Abstract
The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalized -expansion method 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.
1. 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 [1–8]. 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, 9–11]. 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 [15–17]. 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 time-space 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 [23–29]. This article is committed to seeking the new exact solutions for nonlinear time fractional fifth-order KdV via the generalized -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 -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.
2. 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 denote a constant discretization span. Define the forward operator (the symbol means that the left side by the right one).then the fractional difference of order , , of is defined by the expressionand its fractional derivative of order isThe above can be expressed asAn important property and formula of Jumarie’s modified Riemann-Liouville derivative [31] can be cataloged [32] as
3. An Analysis of the Generalized -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 -expansion method for seeking the solutions of nonlinear FDEs.
Step 1. Applying the traveling wave transformation in (6)where and 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 are coefficients to be determined later, and satisfies the following nonlinear ODE:
Step 3. Balancing the higher order derivative term and the higher order nonlinear term of (8), the value of the positive integer is consequently obtained. The detailed expression is the same as the following formula: assuming ,
Step 4. Substituting (9) and (10) into (8), we will obtain a function of ; the parameters , , , , , and can be determined. The general solutions of (9) have been listed as the following.
Type 1 (when ).
Type 2 (when ).
Type 3 (when and ). where is the integrating constant.
4. The Application to the Time Fractional Generalized Fifth-Order KdV Equation
The following is a given space-time fractional generalized fifth-order KdV equation asUtilize the traveling wave transformation (7) of Step 2:where and 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 ; (9) can be written aswhere , , and are constants to be determined later and satisfies (10). Substituting (19) into (18) and setting the coefficients of to zero, we get a system of algebraic equation. Solving (18), we get , , , ,
Case 1 (). where .
Figure 1(a) shows the dark solitary wave solutions of at , , , and in the interval and time in the interval .
Figure 1(b) shows the solitary wave solutions of at , , , and in the interval and time in the interval .
(a)
(b)
Case 2 (). where .
Figure 2(a) shows the periodic solitary wave solutions of at , , and , in the interval and time in the interval .
Figure 2(b) shows the solitary wave solutions of at , , and in the interval and time in the interval .
(a)
(b)
Case 3 (, ). where is the integrating constant and .
Figure 3 shows the solitary wave solutions of at , and , in the interval and time in the interval .
5. Conclusion
In this article, we obtain the exact traveling wave solutions of space-time fractional generalized fifth-order KdV equation by utilizing the generalized -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 -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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported by the National Natural Science Foundation of China (no. 11271164).