Haar Wavelet Method for Solving Fitz Hugh-Nagumo Equation

Authors: G.Hariharan, K.Kannan

Abstract:

In this paper, we develop an accurate and efficient Haar wavelet method for well-known FitzHugh-Nagumo equation. The proposed scheme can be used to a wide class of nonlinear reaction-diffusion equations. The power of this manageable method is confirmed. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.

Keywords: FitzHugh-Nagumo equation, Haar wavelet method, adomain decomposition method, computationally attractive.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1334255

References:

[1] S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method, Applied Mathematical Modelling 32 (2008) 2706-2714.
[2] H.A. Abdusalam, Analytic and approximate solutions for Nagumo telegraph reaction diffusion equation, Appl. Math. and Comput. 157 (2004) 515-522.
[3] M. Argentina, P. Coullet, V. Krinsky, Head-on collisions of waves in an excitable Fitzhugh-Nagumo system: a transition from wave annihilation to classical wave behavior, J. Theor. Biol.205 (2000) 47.
[4] D.G. Aronson, H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math. 30 (1978) 33-76.
[5] C.Cattani, Haar wavelet spline, J.Interdisciplinary Math.4 (2001) 35-47.
[6] D.Y. Chen, Y. Gu, Cole-Hopf quotient and exact solutions of the generalized Fitzhugh-Nagumo equations, Acta Math. Sci. 19 (1) (1999) 7-14.
[7] C.F.Chen, C.H.Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEEE Proc.Pt.D144 (1)(1997) 87-94.
[8] R. Fitzhugh, Impulse and physiological states in models of nerve membrane, Biophys. J. 1 (1961) 445-466.
[9] P. -L. Gong, J. -X. Xu, S. -J. Hu, Resonance in a noise-driven excitable neuron model, Chaos Solitons Fract. 13 (2002) 885.
[10] A. Haar Zur theorie der orthogonalen Funktionsysteme.Math. Annal 1910:69:331- 71.
[11] G.Hariharan, K.Kannan, K.R.Sharma, Haar wavelet method for solving Fisher-s equation, Appl.Math. and Comput. (2009)
[12] G.Hariharan, K.Kannan, K.R.Sharma, Haar wavelet in estimating depth pro£le of soil temperature, Appl. Math. and Comput. (2009).
[13] A.L. Hodgkin, A.F. Huxley, Aquantitive description of membrane current and its application th conduction and excitation in nerve, J. Physiol. 117 (1952) 500.
[14] C.H.Hsiao, W.J.Wang, Haar wavelet approach to nonlinear stiff systems, Math.Comput.Simulat, 57, pp.347-353, 2001.
[15] D.S.Jone, B.D.Sleeman, Differential Equations and Mathematical Biology, Chapman & Hall / CRC, New York,2003.
[16] T. Kawahara, M. Tanaka, Interaction of travelling fronts: an exact solution of a nonlinear diffusion equation, Phys. Lett. A 97 (1983) 311- 314.
[17] U.Lepik, Numerical solution of evolution equations by the Haar wavelet method, J.Appl.Math.Comput.185 (2007) 695-704.
[18] U.Lepik, Numerical solution of differential equations using Haar wavelets, Math.and Comp. Simulation 68(2005) 127-143.
[19] U.Lepik, Application of the Haar wavelet transform to solving integral and differential Equations, Proc.Estonian Acad. Sci.Phys.Math., 2007, 56, 1, 28-46.
[20] H. Li, Y. Guo, New exact solutions to the Fitzhugh-Nagumo equation, Appl. Math. and Comput. 180 (2006) 524-528.
[21] D. Margerit, D. Barkley, Selection of twisted scroll waves in threedimensional excitable media, Phys. Rev. Lett. 86 (1) (2001) 175.
[22] J.S. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE 50 (1962) 2061-2071.
[23] M.C. Nucci, P.A. Clarkson, The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh-Nagumo equation, Phys. Lett. A 164 (1992) 49-56.
[24] H.C. Rosu, O. Cornejo-Perez, Super symmetric pairing of kinks for polynomial nonlinearities, Phys. Rev. E 71 (2005) 1-13.
[25] M. Shih, E. Momoniat, F.M. Mahomed, Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation, J. Math. Phys. 46 (2005) 023503.
[26] A. Slavova, P. Zecca, CNN model for studying dynamics and travelling wave solutions of FitzHugh-Nagumo equation, Journal of Computational and Applied Mathematics 151 (2003) 13-24.
[27] A.M. Wazwaz, The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Applied Mathematics and Computation 188 (2007) 1467.
[28] M.P.Zorzano, L. Vazquez, Emergence of synchronous oscillations in neural networks excited by noise, Physica D 3078 (2003) 1.