scholarly journals The Improved Generalized tanh-coth Method Applied to Sixth-Order Solitary Wave Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
M. Torvattanabun ◽  
J. Simmapim ◽  
D. Saennuad ◽  
T. Somaumchan
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Sixth Order ◽  
Mathematical Tool ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
New Exact Solutions

The improved generalized tanh-coth method is used in nonlinear sixth-order solitary wave equation. This method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. The new exact solutions consisted of trigonometric functions solutions, hyperbolic functions solutions, exponential functions solutions, and rational functions solutions. The numerical results were obtained with the aid of Maple.

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.

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.

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 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.

scholarly journals New solitary Solution for the Kudryashov-Sinelshchikov (KS) equation by Modern Extension of the Hyperbolic Method

2020 ◽  
Vol 25 (2) ◽  
pp. 124
Author(s):  
Ali H. Hazza1 ◽  
Wafaa M. Taha2 ◽  
Raad A. Hameed1 ◽  
, Israa A. Ibrahim1 ◽  
, Israa A . Ibrahim1
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Applied Mathematics ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Mathematical Tool ◽  
3D Graphics ◽  
Hyperbolic Method ◽  
Powerful Mathematical Tool

In the present paper, we apply the modern extension of the hyperbolic tanh function method of nonlinear partial differential equations (NLPDEs) of Kudryashov - Sinelshchikov (KS) equation for obtaining exact and solitary traveling wave solutions. Through our solutions, we gain various functions, such as, hyperbolic, trigonometric and rational functions. Additionally, we support our results by comparisons with other methods and painting 3D graphics of the exact solutions. It is shown that our method provides a powerful mathematical tool to find exact solutions for many other nonlinear equations in applied mathematics   http://dx.doi.org/10.25130/tjps.25.2020.039

scholarly journals The -Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah ◽  
M. Ali Akbar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
Fifth Order

We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG) equation by the -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.

scholarly journals New Exact Solutions for New Model Nonlinear Partial Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-16
Author(s):  
A. Maher ◽  
H. M. El-Hawary ◽  
M. S. Al-Amry
Keyword(s):  
Nonlinear Partial Differential Equation ◽  
Elliptic Functions ◽  
Rational Functions ◽  
Travelling Wave ◽  
Mapping Method ◽  
Travelling Wave Solutions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
New Exact Solutions ◽  
New Form

In this paper we propose a new form of Padé-II equation, namely, a combined Padé-II and modified Padé-II equation. The mapping method is a promising method to solve nonlinear evaluation equations. Therefore, we apply it, to solve the combined Padé-II and modified Padé-II equation. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions, and elliptic functions.

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 New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the(2+1)-Dimensional Evolution Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah
Keyword(s):  
Riccati Equation ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Mapping Method ◽  
Auxiliary Equation ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
Generalized Riccati Equation

The generalized Riccati equation mapping is extended with the basic(G′/G)-expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equationG′(η)=w+uG(η)+vG2(η)is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple.

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