The alternative -expansion method with generalized Riccati equation: Application to fifth order (1+1)-dimensional Caudrey-Dodd-Gibbon equation

2012 ◽  
Vol 7 (5) ◽  
Author(s):  
M. Ali Akbar
Keyword(s):  
Riccati Equation ◽  
Expansion Method ◽  
Generalized Riccati Equation ◽  
Fifth Order

Multiple explicit solutions of the 2D variable coefficients Chafee–Infante model via a generalized expansion method

2021 ◽  
pp. 2150312
Author(s):  
Rodica Cimpoiasu
Keyword(s):  
Riccati Equation ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Time Variable ◽  
Auxiliary Equation ◽  
Natural Phenomena ◽  
Generalized Riccati Equation ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena ◽  
First Time

In this work, we do apply a generalized expansion method to the realistic two-dimensional (2D) Chafee–Infante model with time-variable coefficients which is encountered in physical sciences.The key ideas of this method consist in: (i) to choose a nonlinear wave variable in respect to time-variable into the general finite series solution of the governing model; (ii) to take a full advantage from the general elliptic equation introduced as an auxiliary equation which can degenerate into sub-equations such as Riccati equation, the Jacobian elliptic equations, the generalized Riccati equation. Based upon this powerful technique, we successfully construct for the first time several types of non-autonomous solitary waves as well as some non-autonomous triangular solutions, rational or doubly periodic type ones. We investigate the propagation of non-autonomous solitons and we emphasize as well upon the influence of the variable coefficients. We are providing and analyzing a few graphical representations of some specific solutions. The results of this paper will be valuable in the study of nonlinear physical phenomena. The above- mentioned method could be employed to solve other partial differential equations with variable coefficients which describe various complicated natural phenomena.

Generalized Riccati equation rational expansion method and its application

2007 ◽  
Vol 189 (1) ◽  
pp. 490-499 ◽  
Author(s):  
Ying Zheng ◽  
Yuanyuan Zhang ◽  
Hongqing Zhang
Keyword(s):  
Riccati Equation ◽  
Expansion Method ◽  
Generalized Riccati Equation

scholarly journals Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation

AIP Advances ◽  
2013 ◽  
Vol 3 (5) ◽  
pp. 052104 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah ◽  
Syed Tauseef Mohyud-Din
Keyword(s):  
Riccati Equation ◽  
Mapping Method ◽  
Generalized Riccati Equation ◽  
Fifth Order

GENERALIZED RICCATI EQUATION EXPANSION METHOD AND ITS APPLICATION TO THE (2+1)-DIMENSIONAL BOUSSINESQ EQUATION

2003 ◽  
Vol 14 (04) ◽  
pp. 471-482 ◽  
Author(s):  
YONG CHEN ◽  
BIAO LI ◽  
HONGQING ZHANG
Keyword(s):  
Riccati Equation ◽  
Function Method ◽  
Boussinesq Equation ◽  
Nonlinear Differential Equations ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Generalized Riccati Equation ◽  
Hyperbolic Function Method ◽  
Formal Solutions ◽  
Singular Soliton

Based on the computerized symbolic system Maple and a Riccati equation, a new Riccati equation expansion method for constructing nontraveling wave and coefficient functions' soliton-like solutions is presented by a new general ansätz. The proposed method is more powerful than most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method, and generalized hyperbolic-function method. By using the method, we not only successfully recovered the previously known formal solutions but could also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the (2+1)-dimensional Boussinesq equation and obtain rich new families of the exact solutions, including the nontraveling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, and triangular functions solutions.

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