 Applied Mathematics, 2011, 2, 470-474 doi:10.4236/am.2011.24060 Published Online April 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM An Innovative Solutions for the Generalized FitzHugh-Nagumo Equation by Using the Generalized GG-Expansion Method Sayed Kahlil Elagan1,2, Mohamed Sayed2,3, Yaser Salah Hamed2,3 1Department of Mat hem at ic s, Faculty of Science, Menofia University, Meno u f, Egypt 2Department of Mat hem at ic s, Faculty of Science, Taif University, Taif, Kingdom of Saudi Arabia (KSA) 3Department of Engi neering Mathemati cs , Faculty of Electronic Engineering, Menofia University, Men ouf, Egypt E-mail: {sayed_khalil2000, moh_6_11, eng_yaser_salah}@yahoo.com Received February 24, 201 1; revised Marc h 4, 2011; accepted March 8, 2011 Abstract In this paper, the generalized GG-expansion method is used for construct an innovative explicit traveling wave solutions involving parameter of the generalized FitzHugh-Nagumo equation 1txxuu uuatu , for some special parameter at where =GG satisfies a second order linear differential equation =0GGG  ,  =ptx qt, where pt and qt are functions of t. Keywords: FitzHugh-Nagumo Equation, Generalized GG-Expansion Method, Traveling Wave Solutions 1. Introduction Phenomena in physics and other fields are often des- cribed by nonlinear evolution equations (NLEEs). When we want to understand the physical mechanism of phe- nomena in nature, described by nonlinear evolution equa- tions, exact solutions for the nonlinear evolution equa- tions have to be explored. For example, the wave phe- nomena observed in fluid dynamics [1,2], plasma and elastic media [3,4] and optical fibers [5,6], etc. In the past several decades, many effective methods for obtain- ing exact solutions of NLEEs have been proposed, such as Hirota’s bilinear method , Backlund transfor mation , Painlevé expansion , sine-cosine method , homogeneous balance method , homotopy pertur- bation method [12-14], variational iteration method [15-18], asymptotic methods , non-perturbative me- thods , Adomian decomposition method , tanh- function method [22-26], algebraic method [27-30]. Jacobi elliptic function expansion method [31-33], F-expansion method [34-36] and auxiliary equation method [37-40]. Recently, Wang et al.  introduced a new direct me- thod called the GG-expansion method to look for travelling wave solutions of NLEEs. The GG-expan- sion method is based on the assumptions that the travelling wave solutions can be expressed by a poly- nomial in GG, and that =GG satisfies a se cond order linear ordinary differential equation (LODE): =0GGG, where d=dGG, 22d=dGG , =xVt, V is a constant. The degree of the poly- nomial can be determined by considering the homoge- neous balance between the highest order derivative and nonlinear terms appearing in the given NLEE. The coefficients of the polynomial can be obtained by solving a set of algebraic equations resulted from the process of S. K. ELAGAN ET AL. Copyright © 2011 SciRes. AM 471using the method. By using the GG-expansion method, Wang et al.  successfully obtained more travelling wave solutions of four NLEEs. Very recently, Zhang et al.  proposed a generalized GG-expansion method  to improve the work made in . The main pur- pose of this paper is to use generalized GG-expansion method to solve the generalized FitzHugh-Nagumo equation. The performance of this method is reliable, simple and gives many new solutions, its also standard and computerizable method which enable us to solve complicated nonlinear evolution equations in mathema- tical physics. The paper is organized as follows. In sec- tion 2, we describe briefly the generalized GG-expan- sion method, where =GG satisfies the second order linear ordinary differential equation =0GGG  ,  =ptx qt In section 3, we apply this method to the FitzHugh-Nagumo equation. In section 4, some con- clusions are given. 