EXACT ANALYTIC WAVE SOLUTIONS TO SOME NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS FOR THE SHALLOW WATER WAVE ARISE IN PHYSICS AND ENGINEERING

In this article, we construct a family of closed form traveling wave solutions to the space-time fractional Equal Width equation (EW) and the space-time fractional modified Equal Width equation (mEW) by using newly proposed modified rational fractional (Dξ G/G2)-expansion method and the exp-function method. The considered equations are turned into fractional order ordinary differential equations with the help of a complex fractional transformation along with conformable fractional derivative and then the methods are used to investigate their solutions. The achieved solutions are in terms of trigonometric function, hyperbolic function and rational function which might be used to analyze deeply the physical complex phenomena of real world as they are new and bear much more generality. Two more well-established methods, the (G/G) -expansion method and the rational (G/G) expansion method, are also taken into account to unravel the suggested equations which do not provide any solution. The results reveal that the proposed method is efficient, straightforward and concise which might further be useful tool to examine any other nonlinear evolution equations of fractional order arising in various physical problems.


Introduction
Fractional calculus originating from some speculations of Leibniz and L'Hospital in 1695 has gradually gained profound attention of many researchers for its extensive appearance in various fields of real world. The fractional order nonlinear evolution equations (FNLEEs) and their solutions in closed form play fundamental role in describing, modeling and predicting the underlying mechanisms related to the biology, bio-genetics, physics, solid state physics, condensed matter physics, plasma physics, optical fibers, meteorology, oceanic phenomena, chemistry, chemical kinematics, electromagnetic, electrical circuits, quantum mechanics, polymeric materials, neutron point kinetic model, control and vibration, image and signal processing, system identifications, the finance, acoustics and fluid dynamics [1][2][3][4][5]. Therefore, it has become the core aim in the research area of fractional related problems that how to develop a stable approach for investigating the solutions to FNLEEs in analytical or numerical form. Many researchers have offered different approaches to construct analytic and numerical solutions to NLEEs of fractional order as well as integer order and put them forward for searching traveling wave solutions, such as the exponential decay law [6], the Ibragimov's nonlocal conservation method [7], the reproducing kernel method [8], the Jacobi elliptic function method [9], the ) / ( G G -expansion method and its various modifications [10][11][12][13][14], the Expfunction method [15,16], the sub-equation method [17,18], the first integral method [19,20], the functional variable method [21], the modified trial equation method [22,23], the simplest equation method [24], the Lie group analysis method [25], the fractional characteristic method [26], the auxiliary equation method [27,28], the finite element method [29], the differential transform method [30], the Adomian decomposition method [31,32], the variational iteration method [33], the finite difference method [32], the various homotopy perturbation method [35][36][37][38][39] and the He's variational principle [40] etc. But no method is uniquely substantial to examine the closed form solutions to all kind of FNLEEs. That is why; it is very much indispensable to establish new techniques. In this study, we propose a new technique, called the modified rational fractional ( / 2 ) -expansion method, to construct closed form analytic wave solutions to some FNLEEs in the sense of conformable fractional derivative [41]. This effectual and reliable productive method shows its high performance through providing abundant fresh and general solutions to the suggested equations. The obtained solutions might bring up their importance through the contribution to analyze the inner mechanisms of physical complex phenomena of real world and make an acceptable record in the literature.

Preliminaries and Methodology
Many researchers used this new derivative of fractional order in physical applications due to its convenience, simplicity and usefulness [42][43][44].

Methodology
In this section, we discuss the main steps of the above, mentioned methods to investigate exact analytic solutions of FNLEEs. Consider the FNLEE in the independent variables , 1 , 2 , … , as Q( 1 , … , , 1 , … , , Eq. (1) is turned into the following ordinary differential equation of fractional order with respect to the variable : ( 1 , … , , 1 , … , , 1 , … , , … ) = 0 (3) We may, if possible, take the anti-derivative of Eq. (3) term by term one or more times and integral constant can be set to zero as soliton solutions are sought. Then the following two methods are employed to construct closed form analytic solutions of Eq. (3). 2.2.1. The modified rational fractional ( / 2 )expansion method In this subsection, the main steps of the modified rational fractional ( / 2 ) -expansion method is discussed for finding exact analytic solutions to FNLEEs.
Step 3: Use Eqs. (4) and (5) in Eq. (3) with the value of found in step 2 to obtain a polynomial in ( / 2 ). Set each coefficient of this polynomial to zero and solve them by computer software Maple to gain the values of the unknown parameters available in Eq. (4).
2.2.2. The Exp-function method In this subsection, the main steps of the Exp-function method are discussed for finding exact analytic solutions of nonlinear partial differential equations of fractional order.
Step 1: Consider the wave solution to Eq. (1) as where , , and are positive integers which are known to be further determined, and are unknown constants.
Step 2: Balance the linear term of lowest order of Eq.
(3) with the lowest order nonlinear term to determine the values of and . Similarly, to determine the values of and , balance the linear term of highest order of Eq. (3) with highest order nonlinear term.
Step 3: Substitute Eq. (11) into Eq. (3) with the values of , , and q obtained in step 2, we obtain polynomials in , for any integer . Equating like terms to zero gives a system of algebraic equations for ′ and ′ . Solve this system for ′ and ′ by means of the symbolic computation software, such as Maple.
Step 4: Substitute the values appeared in step 3 into Eq. (11), we obtain traveling wave solutions of Eq. (1) in closed form.

Formulation of the solutions
In this section, the suggested methods are employed to ravel the space-time fractional EW equation and the space-time fractional mEW equation for their solutions in closed form. 3. 1. The space-time fractional EW equation Consider the space-time fractional EW equation + 2 − 3 = 0 (12) which is used as a model in partial differential equation and handles the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. The wave variable ( , ) = ( ), = 1/ + 1/ (13) reduces Eq. (12) to the following fractional order ordinary differential equation with respect to the variable : where 0 and are arbitrary constants.

Results and Discussions:
The modified rational fractional ( / 2 )-expansion method has been proposed and used for finding exact analytic solutions to the space-time fractional EW equation which provided twelve solutions in terms of trigonometric, hyperbolic and rational functions while the space-time fractional mEW equation through the same method made available eighteen solutions in the same form. The exp-function method has also been applied to the suggested equations which gave four solutions for the first equation and eight for the second. Hosseini, and Ayati applied Kudryashov method to the space-time fractional EW equation and the space-time fractional mEW equation and obtained only four solutions (Appendix A, B) in terms of hyperbolic function for each equation [45]. We have applied two well-established methods, the ( ′ / ) -expansion method [46] and the rational ( ′ / )-expansion method [47], to investigate the exact solutions of the considered equations but we haven't got any solutions.

Conclusion
The core aim of this study was to make available further general and fresh closed form analytic solutions to the space-time fractional EW equation and the space-time fractional mEW equation through the proposed modified rational fractional ( / 2 ) -expansion method and the exp-function method. Both methods have successfully presented attractive solutions to the suggested equations though the performance of first one is perceptible. So far we know the achieved solutions are not available in the literature and might create a milestone in research area. We have also utilized the ( ′ / ) -expansion method and the rational ( ′ / )expansion method to unravel the suggested equations but not found any solution. Therefore, it may be claimed that the modified rational fractional ( / 2 ) -expansion method in deriving the closed form analytical solutions is simple, straightforward and productive. This method may be taken into account for further implementation to investigate any fractional order nonlinear evolution equations arising in various fields of science and engineering. fractional third-order evolution equation: Symmetry analysis, explicit solutions and conservation laws, J. Comp. Nonlin. Dynam., 13 (2018), 021011.