scholarly journals New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using the Multiple Simplest Equation Method

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Yun-Mei Zhao
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Water Wave ◽  
Type Equation ◽  
Wave Model ◽  
Rational Functions ◽  
Equation Method ◽  
Number Of Solutions ◽  
New Exact Solutions ◽  
Simplest Equation Method

In our work, a generalized KdV type equation of neglecting the highest-order infinitesimal term, which is an important water wave model, is discussed by using the simplest equation method and its variants. The solutions obtained are general solutions which are in the form of hyperbolic, trigonometric, and rational functions. These methods are more effective and simple than other methods and a number of solutions can be obtained at the same time.

scholarly journals Exact Solutions for a New Nonlinear KdV-Like Wave Equation Using Simplest Equation Method and Its Variants

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yinghui He ◽  
Yun-Mei Zhao ◽  
Yao Long
Keyword(s):  
Thin Film ◽  
Wave Equation ◽  
Exact Solutions ◽  
Wave Motion ◽  
Wave Equations ◽  
Equation Method ◽  
Nonlinear Wave Equations ◽  
New Exact Solutions ◽  
Simplest Equation Method ◽  
Film Mass Transfer

The simplest equation method presents wide applicability to the handling of nonlinear wave equations. In this paper, we focus on the exact solution of a new nonlinear KdV-like wave equation by means of the simplest equation method, the modified simplest equation method and, the extended simplest equation method. The KdV-like wave equation was derived for solitary waves propagating on an interface (liquid-air) with wave motion induced by a harmonic forcing which is more appropriate for the study of thin film mass transfer. Thus finding the exact solutions of this equation is of great importance and interest. By these three methods, many new exact solutions of this equation are obtained.

scholarly journals New Exact Solutions for a Higher Order Wave Equation of KdV Type Using MultipleG′/G-Expansion Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yinghui He
Keyword(s):  
Exact Solutions ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Elliptic Functions ◽  
Wave Model ◽  
Rational Functions ◽  
Expansion Method ◽  
Nonlinear Wave Equations ◽  
Mathematical Tool ◽  
New Exact Solutions

TheG′/G-expansion method is a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems. In our work, exact traveling wave solutions of a generalized KdV type equation of neglecting the highest order infinitesimal term, which is an important water wave model, are discussed by theG′/G-expansion method and its variants. As a result, many new exact solutions involving parameters, expressed by Jacobi elliptic functions, hyperbolic functions, trigonometric function, and the rational functions, are obtained. These methods are more effective and simple than other methods and a number of solutions can be obtained at the same time. The related results are enriched.

New Exact Solutions for Two Nonlinear Equations

2006 ◽  
Vol 61 (1-2) ◽  
pp. 1-6 ◽  
Author(s):  
Zonghang Yang
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Exact Solutions ◽  
Water Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
New Exact Solutions ◽  
Complex Phenomena ◽  
Sivashinsky Equation

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of science, for example the Korteweg-de Vries-Kuramoto-Sivashinsky equation (KdV-KS equation) and the Ablowitz-Kaup-Newell-Segur shallow water wave equation (AKNS-SWW equation). To our knowledge the exact solutions for the first equation were still not obtained and the obtained exact solutions for the second were just N-soliton solutions. In this paper we present kinds of new exact solutions by using the extended tanh-function method.

The new exact solutions of the new coupled Konno-Oono equation by using extended simplest equation method

2018 ◽  
Vol 12 (6) ◽  
pp. 293-301 ◽  
Author(s):  
Montri Torvattanabun ◽  
Papraporn Juntakud ◽  
Adsadawut Saiyun ◽  
Nattawut Khansai
Keyword(s):  
Exact Solutions ◽  
Equation Method ◽  
New Exact Solutions ◽  
Simplest Equation Method

scholarly journals New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using Extended F-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yinghui He ◽  
Yun-Mei Zhao ◽  
Yao Long
Keyword(s):  
Exact Solutions ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Wave Model ◽  
Expansion Method ◽  
Trigonometric Function ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions

The F-expansion method is used to find traveling wave solutions to various wave equations. By giving more solutions of the general subequation, an extended F-expansion method is introduced by Emmanuel. In our work, a generalized KdV type equation of neglecting the highest-order infinitesimal term, which is an important water wave model, is discussed by using the extended F-expansion method. And when the parameters satisfy certain relations, some new exact solutions expressed by Jacobi elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.

scholarly journals New Exact Solutions, Dynamical and Chaotic Behaviors for the Fourth-Order Nonlinear Generalized Boussinesq Water Wave Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Cheng Chen ◽  
Zuolei Wang
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Fourth Order ◽  
Water Wave ◽  
Wave Transformation ◽  
Physical Parameters ◽  
New Exact Solutions ◽  
Dynamical Behaviors ◽  
Cklund Transformation ◽  
Homogeneous Balance Method

Based on the extended homogeneous balance method, the auto-B a ¨ cklund transformation transformation is constructed and some new explicit and exact solutions are given for the fourth-order nonlinear generalized Boussinesq water wave equation. Then, the fourth-order nonlinear generalized Boussinesq water wave equation is transformed into the planer dynamical system under traveling wave transformation. We also investigate the dynamical behaviors and chaotic behaviors of the considered equation. Finally, the numerical simulations show that the change of the physical parameters will affect the dynamic behaviors of the system.

scholarly journals New exact solutions of a generalized shallow water wave equation

2010 ◽  
Vol 82 (2) ◽  
pp. 025003 ◽  
Author(s):  
Bijan Bagchi ◽  
Supratim Das ◽  
Asish Ganguly
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Shallow Water Wave ◽  
New Exact Solutions

scholarly journals Exact Solutions of the Symmetric Regularized Long Wave Equation and the Klein-Gordon-Zakharov Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Isaiah Elvis Mhlanga ◽  
Chaudry Masood Khalique
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Symmetry Approach ◽  
New Exact Solutions ◽  
Long Wave ◽  
Simplest Equation Method ◽  
Regularized Long Wave Equation ◽  
Klein Gordon

We study two nonlinear partial differential equations, namely, the symmetric regularized long wave equation and the Klein-Gordon-Zakharov equations. The Lie symmetry approach along with the simplest equation and exp-function methods are used to obtain solutions of the symmetric regularized long wave equation, while the travelling wave hypothesis approach along with the simplest equation method is utilized to obtain new exact solutions of the Klein-Gordon-Zakharov equations.

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