Closed form Solutions of New Fifth Order Nonlinear Equation and New Generalized Fifth Order Nonlinear Equation via the Enhanced (G’/G)-expansion Method

Closed form solutions of nonlinear evolution equations (NLEEs) are very imperative in order to better understand the inner mechanism and complexity of complex physical phenomena. The enhanced ( / ) G G expansion ′ − method is a effectual and proficient mathematical tool which can be used to discover the closed form solutions of NLEEs arising in mathematical physics, applied mathematics and engineering. In this article, the enhanced ( / ) G G expansion ′ − method is recommended and carry out to investigate the closed form solutions of the new fifth order non-linear equation and the new generalized fifth order non-linear equation. The performance of this method is reliable, proficient and possible to obtain a lot of new exact solutions than the existing other methods.


Introduction
Closed form solutions of nonlinear evolution equations (NLEEs) are getting importance to study of complex phenomena in the field of science and engineering. NLEEs are frequently appear in various fields, such as plasma physics, geophysics, nuclear physics, biomathematics optical fibers, biomechanics, gas dynamics, chemical reactions, geochemistry etc. Closed form solutions of NLEEs and its graphical representation reveal the inner mechanism of complex nonlinear phenomena. Therefore, it is a urgent issue and very important to search for more closed form solutions to NLEEs in order to better realization of the structure of nonlinear phenomena. But till now there is no distinctive method to inspect all kinds NLEEs. As a result diverse groups of mathematicians, physicist and engineers have been working vigorously to develop effective methods for which to solve all NLEEs.

Algorithm of the enhanced ( / )
In this section, we explore the enhanced ( / ) G G expansion ′ − method for finding traveling wave solutions to NLEEs. Let us consider a nonlinear evolution equation in two independent variables x and t in the form: is an unknown function of x and t and R is a polynomial of ( , ) u x t and its partial derivatives which contains the highest degree nonlinear terms. The essential steps concerning this method are offered in the following: Step 1: Initiating a compound variable ξ with the combination of real variables x and t , Where, specify the speed of the traveling wave.
The traveling wave transformations (2.2) permit us in dropping Eq. (2.1) to an ordinary differential equation (ODE) for Where, S is a polynomial in ( ) u ξ and its derivatives with respect to ξ .
Step 2: The solution of Eq. (2.3) can be expressed in the following form: ω . Solving this system of equations supplies the values of the unknown parameters.
Step 5: From the general solution of equation ( into (2.4), we obtain more general and some fresh traveling wave solutions of (2.1).

Applications of the method
In this section, we inspect the closed form solutions of the new fifth order non-linear equation and new generalized fifth order non-linear equation with the help of the enhanced Example 1: In this subsection, we will use the enhanced method search for the exact solution to the following new fifth order non-linear equation of the form [21] ( ) ( ) The equation (3.1) transfer to ODE in the following form using wave transformation (2.2) Integrating (3.2) with respect to twice and taking integration constant to zero, we get balancing the highest-order derivative term u′′′ and the highest-order nonlinear term '2 u yields 1. n = Thus, the solution structure of Eq. (3.3) becomes ( )

Substituting (3.4) with the equation (2.5) into equation (3.3), we attain a polynomial
From this polynomial we get the coefficients ' ( / ) i G G of and ' 2 ' Equating them to zero, we achieve an over-determined system that contains thirty algebraic equations (for simplicity we skip to display them). Solving this system of algebraic equation, we get Set 1:

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Set 2: Set 3:

Set 4:
Set 5: Now substituting solution set 1-5 with equation (2.5) into equation (3.4), we get sufficient traveling wave solution to Eq. (3.1) as follows: When, 0 µ < , we get the hyperbolic solution, Again, for Where, x t ξ µ = ± − Example 2: In this subsection, we will apply the given method in section 2 for the exact solution and then the solitary wave solution to the following generalized new fifth order nonlinear equation of the form [21] ( ) Where, α and β are constant.
Integrating (3.2) with respect to ξ twice and taking integration constant to zero, we attain

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Balancing the highest-order derivative term u''' and the highest-order nonlinear term '2 , u yields

n =
Thus, the solution structure of Eq. (3.26) becomes ( ) Equating the coefficient of these to zero, we achieve a system of algebraic equation which on solving, we get Set 1: Where, Again, for 0, µ > we get the following trigonometric solution: Type-6: ( ) 10 2

Results and Physical Explanations
In this section, we have discussed about the obtained solution of new fifth order non-linear equation and new generalized fifth order non-linear equation. As of the above solution, it has been noticed that  x t − ≤ ≤ is given in Figure 6. For simplicity we ignored the others figures. G G expansion ′ − method against other methods is that the method provides more general and huge amount of new closed form wave solutions. The closed form solutions have its great importance to interpretation the inner mechanism of the complex physical phenomena. Therefore this method is very concise and straightforward to handling and can be applied for finding closed form solutions of other NLEEs arising in science and engineering.