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 ON THE COMPLEX MIXED DARK-BRIGHT WAVE DISTRIBUTIONS TO SOME CONFORMABLE NONLINEAR INTEGRABLE MODELS

Fractals ◽  
2021 ◽  
pp. 2240018
Author(s):  
ARMANDO CIANCIO ◽  
GULNUR YEL ◽  
AJAY KUMAR ◽  
HACI MEHMET BASKONUS ◽  
ESIN ILHAN
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Expansion Method ◽  
Research Paper ◽  
Trigonometric Function ◽  
Integrable Models ◽  
Hyperbolic Function ◽  
Conformable Derivative ◽  
Partial Differential ◽  
Wave Solutions

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted.

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 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.

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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