2. Description the Generalized GG-Expansion Method Suppose that we have the following nonlinear partial differential equation , ,, , , ,=0,txtt xt xxPuuu uuu (2.1) we suppose its solution can be expressed by a polyno- mial GG as follows:   01= , 0,inijiGut ttG  (2.2) where 0 t and  jt are functions of t (=1,2, ,)jn and =,xt is a function of x, t to be determine later, =GGsatisfies the second order linear ordinary differential equatio n  =0,GGG  (2.3) To determine u explicitly we take the fo llowing four steps. Step 1. Determine the integer n by balancing the highest order nonlinear term(s) and the highest order partial derivative of u in Equation (2.1). Step 2. Substitute Equation (2.2) along with Equation (2.3) into Equation (2.1) and collect all terms with the same order of GG together, the left hand side of Equation (2.1) is converted into a polynomial in GG. Then set each coefficient of this polynomial to zero to derive a set of over-determined partial differential equa- tions for 0t, it and . Step 3. Solve the system of all equations obtained in step 2 for 0t, it and  by use of Maple. Step 4. Use the results obtained in above steps to derive a series of f undamental solutions o f Equation (2.3) depending on GG, since the solutions of this equatio n have been well known for us, then we can obtain exact solutions of Equation (2.1). 3. The FitzHugh-Nagumo Equation In this section, we apply the generalized GG-expan- sion method to solve the generalized FitzHugh-Nagumo equation, construct the traveling wave solutions for it as follows: Let us first consider the generalized FitzHugh-Nagumo equation 1txxuu uuatu  (3.1) where at is a function of t. In order to look for the traveling wave solutions of Equation (3.1) we suppose that ,= ,=uxtupt xqt (3.2) Suppose that the solution of Equation (3.1) can b e ex- pressed by a polynomial in GG as follows  01= iniiGut tG  (3.3) Considering the homogeneous balance between xxu and 3u in Equation (3.1) we required that 2=3nn, then =1n. So we can write Equation (3.3) as  0=.Gut tG  (3.4) Substituting Equation (3.4) into Equation (3.1) along with Equation (2.3). We obtain the following equations by comparing coefficients of GG. When =3j then .2=0 3121p (3.5) S. K. ELAGAN ET AL. Copyright © 2011 SciRes. AM 472We solve the equation by setting 1=2p (we could also set 1=2p). The equatio n f or =2j is 2 22211 0111=3 3.px qpa   (3.6) We see from this equation that pt must be a con- stant and then 1t is also constant. Therefore, equa- tion Equation (3.6) simplifies to 2 2221 10111=3 3.qp a (3.7) The equation for =1j is 2221110120101 1=2 232 .qppaa   (3.8) We substitute Equation (3.7) into Equation (3.8) and obtain (after dividing by 1) 22 2101 10 020322322 =0.apap a    (3.9) We solve this equation for a and obtain 222 21010 01032232=.21ppat   (3.10) The equation for =0j is 222301100 00=.qp aa  (3.11) If we substitute Equation (3.7) and Equation (3.10) into Equation (3.11) we obtain 430100 012222 201 12222201 11222 22111212 23312 2342 222=0.ppppppp       (3.12) Now Equation (3.12) is an ordinary differential equa- tion for 0. Therefore, 0 must have a special form in order to be a solution of this equation which means that the function at expressed in terms of 0t by Equation (3.10) must also of a special form. This shows that we cannot solve all the equations if at is an arbitrary function. We can still try to find solutions for some special at. For example, we choose 1=, =1, =0.2p Then 1=1 and Equation (3.12) simplifies to 23000031=0.22 One solution is 01=1 .1tte We find at from Equa tion (3.7) as 13=122tat e (3.13) = 3arctan1.tqth e We choose =1 .Ge Then 1=1 11teuee (3.14) with =3arctan 12txhe is a solution of equation Equation (3.1) when at is given by Equation (3.13). Once can check with the com- puter that u given by Equation (3.14) is really a solu- tion of Equation (3.1). It is shows that this method is powerful in constructing exact solutions of NLEEs. 4. Conclusions This study shows that the generalized GG-expansion method is quite efficient and practically will suited for use in finding exact solutions for the problem considered here. 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