Mathematical Problems in Engineering

Volume 2008 (2008), Article ID 538489, 7 pages

http://dx.doi.org/10.1155/2008/538489

## Exp-Function Method for Solving Huxley Equation

Department of Mathematics, Kunming Teacher's College, Kunming, Yunnan 650031, China

Received 30 August 2007; Revised 13 January 2008; Accepted 24 January 2008

Academic Editor: Oleg Gendelman

Copyright © 2008 Xin-Wei Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Huxley equation is a core mathematical framework for modern biophysically based neural modeling. It is often useful to obtain a generalized solitary solution for fully understanding its physical meanings. There are many methods to solve the equation, but each method can only lead to a special solution. This paper suggests a relatively new method called the Exp-function method for this purpose. The obtained result includes all solutions in open literature as special cases, and the generalized solution with some free parameters might imply some fascinating meanings hidden in Huxley equation.

#### 1. Introduction

The core mathematical framework for modern biophysically based neural modeling was developed half a century ago by Hodgkin and Huxley [1]. In a series of papers published in 1952, they presented the results of an elegant series of electrophysiological experiments in which they investigated the flow of electric current through the surface membrane of the giant nerve fiber of a squid. Huxley equation [2–6] is a nonlinear partial differential equation of second order of the form

This equation is an evolution equation that describes the nerve propagation in biology from which molecular CB properties can be calculated. It also gives a phenomenological description of the behavior of the myosin heads II. This equation has many fascinating phenomena such as bursting oscillation [7], interspike [8], bifurcation, and chaos [9]. A generalized exact solution can gain an insight into these phenomena. There is not a universal method for nonlinear equations. The traditional approaches to this task are the variational iteration method [10–12], the homotopy perturbation method [13, 14], Adomain’s decomposition method [15], and the tanh method [16–18]; however, many methods may sometimes fail or the solution procedure is complex as the degree of nonlinearity increases, for example, calculation of Adomian polynomials in Adomian’s method is terribly tedious. Recently, the study showed that the homotopy perturbation method [19–21] and the variational iteration method [22, 23] can completely overcome the difficulty. If we are really determined to extract physical meanings from analytic formulations of biological processes, we must resort to amelioration of the classical methods using modern mathematical tools. Exp-function method [24–29] is at this moment the most promising candidate theory for this purpose.

#### 2. Basic Idea of Exp-Function Method

Rational approximation for soliton and soliton-like solutions was first proposed by Hirota [30] and further developed by many authors [31, 32]. In this paper, we will apply the Exp-function method to the discussed problem. The basic idea of the Exp-function was proposed in He’s monograph [30]. Some illustrative examples in [24–27] showed that this method is very effective to search for various solitary and periodic solutions of nonlinear equations. Zhu applied the method to some difference-differential equations [28, 29].

Consider the following general partial differential equation: We first unite the independent variables and into one wave variable , leading (2.1) to an ordinary differential equation, The Exp-function method is based on the assumption that traveling wave solutions can be expressed in the following form [30]: where , , , and are positive integers which are unknown to be further determined, and and are unknown constants. To determine the values of and , we balance the linear term of highest order in (2.2) with the highest-order nonlinear term. Similarly to determine the values of and , we balance the linear term of lowest order in (2.2) with the lowest-order nonlinear term.

#### 3. Application to Huxley Equation

Using the wave variable , we have The highest linear term is now given by Balancing the highest order of Exp-function in (3.2), we have , and this gives . Using the same method, we can also obtain that . Wu and He [33] systematically studied the choice of the values of the parameters, and revealing the solution very weakly depends upon the values of the parameters. An illustrating example of Dodd–Bullough–Mikhailov equation was given.

Wu and He considered the following three cases.

*Case 1. *, .
*Case 2. *, .
*Case 3. *, .

All cases led to the equivalent result. Bekir and Boz [34] pointed out that and are valid for most nonlinear partial differential equations.

