scholarly journals New complex exact travelling wave solutions for the generalized-Zakharov equation with complex structures

2016 ◽  
Vol 6 (2) ◽  
pp. 141-150 ◽  
Author(s):  
Haci Mehmet Baskonus ◽  
Hasan Bulut
Keyword(s):  
Complex Structure ◽  
Three Dimensional ◽  
Expansion Method ◽  
Travelling Wave ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Zakharov Equation ◽  
Wolfram Mathematica ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions

In this paper, we apply the sine-Gordon expansion method which is one of the powerful methods to the generalized-Zakharov equation with complex structure. This algorithm yields new complex hyperbolic function solutions to the generalized-Zakharov equation with complex structure. Wolfram Mathematica 9 has been used throughout the paper for plotting two- and three-dimensional surface of travelling wave solutions obtained.

Non-TravellingWave Solutions of the (2+1)-Dimensional Dispersive Long Wave System

2009 ◽  
Vol 64 (9-10) ◽  
pp. 540-552
Author(s):  
Mamdouh M. Hassan
Keyword(s):  
Expansion Method ◽  
Travelling Wave ◽  
Hyperbolic Function ◽  
Wave System ◽  
Jacobi Elliptic Function ◽  
Travelling Wave Solutions ◽  
Rational Solutions ◽  
Long Wave ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

With the aid of symbolic computation and the extended F-expansion method, we construct more general types of exact non-travelling wave solutions of the (2+1)-dimensional dispersive long wave system. These solutions include single and combined Jacobi elliptic function solutions, rational solutions, hyperbolic function solutions, and trigonometric function solutions.

scholarly journals Computational and traveling wave analysis of Tzitzéica and Dodd-Bullough-Mikhailov equations: An exact and analytical study

2021 ◽  
Vol 10 (1) ◽  
pp. 272-281
Author(s):  
Hülya Durur ◽  
Asıf Yokuş ◽  
Kashif Ali Abro
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Three Dimensional ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Stationary Wave ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Travelling Wave Solutions ◽  
Wave Solutions

Abstract Computational and travelling wave solutions provide significant improvements in accuracy and characterize novelty of imposed techniques. In this context, computational and travelling wave solutions have been traced out for Tzitzéica and Dodd-Bullough-Mikhailov equations by means of (1/G′)-expansion method. The different types of solutions have constructed for Tzitzéica and Dodd-Bullough-Mikhailov equations in hyperbolic form. Moreover, solution function of Tzitzéica and Dodd-Bullough-Mikhailov equations has been derived in the format of logarithmic nature. Since both equations contain exponential terms so the solutions produced are expected to be in logarithmic form. Traveling wave solutions are presented in different formats from the solutions introduced in the literature. The reliability, effectiveness and applicability of the (1/G′)-expansion method produced hyperbolic type solutions. For the sake of physical significance, contour graphs, two dimensional and three dimensional graphs have been depicted for stationary wave. Such graphical illustration has been contrasted for stationary wave verses traveling wave solutions. Our graphical comparative analysis suggests that imposed method is reliable and powerful method for obtaining exact solutions of nonlinear evolution equations.

scholarly journals New Improvement of the Expansion Methods for Solving the Generalized Fitzhugh-Nagumo Equation with Time-Dependent Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-35 ◽  
Author(s):  
Jalil Manafian ◽  
Mehrdad Lakestani
Keyword(s):  
Expansion Method ◽  
Travelling Wave ◽  
Trigonometric Function ◽  
Time Dependent ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Function Solution ◽  
Wave Solutions ◽  
Nagumo Equation

An improvement of the expansion methods, namely, the improved tan⁡Φξ/2-expansion method, for solving nonlinear second-order partial differential equation, is proposed. The implementation of the new approach is demonstrated by solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. As a result, many new and more general exact travelling wave solutions are obtained including periodic function solutions, soliton-like solutions, and trigonometric function solutions. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions comparing this method with other methods. The results demonstrate that the new tan⁡Φξ/2-expansion method is more efficient than the Ansatz method and Tanh method applied by Triki and Wazwaz (2013). Recently, this method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. Abundant exact travelling wave solutions including solitons, kink, and periodic and rational solutions have been found. These solutions might play an important role in engineering fields. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear physics.

