Symmetry Groups, Similarity Reductions, and Conservation Laws of the Time-Fractional Fujimoto–Watanabe Equation Using Lie Symmetry Analysis Method

Guo, Baoyong; Dong, Huanhe; Fang, Yong

doi:10.1155/2020/4830684

Cited by 6 publications

(3 citation statements)

References 39 publications

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“…There are many effective techniques available, like the Hirota-bilinear method [ 8 , 9 ], the exp-function approach [ 10 ], the first-integral technique [ 11 ], the -dressing method [ 12 ], the Riemann-Hilbert approach [ 13 , 14 ], the generalized exponential rational function approach [ 15 ], the Ansatz and sub equation theories [ 16 ], the ( G ′/ G )- expansion approach [ 17 ], the Jacobi elliptic function technique [ 18 ], the modified auxiliary expansion technique [ 19 ], Lie symmetry analysis method [ 20 ],direct algebraic approach [ 21 ], the -stepest descent method [ 22 , 23 ], improved Bernoulli sub-equation Function technique [ 24 ], the Sine-Gordon expansion method [ 25 ] and residual power series technique [ 26 ], which effectively established for solving the solution to NLFPDEs.…”

Section: Commencement and Forewordmentioning

confidence: 99%

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique

Zaman¹,

Arefin²,

Akbar

et al. 2023

PLoS ONE

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Nonlinear fractional partial differential equations are highly applicable for representing a wide variety of features in engineering and research, such as shallow-water, oceanography, fluid dynamics, acoustics, plasma physics, optical fiber system, turbulence, nonlinear biological systems, and control theory. In this research, we chose to construct some new closed form solutions of traveling wave of fractional order nonlinear coupled type Boussinesq–Burger (BB) and coupled type Boussinesq equations. In beachside ocean and coastal engineering, the suggested equations are frequently used to explain the spread of shallow-water waves, depict the propagation of waves through dissipative and nonlinear media, and appears during the investigation of the flow of fluid within a dynamic system. The subsidiary extended tanh-function technique for the suggested equations is solved for achieve new results by conformable derivatives. The fractional order differential transform was used to simplify the solution process by converting fractional differential equations to ordinary type differential equations by using the mentioned method. Using this technique, some applicable wave forms of solitons like bell type, kink type, singular kink, multiple kink, periodic wave, and many other types solution were accomplished, and we express our achieve solutions by 3D, contour, list point, and vector plots by using mathematical software such as MATHEMATICA to express the physical sketch much more clearly. Moreover, we assured that the suggested technique is more reliable, pragmatic, and dependable, that also explore more general exact solutions of close form traveling waves.

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Section: Commencement and Forewordmentioning

confidence: 99%

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique

Zaman¹,

Arefin²,

Akbar

et al. 2023

PLoS ONE

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“…Therefore, this new conservation law plays an increasingly important role in solving the conservation laws of FPDEs. More details about conservation laws can be found in [29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

Maarouf

Hilal

2021

Journal of Function Spaces

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The main purpose of this paper is to apply the Lie symmetry analysis method for the two-dimensional time fractional Fokker-Planck (FP) equation in the sense of Riemann–Liouville fractional derivative. The Lie point symmetries are derived to obtain the similarity reductions and explicit solutions of the governing equation. By using the new conservation theorem, the new conserved vectors for the two-dimensional time fractional Fokker-Planck equation have been constructed with a detailed derivation. Finally, we obtain its explicit analytic solutions with the aid of the power series expansion method.

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“…e microwaves are highly penetrative and can work under any weather conditions. For the safety of maritime navigation and offshore platforms, it is of great significance to explore the physical mechanism of the occurrence, evolution, and extinction of freak waves [3][4][5][6]. Moreover, the use of SAR to monitor and predict freak waves is a disaster reduction technology that should be valued [6].…”

Section: Introductionmentioning

confidence: 99%

Computational Simulation and Modeling of Freak Waves Based on Longuet-Higgins Model and Its Electromagnetic Scattering Calculation

Liu

Liang

2020

Complexity

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In order to solve the weak nonlinear problem in the simulation of strong nonlinear freak waves, an improved phase modulation method is proposed based on the Longuet-Higgins model and the comparative experiments of wave spectrum in this paper. Experiments show that this method can simulate the freak waves at fixed time and fixed space coordinates. In addition, by comparing the target wave spectrum and the freak wave measured in Tokai of Japan from the perspective of B-F instability and spectral peakedness, it is proved that the waveform of the simulated freak waves can not only maintain the spectral structure of the target ocean wave spectrum, but also accord with the statistical characteristics of the wave sequences. Then, based on the Kirchhoff approximation method and the modified Two-Scale Method, the electromagnetic scattering model of the simulated freak waves is established, and the normalized radar cross section (NRCS) of the freak waves and their background sea surfaces is analyzed. The calculation results show that the NRCS of the freak waves is usually smaller than their large-scale background sea surfaces. It can be concluded that when the neighborhood NRCS difference is less than or equal to −12 dB, we can determine where the freak waves are.

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Symmetry Groups, Similarity Reductions, and Conservation Laws of the Time-Fractional Fujimoto–Watanabe Equation Using Lie Symmetry Analysis Method

Cited by 6 publications

References 39 publications

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique

Study of the soliton propagation of the fractional nonlinear type evolution equation through a novel technique

Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

Computational Simulation and Modeling of Freak Waves Based on Longuet-Higgins Model and Its Electromagnetic Scattering Calculation

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