Integration of Fractional Differential Equations without Fractional Derivatives

The usual approach to the integration of fractional order differential equations is based on fractional derivatives, mainly on the Caputo derivative and its renowned initial conditions. In this paper, thanks to an elementary counter example, we demonstrate that this usual methodology 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. Based on this model, we solve the previous counter example and propose a methodology which is the generalization of the integer order approach.