scholarly journals A note on improved F -expansion method combined with Riccati equation applied to nonlinear evolution equations

2014 ◽  
Vol 1 (2) ◽  
pp. 140038 ◽  
Author(s):  
Md. Shafiqul Islam ◽  
Kamruzzaman Khan ◽  
M. Ali Akbar ◽  
Antonio Mastroberardino
Keyword(s):  
Riccati Equation ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Wave Equations ◽  
Travelling Wave Solutions

The purpose of this article is to present an analytical method, namely the improved F -expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and 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 Multiple Exact Travelling Solitary Wave Solutions of Nonlinear Evolution Equations

2019 ◽  
pp. 1-13
Author(s):  
M. M. El-Horbaty ◽  
F. M. Ahmed
Keyword(s):  
Riccati Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Travelling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Extended Form ◽  
Wave Solutions ◽  
Tanh Method

An extended Tanh-function method with Riccati equation is presented for constructing multiple exact travelling wave solutions of some nonlinear evolution equations which are particular cases of a generalized equation. The results of solitary waves are general compact forms with non-zero constants of integration. Taking the full advantage of the Riccati equation improves the applicability and reliability of the Tanh method with its extended form.

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.

scholarly journals New Application of -Expansion Method for Generalized (2+1)-Dimensional Nonlinear Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Mohammad Najafi ◽  
Maliheh Najafi ◽  
Somayeh Arbabi
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Travelling Wave Solutions ◽  
Soliton Equation ◽  
Wave Solutions

We established -expansion method for (2+1)-dimensional nonlinear evolution equations. This method was used to construct travelling wave solutions of (2+1)-dimensional nonlinear evolution equations. (2+1)-Dimensional breaking soliton equation, (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation, and (2+1)-dimensional Bogoyavlenskii’s Breaking soliton equation are chosen to illustrate the effectiveness of the method.

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