scholarly journals Exact Solutions of Damped Improved Boussinesq Equations by Extended (G′/G)-Expansion Method

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Kai Fan ◽  
Cunlong Zhou
Keyword(s):  
Exact Solutions ◽  
Fluid Dynamic ◽  
Boussinesq Equations ◽  
Expansion Method ◽  
Travelling Wave ◽  
Auxiliary Function ◽  
Existence Of A Solution ◽  
Hydrodynamic Damping ◽  
Kink Wave ◽  
Physical Phenomena

With the help of the auxiliary function method, we solved the improved Boussinesq (IBq) equation with fluid dynamic damping and the modified IBq (IMBq) equation with Stokes damping, and we obtained their three types of travelling wave exact solutions, which is an extension service of the numerical simulation and the existence of a solution. From the waveform diagram of IBq equation with hydrodynamic damping, it can be seen that when the propagation velocity of kink wave changes, the amplitude also changes significantly, and it is also found that the kink isolated waveform is significantly asymmetric due to the increase of damping coefficient v, which may be of some value in explaining some physical phenomena. In addition, the symbolic computing software maple makes our computing work easier.

scholarly journals Travelling Wave Solutions of the Perturbed Wadati-Segur-Ablowitz Equation by (G'/G)-Expansion Method

2018 ◽  
Vol 6 (06) ◽  
Author(s):  
Figen Kangalgil
Keyword(s):  
Exact Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena

The investigation of the exact solutions of NLPDEs plays an im- portant role for the understanding of most nonlinear physical phenomena. Also, the exact solutions of this equations aid the numerical solvers to assess the correctness of their results. In this paper, (G'/G)-expansion method is pre- sented to construct exact solutions of the Perturbed Wadati-Segur-Ablowitz equation. Obtained the exact solutions are expressed by the hyperbolic, the trigonometric and the rational functions. All calculations have been made with the aid of Maple program. It is shown that the proposed algorithm is elemen- tary, e¤ective and has been used for many PDEs in mathematical physics.  

AUTO-BÄCKLUND TRANSFORMATION AND NEW EXACT SOLUTIONS OF THE (2 + 1)-DIMENSIONAL NIZHNIK–NOVIKOV–VESELOV EQUATION

2005 ◽  
Vol 16 (03) ◽  
pp. 393-412 ◽  
Author(s):  
DENGSHAN WANG ◽  
HONG-QING ZHANG
Keyword(s):  
Exact Solutions ◽  
Soliton Solution ◽  
Bäcklund Transformation ◽  
Rational Solution ◽  
Expansion Method ◽  
Backlund Transformation ◽  
Paper Making ◽  
New Exact Solutions ◽  
Physical Phenomena ◽  
Series Expansion Method

In this paper, making use of the truncated Laurent series expansion method and symbolic computation we get the auto-Bäcklund transformation of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation. As a result, single soliton solution, single soliton-like solution, multi-soliton solution, multi-soliton-like solution, the rational solution and other exact solutions of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation are found. These solutions may be useful to explain some physical phenomena.

scholarly journals Novel Exact Solutions of the Extended Shallow Water Wave and the Fokas Equations

2018 ◽  
Vol 22 ◽  
pp. 01022
Author(s):  
Serbay DURAN ◽  
Berat KARAAGAC ◽  
Alaattin ESEN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Expansion Method ◽  
Travelling Wave ◽  
Wave Transformation ◽  
Partial Differential ◽  
Shallow Water Wave

In this study, a Sine-Gordon expansion method for obtaining novel exact solutions of extended shallow water wave equation and Fokas equation is presented. All of the equations which are under consideration consist of three or four variable. In this method, first of all, partial differential equations are reduced to ordinary differential equations by the help of variable change called as travelling wave transformation, then Sine Gordon expansion method allows us to obtain new exact solutions defined as in terms of hyperbolic trig functions of considered equations. The newly obtained results showed that the method is successful and applicable and can be extended to a wide class of nonlinear partial differential equations.

scholarly journals Exact Solutions for the Wick-Type Stochastic Schamel-Korteweg-de Vries Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Xueqin Wang ◽  
Yadong Shang ◽  
Huahui Di
Keyword(s):  
Exact Solutions ◽  
Symbolic Computation ◽  
Vries Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Variable Coefficients ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Hermite Transformation

We consider the Wick-type stochastic Schamel-Korteweg-de Vries equation with variable coefficients in this paper. With the aid of symbolic computation and Hermite transformation, by employing the (G′/G,1/G)-expansion method, we derive the new exact travelling wave solutions, which include hyperbolic and trigonometric solutions for the considered equations.

