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.

scholarly journals Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Emad A.-B. Abdel-Salam ◽  
Eltayeb A. Yousif
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Expansion Method ◽  
Space Time ◽  
Regularized Long Wave Equation ◽  
Klein Gordon ◽  
Fractional Differential ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena ◽  
First Time

The fractional Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, space-time fractional Korteweg-de Vries equation, regularized long-wave equation, Boussinesq equation, and Klein-Gordon equation are considered. As a result, abundant types of exact analytical solutions are obtained. These solutions include generalized trigonometric and hyperbolic functions solutions which may be useful for further understanding of the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The periodic and kink solutions are founded as special case.

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 Solution of Moving Boundary Space-Time Fractional Burger’s Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
E. A.-B. Abdel-Salam ◽  
E. A. Yousif ◽  
Y. A. S. Arko ◽  
E. A. E. Gumma
Keyword(s):  
Moving Boundary ◽  
Expansion Method ◽  
Variable Coefficients ◽  
Space Time ◽  
Hyperbolic Function ◽  
Burger's Equation ◽  
Burger’S Equation ◽  
Hyperbolic Function Solutions ◽  
First Time ◽  
Boundary Space

The fractional Riccati expansion method is used to solve fractional differential equations with variable coefficients. To illustrate the effectiveness of the method, the moving boundary space-time fractional Burger’s equation is studied. The obtained solutions include generalized trigonometric and hyperbolic function solutions. Among these solutions, some are found for the first time. The linear and periodic moving boundaries for the kink solution of the Burger’s equation are presented graphically and discussed.

scholarly journals Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Emad A.-B. Abdel-Salam ◽  
Zeid I. A. Al-Muhiameed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytic Solutions ◽  
Mkdv Equation ◽  
Mapping Method ◽  
Space Time ◽  
Fractional Differential ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena ◽  
First Time

The fractional mapping method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional combined KdV-mKdV equation. Many types of exact analytical solutions are obtained. The solutions include generalized trigonometric and hyperbolic functions solutions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time.

scholarly journals On the Generalized Simplest Equations: Toward the Solution of Nonlinear Differential Equations with Variable Coefficients

2021 ◽  
Author(s):  
Gunawan Nugroho ◽  
Purwadi Agus Darwito ◽  
Ruri Agung Wahyuono ◽  
Murry Raditya
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riccati Equation ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Variable Coefficients ◽  
Higher Order ◽  
First Order ◽  
Generalized Riccati Equation ◽  
Polynomial Differential Equation

The simplest equations with variable coefficients are considered in this research. The purpose of this study is to extend the procedure for solving the nonlinear differential equation with variable coefficients. In this case, the generalized Riccati equation is solved and becomes a basis to tackle the nonlinear differential equations with variable coefficients. The method shows that Jacobi and Weierstrass equations can be rearranged to become Riccati equation. It is also important to highlight that the solving procedure also involves the reduction of higher order polynomials with examples of Korteweg de Vries and elliptic-like equations. The generalization of the method is also explained for the case of first order polynomial differential equation.

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