For simplicity, we set and , so (2.3) reduces to Substituting (3.3) into (3.1), and by the help of Mathematica, we have where

Solving the system (3.5) simultaneously using Mathematica, we obtain the following results.

*Case 1. **Case 2. *
where and are free parameters.*Case 3. *
where is a free parameter.
For Case 1, we obtain the following solution of (1.1) by substituting (3.6) into (3.3):
For Case 2, we have
where or .
Case 3 leads to the
following exact solution:
where or .

To compare our results with those obtained in [16], we set in (3.9). Equation (3.9) becomes In (3.10), if we set , , and , (3.10) becomes In (3.11), if we set and , (3.11) becomes These are kink solutions obtained by the tanh-coth method in [16].

In (3.9), (3.10), and (3.11), if we set parameters as follows: (1) (2) and (3) and , respectively, we have These are the traveling solutions obtained by the tanh-coth method in [16].

The other three traveling solutions , , and in [16] are equivalent with , , and , respectively.

He and Wu [26] compared the method with the Sinh-function and the Tanh-function methods, and found a single generalized solitonary solution including all solutions obtained by Yomba using the subequation method.

#### 4. Discussions and Conclusions

The Exp-function method leads to generalized solitary solutions with some free parameters involving the known solutions in open literature. The free parameters might imply some physically meaningful results in biological process. Considering the generalized solution expressed in (3.9), in case , it turns out to a special solution in [16]; the physical understating of the special solution was given in [16]. Of course we can set equal to other values, resulting in different solitary shapes. The free parameter might be also relative to initial conditions.

