Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order KdV Equation

The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallowwater under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalized exp(−Φ(ξ))-expansionmethod is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations.


Introduction
Nonlinear fractional differential equations (FDEs) as a special category of nonlinear partial differential equations (PDEs) have its variety of applications in physics, biology, chemistry, fluid flow, electrical networks, signal and image processing, acoustics, and so on [1][2][3][4][5][6][7][8].Owing to widely applications and further properties in various fields of natural sciences, seeking the solutions of fractional PDEs has drawing the attention of scholars.Analyzing their solutions can help us understand and explain the nonlinear phenomena.
The generalized KdV equation is an important mathematical model used to describe long wave motion in shallow water, one-dimensional nonlinear lattice, hydrodynamics, quantum mechanics, plasma physics, and optics [1,3,4,[9][10][11].The generalized KdV equation with higher order nonlinearity is put forward for internal solitary waves in a density and current stratified shear flow with a free surface.As a classical model, long has generated the steady-state version of the KdV equation [12] and an integral expression for the coefficients of the KdV equation in fluid is given by Benney [13].
Up to present, some effective methods have been put forward to search for exact solutions of the fractional KdV equation.Saha Ray and Gupta use the two-dimensional Legendre wavelet method to obtain the traveling wave solutions of the fractional seventh-order KdV equation [14].
In [10], the author applies the modified fractional subequation method to obtain the exact solution of the fractional coupled KdV equation [10].(  /)-expansion method is given to obtain the solitary wave solutions of the fractional KdV equation in [15][16][17].Applying the numerical technique based on the generalized Taylor series formula is original and convenient to obtain explicit and approximate solutions of the nonlinear fractional KdV-Burgers equation with timespace fractional derivatives [18].Application of modified sine-cosine method to solve the fractional fifth-order KdV equation's traveling wave solutions is shown in [19].In addition, the author draws a comparison between the generalized Kudryashov method and exp-function method to solve the exact solution of fractional KdV equation [20,21].Lie group analysis method [22] is also common and fundamental and much more [23][24][25][26][27][28][29].This article is committed to seeking the new exact solutions for nonlinear time fractional fifth-order KdV via the generalized exp(−Φ())-expansion method [30].
The whole paper consists of five sections and the detailed contents and structure are as follows.An introduction is shown in Section 1.In Section 2, the definition of Jumarie's modified Riemann-Liouville derivatives [31,32] is introduced.In Section 3, an analysis of the generalized exp(−Φ())expansion method is formulated.The exact solutions of the fractional fifth-order KdV equation are obtained in Section 4. Finally, conclusions have been drawn in Section 5.

The Definition of Jumarie's Modified Riemann-Liouville Derivatives
In this section, we introduce the definition of the modified Riemann-Liouville derivative by Jumarie.Let  :  → ,  → (), denote a continuous (but not necessarily differentiable) function, and let ℎ > 0 denote a constant discretization span.Define the forward operator FW(ℎ) (the symbol fl means that the left side by the right one).
FW (ℎ)  () fl  ( + ℎ) ; then the fractional difference of order , 0 <  ≤ 1, of () is defined by the expression and its fractional derivative of order  is The above can be expressed as An important property and formula of Jumarie's modified Riemann-Liouville derivative [31] can be cataloged [32] as (−) ,  > 0. (5)

An Analysis of the Generalized 𝑒𝑥𝑝(−Φ(𝜉))-Expansion Method
In this section, Let us consider the following nonlinear FDEs: where  = (, ) is an unknown function and    ,    ,    ,     are Jumarie's modified Riemann-Liouville derivatives of  and  is a function involving nonlinear terms and higher order derivatives.Then we propose the following four steps of the generalized exp(−Φ())-expansion method for seeking the solutions of nonlinear FDEs.
Step 1. Applying the traveling wave transformation in ( 6) where  and V are constants.Equation ( 6) is reduced to a nonlinear ODE for  = () with the help of ( 7) where  is a function of  = () and its derivatives with respect to .
Step 2. Assuming (8) has the following traveling wave solution, where   (1 ≤  ≤ ) are coefficients to be determined later, and Φ = Φ() satisfies the following nonlinear ODE: Step 3. Balancing the higher order derivative term and the higher order nonlinear term of ( 8 ( Step 4. Substituting ( 9) and ( 10) into (8), we will obtain a function of exp(−Φ())  ; the parameters   (1 ≤  ≤ ), , V, , , and  can be determined.The general solutions of ( 9) have been listed as the following.

The Application to the Time Fractional Generalized Fifth-Order KdV Equation
The following is a given space-time fractional generalized fifth-order KdV equation as Utilize the traveling wave transformation (7) of Step 2: where  and V are constants.Then (15) can be reduced to following nonlinear ODE: Integrating once with the respect  and letting the integration constant equal to zero [21], then we get (15) for simplicity, Owing to the balancing principle, let  = 2; (9) can be written as where  0 ,  1 , and  2 are constants to be determined later and Φ() satisfies (10).Substituting ( 19) into (18) and setting the coefficients of (exp(−Φ()))  ( = 0, 1, 2, 3, 4, 5, 6) to zero, we get a system of algebraic equation.Solving (18), we get Case 1 ( = 1).

Conclusion
In this article, we obtain the exact traveling wave solutions of space-time fractional generalized fifth-order KdV equation by utilizing the generalized exp(−Φ())-expansion method along with Jumarie's modified Riemann-Liouville derivatives.Three types of exact solutions are originated in terms of the hyperbolic, trigonometric, and rational function with some parameters and they have great potential which can been further researched.The generalized exp(−Φ())-expansion method has some advantages, such as efficiency, conciseness, and briefness.In addition, it is obvious that this method is rather well-organized and practically applied to many other nonlinear fractional PDEs from natural sciences.

Figure 1 :Figure 2 :
Figure 1: Exact traveling wave solutions of (20) are plotted in different shape.