Applied Mathematics and Computation 217 (2010) 937–938
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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
A note on the (G0 /G)-expansion method again
_
Ismail
Aslan
_
Department of Mathematics, Izmir Institute of Technology, Urla, Izmir
35430, Turkey
a r t i c l e
i n f o
a b s t r a c t
Keywords:
(G0 /G)-expansion method
Simplest equation method
Nonlinear ordinary differential equation
Exact solution
Nonlinear evolution equation
We report an observation on two recent analytic methods; the (G0 /G)-expansion method
and the simplest equation method.
Ó 2010 Elsevier Inc. All rights reserved.
Let us consider a nonlinear ordinary differential equation in the form
Pðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0;
ð1Þ
0
where u = u(z) is an unknown function, P is a polynomial in its arguments. According to the (G /G)-expansion method [1], to
solve Eq. (1), we assume its solutions u(z) in the finite series form
uðzÞ ¼
0 i
N
X
G ðzÞ
ai
;
GðzÞ
i¼0
ai ¼ const:;
aN – 0;
ð2Þ
where G(z) is the solution of the auxiliary linear ordinary differential equation
G00 ðzÞ þ kG0 ðzÞ þ lGðzÞ ¼ 0;
ð3Þ
where k and l are real constants.
Theorem. Let FðzÞ and GðzÞ be two functions such that GðzÞ ¼ FðzÞ expðkz=2Þ. Then, GðzÞ satisfies Eq. (3) if and only if FðzÞ
satisfies the differential equation
F 00 ðzÞ þ aFðzÞ ¼ 0;
a ¼ ð4l k2 Þ=4:
ð4Þ
0
0
Moreover, the fraction G ðzÞ=GðzÞ can be expressed in terms of the fraction F ðzÞ=FðzÞ, namely,
G0 ðzÞ F 0 ðzÞ k
¼
:
GðzÞ
FðzÞ 2
ð5Þ
Corollary. The ansatz (2) can be reduced to the special case
uðzÞ ¼
0 i
N
X
F ðzÞ
bi
;
FðzÞ
i¼0
bi ¼ const:;
bN – 0;
where FðzÞ satisfies the auxiliary Eq. (4).
E-mail address: [email protected]
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2010.05.097
ð6Þ
_ Aslan / Applied Mathematics and Computation 217 (2010) 937–938
I.
938
Obviously, this theorem reduces the study of Eq. (3) to the special case G00 (z) + lG(z) = 0. To be more specific, solving this
special equation is tantamount to solving Eq. (3). Hence, the constant k in the (G0 /G)-expansion method can be taken as zero
for simplicity. For application purposes, the ansatz (6) is much more practical than the ansatz (2) in the sense that it minimizes the number of parameters by combining the constants k and l.
On the other hand, to solve Eq. (1) by means of the simplest equation method [2,3], we assume its solutions u(z) in the
form
uðzÞ ¼ A0 þ A1 YðzÞ þ þ An ðYðzÞÞN þ B1
0 0 N
Y ðzÞ
Y ðzÞ
þ þ Bn
;
YðzÞ
YðzÞ
ð7Þ
where Y(z) is the general solution of the simplest equation and Ai, Bi are arbitrary constants. As a special example, one can
consider the following simplest equation
Y 0 ðzÞ þ ðYðzÞÞ2 þ kYðzÞ þ l ¼ 0;
ð8Þ
where k and l are real constants.
Now, let us take Y(z) = G0 (z)/G(z) in both (7) and (8). Then Eq. (8) turns into Eq. (3). Since
0 1
Y 0 ðzÞ
G0 ðzÞ
G ðzÞ
¼ k l
;
YðzÞ
GðzÞ
GðzÞ
ð9Þ
the ansatz (7) becomes
0 N
0 1 !
G0 ðzÞ
G ðzÞ
G0 ðzÞ
G ðzÞ
þ þ þ An
l
uðzÞ ¼ A0 þ A1
þ B1 k GðzÞ
GðzÞ
GðzÞ
GðzÞ
0 1 !N
G0 ðzÞ
G ðzÞ
þ Bn k l
;
GðzÞ
GðzÞ
ð10Þ
which can be rewritten as
uðzÞ ¼
0 i
N
X
G ðzÞ
ci
;
GðzÞ
i¼N
ci ¼ const:
ð11Þ
The ansatz (11) together with Eq. (3) is known as the extended (or improved) (G0 /G)-expansion method.
Recently, Kudryashov [4] demonstrated that (G0 /G)-expansion method is equivalent to the tanh-method. We think that,
for a new equation or previously unstudied problem, one should not receive a criticism for using one of the equivalent methods. It should be considered as a personal choice. Besides, it is a well-known fact that the simplest equation method was
introduced to the research community earlier than the (G0 /G)-expansion method. The discussion made above convinced
us that the later one is a specific form of the former one.
References
[1] M. Wang, X. Li, J. Zhang, The (G0 /G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys.
Lett. A 372 (2008) 417–423.
[2] N.A. Kudryashov, Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Solitons Fractals 24 (2005) 1217–1231.
[3] N.A. Kudryashov, Exact solitary waves of the Fisher equation, Phys. Lett. A 342 (2005) 99–106.
[4] N.A. Kudryashov, A note on the (G0 /G)-expansion method, Appl. Math. Comput. (2010), doi:10.1016/j.amc.2010.03.071.