scholarly journals Travelling Wave Solutions of the Schrödinger-Boussinesq System

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Adem Kılıcman ◽  
Reza Abazari
Keyword(s):  
Soliton Solutions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Trigonometric Function ◽  
Boussinesq System ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Physical Problems ◽  
Hyperbolic Function Solutions ◽  
Trigonometric Function Solutions

We establish exact solutions for the Schrödinger-Boussinesq Systemiut+uxx−auv=0,vtt−vxx+vxxxx−b(|u|2)xx=0, whereaandbare real constants. The (G′/G)-expansion method is used to construct exact periodic and soliton solutions of this equation. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. As a result, hyperbolic function solutions and trigonometric function solutions with parameters are obtained. These solutions may be important and of significance for the explanation of some practical physical problems.

scholarly journals A Generalized and Improved(G′/G)-Expansion Method for Nonlinear Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
M. Ali Akbar ◽  
Norhashidah Hj. Mohd. Ali ◽  
E. M. E. Zayed
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Rational Functions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Wave Solutions

A generalized and improved(G′/G)-expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation and the strain wave equation in microstructured solids. Abundant exact travelling wave solutions of these equations are obtained, which include the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also it is shown that the proposed method is efficient for solving nonlinear evolution equations in mathematical physics and in engineering.

Travelling Wave Solutions for the Burgers Equation and the Korteweg-de Vries Equation with Variable Coefficients Using the Generalized (G´/G)-Expansion Method

2010 ◽  
Vol 65 (12) ◽  
pp. 1065-1070 ◽  
Author(s):  
Elsayed M. E. Zayed ◽  
Mahmoud A. M. Abdelaziz
Keyword(s):  
Burgers Equation ◽  
Kdv Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
De Vries

In this article, a generalized (G´/G)-expansion method is used to find exact travelling wave solutions of the Burgers equation and the Korteweg-de Vries (KdV) equation with variable coefficients. As a result, hyperbolic, trigonometric, and rational function solutions with parameters are obtained. When these parameters are taking special values, the solitary wave solutions are derived from the hyperbolic function solution. It is shown that the proposed method is direct, effective, and can be applied to many other nonlinear evolution equations in mathematical physics.

New travelling wave solutions for coupled fractional variant Boussinesq equation and approximate long water wave equation

2015 ◽  
Vol 25 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Limei Yan
Keyword(s):  
Wave Equation ◽  
Water Wave ◽  
Boussinesq Equation ◽  
Travelling Wave ◽  
Hyperbolic Function ◽  
Travelling Wave Solutions ◽  
Rational Solutions ◽  
Content Type ◽  
Wave Solutions ◽  
Hyperbolic Function Solutions

Purpose – The purpose of this paper is to apply the fractional sub-equation method to research on coupled fractional variant Boussinesq equation and fractional approximate long water wave equation. Design/methodology/approach – The algorithm is implemented with the aid of fractional Ricatti equation and the symbol computational system Mathematica. Findings – New travelling wave solutions, which include generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions, for these two equations are obtained. Originality/value – The algorithm is demonstrated to be direct and precise, and can be used for many other nonlinear fractional partial differential equations. The fractional derivatives described in this paper are in the Jumarie's modified Riemann-Liouville sense.

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.  

scholarly journals Equilibrium and travelling-wave solutions of plane Couette flow

2009 ◽  
Vol 638 ◽  
pp. 243-266 ◽  
Author(s):  
J. F. GIBSON ◽  
J. HALCROW ◽  
P. CVITANOVIĆ
Keyword(s):  
Couette Flow ◽  
Three Dimensional ◽  
Physical Space ◽  
Space Velocity ◽  
Travelling Wave ◽  
Isotropy Group ◽  
Travelling Wave Solutions ◽  
Plane Couette Flow ◽  
Equilibrium Solutions ◽  
Wave Solutions

We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new travelling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their three-dimensional physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low-Re turbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space on to suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

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