The wave patterns to the time-space fractional Casimir equation for the Ito system

2021 ◽  
pp. 2150396
Author(s):  
Damin Cao ◽  
Wei Xu ◽  
Fajiang He
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Soliton Solution ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Graphical Representations ◽  
Wave Solutions ◽  
Time Space ◽  
Wave Patterns

In this paper, the time-space fractional Casimir equation for the Ito system with conformal fractional derivative is taken into consideration and the corresponding traveling wave solutions are given and the effects of the fractional order to the peakon soliton solution are also discussed and analyzed. In addition, some graphical representations are also provided to show the properties of the solution directly.

Explicit, periodic and dispersive soliton solutions to the conformable time-fractional Wu–Zhang system

2021 ◽  
pp. 2150417
Author(s):  
Kalim U. Tariq ◽  
Mostafa M. A. Khater ◽  
Muhammad Younis
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Conformable Fractional Derivative ◽  
Physical Behavior ◽  
Wave Solutions

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified.

scholarly journals The first integral method and traveling wave solutions to Davey–Stewartson equation

2012 ◽  
Vol 17 (2) ◽  
pp. 182-193 ◽  
Author(s):  
Hossein Jafari ◽  
Atefe Sooraki ◽  
Yahya Talebi ◽  
Anjan Biswas
Keyword(s):  
Exact Solutions ◽  
Soliton Solution ◽  
Commutative Algebra ◽  
Traveling Wave ◽  
Integral Method ◽  
Traveling Wave Solutions ◽  
First Integral ◽  
First Integral Method ◽  
Wave Solutions

In this paper, the first integral method will be applied to integrate the Davey–Stewartson’s equation. Using this method, a few exact solutions will be obtained using ideas from the theory of commutative algebra. Finally, soliton solution will also be obtained using the traveling wave hypothesis.

Bifurcations and Exact Solutions for a Class of MKdV Equations with the Conformable Fractional Derivative via Dynamical System Method

2020 ◽  
Vol 30 (01) ◽  
pp. 2050004 ◽  
Author(s):  
Jianli Liang ◽  
Longkun Tang ◽  
Yonghui Xia ◽  
Yi Zhang
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Leibniz Rule ◽  
Wave Transformation ◽  
Conformable Fractional Derivative ◽  
Simplest Equation Method

In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation [Formula: see text] [Formula: see text], we reduce the PDE to an ODE which depends on the fractional order [Formula: see text], then the analysis depends on the order [Formula: see text]. Moreover, as [Formula: see text], the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order [Formula: see text]. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative.

scholarly journals Exact Solutions to Time-Fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 742 ◽  
Author(s):  
Tao Liu
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Equation Method ◽  
Singular Solutions ◽  
Trial Equation Method ◽  
Wave Solutions ◽  
Fifth Order ◽  
Fkdv Equation

We study a fifth order time-fractional KdV equation (FKdV) under meaning of the conformal fractional derivative. By trial equation method based on symmetry, we construct the abundant exact traveling wave solutions to the FKdV equation. These solutions show rich evolution patterns including solitons, rational singular solutions, periodic and double periodic solutions and so forth. In particular, under the concrete parameters, we give the representations of all these solutions.

Traveling-wave solutions of the Klein–Gordon equations with M-fractional derivative

Pramana ◽  
2022 ◽  
Vol 96 (1) ◽  
Author(s):  
Alphonse Houwe ◽  
Hadi Rezazadeh ◽  
Ahmet Bekir ◽  
Serge Y Doka
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
Klein Gordon

scholarly journals Traveling Wave Solutions to a Mathematical Model of Fractional order (2+1)-Dimensional Breaking Soliton Equation

Fractals ◽  
2021 ◽  
Author(s):  
Umair Ali ◽  
Azhar Iqbal ◽  
Ilyas Khan ◽  
Kashif Kamran ◽  
Shabbir Muhammad ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Fractional Order ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Soliton Equation ◽  
Wave Solutions
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