On finite series solutions of conformable time-fractional Cahn-Allen equation

Abstract The aim of this article is to derive new exact solutions of conformable time-fractional Cahn-Allen equation. We have achieved this aim by hyperbolic function and expa function methods with the aid of symbolic computation using Mathematica. This idea seems to be very easy to employ with reliable results. The time fractional Cahn-Allen equation is reduced to respective nonlinear ordinary differential equation of fractional order. Also, we have depicted graphically the constructed solutions.


Introduction
Fractional di erential equation may be considered as the missing part of the classical di erential equations. In recent years, many authors have studied the nonlinear fractional di erential equations for example see [1][2][3][4][5][6][7][8] because these equations express many complex nonlinear physical phenomena and dynamic forms in physics. Several de nitions of fractional derivative have been presented to the literature, amongst are Atangana Baleanu operator, Caputo-Fabrizio and conformable derivative.
In this research, we apply two methods on conformable time-fractional Cahn-Allen equation to scrutinize the new explicit exact solutions [9][10][11][12] that may read as We employ the expa function approach [13][14][15] and the hyperbolic function approach [16][17][18] via traveling wave transformation with the conformable derivative. The rest of the article is arranged as follows: In Section 2 some preliminaries and notations dealing with the fractional calculus theory are brie y described. Section 3 presents description of methods. The exact solutions of the nonlinear Cahn-Allen equation are constructed in Section 4. In Section 5 some graphical representation for solutions are showed. Section 6 presents the summary of the obtained results.

Conformable derivative
The latter part presents the results of the current study graphically.
We recall the conformable derivative with some of its properties [19]. De nation 1 Suppose h : R > → R be a function. Then, for all t > , is known as α, < α ≤ order conformable fractional derivative of p. The followings are some useful properties: D α t (a p + b g) = aD α t (p) + bD α t (g), for all a, b ∈ R D α t (p g) = p D α t (g) + g D α t (p) Let p : R > → R be an α-di erentiable function, g be a di erentiable function de ned in the range of p.
On the top of that, the following rules hold.

Description of methods
The present subsection provides a brief explanation for two reliable techniques in engendering new exact solutions to nonlinear conformable time-fractional equation.
For this purpose, suppose that we have a nonlinear conformable time FDE that can be presented in the form The FDE (2) can be changed into the following nonlinear ODE of integer order with the use of following wave transformation where ℘ is a polynomial in U(η) and its total derivatives with = d/dη while k and k are nonzero arbitrary constants.

. The exp a function approach
Let us try to search a non-trivial solution for (3) in the following form [13][14][15]20] where A i and B i , for ( ≤ i ≤ N), are found later and N is a free positive constant. Replacing (5) in the nonlinear (3), yields ℘(a η ) = q + q a η + ... + qτ a τη = .
Setting (5) to be zero, results give a set of nonlinear equations as follows: by solving the generated set (7), we acquire non-trivial solutions of the nonlinear PDE (2).

Application to time-fractional Cahn-Allen equation
Using the transformation (4) in (1), we get Through balancing the terms U and U , we select N = , the nontrivial solution (5) reduces to: By setting the above solution in (9) We now again consider (9) to solve by utilizing the hyperbolic function approach. Case1: dρ dξ = sinh(ρ) Through homogenous balancing principle, the terms U and U gives N = and the non-trivial solution (8) becomes By setting the above non-trivial solution (19) in (9) and equating the coe cients to zero in the resultant equation, we reach a set of nonlinear polynomial equations.
On solving the obtained set of equations yields the following sets of solutions which will give us the new exact solutions of (1).
We now write some new exact solutions using (20) to (31) as follows. From (19) and (20) we get, (32) Now from Eqs. (19) and (24) we get, Now from Eqs. (19) and (28) we get, (34) The other solutions can be formulated on the similar way. Case2: dρ dξ = cosh(ρ) and for N = , the non-trivial solution (8) becomes By setting the above non-trivial solution in (9) and equating the coe cients to zero in the resultant equation, we reach the following set of polynomial equations.
On solving this system of equations, we obtain the following sets of solutions that will give us the new exact solutions of (1)

Some graphical illustrations
In this section, we give some graphical illustrations of the acquired solutions of our equations. The 2-dimensional and 3-dimensional plots of certain solutions are presented as follows.

Conclusion
In this paper an e cient method was built up to solve conformable time-fractional Cahn-Allen equation. This idea is based on the idea of hyperbolic function and expa function, which is a known method for solving di usion equations. These methods are very powerful with minimum algebraic work. The computations are done using Maple 18.