Abstract
In this study, the generalized Atangana’s fractional BBM–Burgers equation (GBBM-B) with the dissipative term is investigated by utilizing the modified sub-equation method and the new G'/(bG' + G + a)-expansion method; with the aid of symbolic computations, many types of new exact explicit solutions including solitary wave solutions, trigonometric function periodic solutions, and the rational function solutions are obtained. Some 3D and 2D plots of these solutions are simulated, which show the novelty and visibility of the propagation behavior and dynamical structure of the corresponding equation. Moreover, with the selection of different values on the parameters and orders, we can deduce many types of exact solutions in special cases. We also discussed the changes and characteristics of these solutions, which can help us further understand the inner structure of this equation. The obtained solutions indicate that the approach is easy and effective for nonlinear models with high-order dispersion terms.
1 Introduction
As is known, calculus was founded by Newton and Leibniz at the end of the 1660s, and fractional order calculus has gradually become one of the new special fields in natural sciences and mathematical physics since 1695 [1]. In recent years, due to the wide application of fractional order calculus in nonlinear partial differential equations (PDEs), especially fractional PDEs [2–4], many nonlinear phenomena come down to fractional models, such as ecological and economic systems [5], two-scale thermal science [6], mechanics [7], chaotic oscillations [8], atmospheric science [9], and optical fiber [10–12]. Searching for exact explicit solutions of these nonlinear fractional PDEs plays a significant role in the study of the dynamics of those phenomena. Until now, many powerful methods for this subject have been offered, such as the Darboux transformation [13], Bäcklund transformation method [14], and Hirota bilinear method [15], which can be used to find N-soliton solutions. The improved F-expansion method [16], projective Riccati equation method [17], sine-Gordon method [18], Jacobi elliptic function expansion method [19], G'/G-expansion method [20], (G'/G,1/G)-expansion method [21], improved (m + G'/G)-expansion method [22], improved G'/-expansion method [23], exp (−φ(ξ)) technique [24], homogeneous balance method [25], first integral method [26], inverse scattering transformation [27], and Lie symmetry method [28], etc [29–34] can be used to find Jacobi periodic solutions, solitary wave solutions, and trigonometric function solutions of these models. Until now, there are many types of definitions for the fractional derivative, and the most classic definitions are as follows:
Riemann–Liouville fractional derivative [35]:
Caputo fractional derivative [36]:
Jumarie’s fractional derivative [37]:
Ji-Huan He’s fractional derivative [38]:
Furthermore, the Atangana–Baleanu derivative [39], M-fractional derivative [40], conformable fractional derivative [41], and Atangana’s fractional derivative [42, 43] which will be utilized in this article, are built recently.
In this paper, we consider the generalized Atangana’s fractional BBM–Burgers equation with the dissipative term in the following form [44–47]:where are the Atangana’s fractional derivative [42]The coefficients are real constants; when or , Eq. 1 is related to the well-known BBM equation or fractional BBM equation, which was proposed by Benjamin–Bona–Mahony and describes approximately the unidirectional propagation of a long wave in certain nonlinear dispersive systems as a refinement of the KdV equation [48–51]. When or , Eq. 1 is related to the well-known Burgers equation [52, 53]. Some related research studies about Eq. 1 can be found in [45, 54, 55].
Next, we review some basic definitions and properties of the Atangana fractional derivative which are used further in this paper [42, 43].
Definition: For a function we defined the Atangana fractional derivative operator and integral operator of of the order as [42, 43]
Also, we have the following important properties [42, 43]:
The rest of the paper is organized as follows. In Section 2, we introduce the modified sub-equation method [56–59] and the new G'/(bG'+G+a)-expansion method, while in Section 3, some exact solutions of the GBBM–Burgers equation are found and discussed by utilizing the proposed methods. Finally, the conclusion is presented in Section 4.
2 Description of the two methods
2.1 The modified sub-equation method
Consider the following Atangana’s fractional differential equation:
We use the following wave transformation [60]:where the constant means the wave number which can reflect the frequency and is the wave speed. Thus, Eq. 2 reduces to an ordinary differential equation:
Assume that Eq. 4 has the following solution:where is a balance number, , and and the variable function are determined later. The function satisfies the Riccati equation defined by
Equation 6 gives the following solutions:When , we can obtain the results mentioned in [56–59].
Substituting Eqs 6, 5 into Eq. 4, collecting the coefficients of to zero yields algebraic equations (AEs) for and . Utilizing mathematical software to solve the AEs, we can obtain the solutions of Eq. 4.
2.2 The G'/(bG' + G + a)-expansion method
With similar steps to technique Section 2.1, we give the main steps of this method.
Step 1Assume that Eq. 4 has the following solution:where , and and the variable function are determined later. The parameters and are arbitrary constants, and is a solution of the following auxiliary ODE:where are two arbitrary real numbers. We can find the following constrained condition:Equation 9 gives the following solutions:
Case 1When , we have , and are arbitrary constants that satisfy as in case 2; thus,
Case 2When , we have ,
Step 2Substituting Eqs 7, 9 into Eq. 4 and setting the coefficients of zero yield a set of AEs for and . After solving the AEs and substituting each of the solutions along with Eqs 7, 3 into Eq. 2, we can obtain the solutions of Eq. 2.In the following section, we will use these two methods to solve the GBBM–Burgers equation.
