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.

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.

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 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.

A variety of wave amplitudes for the conformable fractional (2 + 1)-dimensional Ito equation

2021 ◽  
pp. 2150254
Author(s):  
Emad A. Az-Zo’bi ◽  
Wael A. Alzoubi ◽  
Lanre Akinyemi ◽  
Mehmet Şenol ◽  
Basem S. Masaedeh
Keyword(s):  
Mapping Method ◽  
Jacobi Elliptic Function ◽  
Conformable Derivative ◽  
Hyperbolic Functions ◽  
Fractional Complex Transform ◽  
New Exact Solutions ◽  
Higher Dimensional ◽  
Fractional Orders ◽  
Ito Equation ◽  
Ordinary Derivative

The conformable derivative and adequate fractional complex transform are implemented to discuss the fractional higher-dimensional Ito equation analytically. The Jacobi elliptic function method and Riccati equation mapping method are successfully used for this purpose. New exact solutions in terms of linear, rational, periodic and hyperbolic functions for the wave amplitude are derived. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. Numerical simulations of some obtained solutions with special choices of free constants and various fractional orders are displayed.

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.  

Travelling And Periodic Wave Solutions Of Some Nonlinear Wave Equations

2004 ◽  
Vol 59 (7-8) ◽  
pp. 389-396 ◽  
Author(s):  
A. H. Khater ◽  
M. M. Hassan
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Nonlinear Wave Equations ◽  
Travelling Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Pacs 02.30

We present the mixed dn-sn method for finding periodic wave solutions of some nonlinear wave equations. Introducing an appropriate transformation, we extend this method to a special type of nonlinear equations and construct their solutions, which are not expressible as polynomials in the Jacobi elliptic functions. The obtained solutions include the well known kink-type and bell-type solutions as a limiting cases. Also, some new travelling wave solutions are found. - PACS: 02.30.Jr; 03.40.Kf

Three Kind of Triangle Rational Functions and Applications for Discrete Nonlinear Equations

2011 ◽  
Vol 317-319 ◽  
pp. 2168-2171
Author(s):  
Xiu Rong Guo ◽  
Zheng Tao Liu ◽  
Mei Guo
Keyword(s):  
Nonlinear Equations ◽  
Difference Equations ◽  
Soliton Solutions ◽  
Rational Functions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Formal Solutions

In order to efficiently search for new soliton solutions to differential-difference equations (DDEs), three kinds of triangle rational functions are first introduced. Then a kind of formal solutions of DDEs are presented which are expressed by a unified nonlinear combination of the three kinds of triangle rational functions. As illustrative examples, the periodic travelling-wave solutions of the discrete modified KdV(mKdV) equations are obtained.

scholarly journals On solutions of the combined KdV-nKdV equation

2019 ◽  
Vol 30 (2) ◽  
pp. 33
Author(s):  
Mohammed Allami ◽  
A. K. Mutashar ◽  
A. S. Rashid
Keyword(s):  
Vries Equation ◽  
Special Functions ◽  
Elliptic Functions ◽  
Rational Functions ◽  
Trigonometric Functions ◽  
Exponential Functions ◽  
Jacobian Elliptic Functions ◽  
Negative Order ◽  
De Vries ◽  
Wave Reduction

The aim of this work is to deal with a new integrable nonlinear equation of wave propagation, the combined of the Korteweg-de vries equation and the negative order Korteweg-de vries equation (combined KdV-nKdV) equation, which was more recently proposed by Wazwaz. Upon using wave reduction variable, it turns out that the reduced combined KdV-nKdV equation is alike the reduced (3+1)-dimensional Jimbo Miwa (JM) equation, the reduced (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (PYTSF) equation and the reduced (3 + 1)¬dimensional generalized shallow water (GSW) equation in the trav¬elling wave. In fact, the four transformed equations belong to the same class of ordinary differential equation. With the benefit of a well known general solutions for the reduced equation, we show that sub¬jects to some scaling and change of parameters, a variety of families of solutions are constructed for the combined KdV-nKdV equation which can be expressed in terms of rational functions, exponential functions and periodic solutions of trigonometric functions and hyperbolic func¬tions. In addition to that the equation admits solitary waves, and double periodic waves in terms of special functions such as Jacobian elliptic functions and Weierstrass elliptic functions.

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.

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