Exact Travelling Wave Solution of Fractional Space-time Zakharov–Kuznetsov Equation

Objectives: To find the new exact travelling wave solutions of fractional order space-time Zakharov– Kuznetsov (ZK) equation of ion-acoustic waves. Methods/Statistical: The fractional derivative is defined in the sense of modified RiemannLiouville derivative to convert the fractional space-time ‎ZK equation to nonlinear ordinary differential equation‎.‎ The two proposed techniques the (ω'/ω)-expansion and extended (ω'/ω)-expansion methods are employed for constructing the new exact travelling wave solutions of ZK equation. Findings/Results: The obtained new exact travelling wave solutions of ZK equation include the hyperbolic function, rational function, and trigonometric function. ‎Application: The results reveal that the two used methods here are effective and powerful methods and might be accustomed to ‎establish the other solutions of nonlinear fractional partial differential equations that arising in ‎nonlinear science.‎ *Author for correspondence


Introduction
The fractional calculus has a wide array namely, nonlinear optic, solid state physics, fluid flow, plasma physics, control theory, signal processing, systems identifications, biology 1-28 and so on. During this paper the target of our aim by means that two proposed methods, namely and extended -expansions methods to solve the following fractional space-time in nonlinear of ion-acoustic waves arising in plasma physics as [29][30][31] .
This paper has been outlined as follows: Section 2represents the modified Riemann-Liouville derivative and it's a few properties. In sections 3A, 3B two proposed techniques are explained. In sections 4, 5 the obtained solutions of the reduced Eq. (1.5) have been presented. In last section the conclusion is given.

The Modified Riemann-Liouville Derivative and Some Properties
In this section, the Jumarie's Riemann-Liouville derivative is outlined as follows 1-2 With the properties: where a, b are constants.

Methodology of the -Expansion Technique
In what follows, we explain the algorithm of theexpansion method as follows: Step 1: Assume the general fractional space-time NPDE as where, is an unknown functionin two independent variables and, is a polynomial in terms of , and its derivatives and including the nonlinear terms. Step 2: Consider the travelling Wavetrans form as: Step 3: We express the solution of ODE (3.3a) in the form: where, satisfies where, i β , λ and μ can be specified later on, Step 4: Inserting Eq.

The Extended −Expansion Technique
In what follows, the proposed technique 20 is described. For a give general partial differential equation as We express the solution ( ) V η as finite series in the where, N is the balancing number which is positive integer, it is evaluated by considering the homogeneous balance. ω=ω(η) satisfies (3.5b) where, D, E and F are real parameters should be determined later.

Application of the (ω'/ω) − Expansion Technique to the ZK Equation
Apply the (ω'/ω) −expansion technique to extract the exact solution for ZK equation (1.5). Assume that the homogenous balancing between V 3 with VV", we get 2 N = . Then the solutions of Eq. (1.5) is defined as: Satisfies Eq. (3.5a), are constants should be calculated later on.
Inserting Eq. (4.1) into Eq. (1.5) and equating the same power of (ω'/ω) to be zero. We get the set of algebraic equations for 0 1, 2 , , , , , k l w β β β . By solving this set of algebraic equations, we have

Application of Extended − Expansion Method to the ZK Equation
Here, to solve Eq.(1.5) via the extended −expansion method. Consider the balancing with , wed educe 2 N = .Therefore, the solution of Eq.(1.5) becomes , ,

Conclusion
In this study, the-expansion and the extended-expansion methods have been applied for obtaining the new exactsolutions of ZK equation of fractional order arising in physics. The obtained solutions can be expressed as trigonometric, hyperbolic and rational functions. The fractional transformation is used to convert the partial differential equation into the ordinary differential equation. The solutions reported here have not been published elsewhere. Finally, the two planned methods are direct, concise, elementary, and effective and might be used for solving other nonlinear partial differential equation fractional order. This is often our duty in the future.

Acknowledgement
This Paper has been sponsored by the Research Sponsorship Program, University of Bisha, Kingdom of Saudi Arabia: grand Number (UB-016-1438).