A Plea for the Integration of Fractional Differential Systems: The Initial Value Problem
Language
EN
Article de revue
This item was published in
Fractal and Fractional. 2022-09-28, vol. 6, n° 10, p. 550
English Abstract
The usual approach to the integration of fractional order initial value problems is based on the Caputo derivative, whose initial conditions are used to formulate the classical integral equation. Thanks to an elementary ...Read more >
The usual approach to the integration of fractional order initial value problems is based on the Caputo derivative, whose initial conditions are used to formulate the classical integral equation. Thanks to an elementary counter example, we demonstrate that this technique leads to wrong free-response transients. The solution of this fundamental problem is to use the frequency-distributed model of the fractional integrator and its distributed initial conditions. Using this model, we solve the previous counter example and propose a methodology which is the generalization of the integer order approach. Finally, this technique is applied to the modeling of Fractional Differential Systems (FDS) and the formulation of their transients in the linear case. Two expressions are derived, one using the Mittag–Leffler function and a new one based on the definition of a distributed exponential function.Read less <
Keywords
Initial value problem
Fractional differential systems
Fractional integral equation
Infinite state approach
Riemann–Liouville integral
Frequency distributed exponential function