#### References

- A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,”
*The Journal of Physiology*, vol. 117, no. 4, pp. 500–544, 1952. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Wang, L. Q. Chen, and X. Y. Fei, “Bifurcation control of the Hodgkin-Huxley equations,”
*Chaos, Solitons & Fractals*, vol. 33, no. 1, pp. 217–224, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Wang, L. Chen, and X. Fei, “Analysis and control of the bifurcation of Hodgkin-Huxley model,”
*Chaos, Solitons & Fractals*, vol. 31, no. 1, pp. 247–256, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “A modified Hodgkin-Huxley model,”
*Chaos, Solitons & Fractals*, vol. 29, no. 2, pp. 303–306, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - J.-H. He and X.-H. Wu, “A modified Morris—Lecar model for interacting ion channels,”
*Neurocomputing*, vol. 64, pp. 543–545, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-H. He, “Resistance in cell membrane and nerve fiber,”
*Neuroscience Letters*, vol. 373, no. 1, pp. 48–50, 2005. View at Publisher · View at Google Scholar · View at PubMed - L. X. Duan and Q. S. Lu, “Bursting oscillations near codimension-two bifurcations in the Chay Neuron model,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 7, no. 1, pp. 59–64, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Q. Liu, T. Fan, and Q. S. Lu, “The spike order of the winnerless competition (WLC) model and its application to the inhibition neural system,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 6, no. 2, pp. 133–138, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. J. Zhang, J. X. Xu, H. Yao, and R.-X. Wei, “Mechanism of bifurcation-dependent coherence resonance of an excitable neuron model,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 7, no. 4, pp. 447–450, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He and X.-H. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,”
*Chaos, Solitons & Fractals*, vol. 29, no. 1, pp. 108–113, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “Some asymptotic methods for strongly nonlinear equations,”
*International Journal of Modern Physics B*, vol. 20, no. 10, pp. 1141–1199, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Batiha, M. S. M. Noorani, and I. Hashim, “Numerical simulation of the generalized Huxley equation by He's variational iteration method,”
*Applied Mathematics and Computation*, vol. 186, no. 2, pp. 1322–1325, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “Addendum: new interpretation of homotopy perturbation method,”
*International Journal of Modern Physics B*, vol. 20, no. 18, pp. 2561–2568, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - S. H. Hashemi, H. R. Mohammadi, Daniali, and D. D. Ganji, “Numerical simulation of the generalized Huxley equation by He's homotopy perturbation method,”
*Applied Mathematics and Computation*, vol. 192, no. 1, pp. 157–161, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A.-M. Wazwaz and A. Gorguis, “An analytic study of Fisher's equation by using Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 154, no. 3, pp. 609–620, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Abdul-Majid Wazwaz, “Analytic study on Burgers, Fisher, Huxley equations and combined forms of these equations,”
*Applied Mathematics and Computation*, vol. 195, no. 2, pp. 754–761, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-L. Zhang, M.-L. Wang, and X.-R. Li, “The subsidiary elliptic-like equation and the exact solutions of the higher-order nonlinear Schrödinger equation,”
*Chaos, Solitons & Fractals*, vol. 33, no. 5, pp. 1450–1457, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - C. Dai, G. Zhou, and J. Zhang, “Exotic localized structures based on variable separation solution of $(2+1)$-dimensional KdV equation via the extended tanh-function method,”
*Chaos, Solitons & Fractals*, vol. 33, no. 5, pp. 1458–1467, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - A. Ghorbani and J. Saberi-Nadjafi, “He's homotopy perturbation method for calculating Adomian polynomials,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 2, pp. 229–232, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. K. Ghori, M. Ahmed, and A. M. Siddiqui, “Application of homotopy perturbation method to squeezing flow of a Newtonian fluid,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 2, pp. 179–184, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Rana, A. M. Siddiqui, Q. K. Ghori et al., “Application of He's homotopy perturbation method to Sumudu transform,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 2, pp. 185–190, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Yusufoglu, “Variational iteration method for construction of some compact and noncompact structures of Klein-Gordon equations,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 2, pp. 153–158, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Tari, D. D. Ganji, and M. Rostamian, “Approximate solutions of K (2,2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 2, pp. 203–210, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-H. Wu and J.-H. He, “Solitary solutions, periodic solutions and compacton-like solutions using the exp-function method,”
*Computers & Mathematics with Applications*, vol. 54, no. 7-8, pp. 966–986, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.H. He and L.N. Zhang, “Generalized solitary solution and compacton-like solution of the Jaulent–Miodek equations using the Exp-function method,”
*Physics Letters A*, vol. 372, no. 7, pp. 1044–1047, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,”
*Chaos, Solitons & Fractals*, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He and M. A. Abdou, “New periodic solutions for nonlinear evolution equations using Exp-function method,”
*Chaos, Solitons & Fractals*, vol. 34, no. 5, pp. 1421–1429, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-D. Zhu, “Exp-function method for the hybrid-lattice system,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 3, pp. 461–464, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-D. Zhu, “Exp-function method for the discrete mKdV lattice,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 3, pp. 465–468, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He,
*Non-Perturbative Methods for Strongly Nonlinear Problems*, Dissertation. de-Verlag im Internet GmbH, Berlin, Germany, 2006. View at Zentralblatt MATH · View at MathSciNet - R. K. Bullough and P. J. Caudre, Eds.,
*Solitons*, R. K. Bullough and P. J. Caudre, Eds., vol. 17 of*Topics in Current Physics*, Springer, Berlin, Germany, 1980. View at Zentralblatt MATH · View at MathSciNet - F. Lambert and M. Musette, “Solitary waves, padeons and solitons,” in
*Padé Approximation and Its Applications Bad Honnef 1983*, vol. 1071 of*Lecture Notes in Mathematics*, pp. 197–212, Springer, Berlin, Germany, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-H. Wu and J.-H. He, “Solitary solutions, periodic solutions and compacton-like solutions using the exp-function method,”
*Computers & Mathematics with Applications*, vol. 54, no. 7-8, pp. 966–986, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Bekir and A. Boz, “Exact solutions for a class of nonlinear partial differential equations using exp-function method,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 4, pp. 505–512, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet