scholarly journals Generalized Variational Principle for the Fractal (2 + 1)-Dimensional Zakharov–Kuznetsov Equation in Quantum Magneto-Plasmas

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1022
Author(s):  
Yan-Hong Liang ◽  
Kang-Jia Wang
Keyword(s):  
Variational Principle ◽  
Variational Formulation ◽  
Inverse Method ◽  
Fractal Theory ◽  
Traveling Wave Solutions ◽  
Generalized Variational Principle ◽  
Fractal Derivative ◽  
Fractal Space ◽  
First Time ◽  
Symmetric Theory

In this paper, we propose the fractal (2 + 1)-dimensional Zakharov–Kuznetsov equation based on He’s fractal derivative for the first time. The fractal generalized variational formulation is established by using the semi-inverse method and two-scale fractal theory. The obtained fractal variational principle is important since it not only reveals the structure of the traveling wave solutions but also helps us study the symmetric theory. The finding of this paper will contribute to the study of symmetry in the fractal space.

scholarly journals Variational principle for nonlinear fractional wave equation in a fractal space

2021 ◽  
pp. 18-18
Author(s):  
Shao-Wen Yao
Keyword(s):  
Wave Equation ◽  
Variational Principle ◽  
Inverse Method ◽  
Scale Method ◽  
Fractional Wave Equation ◽  
Fractal Derivative ◽  
Fractal Space

The fractal derivative is adopted to describe the nonlinear fractional wave equation in a fractal space. A variational principle is successfully established by the semi-inverse method. The two-scale method and He?s exp-function are usedto solve the equation, and a good result is obtained.

scholarly journals Variational principle of the Whitham-Broer-Kaup equation in shallow water wave with fractal derivatives

2021 ◽  
pp. 19-19
Author(s):  
Wei-Wei Ling ◽  
Pin-Xia Wu
Keyword(s):  
Variational Principle ◽  
Shallow Water ◽  
Solitary Waves ◽  
Water Waves ◽  
Water Wave ◽  
Variational Formulation ◽  
Inverse Method ◽  
Solution Structure ◽  
Shallow Water Waves ◽  
Fractal Space

The Whitham-Broer-Kaup equation exists widely in shallow water waves, but unsmooth boundary seriously affects the properties of solitary waves and has certain deviations in scientific research. The aim of this paper is to introduce its modification with fractal derivatives in a fractal space and to establish a fractal variational formulation by the semi-inverse method. The obtained fractal variational principle shows conservation laws in an energy form in the fractal space and also hints its possible solution structure.

scholarly journals On the semi-inverse method and variational principle

2013 ◽  
Vol 17 (5) ◽  
pp. 1565-1568 ◽  
Author(s):  
Xue-Wei Li ◽  
Ya Li ◽  
Ji-Huan He
Keyword(s):  
Heat Conduction ◽  
Variational Principle ◽  
Inverse Method ◽  
Generalized Variational Principle ◽  
Open Forum ◽  
One Dimensional ◽  
History Of

In this Open Forum, Liu et al. proved the equivalence between He-Lee 2009 variational principle and that by Tao and Chen (Tao, Z. L., Chen, G. H., Thermal Science, 17(2013), pp. 951-952) for one dimensional heat conduction. We confirm the correction of Liu et al.?s proof, and give a short remark on the history of the semi-inverse method for establishment of a generalized variational principle.

New Solitary Wave Solution for the Boussinesq Wave Equation Using the Semi-Inverse Method and the Exp-Function Method

2009 ◽  
Vol 64 (11) ◽  
pp. 709-712 ◽  
Author(s):  
Wenjun Liu
Keyword(s):  
Wave Equation ◽  
Variational Principle ◽  
Solitary Wave ◽  
Wave Solution ◽  
Variational Formulation ◽  
Function Method ◽  
Inverse Method ◽  
Solitary Wave Solution ◽  
Solitary Solutions

Using the semi-inverse method, a variational formulation is established for the Boussinesq wave equation. Based on the obtained variational principle, solitary solutions in the sech-function and expfunction forms are obtained

Solitary Solutions of the Boiti-Leon-Manna-Pempinelli Equation Using He’S Variational Method

2008 ◽  
Vol 63 (10-11) ◽  
pp. 634-636 ◽  
Author(s):  
Zhao-Ling Tao
Keyword(s):  
Variational Method ◽  
Traveling Wave ◽  
Variational Formulation ◽  
Inverse Method ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
Solitary Solutions

A variational formulation is established for the Boiti-Leon-Manna-Pempinelli equation using He’s semi-inverse method; three kinds of traveling wave solutions are obtained.

scholarly journals A fractal variational theory of the Broer-Kaup system in shallow water waves

2021 ◽  
pp. 87-87
Author(s):  
Wei-Wei Ling ◽  
Pin-Xia Wu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Variational Formulation ◽  
Inverse Method ◽  
Variational Theory ◽  
Shallow Water Waves ◽  
Solution Structures ◽  
Fractal Space ◽  
Variational Formulations

The Broer-Kaup equation is one of many equations describing some phenomena of shallow water wave. There are many errors in scientific research because of the existence of the non-smooth boundaries. In this paper, we generalize the Broer-Kaup equation to the fractal space and establish fractal variational formulations through the semi-inverse method. The acquired fractal variational formulation reveals conservation laws in an energy form in the fractal space and suggests possible solution structures of the morphology of the solitary waves.

A VARIATIONAL FORMULATION FOR ANISOTROPIC WAVE TRAVELING IN A POROUS MEDIUM

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950047 ◽  
Author(s):  
YAN WANG ◽  
JIANYE AN ◽  
XIAOQIAN WANG
Keyword(s):  
Porous Medium ◽  
Variational Principle ◽  
Energy Conservation ◽  
Conservation Law ◽  
Variational Formulation ◽  
Coastal Protection ◽  
Fractal Derivative ◽  
Energy Conservation Law

An anisotropic wave in a porous medium is a hot topic in the coastal protection. A fractal derivative model is established, and a variational principle is established for the anisotropic wave traveling. The variational principle reveals an energy conservation law during the traveling process.

scholarly journals A powerful and simple frequency formula to nonlinear fractal oscillators

2020 ◽  
pp. 146134842094783
Author(s):  
Kang-Le Wang ◽  
Chun-Fu Wei
Keyword(s):  
Variational Principle ◽  
Conservation Laws ◽  
Nonlinear Oscillator ◽  
Transform Method ◽  
Inverse Transform ◽  
Simple Tool ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Approximate Frequency

In this work, a fractal nonlinear oscillator is successfully established by fractal derivative in a fractal space, and its variational principle is obtained by semi-inverse transform method. The variational principle can provide conservation laws in an energy form. The approximate frequency of the fractal oscillator is found by a simple fractal frequency formula. An example shows the fractal frequency formula is a powerful and simple tool to fractal oscillators.

Variational Approach for the Lane-Emden Equation

2008 ◽  
Vol 63 (10-11) ◽  
pp. 637-640
Author(s):  
Lan Xu ◽  
Nan Zhang
Keyword(s):  
Variational Principle ◽  
Variational Formulation ◽  
Variational Approach ◽  
Inverse Method ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Emden Equation ◽  
Good Agreement

A variational principle for the Lane-Emden equation is established by He’s semi-inverse method. Based on the established variational formulation, approximate solutions can be easily obtained by the Ritz method. The obtained solutions are in good agreement with the exact ones. The results show that the variational approach is very effective and convenient for solving the Lane-Emden equation.

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