New Hyperbolic Function Solutions for Some Nonlinear Partial Differential Equation Arising in Mathematical Physics

In this study, we investigate some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin–Bona–Mahony equation and the (2 + 1)-dimensional cubic Klein–Gordon equation by using the generalized Kudryashov method. After we submitted the general properties of the generalized Kudryashov method in Section 2, we applied this method to these problems to obtain some new analytical solutions, such as rational function solutions, exponential function solutions and hyperbolic function solutions in Section 3. Afterwards, we draw twoand three-dimensional surfaces of analytical solutions by using Wolfram Mathematica 9.


Introduction
The nonlinear partial differential equations (NLPDEs) going on to attract attention in the last 20 years have been dense and have wide historical application areas.They have a fundamental role in various applications fields such as science, engineering, and information theory, which is very important in entropy and applied sciences.Moreover, they have always been used to explain many natural phenomena from population, climate changes, and earthquakes to chemical structures of atoms and so on.Much
When it comes to classification of this study, we submit the general structure of the new generalized Kudryashov method (GKM) in Section 2.Then, in Section 3, we successfully apply the GKM to obtain some new analytical solutions to the (1 + 1)-dimensional nonlinear Dispersive Modified Benjamin-Bona-Mahony equation (DMBBME) [14] defined by: 2 0, in which α is constant and the (2 + 1)-dimensional nonlinear cubic Klein-Gordon equation (cKGE) [15] defined by: 3 0, where α and β are non-zero and constants.
The first studies on the (1 + 1)-dimensional nonlinear DMBBME have been conducted by Benjamin, Bona and Mahony (1972) as an improvement of the Korteweg-de Vries equation for modeling long surface gravity waves of small amplitude, propagating uni-directionally in (1 + 1)-dimensions [16].It can characterize the hydromagnetic waves in cold plasma, acoustic waves in inharmonic crystals and acoustic gravity waves in compressible fluids [14,17,18].
The (1 + 1)-dimensional nonlinear cKGE is a different version of the Schrödinger equation [19].Furthermore, this equation represents the quantum amplitude for finding a point particle in various places and the relativistic wave function [19].
Rudolf Clausius introduced "Entropy" in 1865 as a state function in thermodynamics [20,21].Some leading researchers such as Ludwig Boltzmann, Josiah W. Gibbs, and James C. Maxwell have given physical and geometrical interpretations of entropy [20,21].
The inspiring quotations noted by Pynchon were submitted to the literature in 1966.Being in the inspiring quotations noted by Pynchon, entropy has been investigated by many researchers as a concept in many scientific fields.A lot of physical problems are described by using the general structures of entropy and partial differential equations.In this sense, Parsani et al. have investigated the entropy concept for the three-dimensional compressible Navier-Stokes equations [22].Zhao et al. studied crowd macro state detection using entropy model [23].Volker Elling introduced a paper entitled "Relative entropy and compressible potential flow" to the literature in 2015 [24].Chen et al. observed the analysis of entropy generation in nanofluid [25].Lv et al. published a manuscript entitled Entropy-Bounded Discontinuous Galerkin Scheme for Euler Equations in 2015 [26].Broadbridge has researched the entropy diagnostics for fourth order partial differential equations in conservation form [27].

General Structure of the Method
It would be useful to remember the general structure of GKM [28][29][30][31].Therefore, we consider the following NLPDEs
Step 1.First of all, we must consider the travelling wave transformation of Equation (3) as following: in which k and w are arbitrary constants and not zero.Taking necessary derivations of Equation (4), Equation (3) converts into a nonlinear ordinary differential equation (NLODE) as the following: Step 2. Let's consider trial function of analytical solution for Equation (5) as following: where and the function Q is a special solution which is the general Riccati equation [12,28] defined by Studies on existence of solutions of Equation (7) characterized by Riccati differential equation have already been submitted to the literature [32].Taking into consideration Equation (7), we obtain ( ) ( ) )( ) Step 3.Under the terms of the proposed method, we suppose that analytical solution of Equation ( 5) can be written as following: To calculate the values of M and N in Equation (10), being the pole order for the general solution of Equation ( 5), we consider the principle of balance between highest order derivations and highest order power nonlinear terms in Equation (5).Then, we can take some values for M and N to obtain new analytical solutions to the Equation (5).
Step 4. Replacing Equation (6) into Equation (5), it provides us a polynomial ( ) Q establishing the coefficients of ( ) R Q to zero, we acquire a system of algebraic equations.Solving this system by using various computer programs such as Maple, Matlap and Mathematica, we can find the values of In this way, we attain the travelling wave solutions to Equation (5).

