New Exact Traveling Wave Solutions of (2 + 1)-Dimensional Time-Fractional Zoomeron Equation

In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.


Introduction
Fractional partial differential equations (FPDEs) have a wide of applications in different fields, such as biology, physics, signal processing, fluid mechanics, and electromagnetic, and so on. In recent decades, many effective methods have been presented to obtain the exact traveling wave solutions of FPDEs, for example,  [9], the F-expansion method [10] [18] [19] [20] [21].
In this paper, we use the complex traveling wave transformation to deduce (2 + 1)-dimensional conformable time-fractional Zoomeron equation into ordinary differential equation. Furthermore, inspired by the reference [22], we introduce the new mapping approach and the new extended auxiliary equation approach [23] [24] [25] to investigate the exact solutions of (2 + 1)-dimensional timefractional Zoomeron equation [20]: xy xy when 1 α = , Equation (1) reduces to the (2 + 1)-dimensional Zoomeron equation [26]. Aksoy E. [27] obtained two types of exact analytical solutions including hyperbolic function solutions and trigonometric function solutions by using sub-equation and generalized Kudryashov methods in Equation (1). Hosseini K. [28] obtained several new wave form solutions of Equation (1) (1), the aim of this investigation is to establish more general solutions and some new solutions using the two methods mentioned above.
The organization of this paper is as follows: In Section 2, we introduce the conformable fractional derivative. In Section 3, we introduce the new mapping approach and the new extended auxiliary equation approach to investigate the solutions of (2 + 1)-dimensional time-fractional Zoomeron equation, and analyze the dynamic behaviors of the solutions in Section 4. Finally, we give some conclusions in Section 5.

The Conformable Fractional Derivative
In this section, we introduce the conformable fractional derivative [20] [21].

Description of the Methods
Suppose that a nonlinear fractional differential equation with the conformable time-fractional derivative: where H is a polynomial of ( ) , u x t and its partial conformable derivatives including the highest order derivative and the nonlinear term.
We use the complex traveling wave transformation where , , k h l are non-zero arbitrary constants. Equation (1) converts into a nonlinear ordinary differential equation: where P is a polynomial of ( ) , u x t and its partial derivatives,

The New Mapping Approach
We suppose the solution of Equation ( where , , r p q and s are arbitrary constants, the solutions of Equation (10) given by reference [23] with 0 s ≠ .

The New Extended Auxiliary Equation Approach
We suppose the solution of Equation (8) as follow: are constants and the positive integer N can be determined by balancing the highest order derivative and the nonlinear term in Equation (8).
( ) F ξ satisfies the following equation: are constants and 6 0 c ≠ . Equation (12) has the following solutions: where the function

Applications
We substitute Equation (7) into Equation (1), which deduce the nonlinear ordinary differential equation: We integrate Equation (14) twice, then we have where the prime denotes the derivative with respect to ξ , the second constant of integration is zero. Balancing the highest order derivative term u′′ and the highest order nonlinear term 3 u , we get

Application of the New Mapping Approach
We assume that the solution of Equation (9)   u a a a ξ ϕ ξ ϕ ξ = + + (16) where 0 1 2 , , a a a are constants.
Substituting Equation (16) and its derivatives and Equation (10) into Equation (15), yields a system of equations of ( ) i ϕ ξ , then setting the coefficients of ( )( ) to zero, we can deduce the following set of algebraic polynomials with the respect 0 1 2 , , a a a : Solving the above algebraic equations, we obtain the following two results: Substituting Equation (18) and the solutions in reference [23] into Equation (16), qh l k q h l k la a a a a p q q s la la h l k Substituting Equation (21) and the solutions in reference [23] into Equation (16), where

Application of the New Extended Auxiliary Equation Approach
We assume the solution of Equation (11) as follow: where 0 1 2 , , a a a are constants.

Dynamical Behaviors
In this section, we analyze the dynamic behaviors of the solutions in (2 + 1)-dimensional time-fractional Zoomeron equation. Figure 1 and Figure 2 are the 3D and 2D graphs of the solutions (19) and (20), (22) and (23)

Conclusion
In conclusion, (2 + 1)-dimensional time-fractional Zoomeron equation has been investigated by the new mapping approach and the new extended auxiliary equation approach. Singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation have been obtained, where Jacobi elliptic function solutions are new solutions. When 1 m → or 0 m → , the Jacobi elliptic function solutions degenerate into the hyperbolic function solutions and the periodic function solutions. Consequently, it is obvious that the application of these two methods is effective to the time-fractional equations.