3 Exact solutions to the GBBM–Burgers equation
3.1 Using the modified sub-equation method
Substituting Eq. 3 into Eq. 1 and integrating Eq. 1 once, we havewhere is the integral constant. By balancing the highest derivative term with the nonlinear terms in Eq. 10, we obtain . Therefore, we assume that Eq. 10 has the following solutions:where are constants to be determined later. Substituting Eqs 11, 6 into Eq. 10, collecting the coefficients of to zero, we have
Solving the aforementioned AEs, we have the following cases:
Case 1
Case 2We can obtain the following traveling wave solutions.
Family 1
FIGURE 1
3D plot, 2D plot, and contour plot of with .
FIGURE 2
3D plot, 2D plot, and contour plot of with .
Family 2
FIGURE 3
3D plot, 2D plot, and contour plot of with .
FIGURE 4
3D plot, 2D plot, and contour plot of with .
Family 3
Set 3If we select . The numerical simulation of rational function is shown in Figure 5.
FIGURE 5
3D plot, 2D plot, and contour plot of with .
3.2 Using the G'/(bG' + G + a)-expansion method
We assume that Eq. 10 has the following solutions:where and are constants to be determined later; if we select , substituting Eqs 9, 12 into Eq. 10 and setting the coefficients of zero yields
We can deduce the following solutions with the aid of mathematical software.
Case 1
Case 2
Case 3
Case 4We can determine the following solutions.
Family 4
For case 1, we have
Set 4whereThe numerical simulation of is shown in Figure 6, where we select
FIGURE 6
3D plot, 2D plot, and contour plot of with .
For case 2, we have
Set 5whereThe numerical simulation of is shown in Figure 7, where we selectFor case 3, we have
FIGURE 7
3D plot, 2D plot, and contour plot of with
Set 6whereThe numerical simulation of is shown in Figure 8, where we selectFamily 5 For case 4, we have
FIGURE 8
3D plot, 2D plot, and contour plot of with
Set 7whereThe numerical simulation of with the fractional order is shown in Figure 9, where we selectClearly, if we select the special value of in , we can obtain the tanh, coth, tan, and cot-type solutions; without loss of generality, we selectThus, we obtain the following solutions:If we selectThus,We haveThe simulation of is shown in Figure 10.
FIGURE 9
3D plot, 2D plot, and contour plot of with
FIGURE 10
3D plot, 2D plot, and contour plot of with
3.3 Results and discussion
After utilizing the modified sub-equation method and the new G'/(bG'+G+a)-expansion method, we obtain many types of exact solutions of Eq. 1, and some structures of these solutions are simulated in Figures 1–10. Visualization can help us better understand the dynamic behavior and propagation property of these solutions. For example, the bell-shape-like solitary wave solution of Eq. 1 is shown in Figure 1, and we find that there are two asymptotes on either side of the peak for , while the bell-shape soliton solution has only one. The shape of trigonometric function solutions has a break when , which is shown in Figure 2. The same phenomena happen for and which are simulated in Figures 5, 8. The simulations of periodic solutions , , and are shown in Figures 3, 4, 9 for or We can find that the waveform of a single period widens as the order decreases, and the changes of are simulated in Figure 11. The kink soliton solution and the solitary wave solution for the fractional order are simulated in Figures 6, 7. From Figure 10, we find the solution has one peak, one valley, and two asymptotes. These different propagation patterns can probably explain the different phenomena for this model.
FIGURE 11
Changes in the waveform of with different values of
4 Conclusion
In conclusion, many types of new exact solutions for the Atangana fractional GBBM–Burgers equation with the dissipative term have been found after utilizing the modified sub-equation method and the new G'/(bG' + G + a)-expansion method. Some propagation behavioral patterns of these solutions are discussed and simulated, the graphs of which show that these solitary wave solutions, trigonometric function periodic solutions, and rational function solutions are propagated through different patterns. The two efficient and significant methods can be used for many other nonlinear models such as the vmKdV equation, Ginzburg–Landau equation, and NLS-KDV equation. However, it is still worth researching whether the method can be used in a system with high dimensions and high order. Finally, all these solutions obtained in the present article have been checked by mathematical software.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
BH: completed the study, carried out the tests, and drafted the manuscript.
Funding
This work is supported by the practical innovation training program projects for the university students of Jiangsu Province (Grant No. 202211276054Y), natural science research projects of Institutions in Jiangsu Province (Grant No. 18KJB110013), and the Nanjing Institute of Technology (Grant Nos. ZKJ201513 and YZKC2019086).
Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
generalized BBM–Burgers equation, Atangana’s fractional derivative, dissipative term, modified sub-equation method, G'/(bG' + G + a)-expansion method, exact solutions
Citation
Hong B (2022) Assorted exact explicit solutions for the generalized Atangana’s fractional BBM–Burgers equation with the dissipative term. Front. Phys. 10:1071200. doi: 10.3389/fphy.2022.1071200
Received
15 October 2022
Accepted
26 October 2022
Published
25 November 2022
Volume
10 - 2022
Edited by
Fei Yu, Changsha University of Science and Technology, China
Reviewed by
Ji-Huan He, Soochow University, China
Kangsheng Zhao, Fudan University, China
Rongfei Xu, Nanjing Normal University, China
Updates
Copyright
© 2022 Hong.
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*Correspondence: Baojian Hong, hbj@njit.edu.cn
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.