Traveling wave solutions in closed form for some nonlinear fractional evolution equations related to conformable fractional derivative

Abstract: Fractional order nonlinear evolution equations involving conformable fractional derivative are formulated and revealed for attractive solutions to depict the physical phenomena of nonlinear mechanisms in the real world. The core aim of this article is to explore further new general exact traveling wave solutions of nonlinear fractional evolution equations, namely, the space time fractional (2+1)-dimensional dispersive long wave equations, the (3+1)-dimensional space time fractional mKdV-ZK equation and the space time fractional modified regularized long-wave equation. The mentioned equations are firstly turned into the fractional order ordinary differential equations with the aid of a suitable composite transformation and then hunted their solutions by means of recently established fractional generalized (DDξξGG GG ⁄ )-expansion method. This productive method successfully generates many new and general closed form traveling wave solutions in accurate, reliable and efficient way in terms of hyperbolic, trigonometric and rational. The obtained results might play important roles for describing the complex phenomena related to science and engineering and also be newly recorded in the literature for their high acceptance. The suggested method will draw the attention to the researchers to establish further new solutions to any other nonlinear evolution equations.


Introduction
The physical phenomena of real world can effectively be modeled by making use of the theory of derivatives and integrals of fractional order. In this sense, the nonlinear fractional order evolution equations (NLFEEs) have recently become a burning topic to the researchers for searching their exact traveling wave solutions to depict the physical phenomena due to the nonlinear mechanisms arisen in various fields. The use of nonlinear equations is extensive as the nonlinearity exists everywhere in the world. NLFEEs have been attracted great interest due to their frequent appearance in many applications such as in biology, physics, chemistry, electromagnetic, polymeric materials, neutron point kinetic model, control and vibration, image and signal processing, system identifications, the finance, acoustics and fluid dynamics [1][2][3]. Many researchers have offered different approaches to construct analytic and numerical solutions to NLFEEs as well as NLEEs and put them forward for searching traveling wave solutions, such as the exponential decay law [4], the Ibragimov's nonlocal conservation method [5], the reproducing kernel method [6], the Jacobi elliptic function method [7], the ) / ( G G′ -expansion method and its various modifications [8][9][10][11][12], the Exp-function method [13,14], the sub-equation method [15,16], the first integral method [17,18], the functional variable method [19], the modified trial equation method [20,21], the simplest equation method [22], the Lie group analysis method [23], the fractional characteristic method [24], the auxiliary equation method [25,26], the finite element method [27], the differential transform method [28], the Adomian decomposition method [29,30], the variational iteration method [31], the finite difference method [32], the various homotopy perturbation method [33][34][35][36][37] and the He's variational principle [38] etc. But no method is uniquely appreciable to investigate the exact solutions to all kind of NLFEEs. That is why; it is very much needed to introduce new method. In this study, we implemented recently established effectual and reliable productive method, called the fractional generalized ) / ( G G D α ξ -expansion method to construct closed form analytic wave solutions to some NLFEEs in the sense of conformable fractional derivative [39]. The results obtained throughout the article have been compared with those existing in the literature and shown that the achieved solutions are new and much more general. We have finally concluded that the solutions might bring up their importance through the contribution and be recorded in the literature.

Conformable fractional derivative
A new and simple definition of derivative for fractional order introduced by Khalil et al. [39] is called conformable fractional derivative. This definition is analogous to the ordinary derivative This integral represents usual Riemann improper integral.
The conformable fractional derivative satisfies the following useful properties [39]: Many researchers used this new derivative of fractional order in physical applications due to its convenience, simplicity and usefulness [40][41][42].

The fractional generalized
Consider a nonlinear partial differential equation of fractional order in the independent variables where F is a polynomial in ) ..., , , and d are arbitrary constants with at least one of n a and n b is nonzero and , which can be derived by using conformable fractional derivative, we can easily obtain the following solutions to Eq. (2.2.5): Family 4: When

Formulation of the solutions
In this section, the closed form traveling wave solutions to the suggested equations are examined.

The space time fractional (2+1)-dimensional dispersive long wave equations
Consider the space time fractional (2+1)-dimensional dispersive long wave equations where 1 0 ≤ < α ; the notation α α x u ∂ ∂ denotes the α -order partial derivative of u with respect to x and the other notations are so. This system of equations was first obtained by Boiti et al. [43] as compatibility condition for a weak Lax pair.
where k , l and w are constants, with the aid of chain rule  Set-1: Eqs. (3.1.8), (3.1.9) together with Eqs. (2.2.7)-(2.2.11) make available the following three types solutions in terms hyperbolic function, trigonometric function and rational function as: When In When Consider the arbitrary constants as 0 1 ≠ C and 0 2 = C , then Following the same procedure as above for Eqs.
The choices for arbitrary constants as When The choices for arbitrary constants as When In particular, if When When Making use of Eq. (3.2.9) as above will also provide further new and general exact traveling wave solutions in terms of hyperbolic, trigonometric and rational. For convenience of readers we have not record these all solutions in this study. Guner et al. [45] obtained only four solutions by ) / ( G G′ expansion method where as our applied fractional generalized ) / ( G G D α ξ -expansion method has ensured many solutions which are further new and general. To the best of our knowledge, these solutions have not been visible in any earlier study.

The space time fractional modified regularized long-wave equation
The following nonlinear space-time fractional modified regularized long-wave equation is considered to be examined for further exact traveling wave solutions: where τ δ , and η are constants. This equation proposed by Benjamin et al. to describe approximately the unidirectional propagation of long waves in certain dispersive systems is supposed to be alternative to the modified KdV equation. Eq. (3.3.1) has been modeled to demonstrate some physical phenomena like transverse waves in shallow water and magneto hydrodynamic waves in plasma and photon packets in nonlinear crystals [46][47][48].
The fractional complex transformation Integrating Eq. (3.3.3) and setting integral constant to zero gives Taking homogeneous balance between highest order linear term and highest nonlinear term from Eq.
where at least one of 1 a and 1 b is nonzero.
Case 2: When Case 4: When For particular values of the arbitrary constants as Case 5: When  [49] and the improved fractional Riccati expansion method [50] are only in terms of hyperbolic, where as we achieved those in terms of hyperbolic function, trigonometric function and rational function in explicitly general form. We have not recorded these results to avoid the annoyance of the readers. On comparison, our solutions are general and much more in number than those of [49,50].

Graphical representations of the solutions
The complex physical mechanism of real world can be illustrated by means of graphical representations. The graphs (Figures 1-3) drown for the exact solutions obtained in this study has been appeared in different shape like kink type soliton, bell shape soliton, singular bell shape soliton, anti bell shape soliton, periodic solution, singular periodic solution etc. We have recorded here only few graphs rather than all for making it easily readable.

Conclusion
This article has been put in writing further new and general traveling wave solutions in closed form to the space time fractional (2+1)-dimensional dispersive long wave equations, the (3+1)dimensional space time fractional mKdV-ZK equation and the space time fractional modified regularized long-wave equation. The solutions have successfully constructed in terms of hyperbolic function, trigonometric function and rational function by the newly established fractional generalized ) / ( G G D α ξ -expansion method. To the best of our knowledge, these results are not available in the literature. The obtained solutions might play important roles to analyze the mechanisms of complex physical phenomena of the real world. The performance of the suggested method is highly appreciable for its easiest productive behavior and worthy for revealing rare solutions to more fractional order nonlinear evolution equations. Since each nonlinear equation has its own anomalous characteristic, the future research might be how the suggested method is compatible for revealing the solutions to other fractional nonlinear evolution equations.