**7. Conclusions**

In this paper, it has been proved, thanks to an elementary counter example, that the solution of the FDE initial value problem cannot be provided by the well-known Caputo derivative approach. The origin of this error relies on ignoring the dynamics of the fractional integrator, which has caused confusion between pseudo initial conditions of the Caputo derivative and the initialization function of the Riemann–Liouville integral.

Furthermore, it has been demonstrated that it is necessary to take into account two complementary models of the fractional integrator: the classical model used to express the pseudo state variable *x*(*t*) is an input/output model, whereas the distributed model is an infinite-dimension state variable model, which is adapted to the formulation of free responses thanks to the initial values of its distributed state variable *z*(*ω*, *t*), as proved by the counter-example.

These two models of the fractional integrator generate two complementary approaches to the modeling of FDE and FDS one being focused on the pseudo-state variable, whereas the other permits to express internal transients linked to the distributed state variable. Two expressions of their free responses have been formulated in the linear case, one with the help of a convolution of the Mittag–Leffler function with the initialization function, and the other thanks to the definition of a distributed exponential function, which is a straightforward generalization of the integer-order case.

Beyond the misuse of fractional derivatives to solve the initial-value problem, the main conclusion of this paper is that any fractional-order system or one characterized by a long memory phenomenon is an infinite dimension system, whatever its input/output representation, linear or nonlinear.

Consequently, the use of an internal distributed representation is not an option but is necessary to correctly express initialization problems and dynamical transients.

**Author Contributions:** N.M. and J.-C.T. contributed equally to the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
