scholarly journals The Periodic Solutions to Kawahara Equation by Means of the Auxiliary Equation with a Sixth-Degree Nonlinear Term

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Zehra Pınar ◽  
Turgut Öziş
Keyword(s):  
Exact Solutions ◽  
Nonlinear Equations ◽  
Nonlinear Term ◽  
Travelling Wave ◽  
Nonlinear Ordinary Differential Equation ◽  
Auxiliary Equation ◽  
Travelling Wave Solutions ◽  
Kawahara Equation ◽  
First Order ◽  
Different Types

It is well known that different types of exact solutions of an auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term are presented to obtain novel exact solutions of the Kawahara equation. By the aid of the solutions of the original auxiliary equation, some other physically important nonlinear equations can be solved to construct novel exact solutions.

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 Extended Auxiliary Equation Method to the perturbed Gerdjikov-Ivanov Equation

2018 ◽  
Author(s):  
Jamilu Sabi'u ◽  
Hadi Rezazadeh ◽  
Mostafa Eslami ◽  
Alper Korkmaz
Keyword(s):  
Travelling Wave ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Travelling Wave Solutions ◽  
Linear Optics ◽  
Extended Form ◽  
Auxiliary Equation Method ◽  
First Order ◽  
Non Linear Optics ◽  
New Form

We apply utilized the extended form of the auxiliary equation method to obtain extensively reliable exact travelling wave solutions of perturbed Gerdjikov–Ivanov equation (GIE) that is widely used as a model in the field theory of quanta and non-linear optics. The method is based on a simple first order second degree ODE. The new form of the approach gives more solutions to the governing equation efficiently.

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 A simple and direct method for generating travelling wave solutions for nonlinear equations

2008 ◽  
Vol 323 (5) ◽  
pp. 1150-1167 ◽  
Author(s):  
D. Bazeia ◽  
Ashok Das ◽  
L. Losano ◽  
A. Silva
Keyword(s):  
Nonlinear Equations ◽  
Direct Method ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions

New Non-Travelling Wave Solutions of Calogero Equation

2016 ◽  
Vol 8 (6) ◽  
pp. 1036-1049 ◽  
Author(s):  
Xiaoming Peng ◽  
Yadong Shang ◽  
Xiaoxiao Zheng
Keyword(s):  
Nonlinear Equations ◽  
Symbolic Computation ◽  
Travelling Wave ◽  
Explicit Solutions ◽  
Travelling Wave Solutions ◽  
Variable Separation ◽  
Test Technique ◽  
Reduced Equations ◽  
Wave Solutions ◽  
Variable Separation Approach

AbstractIn this paper, the idea of a combination of variable separation approach and the extended homoclinic test approach is proposed to seek non-travelling wave solutions of Calogero equation. The equation is reduced to some (1+1)-dimensional nonlinear equations by applying the variable separation approach and solves reduced equations with the extended homoclinic test technique. Based on this idea and with the aid of symbolic computation, some new explicit solutions can be obtained.

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.

Analytic bounded travelling wave solutions of some nonlinear equations II

2009 ◽  
Vol 41 (2) ◽  
pp. 803-810
Author(s):  
Eugenia N. Petropoulou ◽  
Panayiotis D. Siafarikas ◽  
Ioannis D. Stabolas
Keyword(s):  
Nonlinear Equations ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions
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