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.

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.

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 Comparison between the (G’/G) - expansion method and the modified extended tanh method

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 88-94 ◽  
Author(s):  
Şamil Akçaği ◽  
Tuğba Aydemir
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
The Other ◽  
Extended Tanh Method ◽  
New Exact Solutions ◽  
Tanh Method

AbstractIn this paper, firstly, we give a connection between well known and commonly used methods called the $\left( {{{G'} \over G}} \right)$ -expansion method and the modified extended tanh method which are often used for finding exact solutions of nonlinear partial differential equations (NPDEs). We demonstrate that giving a convenient transformation and formula, all of the solutions obtained by using the $\left( {{{G'} \over G}} \right)$ - expansion method can be converted the solutions obtained by using the modified extended tanh method. Secondly, contrary to the assertion in some papers, the $\left( {{{G'} \over G}} \right)$-expansion method gives neither all of the solutions obtained by using the other method nor new solutions for NPDEs. Namely, while the modified extended tanh method gives more solutions in a straightforward, concise and elegant manner without reproducing a lot of different forms of the same solution. On the other hand, the $\left( {{{G'} \over G}} \right)$-expansion method provides less solutions in a rather cumbersome form. Lastly, we obtain new exact solutions for the Lonngren wave equation as an illustrative example by using these methods.

scholarly journals The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Yun-Mei Zhao ◽  
Ying-Hui He ◽  
Yao Long
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Equation Method ◽  
Simplest Equation Method ◽  
Klein Gordon ◽  
Schrödinger's Equation ◽  
Wave Reduction

A good idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the elliptic-like equations are derived using the simplest equation method and the modified simplest equation method, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained. For example, the perturbed nonlinear Schrödinger’s equation (NLSE), the Klein-Gordon-Zakharov (KGZ) system, the generalized Davey-Stewartson (GDS) equations, the Davey-Stewartson (DS) equations, and the generalized Zakharov (GZ) equations are investigated and the exact solutions are presented using this method.

scholarly journals A FAMILY OF THE SPIRAL SOLUTIONS OF THE NONLINEAR KLEIN‐GORDON EQUATION

1998 ◽  
Vol 3 (1) ◽  
pp. 98-103
Author(s):  
V. V. Gudkov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Klein Gordon ◽  
Klein Gordon Equation

A family of the functions, intended for a construction the exact travelling wave solutions of nonlinear partial differential equations, is given. Exact solutions of the Klein‐Gordon equation with a special potential are obtained. The behavior of complex and hypercomplex solutions of the second order is presented.

scholarly journals Exact Solutions of the Space Time Fractional Symmetric Regularized Long Wave Equation Using Different Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Özkan Güner ◽  
Dursun Eser
Keyword(s):  
Exact Solutions ◽  
Expansion Method ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Long Wave ◽  
Functional Variable ◽  
Liouville Derivative ◽  
Functional Variable Method ◽  
Regularized Long Wave Equation ◽  
Fractional Differential

We apply the functional variable method, exp-function method, and(G′/G)-expansion method to establish the exact solutions of the nonlinear fractional partial differential equation (NLFPDE) in the sense of the modified Riemann-Liouville derivative. As a result, some new exact solutions for them are obtained. The results show that these methods are very effective and powerful mathematical tools for solving nonlinear fractional equations arising in mathematical physics. As a result, these methods can also be applied to other nonlinear fractional differential equations.

Rosenau-KdV Equation Coupling with the Rosenau-RLW Equation: Conservation Laws and Exact Solutions

2017 ◽  
Vol 18 (6) ◽  
pp. 451-456 ◽  
Author(s):  
Ben Muatjetjeja ◽  
Abdullahi Rashid Adem
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Vries Equation ◽  
Kdv Equation ◽  
Long Wave ◽  
Rlw Equation ◽  
Kudryashov Method ◽  
Regularized Long Wave Equation ◽  
De Vries

AbstractWe compute the conservation laws for the Rosenau-Kortweg de Vries equation coupling with the Regularized Long-Wave equation using Noether’s approach through a remarkable method of increasing the order of the Rosenau-KdV-RLW equation. Furthermore, exact solutions for the Rosenau- KdV-RLW equation are acquired by employing the Kudryashov method.

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