scholarly journals Solving fractional differential equations by the ultraspherical integration method

2021 ◽  
Vol 40 ◽  
pp. 1-14
Author(s):  
Ali Khani ◽  
S. Panahi

In this paper, we present a numerical method to solve a linear fractional differential equations. This new investigation is based on ultraspherical integration matrix to approximate the highest order derivatives to the lower order derivatives. By this approximation the problem is reduced to a constrained optimization problem which can be solved by using the penalty quadratic interpolation method. Numerical examples are included to confirm the efficiency and accuracy of the proposed method.

Author(s):  
Tadeusz Jankowski

AbstractUsing the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives $D^{q}_{T}x$ and $D^{q}_{T}y$. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.


Author(s):  
Carl F. Lorenzo ◽  
Rachid Malti ◽  
Tom T. Hartley

A new method for the solution of linear constant coefficient fractional differential equations of any commensurate order based on the Laplace transforms of the fractional meta-trigonometric functions and the R-function is presented. The new method simplifies the solution of such equations. A simplifying characterization that reduces the number of parameters in the fractional meta-trigonometric functions is introduced.


Author(s):  
Nguyen Cong ◽  
Doan Son ◽  
Hoang Tuan

AbstractOur aim in this paper is to investigate the asymptotic behavior of solutions of linear fractional differential equations. First, we show that the classical Lyapunov exponent of an arbitrary nontrivial solution of a bounded linear fractional differential equation is always nonnegative. Next, using the Mittag-Leffler function, we introduce an adequate notion of fractional Lyapunov exponent for an arbitrary function. We show that for a linear fractional differential equation, the fractional Lyapunov spectrum which consists of all possible fractional Lyapunov exponents of its solutions provides a good description of asymptotic behavior of this equation. Consequently, the stability of a linear fractional differential equation can be characterized by its fractional Lyapunov spectrum. Finally, to illustrate the theoretical results we compute explicitly the fractional Lyapunov exponent of an arbitrary solution of a planar time-invariant linear fractional differential equation.


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