Haar Wavelet Method for Solving Fitz Hugh-Nagumo Equation
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:56767", author = "G.Hariharan and K.Kannan", title = "Haar Wavelet Method for Solving Fitz Hugh-Nagumo Equation", 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.", volume = "4", number = "7", pages = "919-5", }
