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 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 Exact Solutions of the Space-Time Fractional Bidirectional Wave Equations Using the(G′/G)-Expansion Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Wei Li ◽  
Huizhang Yang ◽  
Bin He
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Wave Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Space Time ◽  
Promising Tool ◽  
Liouville Derivative ◽  
Fractional Differential

Based on Jumarie’s modified Riemann-Liouville derivative, the fractional complex transformation is used to transform fractional differential equations to ordinary differential equations. Exact solutions including the hyperbolic functions, the trigonometric functions, and the rational functions for the space-time fractional bidirectional wave equations are obtained using the(G′/G)-expansion method. The method provides a promising tool for solving nonlinear fractional differential equations.

scholarly journals An analytical approach to obtain exact solutions of some space-time conformable fractional differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Hossein Jafari
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytical Approach ◽  
Expansion Method ◽  
Space Time ◽  
Wave Transformation ◽  
Type Solution ◽  
Polynomial Type ◽  
Fractional Differential ◽  
Duffing Model

Abstract In this paper, the sine-Gordon expansion method is used to obtain analytical solutions of the conformable space-time generalized reaction Duffing model and conformable space-time Eckhaus equation with the aid of symbolic computation. These equations can be reduced into ordinary differential equations (ODEs) using a suitable wave transformation with a predicted polynomial-type solution.

Auxiliary Equation Method for Fractional Differential Equations with Modified Riemann–Liouville Derivative

2016 ◽  
Vol 17 (7-8) ◽  
pp. 413-420 ◽  
Author(s):  
Arzu Akbulut ◽  
Melike Kaplan ◽  
Ahmet Bekir
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Auxiliary Equation Method ◽  
Liouville Derivative ◽  
Regularized Long Wave Equation ◽  
Klein Gordon ◽  
Fractional Differential

Abstract:In this work, the auxiliary equation method is applied to derive exact solutions of nonlinear fractional Klein–Gordon equation and space-time fractional Symmetric Regularized Long Wave equation. Consequently, some exact solutions of these equations are successfully obtained. These solutions are formed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions founded by the suggested method indicate that the approach is easy to implement and powerful.

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