Applied Mathematics, 2011, 2, 470-474
doi:10.4236/am.2011.24060 Published Online April 2011 (http://www.SciRP.org/journal/am)
An Innovative Solutions for the Generalized
FitzHugh-Nagumo Equation by Using the Generalized




G
G-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
Received February 24, 201 1; revised Marc h 4, 2011; accepted March 8, 2011
Abstract
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
txx
uu uuatu ,
for some special parameter

at where


=GG

satisfies a second order linear differential equation
=0GGG


 
 ,


=
p
tx qt  , where


p
t and


qt are functions of t.
Keywords: FitzHugh-Nagumo Equation, Generalized G
G




-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 [7], Backlund transfor mation
[8], Painlevé expansion [9], sine-cosine method [10],
homogeneous balance method [11], homotopy pertur-
bation method [12-14], variational iteration method
[15-18], asymptotic methods [19], non-perturbative me-
thods [20], Adomian decomposition method [21], 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. [41] introduced a new direct me-
thod called the G
G




-expansion method to look for
travelling wave solutions of NLEEs. The G
G




-expan-
sion method is based on the assumptions that the
travelling wave solutions can be expressed by a poly-
nomial in G
G




, and that

=GG

satisfies a se cond
order linear ordinary differential equation (LODE):
=0GGG





, where

d
=d
G
G


,


2
2
d
=d
G
G


 ,
=
x
Vt


, 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. 471
using the method. By using the G
G




-expansion method,
Wang et al. [41] successfully obtained more travelling
wave solutions of four NLEEs. Very recently, Zhang et al.
[42] proposed a generalized G
G




-expansion method
[42] to improve the work made in [41]. The main pur-
pose of this paper is to use generalized G
G




-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 G
G




-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




G
G-Expansion Method
Suppose that we have the following nonlinear partial
differential equation

, ,, , , ,=0,
txtt xt xx
Puuu uuu (2.1)
we suppose its solution can be expressed by a polyno-
mial G
G




as follows:
  
0
1
= , 0,
i
n
ij
i
G
ut tt
G
 






 (2.2)
where

0 t
 and

jt
 are functions of t
(=1,2, ,)jn and

=,
x
t  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 G
G




together, the left hand side of
Equation (2.1) is converted into a polynomial in G
G




.
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 G
G




, 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 G
G




-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

 
1
txx
uu 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 G
G




as follows
 
0
1
=
i
n
i
i
G
ut t
G
 






 (3.3)
Considering the homogeneous balance between
x
x
u
and 3
u in Equation (3.1) we required that 2=3nn

,
then =1n. So we can write Equation (3.3) as
 
0
=.
G
ut t
G
 




 (3.4)
Substituting Equation (3.4) into Equation (3.1) along
with Equation (2.3). We obtain the following equations
by comparing coefficients of G
G




. When =3j then
.2=0 3
1
2
1p (3.5)

S. K. ELAGAN ET AL. 472
We solve the equation by setting 1=2p

(we could
also set 1=2p
 ). The equatio n f or =2j is

2 222
11 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
2 222
1 10111
=3 3.qp a 

 (3.7)
The equation for =1j is
222
11101
2
0101 1
=2 2
32 .
qpp
aa

 



  (3.8)
We substitute Equation (3.7) into Equation (3.8) and
obtain (after dividing by 1

)
22 2
101 10 0
2
0
3223
22 =0.
ap
ap a

  


  (3.9)
We solve this equation for a and obtain

222 2
1010 0
10
32232
=.
21
pp
at

 

 
 (3.10)
The equation for =0j is
2223
01100 00
=.qp aa

 

 (3.11)
If we substitute Equation (3.7) and Equation (3.10)
into Equation (3.11) we obtain





43
0100 01
2222 2
01 1
22222
01 11
222 22
111
212 2
3312 2
342 2
22=0.
pp
ppp
pp
 
 

 
 
 

 

(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.
2
p 
Then 1=1

and Equation (3.12) simplifies to
23
0000
31
=0.
22


One solution is

01
=1 .
1t
te  
We find


at from Equa tion (3.7) as

13
=1
22
t
at e

 (3.13)


= 3arctan1.
t
qth e
We choose


=1 .Ge




Then
1
=1 1
1t
e
ue
e






 (3.14)
with

=3arctan 1
2
t
xhe
 
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
G
G




-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


at of the generalized FitzHugh-
Nagumo equation are obtained, from which some expo-
nential 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-Nagum o equation.
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