An Innovative Solutions for the Generalized FitzHugh-Nagumo Equation by Using the Generalized G G-Expansion Method

In this paper, the generalized G G        -expansion method is used for construct an innovative explicit traveling wave solutions involving parameter of the generalized FitzHugh-Nagumo equation       1 t xx u u u u a t u     , for some special parameter   a t where   = G G  satisfies a second order linear differential equation = 0 G G G       ,     = p t x q t   , where   p t and   q t are functions of t .

for some special parameter   a t where
Recently, Wang et al. [41] introduced a new direct me- -expansion method to look for travelling wave solutions of NLEEs.The sion method is based on the assumptions that the travelling wave solutions can be expressed by a poly- , and that The degree of the polynomial can be determined by considering the homogeneous 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 using the method.By using the G G Wang et al. [41] successfully obtained more travelling wave solutions of four NLEEs.Very recently, Zhang et al. [42] proposed a generalized -expansion method [42] to improve the work made in [41].The main purpose of this paper is to use generalized 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 mathematical physics.The paper is organized as follows.In section 2, we describe briefly the generalized In section 3, we apply this method to the FitzHugh-Nagumo equation.In section 4, some conclusions are given.

Expansion Method
Suppose that we have the following nonlinear partial differential equation

 
, , , , , , = 0, we suppose its solution can be expressed by a polyno- as follows: To determine u explicitly we take the following 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 together, the left hand side of Then set each coefficient of this polynomial to zero to derive a set of over-determined partial differential equations for  and  .
Step 3. Solve the system of all equations obtained in step 2 for  and  by use of Maple.
Step 4. Use the results obtained in above steps to derive a series of fundamental solutions of Equation (2.3) , since the solutions of this equation have been well known for us, then we can obtain exact solutions of Equation (2.1).

The FitzHugh-Nagumo Equation
In this section, we apply the generalized -expansion 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 where   a t is a function of t .In order to look for the traveling wave solutions of Equation (3.1) we suppose that Suppose that the solution of Equation (3.1) can be ex-

3). We obtain the following equations by comparing coefficients of
We solve the equation by setting 1 = 2p  (we could also set 1 = 2p

 
).The equation for = 2 j is   We see from this equation that   p t must be a constant and then    is also constant.Therefore, equation Equation (3.6) simplifies to The equation for = 1 j is We substitute Equation (3.7) into Equation (3.8) and obtain (after dividing by 1 We solve this equation for a and obtain The equation for = 0 j is Now Equation (3.12) is an ordinary differential equation for 0  .Therefore, 0  must have a special form in order to be a solution of this equation which means that the function   a t expressed in terms of   0 t  by Equation (3.10) must also of a special form.This shows that we cannot solve all the equations if   a t is an arbitrary function.
We can still try to find solutions for some special   a t .For example, we choose We find   a t from Equation (3.7) as We choose is a solution of equation Equation (3.1) when   a t is given by Equation (3.13).Once can check with the computer that u given by Equation (3.14) is really a solution of Equation (3.1).It is shows that this method is powerful in constructing exact solutions of NLEEs.

Conclusions
This study shows that the generalized -expansion method is quite efficient and practically will suited for use in finding exact solutions for the problem considered here.New and more general excat solutions with arbitrary function   a t of the generalized FitzHugh-Nagumo equation are obtained, from which some exponential function solutions are also derived when setting the arbitrary function as special values.We construct an innovative explicit traveling wave solutions involving parameter of the generalized FitzHugh-Nagumo equation.