The alternative system initial conditions versus the derivative initial conditions is focused in this paper. It is shown that Riemann-Liouville and Caputo initial conditions result from the corresponding derivative and not necessarily from the system at hand. To setup the correct system initialization, a formulation generalizing the integer order approach is presented. This is based on a generalization to the fractional environment of the well known jump formula. The obtained scheme is very general and does not depend on any transform. Besides, it can also be used in the time variant case. The Riemann-Liouville and Caputo initial conditions are interpreted in terms of this general framework and deduced equations where they are correct.
Coito, F. J. A. V. D., Ortigueira, M. D., & DEE Group Author (2010). System initial conditions vs derivative initial conditions. Computers & Mathematics With Applications, 59(5), 1782-1789. https://doi.org/10.1016/j.camwa.2009.08.036