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.

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 Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Ai-Min Yang ◽  
Xiao-Jun Yang ◽  
Zheng-Biao Li
Keyword(s):  
Differential Equations ◽  
Series Expansion ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Expansion Method ◽  
Cantor Sets ◽  
Diffusion Equations ◽  
And Diffusion ◽  
Series Expansion Method

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

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