- Research
- Open Access
Conservation laws, soliton-like and stability analysis for the time fractional dispersive long-wave equation
- Abdullahi Yusuf^{1, 2}Email authorView ORCID ID profile,
- Mustafa Inc^{1},
- Aliyu Isa Aliyu^{1, 2} and
- Dumitru Baleanu^{3, 4}
https://doi.org/10.1186/s13662-018-1780-y
© The Author(s) 2018
- Received: 25 July 2018
- Accepted: 23 August 2018
- Published: 12 September 2018
Abstract
In this manuscript we investigate the time fractional dispersive long wave equation (DLWE) and its corresponding integer order DLWE. The symmetry properties and reductions are derived. We construct the conservation laws (Cls) with Riemann–Liouville (RL) for the time fractional DLWE via a new conservation theorem. The conformable derivative is employed to establish soliton-like solutions for the governing equation by using the generalized projective method (GPM). Moreover, the Cls via the multiplier technique and the stability analysis via the concept of linear stability analysis for the integer order DLWE are established. Some graphical features are presented to explain the physical mechanism of the solutions.
Keywords
- Time fractional PDEs
- RL fractional derivative
- Cls
- Solitons
- Stability analysis
1 Introduction
Fractional calculus has mesmerizing features due to its pragmatic applications in various areas of science, social science, finance, and engineering to mention a few. Owing to this, a lot of meaningful definitions that have to do with fractional derivatives have been proposed by different authors in order to fully explain the memory effect [1–4]. Among the existing derivatives, we mention Grunwald–Letnikov, Marchaud, Riemann–Liouville, Hadamard, modified Riemann–Liouville, and Caputo [5–10].
Recently, a new definition of derivative has been introduced, and it is called the conformable derivative. The newly introduced conformable derivative satisfies a lot of characteristics such as product and quotient formulas, and it is used to model some physical problems [11]. Several authors have utilized this definition in the real world problems [12–16].
Cls have originated from the pragmatic phenomena such as energy, mass, and momentum [17]. The Cls have been utilized for developing numerical techniques, proving the existence and uniqueness of solutions [18], analysis of the internal characteristics like recursion operators, bi-Hamiltonian structures [19]. It should be noted that there have been numerous generalizations of Noether’s theorem and Euler–Lagrange’s [20] associating to several definitions of fractional derivative to establish Cls for fractional nonlinear PDEs possessing fractional Lagrangians [21–23]. Furthermore, nonlinear physical phenomena may be explained through establishing exact solutions. This has brought a strong motivation to authors to obtain exact solutions using different schemes [24–51].
In the present paper, we investigate the Cls and soliton-like solutions of the time fractional DLWS with RL and conformable derivatives, respectively. Moreover, we compute the Cls via the multiplier technique and the stability analysis via the concept of linear stability of the integer order DLWS.
2 Basic tools
3 The models
Theorem 1
Proof
Similar steps can be found in [41]. □
3.1 Conservation laws for Eq. (6)
4 Soliton-like solutions
5 Conservation laws for Eq. (7) by multiplier
5.1 Stability analysis to Eq. (7)
6 Conclusion
We investigated time fractional DLWE and its corresponding integer order. The symmetry properties and reductions were derived. We constructed the Cls with RL for the time fractional DLWE via new conservation theorem. The conformable derivative was employed to establish soliton-like solutions for the time fractional DLWE by using the generalized projective method (GPM). Moreover, the Cls via the multiplier technique and the stability analysis via the concept of linear stability analysis for the integer order DLWE were established. Some graphical features for the obtained results were also presented.
Declarations
Funding
Not applicable.
Authors’ contributions
All authors read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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