ON THE EXACT SOLUTIONS TO CONFORMABLE EQUAL WIDTH WAVE EQUATION BY IMPROVED BERNOULLI SUB-EQUATION FUNCTION METHOD

In this paper, we consider conformable equal width wave (EW) equation in order to construct its exact solutions. This equation plays an important role in physics and gives an interesting model to define change waves with weak nonlinearity. The aim of this paper is to present new exact solutions to conformable EW equation. For this purpose, we use an effective method called Improved Bernoulli Sub-Equation Function Method (IBSEFM). Based on the values of the solutions, the 2D and 3D graphs and contour surfaces are plotted with the aid of mathematics software. The obtained results confirm that IBSEFM is a powerful mathematical tool to solve nonlinear conformable partial equations arising in mathematical physics.


Introduction
Fractional differential equations are the generalization of classical differential equations with integer order.So, in recent years, fractional differential equations become the field of physicists and mathematicians who investigate the expediency of such non-integer order derivatives in different areas of physics and mathematics.These equations became a useful tool for describing numerous nonlinear phenomena of physics such as heat conduction systems, nonlinear chaotic systems, viscoelasticity, plasma waves, acustic gravity waves, diffusion processes [1][2][3].Many numerical and analytical methods were developed and successfully employed to solve these equations such as modified Kudryashov method [4], homotopy perturbation method [5], new extended direct algebraic method [6], fractional Riccati expansion method [7], modified extended tanh method [8].
During the last few years, a straightforward definition of conformable derivative is given [9].The conformable derivative operator is compatible to many real-world problems and provides some properties of classical calculus such as derivative of quotient or product of two functions, the chain rule [10].During the last few years, many of techniques applied to find exact solutions to conformable nonlinear partial differential equations [11][12][13][14][15][16].
In this paper, we consider the following conformable EW equation: where , pl are real parameters, u is a function of independent variables.The operator t D  represents conformal derivative operator defined only for positive region of t [10].Before consideration of the solution procedure, let us give some properties of conformable derivative.
The conformable derivative of order  with respect to the independent variable for a function

 
= ( ) : 0, .y y t  Theorem 1.1.Assume that the order of the derivative is   0,1  and suppose that = ( ) u u t and = ( ) Conformable differential operator satisfies some critical fundamental properties like the chain rule, Taylor series expansion and Laplace transform.
Theorem 1.2.Let = ( ) u u t be an   conformable differentiable function and assume that The proofs of Theorems 1 and 2 are given in [17] and [9], respectively.The rest part of the paper is organized as follows.In Section 1, descripton of the IBSEFM is given.In Section 2, the application of IBSEFM is mentioned.Finally, this study is completed by providing conclusions in the last section.

Description of the IBSEFM
In this section, we give the fundamental properties of the IBSEFM.This method is direct, significant, advanced algebraic method to establish reliable exact solutions for both nonlinear and nonlinear fractional partial differential equations [11,[18][19][20].
Let us describe five main steps of the IBSEFM.
Step 1.Let us take into account the following conformable partial differantial equation of the form where () t D  is the conformable fractional derivate operator, ( , ) v x t is an unknown function, P is a polynomial and its partial derivatives contain fractional derivatives.The aim is to convert conformable nonlinear partial differential equation with a suitable fractional transformation into the ordinary differantial equation.The wave transformation is Using the properties of conformable fractional derivate, we convert (2) into an ODE of the form ( , , ,...) = 0.
' '' N V V V (4) If we integrate (4) term to term, we acquire integration constant(s), which can be determined later.
Step 3. The positive nonzero integers ,, m n M are found by balance principle that is both nonlinear term and the highest order derivative term of (4).
i is  Step 5. When we solve (4), we get the following two cases with respect to  and : d Using a complete discrimination system of () F  , we obtain the analytical solutions to (4) via mathematics software and categorize the exact solutions to (4).To achieve better results, we can plot two and three dimensional figures of analytical solutions by considering proper values of parameters.

Application of the IBSEFM
Take the travelling wave transformation where q and m are nonzero constants to be determined.Then Equation (1) turns into the following ordinary differential equation: When we apply the balance principle for the terms 2 u and '' u , the result for , mn and M is = 1.M m n  By balancing the order between the nonlinear term and highest order derivative in Equation (10), we obtain = 3, = 3 Mn and =1 m , then we get Substituting ( 3)-( 11) in (10), we get the algeabraic equation system according to F : Putting (13) along with ( 3)-( 11) in ( 7), we acquire the exponential function solution to the EW equation as follows: for 3D surface, 0 < < 3, x 0 < < 3 t for contourplot and where 40 , , , , 0 d l q a b  .Substituting (14) along with (3)-( 11) in (7), we obtain solution to (1)   where 40 , , , , , , 0 l m p q a b   .When we put (15) along with (3)-( 11) in (7), and obtain the complex solution to (1) as follows:  

Conclusion
In this paper, the IBSEFM method is applied for the conformable EW equation.Using a wave transformation, we convert the conformable differential equation into the ordinary differential equation, which can be solved according to the IBSEFM.By means of this method, exact solutions are obtained.The contourplot, 3D and 2D surfaces of all solutions obtained by IBSEFM under the suitable values of parameters are plotted to show the main characteristic physical properties of the solutions with the help of mathematics software.According to the results, one can see that the formats of travelling wave solutions in two and three dimensional surfaces are similar to the physical meaning of results.
The solutions are also solitary wave solutions.Also, it is clear that more steps are developed and better approximations are obtained.The conclusions show that the IBSEFM is simple, effective and powerful.Thus, in mathematical physics, it is applicable to solve other conformable partial differential equations.We claim that the IBSEFM method is practically well suited, since it can be adopted to a wide range of nonlinear differential equations.Eventually, this method is influential and suitable for solving other types of nonlinear differential equations in which the balance principle is satisfied.
will be determined later.The numbers 0, 0 mn  are chosen arbitrary constants to balance principle.Consider the Bernoulli differential equation of the form Then we solve the system of equations of F and, in each case, substitute the obtained coefficients to get the new solution(s) ( , ) u x t .Solving the system by Wolfram Mathematica software, we obtain the coefficients as follows.