On The Exact Solutions to Conformable Time Fractional Equations in EW Family Using Sine-Gordon Equation Approach

New exact solutions to conformable time fractional EW and modified EW equations are constructed by using Sine-Gordon expansion approach. The fractional traveling wave transform and homogeneous balance have significant roles in the solution procedure. The predicted solution is of the form of some finite series of multiplication of powers of cos and sin functions. The relation among trigonometric and hyperbolic functions in sense of Sine-Gordon expansion gives opportunity to construct the solutions in terms of hyperbolic functions.


Introduction
In the recent several decades, many developments in computer algebra field have been witnessed.These developments have also lead to solve many nonlinear PDE problems symbolically due to faster symbolic algebraic manipulations when compared with study with pencil.Moreover, more methods have been introduced and implemented to lots of nonlinear PDEs.The prominent ones of those methods are various types of simple ansatz techniques with positive integer powers [1][2][3].First integral method [4][5][6] is an alternative technique to simple ansatzes to construct exact solutions to nonlinear PDEs.Exp function method [7,8] assumes the predicted solutions as a finite series of some particular functions.(G /G) expansion method [9,10] is an alternative that approaches the solution with a finite power series of a function satisfying a particular ODE.Trigonometric and hyperbolic type solutions to nonlinear PDEs can be determined by implementation of sine-cosine approach [11][12][13].Later on, the implementations of those methods have been extended to solutions of fractional nonlinear PDEs [14][15][16][17][18][19][20][21][22][23].Existence of some particular traveling wave transforms has given opportunity to exact solutions to fractional PDEs with nonlinear terms.Physical interpretation of fractional derivatives has been comprehended deeply by examining plots of solutions in both time fractional and spacetime fractional cases.Even though there exist various definitions of fractional derivative in the literature, we focus time fractional equal width equation (fEWE) and time fractional modified equal width equation (fmEWE) where u is function of the independent variables t and x, p and q are real parameters.One should note that D γ t is conformal fractional derivative operator defined in [24].The original of the fEWE with integer order appeared in the study of Morrison et al. [25].This equation (when γ = 1) has a simple relation with the well-known RLW or BBM equation.The integer ordered EWE has singular wave solutions expressed in forms of powers of sech function [25].
2 Fractional Derivative in Conformable Sense γ.th order conformable derivative of a function y = y(t) is defined as where y = y(t) : [0, ∞) → R [24].This recent definition of fractional derivative satisfies the following properties.
Theorem 1 Let u = u(t) and y = y(t) are γ-differentiable for all positive t.Then, Many significant properties like the chain rule, Laplace transform and Taylor series expansion are valid for this definition of fractional derivative operator [28].
Theorem 2 Let u = u(t) be an γ-conformable differentiable function.Then, where y = y(t) is defined in the range of u(t) and differentiable in the classical sense.

Sine-Gordon Equation (SGE) Approach
Consider the one dimensional SGE of the form where u = u(x, t).The traveling wave transform in one dimension u(x, t) = U (η) with η = a(x − νt α /α) reduces the SGE to the ODE where ν indicates the velocity of the wave described in the traveling wave transform [29].Some algebra and substitutions converts the transformed equation to where C is integration constant and assumed zero.Assume that H(η) = U (η)/2 and m 2 /(a 2 (1 − ν 2 )) = 1.Then, (7) takes the form Thus, (8) gives the following relations where c = 0 is integral constant.On the other hand, the governing fractional PDE is reduced to an ODE of the form Ω(U, U , U , . ..) = 0 (12) by using the fractional traveling wave transform u(x, t) = U (η) with the transform variable η = a(x − νt γ /γ).Then, a predicted solution to (12) of the form is constructed.This solution can be expressed in terms of w as due to the relations ( 9) - (10).The solution process begins by finding s by the help of homogenous balance procedure for (12).The determination of s gives the exact power of the solution in series of trigonometric and hyperbolic functions.
After substitution of the predicted solution ( 14) into (12) and some algebra, the coefficients of powers of sin H cos H are assumed as zero.Thus, an algebraic system of equations are obtained.This system is solved for the predicted solution coefficients A 0 , A 1 , B 1 , . . .and one or both of transform coefficients a, ν.If one can find a solution satisfying the conditions given above, then, the solutions are constructed by using ( 9) -( 10) and η.The final forms of the solutions are written by substitution of x and t with the relations of transform parameters a and ν instead of η.

Solutions to the conformable time fractional EWE
The time fractional EWE of the form ( 1) is reduced to the ODE where C is constant by the use of fractional traveling wave transform.The balance between U 2 and U gives s = 2. Thus, the predicted solution is constructed as Substitution of the predicted solution ( 16) into (15) gives Following usage of some trigonometric identities and simplifications, the coefficients of multiplications of powers of sin and cos functions are equated to zero.The solution of this resultant algebraic system of equations for A i , i = 1, 2, 3, B j , j =

Solutions to the conformable time fractional mEWE
The fractional traveling wave transform reduces the mEWE to where C is integration constant.The homogeneous balance between U 3 and U gives s = 1.Thus, the predicted solution takes the form in terms of w.Substitution of this solution into (22)gives ) Using some trigonometric identities and simplifications, we equate the coefficients of powers of multiplications of cos and sin functions and the constants to zero.Thus, we find a system of algebraic equations.Solution of this system for a, ν, A 0 , A 1 , B 1 and C gives for C = 0 and arbitrary ν.Thus, the solutions to (2) are constructed as and A particular form of the solution u 9 (x, t) is determined as by using the parameters as ν = 5, q = −1, p = 1.This solution models propagation of an initial positive pulse along x-axis.As time proceeds, the initial pulse travels by keeping its shape and height.This particular solution is depicted for various values of γ in   SGE method was implemented to extract the exact solutions to conformable fEWE and fmEWE equations.Fractional traveling wave transform reduces the governing equation to some ODEs.The homogeneous balance principle has a significant role to determine the degree of the series of the solutions.The solutions consist of multiplications of powers of sech and tanh functions.Once the degree and structure of predicted solution was determined, it was directly substituted into the resultant ODE.The algebraic system constructed by equating the coefficients to zero was solved to determine the coefficients of the predicted solution and the parameters of fractional traveling wave transform.Illustrations of some particular forms of the solutions indicate the effects of γ to the initial pulses.

= 1 Figure 1 :
Figure 1: The solution u 5 (x, t) for various derivative orders Fig 2(a)-2(d) in some finite domain of x and t.When γ is 0.20 and 0.40, we observe that the pulse moves faster at the beginning, but slows down as time goes, Fig 2(a)-2(b).The change of velocity is hardly observed for γ = 0.70, Fig 2(c).The velocity is constant with the choice γ = 1, Fig 2(d).

Figure 2 :
Figure 2: The solution u 9 (x, t) for various derivative orders