scholarly journals Exact Solutions of Equation Generated by the Jaulent-Miodek Hierarchy by(G’/G)-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Wafaa M. Taha ◽  
M. S. M. Noorani
Keyword(s):  
Exact Solutions ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Solitary Wave Solutions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Wave Solutions ◽  
Tanh Method ◽  
Dimensional Equation

The(G’/G)-expansion method is proposed for constructing more general exact solutions of the nonlinear(2+1)-dimensional equation generated by the Jaulent-Miodek Hierarchy. As a result, when the parameters are taken at special values, some new traveling wave solutions are obtained which include solitary wave solutions which are based from the hyperbolic functions, trigonometric functions, and rational functions. We find in this work that the(G’/G)-expansion method give some new results which are easier and faster to compute by the help of a symbolic computation system. The results obtained were compared with tanh method.

The G′/G-Expansion Method for Solutions of Evolution Equations

2014 ◽  
Vol 940 ◽  
pp. 425-428
Author(s):  
Chun Huan Xiang ◽  
Bo Liang ◽  
Hong Lei Wang
Keyword(s):  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Nonlinear Problems ◽  
Rational Functions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Wave Solutions

The investigation about traveling wave solutions of nonlinear equations is an important and interesting subject because they play important role in understanding the nonlinear problems. By using the (G′/G)-expansion method proposed recently, we construct the travelling wave solutions involving parameters for the Hirota and Satsuma equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The numerical simulation figures are shown.

scholarly journals The Extended MultipleG′/G-Expansion Method and Its Application to the Caudrey-Dodd-Gibbon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Huizhang Yang ◽  
Wei Li ◽  
Biyu Yang
Keyword(s):  
Exact Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Special Values ◽  
Wave Solutions ◽  
Complexiton Solutions ◽  
Hyperbolic Function Solutions

An extended multipleG′/G-expansion method is used to seek the exact solutions of Caudrey-Dodd-Gibbon equation. As a result, plentiful new complexiton solutions consisting of hyperbolic functions, trigonometric functions, rational functions, and their mixture with arbitrary parameters are effectively obtained. When some parameters are properly chosen as special values, the known double solitary-like wave solutions are derived from the double hyperbolic function solutions.

scholarly journals The(G′/G)-Expansion Method and Its Application for Higher-Order Equations of KdV (III)

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Huizhang Yang ◽  
Wei Li ◽  
Biyu Yang
Keyword(s):  
Traveling Wave ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Higher Order ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Wave Solutions ◽  
Linear Differential

New exact traveling wave solutions of a higher-order KdV equation type are studied by the(G′/G)-expansion method, whereG=G(ξ)satisfies a second-order linear differential equation. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. The property of this method is that it is quite simple and understandable.

scholarly journals Applications of the Novel (G′/G)-Expansion Method for a Time Fractional Simplified Modified Camassa-Holm (MCH) Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Muhammad Shakeel ◽  
Qazi Mahmood Ul-Hassan ◽  
Jamshad Ahmad
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
The Novel ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Liouville Derivative ◽  
Free Parameters

We use the fractional derivatives in modified Riemann-Liouville derivative sense to construct exact solutions of time fractional simplified modified Camassa-Holm (MCH) equation. A generalized fractional complex transform is properly used to convert this equation to ordinary differential equation and, as a result, many exact analytical solutions are obtained with more free parameters. When these free parameters are taken as particular values, the traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. Moreover, the numerical presentations of some of the solutions have been demonstrated with the aid of commercial software Maple. The recital of the method is trustworthy and useful and gives more new general exact solutions.

scholarly journals Exact Solutions of the Kudryashov-Sinelshchikov Equation Using the MultipleG′/G-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yinghui He ◽  
Shaolin Li ◽  
Yao Long
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Rational Functions ◽  
Expansion Method ◽  
Jacobi Elliptic Functions ◽  
Hyperbolic Functions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

Exact traveling wave solutions of the Kudryashov-Sinelshchikov equation are studied by theG′/G-expansion method and its variants. The solutions obtained include the form of Jacobi elliptic functions, hyperbolic functions, and trigonometric and rational functions. Many new exact traveling wave solutions can easily be derived from the general results under certain conditions. These methods are effective, simple, and many types of solutions can be obtained at the same time.

Investigation of solitary wave solutions for the (3 + 1)-dimensional Zakharov–Kuznetsov equation

2019 ◽  
Vol 33 (29) ◽  
pp. 1950350 ◽  
Author(s):  
Asif Yokus ◽  
Bülent Kuzu ◽  
Uğur Demiroğlu
Keyword(s):  
Exact Solution ◽  
Truncation Error ◽  
Numerical Solutions ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Wave Solutions ◽  
Package Program

In this paper, the new traveling wave solutions containing the trigonometric functions, hyperbolic functions and rational functions of [Formula: see text]-dimensional Zakharov–Kuznetsov equation are obtained. The graphs of the solution functions are presented by giving specific values to the constants. Numerical solutions are obtained by using finite difference method with new initial condition. Von Neumann’s Stability, Consistency and Linear Stability analysis of the equation are performed and [Formula: see text], [Formula: see text] norm errors are also examined with the truncation error. The exact solution obtained is presented via numerical solutions and absolute error graphs, and the analysis of exact solution and the numerical solutions are performed. Complex operations and graphical drawings were made using the computer package program.

The (G'/G,1/G)-Expansion Method for Solving the (2+1)-Dimensional Breaking Soliton Equations

2013 ◽  
Vol 787 ◽  
pp. 1006-1010
Author(s):  
Yun Jie Yang ◽  
Yun Mei Zhao ◽  
Yan He
Keyword(s):  
Rational Functions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Soliton Equations ◽  
Wave Solutions

In this paper, the-expansion method is applied to construct more general exact travelling solutions of the (2+1)-dimensional breaking soliton equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions.

scholarly journals Exact Solutions of the Generalized Benjamin-Bona-Mahony Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-5 ◽  
Author(s):  
Xun Liu ◽  
Lixin Tian ◽  
Yuhai Wu
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Traveling Wave ◽  
Arbitrary Order ◽  
Traveling Wave Solutions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Weierstrass Elliptic Function ◽  
Wave Solutions

We apply the theory of Weierstrass elliptic function to study exact solutions of the generalized Benjamin-Bona-Mahony equation. By using the theory of Weierstrass elliptic integration, we get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations which have arbitrary-order nonlinearity.

scholarly journals Conformal Fractional ((D_ξ^α G(ξ))/G(ξ) )- Expansion Method and Its Applications for Solving the Nonlinear Fractional Sharma-Tasso-Olver equation

2021 ◽  
Vol 2 (5) ◽  
pp. 1-8
Author(s):  
Alaaeddin Amin Moussa ◽  
Lama Abdulaziz Alhakim
Keyword(s):  
Partial Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Wave Solutions ◽  
Special Cases

In this article, we generalize the ((G^' (ξ))/G(ξ) )- expansion method which is one of the most important methods to finding the exact solutions of nonlinear partial differential equations. The new generalized method, named conformal fractional ((D_ξ^α G(ξ))/G(ξ) )-expansion method, takes advantage of Katugampola’s fractional derivative to create many useful traveling wave solutions of the nonlinear conformal fractional Sharma-Tasso-Olver equation. The obtained solutions have been articulated by the hyperbolic, trigonometric and rational functions with arbitrary constants. These solutions are algebraically verified using Maple and their physical characteristics are illustrated in some special cases.

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.  

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