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.

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.

Symbolic Computation and Non-travelling Wave Solutions of the (2+1)-Dimensional Korteweg de Vries Equation

2005 ◽  
Vol 60 (4) ◽  
pp. 221-228 ◽  
Author(s):  
Dengshan Wang ◽  
Hong-Qing Zhang
Keyword(s):  
Symbolic Computation ◽  
Evolution Equations ◽  
Vries Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
De Vries

Abstract In this paper, with the aid of symbolic computation we improve the extended F-expansion method described in Chaos, Solitons and Fractals 22, 111 (2004) to solve the (2+1)-dimensional Korteweg de Vries equation. Using this method, we derive many exact non-travelling wave solutions. These are more general than the previous solutions derived with the extended F-expansion method. They include the Jacobi elliptic function, soliton-like trigonometric function solutions, and so on. Our method can be applied to other nonlinear evolution 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 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.

Travelling-wave solutions to the Korteweg-de Vries-Burgers equation

1985 ◽  
Vol 101 (3-4) ◽  
pp. 207-226 ◽  
Author(s):  
J. L. Bona ◽  
M. E. Schonbek
Keyword(s):  
Burgers Equation ◽  
Travelling Wave ◽  
Singular Limit ◽  
Travelling Wave Solutions ◽  
Qualitative Properties ◽  
Wave Solutions ◽  
De Vries ◽  
Limiting Behaviour ◽  
Image Position

SynopsisThe existence and certain qualitative properties of travelling-wave solutions to the Korteweg-de Vries-Burgers equation,are established. The limiting behaviour of these waves, when ε tends to zero and when δ tends to zero is examined together with a singular limit wherein both ε and δ tend to zero.

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.

scholarly journals Some New Non-Travelling Wave Solutions of the Fisher Equation with Nonlinear Auxiliary Equation

2018 ◽  
Vol 3 (2) ◽  
pp. 92-101
Author(s):  
Anika Tashin Khan ◽  
Hasibun Naher
Keyword(s):  
Evolution Equations ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Evolution Equations ◽  
Fisher Equation ◽  
Auxiliary Equation ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Functional Forms

We have generated many new non-travelling wave solutions by executing the new extended generalized and improved (G'/G)-Expansion Method. Here the nonlinear ordinary differential equation with many new and real parameters has been used as an auxiliary equation. We have investigated the Fisher equation to show the advantages and effectiveness of this method. The obtained non-travelling solutions are expressed through the hyperbolic functions, trigonometric functions and rational functional forms. Results showing that the method is concise, direct and highly effective to study nonlinear evolution equations those are in mathematical physics and engineering.

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.

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