Journal of Function Spaces
Volume 2018 (2018), Article ID 7035194, 6 pages
https://doi.org/10.1155/2018/7035194

Research Article

A Note on the Fractional Generalized Higher Order KdV Equation

Yongyi Gu

School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, China

Correspondence should be addressed to Yongyi Gu; gdguyongyi@163.com

Received 30 May 2018; Accepted 19 July 2018; Published 23 August 2018

Academic Editor: Xinguang Zhang

Copyright © 2018 Yongyi Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We obtain exact solutions to the fractional generalized higher order Korteweg-de Vries (KdV) equation using the complex method. It has showed that the applied method is very useful and is practically well suited for the nonlinear differential equations, those arising in mathematical physics.

1. Introduction

Nonlinear fractional differential equations (NFDEs) are universally applied in signal processing, electrical networks, acoustics, fluid dynamics, biology, chemistry, physics, etc. For example, the singular behaviours [1–9] and impulsive phenomena [10–19] often exhibit some blow-up properties [20–25] which occur in a lot of complex physical processes. NFDEs have been attracted extensive attention and have been widely investigated [26–38]. Exact solutions of NFDEs play an important role in the study of mathematical physics phenomena. Therefore, seeking exact solutions of NFDEs is an interesting and significant subject.

The fractional generalized higher order KdV equation is a useful model. Applying the generalized exp(-)-expansion method, Lu et al. [39] obtained exact solutions of this equation. In this article, we would like to utilize the complex method [40–43] to seek exact solutions to the fractional generalized higher order KdV equation.

2. Preliminaries

Let ,   be a continuous function and denote a constant discrete span. Define the operator as follows:

then the fractional difference of of order can be expressed as

where , and its fractional derivative of order can be expressed as

The above is expressed as

Further, Jumarie’s modified Riemann-Liouville derivative [44, 45] is given by

then its related NFDE is given by

Let , , , , and

then the degree of is denoted by . We define the differential polynomial as

in which is a finite index set and are constants. The degree of can be denoted by .

The ordinary differential equation (ODE) is given by

where are constants, .

Suppose that the meromorphic solutions of (9) have at least one pole. Let and insert the Laurent series

into (9); if it is determined different Laurent singular parts:

then (9) is said to satisfy the weak condition.

Give two complex numbers such that , let be the discrete subset , and L is isomorphic to . Let the discriminant and

A meromorphic function with double periods , which satisfies the equation

in which , , and , is called the Weierstrass elliptic function.

If a meromorphic function is a rational function of , an elliptic function, or a rational function of , then we say that belongs to the class .

In 2009, Eremenko et al. [46] considered the following -order Briot-Bouquet equation (BBEq):

where and are constant coefficient polynomials. For the -order BBEq, we have the lemma as follows.

Lemma 1 (see [41, 47, 48]). Let , and , and the -order BBEq

satisfies weak condition, then the meromorphic solutions belong to the class . Suppose that for some values of parameters such solutions exist, then other meromorphic solutions should form one-parametric family , . Then each elliptic solution with a pole at can be expressed as

in which are determined by (10), , and .
Each rational function solution can be expressed as

and it has distinct poles of multiplicity .
Each simply periodic solution is a rational function of and can be expressed as

and it has distinct poles of multiplicity .

Lemma 2 (see [48, 49]). Weierstrass elliptic functions have the following addition formula:

When , it can be degenerated to rational functions according to

When , it can also be degenerated to simple periodic functions according to

3. Main Results

The fractional generalized higher order KdV equation [39] is given as

Substituting traveling wave transform

into (22), we get

Integrating (24) yields

where and are constants and is the integral constant.

Theorem 3. If , then the meromorphic solutions of (25) have the following forms.
The rational function solutions

where , , and .
The simply periodic solutions

where , , and .
The elliptic function solutions

where , , and .

Proof. Substituting (10) into (25) we have ,  ,  ,  ,  ,  ,  , and is an arbitrary constant.
Therefore, (25) satisfies the weak condition. In the following, we will show the meromorphic solutions of (25).
By (17), we infer that the indeterminant rational solutions of (25) are

with pole at .
Inserting into (25), we have

then we get ,  , and  .
So we can determine that

where and .
Therefore the rational solutions of (25) are

where , , and .
Let . To derive simply periodic solutions, we substitute into (25) to yield

Substituting

into (33), we obtain that

where and .
Substituting into (35), we achieve simply periodic solutions to (25) with pole at

where and .
Thurs, simply periodic solutions of (25) are

where , , and .
From (16) of Lemma 1, the elliptic solutions of (25) are expressed as

with pole at .
Putting into (25), we get

where and .
So, the elliptic solutions of (25) are

where .
We can apply the addition formula to rewrite it as

where , , and .
Substitute traveling wave transform into the meromorphic solutions of (25) to get traveling wave exact solutions to the fractional generalized higher order KdV equation. So we obtain Theorem 4 as follows.

Theorem 4. If , then traveling wave solutions of (25) have the following forms.
The rational function solutions

where , , and .
The simply periodic solutions

where , , and .
The elliptic function solutions

where , , and .

4. Conclusions

In this note, we have used the complex method to construct exact solutions to the mentioned NFDE. Although we do not show that the meromorphic solutions of the fractional generalized higher order KdV equation belong to the class , we can still obtain the meromorphic solutions to this NFDE and then get its traveling wave exact solutions. The results demonstrate that the applied method is direct and efficient method, which allows us to do tedious and complicated algebraic calculation. We can utilize these ideas to other NFDEs.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the NSF of China (11701111 and 11271090); the NSF of Guangdong Province (2016A030310257); Guangdong Universities (Basic and Applied Research) Major Project (2017KZDXM038); Guangzhou City Social Science Federation “Yangcheng Young Scholars” Project (18QNXR35).

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