-discrete Galerkin approximation of the Fitzhugh-Nagumo equations, Appl. Math. Comput., 1992, 50, 93-114. Google Scholar
[20]	C. Johnson, S. Larsson, V. Thomée and L. B. Wahlbin, Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data, Mathematics of Computation, 1987, 180, 331-357. Google Scholar
[21]	M. M. Khader and K. M. Abualnaja, Galerkin-FEM for obtaining the numerical solution of the linear fractional Klein-Gorden equation, Journal of Applied Analysis and Computation, 2019, 9(1), 261-270. doi: 10.11948/2019.261. CrossRef Google Scholar
[22]	T. Kawahara and M. Tanaka, Interaction of travelling fronts: an exact solution of a nonlinear diffusion equation, Physics Letters A., 1983, 97, 311-314. doi: 10.1016/0375-9601(83)90648-5 CrossRef Google Scholar
[23]	H. Li and Y. Guo, New exact solutions to the Fitzhugh-Nagumo equation, Applied Mathematics and Computation, 2006, 180, 524-528. doi: 10.1016/j.amc.2005.12.035 CrossRef Google Scholar
[24]	J. L. Louis, E. Magenes and P. Kenneth, Non-homogeneous Boundary Value Problems and Applications, Springer Berlin, 1972, 1. Google Scholar
[25]	J. M. S. Lubuma. E. Mureithi and Y. A. Terefe, Analysis and dynamically consistent numerical scheme for the SIS model and related reaction diffusion equation, AIP Conf. Proc., 2011, 168. Google Scholar
[26]	J. M. S. Lubuma, E. Mureithi and Y. A. Terefe, Nonstandard discretization of the SIS Epidemiological model with and without diffusion, Contemporary Mathematics, 2014, 618. Google Scholar
[27]	R. E. Mickens, Nonstandard finite difference models of differential equations, World Scientific, 1994. Google Scholar
[28]	S. M. Moghadas, M. E. Alexander, B. D. Corbett and A. B. Gumel, A positivity-preserving Mickens-type discretization of an epidemic model, The Journal of Difference Equations and Applications, 2003, 9(11), 1037-1051. doi: 10.1080/1023619031000146913 CrossRef Google Scholar
[29]	J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 1962, 50(10), 2061-2070. doi: 10.1109/JRPROC.1962.288235 CrossRef Google Scholar
[30]	J. Nagumo, S. Yoshizawa and S. Arimoto, Bistable transmission lines, Transactions on IEEE Circuit Theory, 1965, 12, 400-412. doi: 10.1109/TCT.1965.1082476 CrossRef Google Scholar
[31]	M. C. Nucci and P. A. Clarkson, The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh-Nagumo equation, Physics Letters A, 1992, 164, 49-56. doi: 10.1016/0375-9601(92)90904-Z CrossRef Google Scholar
[32]	K. C. Partida, On the use of nonstandard finite difference methods, Journal of Difference Equations and Applications, 2005, 11(8), 735-758. doi: 10.1080/10236190500127471 CrossRef Google Scholar
[33]	W. Rudin, Functional Analysis, McGraw-Hill, New York, 1991. Google Scholar
[34]	S. Singh, Mixed-type discontinuous Galerkin approach for solving the generalized Fitzhugh-Nagumo reaction-diffusion model, Int. J. Appl. Comput. Math., 2021, 7, 207. doi. org/10.1007/s40819-021-01153-9. doi: 10.1007/s40819-021-01153-9 CrossRef Google Scholar
[35]	A. C. Scott, Neustor propagation on a tunnel diode loaded transmission line, Proceedings of IEEE, 1963, 51, 240-249. doi: 10.1109/PROC.1963.1715 CrossRef Google Scholar
[36]	J. Shen, Long time stability and convergence for the fully discrete nonlinear Galerkin methods, Appl. Anal., 1990, 38, 201-229. doi: 10.1080/00036819008839963 CrossRef Google Scholar
[37]	M. Shih. E. Momoniat and F. M. Mahomed, Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation, Journal of Mathematical Physics, 2005, 46, 023503. doi: 10.1063/1.1839276 CrossRef Google Scholar
[38]	K. Takashi, The sub-supersolution method for the Fitzhugh-Nagumo type reaction-diffusion system with heterogeneity, Discrete and Continuous Dynamical System, 2018, 38(5), 2441-2465. doi: 10.3934/dcds.2018.101. CrossRef Google Scholar
[39]	R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Amsterdam, North-Holland, 1984. Google Scholar
[40]	R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, Springer-Verlag, New York, 1997. Google Scholar
[41]	R. A. Van Gorder, A variational formulation of the Nagumo reaction-diffusion equation and the Nagumo telegraph equation, Nonlinear Anal. Real World Appl., 2010, 11, 2957-2962. doi: 10.1016/j.nonrwa.2009.10.016 CrossRef Google Scholar
[42]	B. van der Pol. and J. van der Mark, The heartbeat considered as a relaxation oscillation and an electrical model of the heart, In Philosophical Magazine series, 1928, 7(6), 763-775. Google Scholar
[43]	M. F. Wheeler, A priori $L^{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 1973, 10, 723-759. doi: 10.1137/0710062 CrossRef Google Scholar
[44]	W. Zhao and A. Gu, Regularity of pullback attractors and random equilibrium for non-autonomous system stochastic Fitzhugh-Nagumo system on unbounded domain, Journal of Applied Analysis and Computation, 2017, 7(4), 1285-1311. doi: 10.1948/2017.079. CrossRef Google Scholar