Implementation of Method Proposed
In this section, we have obtained the analytical solutions of the (1 + 1) dimensional nonlinear DMBBME and the (2 + 1)-dimensional nonlinear DMBBME cKGE by using GKM.  1) as following: being w is constant and not zero.Substituting Equation (11) into the Equation ( 1) by integrating with respect to η , we can find the NLODE for the (1 + 1)-dimensional nonlinear DMBBME: ( ) When we rearrange Equation ( 6) and Equation ( 9), with the help of balance principle, we obtain the term for suitability This resolution procedure is applied and we obtain results as follows: Case 1: If we take 1 M = and 2 N = for Equation (6), then, we write following equalities: and where 2 1 0, 0 a b ≠ ≠ .When we use Equations ( 14) and ( 16) in Equation ( 12), we get a system of algebraic equations.By solving this system with the help of Wolfram Mathematica 9, we obtain the following coefficients; Substituting Equation (17) and Equation ( 18) into Equation ( 14) along with Q , we obtain the hyperbolic function solutions of the (1 + 1)-dimensional nonlinear DMBBME as following, respectively; and Remark 1.The hyperbolic function solutions Equation (19) and Equation (20) obtained by using GKM are new analytical solutions when we compare it with the solutions obtained by Khan et al. [14].These hyperbolic function solutions have been checked up whether they have verified the (1 + 1)-dimensional nonlinear DMBBME by Wolfram Mathematica 9.
Meanwhile, when the general properties of hyperbolic functions are considered, they can be written as complex trigonometric function solutions: .
To the best of our knowledge, the application of GKM in Equation ( 1) has not been submitted in the literature before.Figure 1, being the surfaces of complex trigonometric function solution Equation (19), was drawn by the same computer program.Of course, it is possible to see different surfaces of solutions according to various values of parameters.
Case 2: If we take 2 M = and 3 N = for Equation ( 6), then, we can write following equalities: ( ) ( ) ≠ .When we use Equations ( 21) and ( 23) in Equation ( 12), we get a system of algebraic equations for Equation (12).By solving this system with the help of Wolfram Mathematica 9, we obtain the following coefficients: Case 2.1: ( ) ( ) ( ) Case 2.4: ( ) Case 2.6: Case 2.7: Case 2.8: Case 2.9: Substituting Equations ( 24)-(32) into Equation ( 14) along with Q , we can obtain some new analytical solutions to the (1 + 1)-dimensional nonlinear DMBBME as the following, respectively.For Case 2.1, we can find the rational exponential function solution to Equation (1) as following:  ( ) ( ) For Case 2.4 and Case 2.5, another rational exponential function solution to Equation ( 1) is found in the following manner: ( ) ( ) For Case 2.8 and Case 2.9, we can get another new complex trigonometric function solution to Equation ( 1):    Remark 2. Travelling wave solutions, Equations ( 33) and ( 35)-( 37), obtained by using GKM are new analytical solutions to the (1 + 1)-dimensional nonlinear DMBBME and analytical solution, Equation (34), is the same as the solution of Khan et al. [14].We have checked up whether they have verified the (1 + 1)-dimensional nonlinear DMBBME with the help of Wolfram Mathematica 9.Then, Figures 2-6 were drawn by the same program.To the best of our knowledge, the application of GKM to Equation (1) has not been submitted to the literature before.
Example 2. Let us consider the (2 + 1)-dimensional nonlinear cKGE defined by: 3 0, 0, 0, in which , α β coefficients are constants and not zero [15].When it comes to convert Equation ( 38) into NLODE, we can perform the travelling wave transformation: in which k and c are constants and not zero.Substituting Equation (39) into Equation (38), we find the NLODE for the (2 + 1)-dimensional nonlinear cKGE: ( ) When we rearrange Equation ( 6) and Equation ( 9), we obtain a relationship between N and M with the help of balance principle as following; 1.

Conclusions
In this present manuscript, we have applied the GKM to the (1 + 1)-dimensional nonlinear DMBBME and the (2 + 1)-dimensional nonlinear cKGE.Then, we obtained some new analytical solutions, such as complex trigonometric function, trigonometric function, and hyperbolic function solutions.We have proven that they have been verified Equation (1) and Equation (2) by using Wolfram Mathematica 9.
Under the terms of this information, it has been observed that GKM has been a powerful tool to obtain the analytical solutions of such differential equations.We think that this method can also be conducted on other nonlinear partial differential equations.

Discussions and Comparisons
Ma and Fuchssteiner have already submitted to the literature a study including systematical approaches to travelling wave solutions including analytical function solutions such as tangent hyperbolic function, rational function and trigonometric function solutions of general Riccati equation [12].
We have obtained the tangent hyperbolic function solution ( ) , u x t for the (1 + 1)-dimensional nonlinear DMBBME by using GKM, which is a special case of the transformed rational function method [8].

Example 1 .
Let's consider the travelling wave solutions for the (1 + 1)-dimensional nonlinear DMBBME and we perform the travelling wave transformation ( ) ( ) NLPDE into NLODE, we can get by carrying out the travelling wave transformation
constants.When we use Equations (42) and (44) in Equation (40), we find a system of algebraic equations.Solving system of algebraic equations using Wolfram Mathematica 9, it gives us the following coefficients: Case 1.1.

4 , 3 , 5 , 6 ,
u x t , hyperbolic function solutions such as ( ) 1 , u x t and ( ) 2 , , u x t exponential rational function solutions such as ( ) u x t and ( ) u x t and complex trigonometric function solutions such as ( ) u x t and ( ) 7