F-Expansion Method and Its Application for Finding New Exact Solutions to the Kudryashov-Sinelshchikov Equation

Based on the F-expansion method, and the extended version of F-expansion method, we investigate the exact solutions of the Kudryashov-Sinelshchikov equation.With the aid ofMaple, more exact solutions expressed by Jacobi elliptic function are obtained. When themodulusmof Jacobi elliptic function is driven to the limits 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained.


Introduction.
In the recent years, the study of nonlinear partial differential equations (NLEEs) modelling physical phenomena has become an important toll.Seeking exact solutions of NLEEs has long been one of the central themes of perpetual interest in mathematics and physics.With the development of symbolic computation packages like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as homogeneous balance method [1,2], auxiliary equation method [3,4], the Expfunction method [5,6], Darboux transformation [7,8], tanhfunction method [9], the modified extended tanh-function [10], and Jacobi elliptic function expansion method [11,12].
The F-expansion method is an effective and direct algebraic method for finding the exact solutions of nonlinear evolution problems [13][14][15], many nonlinear equations have been successfully solved.Later, the further developed methods named the generalized F-expansion method [16,17], the modified F-expansion method [18], the extended F-expansion method [19], and the improved F-expansion method [20] have been proposed and applied to many nonlinear problems.
Recently, Kudryashov and Sinelshchikov [21] introduced the following equation: where , , , ], and  are real parameters.Equation (1) describes the pressure waves in the liquid with gas bubbles taking into account the heat transfer and viscosity [1].It was called the Kudryashov-Sinelshchikov equation [22].
In practice, analysis of propagation of the pressure waves in a liquid with gas bubbles is an important problem.We know that there are solitary and periodic waves in a mixture of a liquid and gas bubbles, and these waves can be described by the Burgers equation, the Korteweg-de Vries (KdV) equation, and the Burgers-Korteweg-de Vries (BKdV) equation [23][24][25][26].The Kudryashov-Sinelshchikov equation is a generalization of the KdV and the BKdV equations.Indeed, assuming that  =  =  = 0, we have the Burgers-Kortewegde-Vries equation.In the case of  =  =  =  = 0, we get the famous Korteweg-de Vries equation.
Recently, the Kudryashov-Sinelshchikov equation has been investigated by different methods and some exact solutions are derived.Ryabov [22] obtained some exact solutions for  = −3 and  = −4 using a modification of the truncated expansion method [27,28].Using the bifurcation theory and the method of phase portraits analysis

The F-Expansion Method and Its Extended Version
In this section, we will give the detailed description of the Fexpansion method and its extended version.Suppose that we have a nonlinear partial differential equation (PDE) for (, ) in the form where  is a polynomial in its arguments.By taking (, ) = (),  =  − , we look for traveling wave solutions of (2) and transform it to the ordinary differential equation (ODE)  (, −  ,   ,  2   , −  ,   , . ..) = 0. ( Suppose that the solution  of (3) can be expressed as a finite series in the form where  0 , and   ,   ( = 1, 2, . . ., ) are constants to be determined later, and () is a solution of the auxiliary LODE where , , and  are constants.If the values of , , and  are known, the Jacobian elliptic function solutions () can be obtained from (5) which can also be found in Table 1.
these overdetermined algebraic equations by symbolic computation, we can determine those parameters explicitly.
Assuming that the constants  0 ,   , and   ( = 1, 2, . . ., ) can be obtained by solving the algebraic equations, then by substituting these constants and the known general solutions into (4) or ( 6), we can obtain the explicit solutions of (2) immediately.

Exact Solutions of the Kudryashov-Sinelshchikov Equation in the Case of ] = 𝛿= 0
In this section, we solve the Kudryashov-Sinelshchikov equation in case of ] =  = 0 by F-expansion method the in order to find the exact solutions of the Kudryashov-Sinelshchikov equation.Using scale transformation: the Kudryashov-Sinelshchikov equation is written in the form where  = / and  = /.We let Under this transformation, (8) can be reduced to the following ordinary differential equation (ODE): By integrating (10) once with respect to , we have where  is an integration constant.

Exact Solutions of the Kudryashov-Sinelshchikov Equation in the Case
In this section, we solve the Kudryashov-Sinelshchikov equation in case of ] ̸ = 0,  ̸ = 0 by the extended version of Fexpansion method.Using transformation (7), we can write the Kudryashov-Sinelshchikov equation in the following form: where  = /,  = /,  = /. where When  → 1, from (60), the exact solution of (47) is

Conclusions
The F-expansion method and its extended version are very effective in solving various NLEEs.For some NLEEs, the F-expansion method can give nontrivial solutions, for some other NLEEs, the extended version of F-expansion method can give nontrivial solutions, and for some particular NLEEs (especially the complete integrable systems), both Fexpansion method and its extended version are feasible for constructing exact solutions.In summary, lots of new exact Jacobian elliptic function solutions and soliton solutions of the Kudryashov-Sinelshchikov equation are proposed by the F-expansion method and its version.The results of [21,22] have been enriched.These exact solutions have been verified by symbolic computation system-Maple.Moreover, the solutions listed in this paper may be of important significance for the explanation of some relevant physical problems.We would like to study the Kudryashov-Sinelshchikov equation further.