scholarly journals Exact Solutions for Some Fractional Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Abdullah Sonmezoglu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Elliptic Function ◽  
Expansion Method ◽  
Jacobi Elliptic Function ◽  
Solitary Wave Solutions ◽  
Liouville Derivative ◽  
Trigonometric Function Solutions ◽  
Fractional Differential ◽  
Jacobi Elliptic Function Expansion

The extended Jacobi elliptic function expansion method is used for solving fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative. By means of this approach, a few fractional differential equations are successfully solved. As a result, some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions are established. The proposed method can also be applied to other fractional differential equations.

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 Extended Jacobi Elliptic Function Expansion Method to the ZK-MEW Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Weimin Zhang
Keyword(s):  
Solitary Wave ◽  
Elliptic Function ◽  
Expansion Method ◽  
Periodic Wave ◽  
Jacobi Elliptic Function ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Function Expansion ◽  
Jacobi Elliptic Function Expansion

The extended Jacobi elliptic function expansion method is applied for Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation. With the aid of symbolic computation, we construct some new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions.

Fractional series expansion method for fractional differential equations

2015 ◽  
Vol 25 (7) ◽  
pp. 1525-1530 ◽  
Author(s):  
Zheng-Biao Li ◽  
Wei-Hong Zhu
Keyword(s):  
Differential Equations ◽  
Series Expansion ◽  
Fractional Differential Equations ◽  
Expansion Method ◽  
Kantorovich Method ◽  
Content Type ◽  
Analytical Technique ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Series Expansion Method

Purpose – The purpose of this paper is to suggest a new analytical technique called the fractional series expansion method for solving linear fractional differential equations (FDEs). Design/methodology/approach – This method is based on the idea of Kantorovich method, convergent series, and the modified Riemann-Liouville derivative. Findings – This work suggests a new analytical technique. The FDEs are described in Jumarie’s sense. Originality/value – It finds a new method for solving linear FDEs. The solution procedure is elucidated by two examples.

Analytical Solutions of Two Space-Time Fractional Nonlinear Models Using Jacobi Elliptic Function Expansion Method

2021 ◽  
pp. 173-188
Author(s):  
Zillur Rahman ◽  
M. Zulfikar Ali ◽  
Harun-Or-Roshid ◽  
Mohammad Safi Ullah
Keyword(s):  
Elliptic Function ◽  
Nonlinear Models ◽  
Applied Mathematics ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Space Time ◽  
Jacobi Elliptic Function ◽  
Function Expansion ◽  
Partial Models ◽  
Jacobi Elliptic Function Expansion

In this manuscript, the space-time fractional Equal-width (s-tfEW) and the space-time fractional Wazwaz-Benjamin-Bona-Mahony (s-tfWBBM) models have been investigated which are frequently arises in nonlinear optics, solid states, fluid mechanics and shallow water. Jacobi elliptic function expansion integral technique has been used to build more innovative exact solutions of the s-tfEW and s-tfWBBM nonlinear partial models. In this research, fractional beta-derivatives are applied to convert the partial models to ordinary models. Several types of solutions have been derived for the models and performed some new solitary wave phenomena. The derived solutions have been presented in the form of Jacobi elliptic functions initially. Persevering different conditions on a parameter, we have achieved hyperbolic and trigonometric functions solutions from the Jacobi elliptic function solutions. Besides the scientific derivation of the analytical findings, the results have been illustrated graphically for clear identification of the dynamical properties. It is noticeable that the integral scheme is simplest, conventional and convenient in handling many nonlinear models arising in applied mathematics and the applied physics to derive diverse structural precise solutions.

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