Analysis, Modeling and Stability of Fractional Order Differential Systems 1

10Infinite State and Fractional Differentiation of Functions

10.1. Introduction

Fractional calculus has concerned the calculation of fractional integrals and derivatives for any kind of functions for a long time (see [OLD 72, MIL 93] and the references therein). Of course, integration of FDEs has also been an important issue, stimulated by physical and engineering applications [SAM 87, POD 99].

In the previous chapters, the infinite state approach was applied to the modeling and analysis of FDEs. In fact, this methodology also applies to other domains of fractional calculus, and particularly to the calculation of the fractional derivative of a function g(t), i.e. to . This derivative depends on the differentiation technique used: Riemann–Liouville, Caputo or Grünwald–Letnikov.

Let us consider the calculation of with the Caputo derivative. In the first step, we demonstrate that the calculation of the Caputo derivative on an infinite interval makes it possible to express the fractional derivative of the function g(t). In the second step, we analyze the calculation of the Caputo derivative on a truncated interval and the initial condition problem arising at t = t₀. Thus, we demonstrate ...