Turkish Journal of Science & Technology Volume 8(2), 121-123, 2013
Complex Solutions for Burgers-Like Equation
M. T. GENÇOĞLUTunceli University, Tunceli Vocational School, Tunceli mtgencoglu@tunceli.edu.tr
(Received: 04.06.2013; Accepted: 21.08.2013) Abstract
In this paper, a direct algebraic method for the complex solutions of the Burgers-Like was implemented. Several complex solutions of the Burgers-Like were found by using this technique.
Key words: Burgers-Like, Direct algebraic method, Complex solutions, Traveling wave solutions
Burgers-Like Denklemi için Karmaşık Çözümler
ÖzetBu çalışmada Burgers-Like denkleminin karmaşık çözümleri için doğrudan cebirsel metodu uygulanmıştır. Bu teknik kullanılarak Burgers-Like denkleminin birkaç tane karmaşık çözümü bulunmuştur.
Anahtar Kelimeler: Burgers-Like, Doğrudan cebirsel metot, Karmaşık çözümler, Hareket eden dalga çözümleri.
1. Introduction
The theory of nonlinear dispersive wave motion has recently undergone much study. Most researchers do not attempt to characterize the general form of nonlinear dispersive wave equations in usually [1,2]. Nonlinear phenomena play a crucial role in applied mathematics and physics. Furthermore, when an original nonlinear equation is directly calculated, the solution will preserve the actual physical characters of solutions [3]. Explicit solutions to the nonlinear equations are of fundamental importance. Various methods for obtaining explicit solutions to nonlinear evolution equations have been proposed. Many explicit exact methods have been introduced in literature [4-18]. Among them is Generalized Miura Transformation, Darboux Transformation, Cole-Hopf Transformation, Hirota’s dependent variable Transformation, the inverse scattering Transform and the Backlund Transformation, tanh method, sine-cosine method, Painlevé method, homogeneous balance method, similarity reduction method, improved tanh method and so on. In fact, recently a direct algebraic approach has been constructed an automated tanh-function method by Parkes and Duffy [12]. The authors present a Mathematica package that deals with complicated algebraic
and outputs directly the required solutions for particular nonlinear equations.
In this study, a direct algebraic method [18] with symbolic computation was implemented to construct new complex solutions for Burgers-Like.
2. An Analysis of the Method and Applications
First it was given a simple description of the direct algebraic method [18]. Then a direct algebraic method will give. For doing this, it can be considered in a two variables general form of nonlinear PDE
u
,
u
t,
u
x,
u
xx,
0
Q
, (1)It is obtained a nonlinear ODE for u
as following by using
,
,
u x t u
ik x ct
,
,
,
,
0
.
'
u
ikc
u
ik
u
k
2u
Q
(2)where
k
and c are real constants,
i
2
1
,.
d
du
u
Muharrem Tuncay Gençoğlu
122 The solution of the equation (2) is expressed with Eq. (3)
0,
n m m mu
a F
(3)n is a positive integer that can be determined by
balancing the highest order derivate and with the highest nonlinear terms in equation. Substituting solution (3) into Eq. (2) yields a set of algebraic equations for
F
m and
m0,1, 2, ,n
then, all coefficients ofF
m have to vanish. After this separated algebraic equation, it could be found coefficientsa
m and
. F
expresses the solution of the auxiliary ordinary differential equation
2
F
b
F
(4) where
d
dF
F
andb
is a constant. Solutions of Eq. (4) are given the paper [18].In this work, the direct algebraic method which is introduced by Zhang [18] will be considered for complex solution of the Burgers-Like.
Example 1. Consider Burgers-Like
1
0
2
t x x xx
u
u
uu
u
(5) with u x t
, u
, ik x ct
then Eq. (5) become1
0
2
icu
iu
iuu
ku
(6) and integrating (6) yields, it was supplied following equation 21
1
0
2
2
icu iu
iu
ku
(7)when balancing u2with
u
then gives n=1. Therefore, it can be chosen0 1
u
a
a F
. (8) Substituting (8) into Eq. (7) yields a set of algebraic equations fora a b
0, ,
1 and c. Thesealgebraic systems can be written as following
2 0 1 0 0 1 0 1 1 2 1 1
0,
2
2
0,
0.
2
2
ia
a bk
ia
ia c
ia
ia a
ia c
ia
a k
(9)The solutions of system (9) can be obtained as following with the aid of Mathematica
2 0 1 21
1,
,
c
,
0.
a
c
a
ik b
k
k
(10)substituting (10) into (8), the following exact complex traveling wave solutions of Eq.(5) have been obtained.
where
b
0
andk
is an arbitrary real constant. These solutions are obtained as following
2
2
1 2 2
1 1
1 c c
u c ik Tan ikx ikct
k k (11)
2
2
2 2 2 1 1 1 c cu c ik Cot ikx ikct
k k (12)
Figure 1. 3D Graph of the exact complex traveling wave solutions of (11) of Eq.(5)
Complex Solutions for Burgers-Like Equation
123
Figure 2. 2D Graph of the exact complex traveling wave solutions of (11) of Eq.(5)
4. Conclusions
In this paper, a direct algebraic method with symbolic computation was implemented to construct new exact complex solutions for Burgers-Like. The method can be used to many other nonlinear equations or coupled ones. In addition, this method is also computarizable, which allows us to perform complicated and tedious algebraic calculation on a computer. It is seen from figure 1 and 2 that the solution of the eq.(5) is solitary wave equation.
5. References
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