World Applied Sciences Journal 19 (12): 1797-1805, 2012
ISSN 1818�4952
© IDOSI Publications, 2012
DOI: 10.5829/idosi.wasj.2012.19.12.1960
Abstract: In this article, He’s Exp-function method (EFM) is used to construct solitary and soliton solutions of nonlinear evolution equation. Westervelt equation is chosen to illustrate the effectiveness of the method. The method is straightforward and concise and its applications are promising. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. The
EFM presents a wider applicability for handling nonlinear wave equations. Also, it is shown that the EFM, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations.
Key words: The Exp-function Method (EFM)

westervelt equation

solitary and soliton solutions
INTRODUCTION
Recently, the study of nonlinear partial differential equations in modelling physical phenomena has become an important tool. The investigation of the travelling wave solutions plays an important role in nonlinear sciences. Nonlinear wave phenomena appear in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics and geochemistry. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations [1]. A variety of powerful methods has been presented, such as the inverse scattering transform [1],
Hirota’s bilinear method [2], homotopy analysis method [3, 4] variational iteration method [5, 6],
Laplace Adomian decomposition method [7], homotopy perturbation method [8, 9], sine-cosine method [10], tanh-function method [11, 12], Bäcklund transformation [13] and so on. We use an effective method for constructing a range of exact solutions for the following nonlinear partial differential equations which was first presented by J. H. He [14]. A new method called the Exp-function method (EFM) is presented to look for traveling wave solutions of nonlinear evolution equations (NLEEs). The Expfunction method has successfully been applied to many situations. For example, He et al . [15] have solved the nonlinear wave equations by using the EFM. Wu and
He [16] have used the EFM to give solitary solutions, periodic solutions and compacton-like solutions.
Authors of [17] have examined the EFM to obtain new periodic solutions. Abdou [18] has obtained generalized solitonary and periodic solutions for nonlinear partial differential equations by the EFM. Boz and Bekir [19] have applied the EFM for the (3+1)-dimensional nonlinear evolution equations. The Exp-function method along with Hirota’s and tanh-coth methods have been applied for solving solitary wave solutions of the generalized shallow water wave equation by
Wazwaz [20]. The EFM has recently been applied by
Manafian and Bagheri [21] for solving the modified
KdV and the generalized KdV equations. Also, in [22]
Manafian has obtained some new solutions with Expfunction method. The Westervelt equation describing the propagation of finite amplitude sound has the following form [23] where u is the acoustic pressure,

0
and c
0
(1.1)
are the ambient density and sound speed,

= 1+(B/2A) is the nonlinearity coefficient for the fluid and B/A is the nonlinearity parameter. The first two terms in Eq. (1.1), the D’Alembertian operator acting on the acoustic pressure, describe linear lossless wave propagation at the small-signal sound speed. The final term describes nonlinear distortion of the wave due to finite-amplitude effects. If the medium is assumed to be a thermoviscous fluid, the Westervelt equation (Eq. (1.1)) takes the following form [24]
(1.2)
Corresponding Author: J. Manafianheris, Department of Mathematics, Islamic Azad University, Ahar Branch, Ahar, I ran
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World Appl. Sci. J., 19 (12): 1797-1805, 2012
The additional term is a loss term, which is due to the thermal conduction and the viscosity of the fluid. Here δ is the diffusivity of sound; in a thermoviscous fluid, the absorption coefficient α is related to δ and ω=2πf by
(1.3)
The absorption coefficient is a constant, specific to a single frequency. It is interesting to point out that Eq. (1.2) has attracted a considerable amount of research work such as in [23-28, 30]. The usual fourth-order finite difference representation of the second partial derivative would lead an unconditionally unstable scheme, a technique given by
Cohen [26] based on the ”modified equation approach” to obtain fourth-order accuracy in time was used. This technique, while improving the accuracy in time, preserves the simplicity of the second-order accurate time-step scheme. This was performed in order that all derivatives have the same order of accuracy as the boundary conditions which are also fourth order. It was found [25] that lower order boundary conditions did not suppress spurious reflections at the computational boundary that could contaminate low amplitude signals. A technique known as the
Complementary Operator Method (COM) was utilized [27] in the development of the absorbing boundary conditions(ABC). This technique was first utilized in electromagnetics where it was shown to yield excellent results
[27]. The COM method is a differential equation-based ABC method and differs from the other common approach of terminating the grid with the use of an absorbing material. An example of this type of boundary condition is the perfectly matched layer(PML) method originally proposed by Berenger [28]. The COM is based on one-way wave equations such as Higdon’s boundary operators [29]. Also, in [30] Norton et al . investigated the Westervelt equation with viscous attenuation versus a causal propagation operator and also have done a numerical comparison.
The article is organized as follows: In Section 2, first we briefly give the steps of the method and apply the method to solve the nonlinear partial differential equations. In Section 3, we employ the EFM to solve the nonlinear
Westervelt equation. Also a conclusion is given in Section 4. Finally some references are given at the end of this paper
BASIC IDEA OF EXP-FUNCTION METHOD
We first consider the nonlinear equation of the form and introduce a transformation where c is constant to be determined later. Therefore Eq. (2.1) is reduced to an ODE as follows
The EFM is based on the assumption that travelling wave solutions as in ([2]) can be expressed in the form where c,d,p and q are positive integers which could be freely chosen, a n
’s and b n
’s are unknown constantsto be determined. To determine the values of c and p, we balance the linear term of highest order in Eq.(2.3) with the highest order nonlinear term. Also to determine the values of d and q, we balance the linear term of lowest order in
Eq. (2.3) with the lowest order nonlinear term.
THE WESTERVELT EQUATION
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World Appl. Sci. J., 19 (12): 1797-1805, 2012
In this section we employ the Exp-function method to the Westervelt equation as follows where are and
And by using the wave variable reduces it to an ODE
In order to determine values of c and p, we balance the linear term of the highest order u

