Summary: Fractional order differentiation is generally considered as the basis of fractional calculus, but the real basis is in fact fractional order integration and particularly the fractional integrator, because definition and properties of fractional differentiation and of fractional differential systems rely essentially on fractional integration. We present the frequency distributed model of the fractional integrator and its finite dimension approximation. The simulation of FDSs, based on fractional integrators, leads to the definition of FDS internal state variables, which are the state variables of the fractional integrators, as a generalization of the integer order case.
The initial condition problem has been an open problem for a long time in fractional calculus. We demonstrate that the frequency distributed model of the fractional integrator provides a solution to this problem through the knowledge of its internal state. Beyond the solution of this fundamental problem, mastery of the integrator internal state allows the analysis and prediction of fractional differential system transients. Moreover, the finite dimension approximation of the fractional integrator provides an efficient technique for practical simulation of FDSs and analysis of their transients, with a particular insight into the interpretation of initial conditions, as illustrated by numerical simulations.
Laplace transform equations and initial conditions of the Caputo and the Riemann-Liouville derivatives are used to formulate the free responses of FDEs. Because usual equations are wrong, the corresponding free responses do not fit with real transients. We demonstrate that revised equations, including the initial state vector of the fractional integrator (used to perform differentiation) provide corrected free responses which match with real transients, as exhibited by numerical simulations.