New Application of the ( G / G )-Expansion Method for Thin Film Equations

and Applied Analysis 3 outset. The method yields a general solution with free parameters which can be identified by the above conditions. (iii) The general solution obtained by (G/G)-expansion method without approximation. (iv) Finally, the solution procedure can be easily implemented in Mathematica or Maple. 3. Application of the (G 󸀠 /G)-Expansion Method 3.1. Exact Traveling Wave Solution of Standard Thin Film Equation (1). Now we consider (1) which arises in the flow of a surface-tension dominated thin liquid film. Substituting (5) into (1) and integrating the result, and for simplicity equating the integration constant equal to zero, we get the following u n u 󸀠󸀠󸀠 − cu = 0. (12) Suppose that the solution of (12) can be expressed by a polynomial in (G󸀠/G) as follows:


Introduction
In this paper we are interested in the so-called standard thin film equation of the form which in general has important applications in geology, biophysics, physics, and engineering (see [1][2][3]).Also known as the lubrication equation [4], it models the spreading motion of the free surface of a thin film on a solid substrate [5].In particular, the function (, ) is the thickness of the fluid film at position  and time .Here the parameter  denotes the kind of flow.In the case  = 1, the equation models the thickness of a thin film in a Hele-Shaw cell [6].
When  = 2, it is the Navier slip thin film equation which arises in the study of wetting films with a free contact line between film and substrate [7].Furthermore, when  = 3, it corresponds to the surface-tension-driven spreading of a thin Newtonian fluid [8].
Our main objective in this paper is to apply the (  /) method to provide closed-form travelling wave solutions of the generalized thin film equations and also standard thin film equation.To the best of our knowledge, this is the first time this method has been applied to such equations.In solving these equations, we found an instance where the related balance numbers are not the usual positive integers (see Zhang [22]).It is also noted that for appropriate parameters new solitary waves solutions are found.We compare our solutions with the solutions previously obtained by Bertozzi and Pugh [23] and King [9], where they proved the existence solution to thin film equation via separation of variables.The closed-form solution obtained via this method is in good agreement with the solutions reported in [9,23].
Our paper is organized as follows: in Section 2, we present the summary of the (  /)-expansion method, in Section 3, we describe the applications of the (  /)-expansion method for two generalization thin film equations, standard thin film equation and a special case, and in Section 4, some conclusions are given.

Summary of the (𝐺 󸀠 /𝐺)-Expansion Method
In this section, we describe the (  /)-expansion method for finding traveling wave solutions of nonlinear partial differential equations (PDEs).Suppose that a nonlinear partial differential equation, say in two independent variables  and , is given by the following: where  = (,) is an unknown function and  is a polynomial in  = (, ) and its various partial derivatives, in which highest-order derivatives and nonlinear terms are involved.The procedure of the (  /)-expansion method can be presented in the following six steps.
Step 1.To find the traveling wave solutions of ( 4), we introduce the wave variable where the constant  is generally termed the wave velocity.Substituting ( 5) into (4), we obtain the following ordinary differential equations (ODE) in  (which illustrates a principal advantage of a traveling wave solution, i.e., a PDE is reduced to an ODE) as follows.
Step 2. If necessary we integrate (6) as many times as possible and set the constants of integration to be zero for simplicity.The solution process for ( 6) is based on the auxiliary conditions that the dependent variable and its first, second, and higher spatial derivatives tend to zero as  → ∞, that is, From these conditions, we can take the constants of integration to be zero.
Step 3. We suppose that the solution of nonlinear partial differential equation can be expressed by a polynomial in (  /) as follows: where  = () satisfies the second-order linear ordinary differential equation as follows: Here the prime denotes the derivative respective to , and   , , and  are real constants with   ̸ = 0. Using the general solutions of (9), we have the following: Step 4. The positive integer  can be accomplished by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (6) as follows: if we define the degree of () as [()] = , then the degree of other expressions is defined by where  is an integer.Therefore, we can get the value of  in (8).
Step 5. Substitute ( 8) into (6) and use (9) and collect all terms with the same order of (  /) together, then set each coefficient of this polynomial to zero which yields a set of algebraic equations for   , , , and .
Step 6. Substitute   , , , and  obtained in Step 5 and the general solution of ( 9) into (8).Next, depending on the sign of the discriminant  2 − 4, we get the solutions of (6).So, we can obtain exact solutions of the given (4).The advantages of the approach taken in this paper are as follows.
(i) It will be more important to seek solutions of higherorder nonlinear equations which can be reduced to ODEs of the order greater than 3.
(ii) In the (  /)-expansion method, there is no need to apply the initial and boundary conditions at the outset.The method yields a general solution with free parameters which can be identified by the above conditions.
(iii) The general solution obtained by (  /)-expansion method without approximation.
(iv) Finally, the solution procedure can be easily implemented in Mathematica or Maple.

Exact Traveling Wave Solution of Standard Thin Film
Equation (1).Now we consider (1) which arises in the flow of a surface-tension dominated thin liquid film.Substituting ( 5) into (1) and integrating the result, and for simplicity equating the integration constant equal to zero, we get the following Suppose that the solution of ( 12) can be expressed by a polynomial in (  /) as follows: where  is a real constant to be determined later and  satisfies (9).Balancing between     and , we get  = −3/.Now it is easy to deduce that With the aid of symbolic computation, substituting (13) along with ( 9) into ( 12), and setting the coefficients of all powers of (  /) to zero, we obtain the following system of nonlinear algebraic equations for , , , , and : The solutions of this system are as follows: where  and  are arbitrary constants.Consequently, we obtain the exact traveling wave solution of (1), where If we set  1 = 0 and  2 = 1 in (17), we obtain the solitary wave solution where  is as above.This is exactly the same solution obtained by Bertozzi and Pugh [23]: when  = ((3/) − 2)((3/) − 1)(3/)  .We remark that if  ≥ 3, the solution of the system using the (  /) cannot be solved due to the no-slip boundary condition on the liquid solid surface, in one space dimension, a similar result reached by Bertozzi and Pugh [24].

Exact Traveling Wave Solution of the Generalized Thin
Film Equation (2).We study nonnegative solutions of the generalized degenerate fourth-order parabolic equation of thin film equation (2).The solution of (2) as found by King [9] is as follows: requiring  > 0 and  > 0 for 8 < (1−) 2 with 3(−+3)/4 <  < 3( +  + 3)/4.We seek the traveling wave solution of (2) in the form (5). Now upon substituting of ( 5) into (2), one gets and by integrating ( 21) and, for simplicity, equating the integration constant which is equal to zero, we get Balancing between     and , we get  = −3/.Then, suppose that (21) has the following formal solution: where  is an unknown constant to be determined later.Substituting (23), along with (9), into (22), and setting the coefficients of (  /)  , ( = 0, 1, . . ., 5) to zero, we obtain a system of nonlinear algebraic equations as follows: where , , and  are arbitrary constants.Hence, we obtain the exact traveling wave solution of (2) as follows: For the comparison between our solution (26) with that of King's as given in (20), first we assume  1 = 0 and  2 = 1 and then we get the same as that of King's (20) if we take  as in (25) in (20).