with the highest order nonlinear term uu

in Eq. (3.2) and get respectively. Balancin g highest order of the EFM in (3.3) and (3.4), we will have 3c+7p = 2c+6p which leads to the result c = p. Similarly to determine the values of d and q, for the terms u

and uu

in Eq. (3.2) by simple calculation, we obtain respectively. Balancing lowest order of the EFM in (3.5) and (3.6), we get which leads to the result d = q.
Case 1: For simplicity, we set b
1
= 1, p = c = 1 and d = q = 1. Then Eq. (2.4) reduces to
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World Appl. Sci. J., 19 (12): 1797-1805, 2012
Substituting (3.8) into Eq. (3.2) and using the well-known Maple software, we have where and C n
’ are coefficients of exp(n  )’. Equating the coefficients of exp(n 
) to be zero, we obtain the following set of algebraic equations for a
1
, a
0
, a
-1
, b
0
, b
-1
, µ and c, as
Solving the system of algebraic equations with the help of Maple gives the following sets of solutions (I) The first set:
If we choose b
-1
= 1 or b
-1
=-1, then the solution (3.13) becomes to
In case k and ω are imaginary numbers, each of the obtained solitonary solutions above can be converted into periodic solution or compact-like solution. Here, we only discuss the solution given by (3.13). If k = iK, ω = iΩ in
(3.13), then we obtain
Then, (3.14) and (3.15) respectively become
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World Appl. Sci. J., 19 (12): 1797-1805, 2012
Case 2: p = c = 2 and d = q = 1
For simplicity, we set b
2
= 1, p = c = 1 and d = q = 1. Then Eq. (2.4) reduces to
Substituting (3.22) into Eq. (3.2) and by using the well-known software Maple, we get where and C n
’ are coefficients of exp(n  )’. Equating the coefficients of exp(n 
) to be zero, we obtain the following set of algebraic equations for a
2
, a
1
, a
0
, a
-1
,b
0
, b
-1
, b
1
, µ and c, as
Solving this system of algebraic equations by using Maple 12, we obtain the following results (I) The first set:
If we choose b
-1
= 1, or b
-1
=-1, then the solution (3.25) respectively gives
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World Appl. Sci. J., 19 (12): 1797-1805, 2012
In case k and ω are imaginary numbers, each of the obtained solitonary solutions above can be converted into periodic solution or compact-like solution. Here, we only discuss the solution given by (3.13). If k = iK, ω = iΩ then
(3.28) and (3.29) respectively become
(II) The second set is: where

= k(x+y)+
 t and a
2
, b
0
, b
-1
are arbitrary constants.
Case 3: p = c = 2 and d = q = 2
Since the values of c and d can be freely chosen, we set p = c = 2 and d = q = 2 and then the trial function (2.4) becomes
There are some free parameters in (3.34). We set b
2
= 1, b
1
= b
-1
= 0 for simplicity, the trial function, (3.32) is simplified as follows
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World Appl. Sci. J., 19 (12): 1797-1805, 2012
Substituting (3.35) into Eq. (3.2) and by using the well-known Maple software, we will have where and C n
’ are coefficients of exp(n  )’. Equating the coefficients of exp(n 
) to be zero, we obtain the following set of algebraic equations for a
2
, a
-2
, a
1
, a
0
,a
-1
, b
0
, b
-2
, k and

c as
By the same manipulation as illustrated above, we obtain
(I) The first set is:
If we choose b
-2
= 1 or b
-2
=-1, then the solution (3.40) respectively gives
But in the general case for each n

0 we have solutions as follow
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World Appl. Sci. J., 19 (12): 1797-1805, 2012
For one dimension space (3.41) and (3.42) respectively give
As illustrated above that the obtained solitonary solution can be converted into periodic solution or compactlike solution if k and ω are chosen as imaginary numbers. If k = iK and ω = iΩ, then (3.39)-(3.44) respectively become
CONCLUSION
In this article, we investigated the nonlinear Westervelt equation. The Exp-function method is a useful method for finding travelling wave solutions of nonlinear evolution equations. This method has been successfully applied to obtain some new generalized solitonary solutions to the nonlinear Westervelt equation. The EFM is more powerful in searching for exact solutions of nonlinear partial differential equations. Also, new results are formally developed in this article. It can be concluded that this method is a very powerful and efficient technique in finding exact solutions for wide classes of problems.
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