Abstract
First, our work is to transform the space-time fractional approximate long water wave equations into nonlinear ordinary differential equations via the traveling wave transformation in the sense of conformable fractional derivative. Second, we simplify the nonlinear ordinary differential equations into an ordinary differential equation with only one variable by integration and some transformations. Finally, we can further get all single traveling wave solutions of the space-time fractional approximate long water wave equations by the complete discrimination system for the four-order polynomial method; these solutions include the hyperbolic function solutions, rational function solutions, and implicit solutions.
1. Introduction
The exact traveling wave solutions of fractional partial differential equations have attracted much attention from mathematicians and engineering experts [1–10]. In recent years, some methods for constructing fractional partial differential equations have been proposed, such as the fractional double function method [11], improved fractional subequation method [12], -expansion method [13], plane dynamic system analysis method [14], and complete discrimination method of polynomial [15]. The complete discrimination method of polynomials was first proposed by Liu [16]. Using this method, the classification of all single traveling wave solutions of fractional partial differential equations can be constructed.
In this study, we consider the space-time fractional approximate long water wave equations [17]:where , . and represent the conformable fractional derivative. (1) is a very important shallow water waves model, which is usually used to describe the propagation of shallow water waves.
The work of this study is as follows. In Section 2, we give the construction steps of traveling wave solutions for fractional partial differential equations. In Section 3, we apply the method in Section 2 to the space-time fractional approximate long water wave equations. In Section 4, we give a conclusion.
2. Preliminaries
First, we consider the following integrated fractional partial differential equations:where , , and are the two polynomial in , , and its various fractional derivatives.
Step 1. Consider the traveling wave transformation as follows:where and are the constants. Using traveling wave transformation, (2) can be simplified toand satisfy the relationwhere is a polynomial of and its derivatives. is a polynomial of , the derivative of and the derivative of .
Step 2. The above nonlinear ordinary differential equation (4) can be reduced to the following ordinary differential form:Here, the integral form of (6) iswhere is an arbitrary constant. Here, we assume that the fourth-order polynomial isand its complete discrimination system isNext, we can obtain the classification of solutions of (6) according to polynomial complete discrimination system (9) ([18–23]).
3. Single Traveling Wave Solutions of System (1)
Consider the traveling wave transformation as follows:where and are the constants.
Substituting (10) into (1), we obtain the ordinary differential equations as follows:
Integrating the second equation of (11) with respect to , we havewhere is the integration constant. Substituting (12) into the first equation of (11), we obtainwhere is the integration constant.
Multiplying both ends of (13) by and integrating once, we obtainwhere is the integration constant.
In order to give the classification of traveling wave solutions of (1), we need to assume that the coefficient of is zero. So, we make the following transformations:
Substituting (15) into (14), it becomeswhere , , .
The solution of (16) can be written in the following integral form:where is an arbitrary constant.
Case 1. If , , ,where .
Substituting (18) into (17) and combining (14), we can get the solution of (1) asFor example, when , , , and , we obtain . Therefore, we draw the three-dimensional diagram of solution through Maple software, as shown in Figure 1.
Case 2. If , , ,Substituting (20) into (17) and combining (14), we can get the solutions of (1) as
Case 3. If , , , ,where .
Next, substituting (22) into (17) and combining (14), when or , we can get the solution of equation (10) asWhen , we can attainFor example, when , , , , , and , we obtain . Therefore, we draw the kink solitary wave solution through Maple software, as shown in Figure 2.
Case 4. If , , ,where , , and are the real numbers, and .
If and or and , the implicit analytical solution of (13) is obtained:If and or when and , the implicit analytical solution of (13) is obtained:If , the implicit analytical solution of (13) is obtained:
Case 5. If , , , ,where and are the real numbers.
Substituting (29) into (17) and combining (14), when and or when and , we can yield
Case 6. If , ,where , , and are the real numbers.
Substituting (31) into (14), we obtain the solution of equation (1):where .
Case 7. If , , ,where , , , and are the real numbers and .
Then, we can obtain the solution of (1):where .
Case 8. If , ,where , , , and are the real constants, and .
Then, we can give the solution of (1):where , , , , , .
Case 9. If , ,where , , , and are the real constants, and .
So, we can give the solution of (1):where , , , , , , , and .
Remark 1. In this section, we obtain all single traveling wave solutions of (1) by using the complete discriminant of fourth-order polynomials. Accordingly, we can obtain the solutions of equation (1) through the relationship (12) and the solutions .
4. Conclusion
The construction of exact solutions of fractional partial differential equations has always been a very important research field. In this study, the fractional partial differential equations are simplified into an ordinary differential equation with only one variable by integration and some transformations. In this way, through the complete discriminant system of polynomials, we can easily obtain the classification of the exact solutions of the ordinary differential equation. We apply the method to the classification of the single traveling wave solutions of the fractional approximate long water wave equations.
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 Scientific Research Funds of Chengdu University (2081920034).