An operational calculus approach to Hilfer–Prabhakar fractional derivatives

We investigate the general Hilfer–Prabhakar fractional derivative operator using the algebraic machinery of Mikusiński’s operational calculus. We also provide some important characterisations in terms of initial conditions for the function spaces used in the fractional Mikusiński method. Armed with an algebraic interpretation of the Hilfer–Prabhakar derivative, we solve a variety of fractional differential equations involving this operator, including multi-order and multi-term ones.