scholarly journals Applications of an Extended -Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
E. M. E. Zayed ◽  
Shorog Al-Joudi
Keyword(s):  
Exact Solutions ◽  
Traveling Wave Solutions ◽  
Boussinesq Equations ◽  
Expansion Method ◽  
Nonlinear Pdes ◽  
Hyperbolic Function ◽  
Soliton Equation ◽  
New Exact Solutions ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions

We construct the traveling wave solutions of the (1+1)-dimensional modified Benjamin-Bona-Mahony equation, the (2+1)-dimensional typical breaking soliton equation, the (1+1)-dimensional classical Boussinesq equations, and the (2+1)-dimensional Broer-Kaup-Kuperschmidt equations by using an extended -expansion method, whereGsatisfies the second-order linear ordinary differential equation. By using this method, new exact solutions involving parameters, expressed by three types of functions which are hyperbolic, trigonometric and rational function solutions, are obtained. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions.

scholarly journals New Exact Jacobi Elliptic Function Solutions for the Coupled Schrödinger-Boussinesq Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Baojian Hong ◽  
Dianchen Lu
Keyword(s):  
Elliptic Function ◽  
Nonlinear Partial Differential Equations ◽  
Boussinesq Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Algebraic Method ◽  
Auxiliary Function ◽  
Jacobi Elliptic Function ◽  
Deformation Method ◽  
Wave Solutions

A general algebraic method based on the generalized Jacobi elliptic functions expansion method, the improved general mapping deformation method, and the extended auxiliary function method with computerized symbolic computation is proposed to construct more new exact solutions for coupled Schrödinger-Boussinesq equations. As a result, several families of new generalized Jacobi elliptic function wave solutions are obtained by using this method, some of them are degenerated to solitary wave solutions and trigonometric function solutions in the limited cases, which shows that the general method is more powerful than plenty of traditional methods and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

scholarly journals A Fresh Look To Exact Solutions of Some Coupled Equations

2018 ◽  
Vol 22 ◽  
pp. 01006
Author(s):  
Berat Karaagac ◽  
Nuri Murat Yagmurlu ◽  
Alaattin Esen ◽  
Selcuk Kutluay
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Wave Transformation ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Coupled Partial Differential Equations ◽  
Coupled Equations ◽  
New Exact Solutions

This manuscript is going to seek travelling wave solutions of some coupled partial differential equations with an expansion method known as Sine- Gordon expansion method. Primarily, we are going to employ a wave transformation to partial differential equation to reduce the equations into ordinary differential equations. Then, the solution form of the handled equations is going to be constructed as polynomial of hyperbolic trig or trig functions. Finally, with the aid of symbolic computation, new exact solutions of the partial differentials equations will have been found.

scholarly journals Solitary Wave Solutions of the Boussinesq Equation and Its Improved Form

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Reza Abazari ◽  
Adem Kılıçman
Keyword(s):  
Exact Solutions ◽  
Small Amplitude ◽  
Boussinesq Equation ◽  
Boussinesq Equations ◽  
Expansion Method ◽  
Solitary Wave Solutions ◽  
Rational Solutions ◽  
Wave Solutions ◽  
Model Equations

This paper presents the general case study of previous works on generalized Boussinesq equations, (Abazari, 2011) and (Kılıcman and Abazari, 2012), that focuses on the application ofG′/G-expansion method with the aid of Maple to construct more general exact solutions for the coupled Boussinesq equations. In this work, the mentioned method is applied to construct more general exact solutions of Boussinesq equation and improved Boussinesq equation, which the French scientistJoseph Valentin Boussinesq(1842–1929) described in the 1870s model equations for the propagation of long waves on the surface of water with small amplitude. Our work is motivated by the fact that theG′/G-expansion method provides not only more general forms of solutions but also periodic, solitary waves and rational solutions. The method appears to be easier and faster by means of a symbolic computation.

MULTIPLE EXACT SOLUTIONS OF THE GENERALIZED TIME FRACTIONAL FOAM DRAINAGE EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050062
Author(s):  
DANDAN SHI ◽  
YUFENG ZHANG ◽  
WENHAO LIU
Keyword(s):  
Exact Solutions ◽  
Expansion Method ◽  
Transformation Method ◽  
Vertical Density Profile ◽  
New Exact Solutions ◽  
Vertical Density ◽  
Foam Drainage Equation ◽  
Physical Phenomena ◽  
Generalized Time ◽  
Foam Drainage

In this paper, we investigate the exact solutions of the generalized time fractional foam drainage equation. The Lie-group scaling transformation method and improved [Formula: see text]-expansion method are adopted here. The equation describes the evolution of the vertical density profile of a foam under gravity. New exact solutions and maple diagrams of the generalized time fractional foam drainage equation can help us better understand the physical phenomena.

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