In Chapter 1, we defined fractional integration as the key tool for modeling and simulation of FDEs. Unfortunately, the definition of Riemann–Liouville integration
does not provide a suitable technique for the numerical simulation of fractional integration, which is not a classical integral, but in fact a convolution integral.
Therefore, in the first step, we relate this operation to a frequency approach based on the Laplace transform of hn(t), thus:
In the second step (Chapter 6), we will relate fractional integration to a time approach, called the infinite state approach, moreover demonstrating that these two approaches are equivalent and complementary.
The synthesis of the fractional integrator thanks to a frequency methodology is in fact based on Oustaloup’s technique [OUS 00] for the synthesis of the fractional differentiator Dn(s) = sn.
Therefore, in this chapter, we present a frequency approximation of In(s) which will be used to derive a modal formulation, basis of the numerical integration algorithm and of the fractional integrator state variables.