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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 264549, 5 pages
A Simplification for Exp-Function Method When the Balanced Nonlinear Term Is a Certain Product
Zhijiang College, Zhejiang University of Technology, Hangzhou 310024, China
Received 24 October 2013; Revised 8 December 2013; Accepted 8 December 2013
Academic Editor: Hossein Jafari
Copyright © 2013 Hong-Zhun Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Exp-function method plays an important role in searching for analytic solutions of many nonlinear differential equations. In this paper, we prove that the balancing procedure in the method is unnecessary when the balanced nonlinear term is a product of the dependent variable under consideration and its derivatives. And in this case, the ansatz of the method can be simplified to be with less parameters so as to be easy to calculate.
In 2006, He and Wu firstly proposed the so-called Exp-function method to search for solitary solutions and periodic solutions of nonlinear partial differential equations (PDE) . This method soon drew the attention of many researchers and was successfully applied to many nonlinear problems [2–31]. Among them, it is worth mentioning that Zhu firstly applied this method to difference-differential equations, which shows that the method is also effective in this case [8, 9]. After Zhu, Dai et al. generalized the Exp-function method to find new exact traveling wave solutions of nonlinear PDE and nonlinear differential-difference equations . Recently, Ma et al. extended the Exp-function method to multiple Exp-function method for constructing multiple wave solutions [32, 33]. He elucidated how to solve fractional differential equations with local fractional derivatives via the fractional complex transformation and the Exp-function method .
For convenience, we first introduce the Exp-function method in brief.
1.1. Outline of the Exp-Function Method
Suppose that we consider a (1+1)-dimensional nonlinear PDE in the form Using traveling wave transformation we get a nonlinear ordinary differential equation (ODE) where the prime, as it is in the following, denotes the derivation with respect to .
The Exp-function method is based on the assumption that the solutions of (3) can be expressed in the following form: where , , , and are positive integers to be determined and and are constants to be specified.
Then we can express the highest order nonlinear and linear terms in (3) in terms of (4). In the resulting terms, determine and through balancing the highest order Exp-function and and by balancing the lowest order one.
Substituting (4) along with the determined , , , and into (3), we can obtain an equation for . Setting all the coefficients of the different powers of to zero leads to a set of algebraic equations for , , , and . Determine values of , , , and by solving this algebraic equations and put these values into (4). Thus we may obtain nontrivial exact traveling wave solutions of (1).
1.2. An Open Problem
Among the Exp-function method, the balancing computation is laborious but prior. However, we observe that the balancing procedure of the Exp-function method in studied examples always leads to the same case and . This fact has partly been proved in [16, 22]. In , Ali has proved it by assuming the highest order linear and nonlinear terms as and (), respectively. In , making use of the same approach as was done by Ebaid proved the fact for nonlinear terms in the form (), (), (), and (), respectively, along with linear term . Ebaid claimed in the abstract and section “Conclusions” of his article that the case and is the only relation that “can be obtained through applying the Exp-function ansatz for all possible cases of nonlinear ODEs.”
“However, one cannot construct a general form for the highest order nonlinear term because there are many possibilities other than the ones considered” ; Aslan and Marinakis concluded that these authors just took some special cases of the nonlinear term into account and hence the problem still remained open. In this paper, we will construct a special case to show that Ebaid's claim is not true; namely, the case and is not the only relation for some special differential equations and hence the problem is still open.
In what follows, we will discuss the relations of , , , and in a more effective and concise way.
2. Main Result
2.1. Some Terminology
To begin with, we recall some terminology in .
A monomial in a collection of variables is a product where the are nonnegative integers.
The total degree of (5) is the sum of the exponents: .
A polynomial is said to be homogeneous if all the monomials appearing in it with nonzero coefficients have the same total degree.
For instance, is a product of and its derivatives and , and its total degree is . is homogeneous.
2.2. Two Introducing Ansatz Function
For convenience, we assume is expressed in the form where , , , and are positive integers to be determined and and are constants to be specified.
The following three formulas will be used in this section:
Before presenting our definitions, we recall an important fact that the constants , , , and in the ansatz of (4) can be assumed to be nonzero during the process of balancing the linear term of highest order with the highest order nonlinear term of certain ODE, and so do the constants , , , and in ansatz (6). Hence, in this paper, we assume that all above eight constants are nonzero.
Therefore, for ansatz (4), we can define the ansatz functions and as follows: In particular, we define for arbitrary constant .
2.3. Properties of the Ansatz Functions
2.4. Theorem and Proof
In this section, we assume the balanced nonlinear term is a product of dependent variable and its derivatives; namely, where () are nonnegative integers. The fact that the product (16) is a nonlinear term implies . In other words, the total degree of (16) is at least 2.
Theorem 1. Suppose that the balanced nonlinear term in (3) is a product of and its derivatives in the form of (16) and the balancing linear term is , where is a nonnegative integer; then the Exp-function ansatz (4) admits and .
Proof. By contradiction, suppose that . Then we have
Balancing linear and nonlinear terms requires
So we obtain
Since , we arrive at the result . This is a contradiction.
The result can be obtained in a similar way; here we omit the details.
Remark. Since our forms of the linear and nonlinear terms are in a more general setting, Theorem 1 covers the results presented by Ali and Ebaid, respectively.
2.5. A Simplification for the Exp-Function Method
According to Theorem 1, if the balanced nonlinear term is a product, ansatz (4) can be reduced to the following equivalent form: where is a free positive integer with and and are constants to be specified. Ansatz (20) is concise and easy to calculate and makes the Exp-function method more straightforward.
For example, Naher et al. applied the Exp-function method to constructing the traveling wave solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation in the form  The traveling wave transformation carries (21) into an ODE Since the nonlinear term is a product of and , we can immediately assume ansatz (4) in the form which is equivalent to the case and and in accord with Naher's (namely, (3.8) in ).
3. A Counter-Example
In , Ebaid claimed in his abstract that the case and was the only relation that could be obtained by applying this method to any nonlinear ODE. In this section, we construct a counter-example to show that the case and is not the only relation. In fact, the claim does not applied to each nonlinear homogeneous ODE.
For example, we can create the following ODE: which can be rewritten as We have From (27), the relations and hold automatically. Hence by balancing nonlinear term and linear term , we cannot determine the relations of , , , and . That is to say, all of them are free constants. Hence it is possible that the relation and holds. In other words, the relation and is not all-inclusive.
In summary, we present an entire novel approach to prove that the balancing procedure in Exp-function method is unnecessary when the balanced nonlinear term is a product of the dependent variable under consideration and its derivatives. Our results cover the results presented by Ali and Ebaid. We believe that our work can serve as an answer to the open problem proposed by Aslan et al.
Many thanks are due to the helpful comments and suggestions from the anonymous referees and support from the NSF of China (11201427) and the College Foundation of Zhijiang College of Zhejiang University of Technology (YJJ0819).
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