The Exp-function Method for the Riccati Equation and Exact Solutions of Dispersive Long Wave Equations

2008 ◽  
Vol 63 (10-11) ◽  
pp. 663-670 ◽  
Author(s):  
Sheng Zhang ◽  
Wei Wang ◽  
Jing-Lin Tong
Keyword(s):  
Exact Solutions ◽  
Riccati Equation ◽  
Function Method ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
New Exact Solutions ◽  
Long Wave

In this paper, the Exp-function method is used to seek new generalized solitonary solutions of the Riccati equation. Based on the Riccati equation and one of its generalized solitonary solutions, new exact solutions with three arbitrary functions of the (2+1)-dimensional dispersive long wave equations are obtained. Compared with the tanh-function method and its extensions, the proposed method is more powerful. It is shown that the Exp-function method provides a straightforward and important mathematical tool for solving nonlinear evolution equations in mathematical physics.

Application of the Exp-Functionmethod to the Riccati Equation and New Exact Solutions with Three Arbitrary Functions of Quantum Zakharov Equations

2008 ◽  
Vol 63 (10-11) ◽  
pp. 646-652 ◽  
Author(s):  
Mohamed A Abdou ◽  
Essam M. Abulwafa
Keyword(s):  
Exact Solutions ◽  
Riccati Equation ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
New Exact Solutions ◽  
Generalized Riccati Equation ◽  
Solitary Solutions

The Exp-function method with the aid of the symbolic computational system is used for constructing generalized solitary solutions of the generalized Riccati equation. Based on the Riccati equation and its generalized solitary solutions, new exact solutions with three arbitrary functions of quantum Zakharov equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.

scholarly journals A modified generalized projective Riccati equation method

2016 ◽  
Vol 12 (6) ◽  
pp. 6318-6334
Author(s):  
Luwai Wazzan ◽  
Shafeek A Ghaleb
Keyword(s):  
Exact Solutions ◽  
Riccati Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Evolution Equations ◽  
Equation Method ◽  
New Exact Solutions ◽  
Special Cases ◽  
Projective Riccati Equation Method

A modification of the generalized projective Riccati equation method is proposed to treat some nonlinear evolution equations and obtain their exact solutions. Some known methods are obtained as special cases of the proposed method. In addition, the method is implemented to find new exact solutions for the well-known Dreinfelds-Sokolov-Wilson system of nonlinear partial differential equations.

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.

Application of the Subequation Method to Some Differential Equations of Time-Fractional Order

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Ahmet Bekir ◽  
Esin Aksoy
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
New Exact Solutions

The main goal of this paper is to develop subequation method for solving nonlinear evolution equations of time-fractional order. We use the subequation method to calculate the exact solutions of the time-fractional Burgers, Sharma–Tasso–Olver, and Fisher's equations. Consequently, we establish some new exact solutions for these equations.

New Exact Solutions to a Class of Nonlinear Evolution Equations with Symbolic Computations

2013 ◽  
Vol 645 ◽  
pp. 312-315
Author(s):  
Jian Ya Ge ◽  
Tie Cheng Xia
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Algebraic Method ◽  
Inverse Scattering Method ◽  
Nonlinear Evolution Equations ◽  
Scattering Method ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions

Searching for exact solutions to nonlinear evolution equations is an important topic in mathematical physics and engineering. Many methods of finding exact solutions have been presented such as the inverse scattering method, algebraic method and so on. In this paper, by using Fan sub-equation method with the help of Maple, several meaningful solutions are obtained including bell shape solutions, trigonometric function solutions, twist shape solutions and Jacobi elliptic function solutions for a class of nonlinear evolution equation. This method can be applied to other nonlinear partial differential equations.

Functional variable method to study nonlinear evolution equations

2013 ◽  
Vol 3 (3) ◽  
Author(s):  
Mostafa Eslami ◽  
Mohammad Mirzazadeh
Keyword(s):  
Exact Solutions ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Variable Method ◽  
Nonlinear Wave Equations ◽  
Wave Solutions ◽  
Functional Variable ◽  
Functional Variable Method

AbstractThe functional variable method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations. This method presents a wider applicability for handling nonlinear wave equations. In this paper, the functional variable method is used to construct exact solutions of Davey-Stewartson equation, generalized Zakharov equation, K(m, n) equation with generalized evolution term, (2 + 1)-dimensional long-wave-short-wave resonance interaction equation and nonlinear Schrödinger equation with power law nonlinearity. The obtained solutions include solitary wave solutions, periodic wave solutions.

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