Solitons and Periodic Solutions of the Fisher Equation with Nonlinear Ordinary Differential Equation as Auxiliary Equation

Abstract In this article the new extension of the generalized and improved ( '/ ) G G -expansion method has been used to generate many new and abundant solitons and periodic solutions, where the nonlinear ordinary differential equation has been used as an auxiliary equation, involving many new and real parameters. We choose the Fisher Equation in order to explain the advantages and effectives of this method. The illustrated results belongs to hyperbolic functions, trigonometric functions and rational functional forms which show that the implemented method is highly effective for investigating nonlinear evolution equations in mathematical physics and engineering science.


Introduction
In physical sciences all essential equations are nonlinear and these are often complicated to interpret. So the exact solutions of nonlinear evolution equations (NLEEs) have turned out to be a chief concern for researchers. NLEE is one of the most powerful and important modelled equations among all equations in nonlinear sciences and it plays a vital role in the field of scientific work of engineering sciences such as chemical kinematics, fluid mechanics, chemistry, biology, nonlinear optics, optical fibers, plasma physics, solid state physics, biophysics, geochemistry, quantum mechanics, chemical physics, condensed matter physics, high-energy physics and so on. As they reveal a lot of physical information which help to understand the operation of the physical model better, that is why the explicit solutions of NLEEs play important role in the study of physical phenomena and remains a crucial field for researchers in the ongoing investigation.
For the past few decades, a vast research has been going on to construct explicit solutions of NLEEs, which are used as models in order to describe many important and problematics physical phenomena in various fields of science. So to figure out the exact solutions of NLEEs substantial work are being made by mathematicians and scientists and have developed effective and convincing methods such as the Hirota's bilinear transformation method [1], the tanh-function method [2,3], the exp-function method [4,5], the F-expansion method [6], the Jacobi elliptic function method [7], the homogeneous balance method [8], the homotopy perturbation method [9], the tanh-coth method [10], the direct algebraic method [11], the Backlund transformation method [12], and others [13,14,15,16].
Later in 2008, Wang et al [17] introduced a new method called the ( ) G G ′ -expansion method for finding the solutions of traveling waves of NLEEs. This ( ) G G ′ expansion method shows that it is one of the most powerful and effective method to solve NLEEs since it gives a clear and short to the point results in terms of hyperbolic functions, trigonometric functions and rational functions which is why scientists have carried out a lot of researches to construct traveling wave solutions via this method [18][19][20][21].
Further research of ( ) G G ′ -expansion method has been carried out by many researchers to show the possible productivity of the application. For example-Zhang et al [22] expanded the original ( ) G G ′ -expansion method and named as the improved ( ) Using this method many researches have been carried out in order to find travelling wave solutions for NLPDEs [23][24][25][26][27][28][29][30]. Then Akber et al [31] introduced the generalized and improved ( ) where P is the polynomial and here ( , ) u u x t = is an unknown function. In the polynomial P contains different partial derivatives of the function u itself wherein involves the highest order derivatives and the highest nonlinear terms. Now the prime process of this method is being discussed in steps below.
Step 1: Suppose that, where the constant term W is known as the speed of wave, is substituted in Eq. (2.1), which allows a PDE to convert an ODE with respect to ξ .
( , , , ...) 0 Q u u u u ′ ′′ ′′′ = (2.3) Step 2: Eq. (2.3) is being integrated term by term and if needed it can be integrated more than once and the integral constants may be set to zero to make easy to solve. Now the integrated travelling wave solution of Eq. (2.3) can be represented as.

Application of the Method
Let us consider the Fisher equation to investigate and construct new wave solutions by executing new extension of the generalized and improved ( ) [34]. The Fisher equation: By using the wave transformation of Eq. (2.2), into Eq. (3.1), the above equation transforms into the following NLODE: Now by taking the homogeneous balance between the nonlinear term 2 u and the highest order derivative term u′′ in Eq. (3.2), we obtain the value for N i.e. 2 N = . Therefore the solution of Eq. (3.2) can be written in the form:   ,  ,  96  6  1  ,  1536  ,  1536  1  3  6  16 , , ,

Discussions
Various methods have been used to investigate for the solutions of Fisher Equation, such as Kudryashov [35] investigated by using simplest equation method, Wazwaz et al. [36] studied by using the Adomain decomposition method, in Ref. [37] Öziş et al. implemented by the Exp-function method, the homotopy analysis method executed by Tan et al. [38], and Ablowitz et al. [39] investigated solutions for a special wave speed. To the best our awareness the Fisher Equation has not been investigated by the new generalized and improved ( )

Graphical Representations
With the help of the computational software, Maple, we have illustrated some of the obtained solutions for travelling waves solutions in below.