Closed-Form Discretization of Fractional-Order Differential and Integral Operators

2019 ◽  
pp. 1-17
Author(s):  
Reyad El-Khazali ◽  
J. A. Tenreiro Machado
Keyword(s):  
Closed Form ◽  
Fractional Order ◽  
Integral Operators

scholarly journals Exact Solution of Impulse Response to a Class of Fractional Oscillators and Its Stability

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Ming Li ◽  
S. C. Lim ◽  
Shengyong Chen
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Impulse Response ◽  
Fractional Order ◽  
System Analysis ◽  
Stability Behavior ◽  
Single Degree ◽  
Fractional Oscillator ◽  
The Stability ◽  
Typical Model

Oscillator of single-degree-freedom is a typical model in system analysis. Oscillations resulted from differential equations with fractional order attract the interests of researchers since such a type of oscillations may appear dramatic behaviors in system responses. However, a solution to the impulse response of a class of fractional oscillators studied in this paper remains unknown in the field. In this paper, we propose the solution in the closed form to the impulse response of the class of fractional oscillators. Based on it, we reveal the stability behavior of this class of fractional oscillators as follows. A fractional oscillator in this class may be strictly stable, nonstable, or marginally stable, depending on the ranges of its fractional order.

Analytic Solution of Fractional-Order Heat- and Wave-Like Equations Using Generalized n-dimensional Differential Transform Method

2011 ◽  
Vol 66 (10-11) ◽  
pp. 581-590 ◽  
Author(s):  
Vipul K. Baranwal ◽  
Ram K. Pandey ◽  
Manoj P. Tripathi ◽  
Om P. Singh
Keyword(s):  
Closed Form ◽  
Fractional Order ◽  
Analytic Solution ◽  
Accurate Solution ◽  
Differential Transform Method ◽  
Transform Method ◽  
Differential Transform ◽  
User Friendly

In this paper, we have introduced a generalized n-dimensional differential transform method to propose a user friendly algorithm to obtain the closed form analytic solution for n-dimensional fractional heat- and wave-like equations. Three examples are given to establish the simplicity of the algorithm. In Example 5.3, we show that ten terms of the series representing the solution, even in fractional order, give a very accurate solution.

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