Putting an End to the Physical Initial Conditions of the Caputo Derivative: The Infinite State Solution

In this paper, a counter-example based on a realistic initial condition invalidates the usual approach related to the so-called physical initial condition of the Caputo derivative used to solve fractional-order Cauchy problems. Due to Infinite State representation, we prove that the initial condition of the Caputo derivative has to take into account the distributed states of an associated fractional integrator. Then, we prove that the free response of the counter-example requires the knowledge of the associated fractional integrator free response, and a realistic solution is proposed for the convolution problem based on the Mittag–Leffler function. Moreover, a simple and efficient technique based on Infinite State representation is proposed to solve the previous free response problem. Finally, numerical simulations demonstrate that the usual Caputo technique is based on an unrealistic initial condition without any physical meaning.