Mikusinski's operational calculus for Prabhakar fractional calculus

Abstract

The operational calculus method of Mikusinski is a powerful yet underappreciated theory, enabling differential equations to be solved using abstract algebra. We interpret the fractional integrals and derivatives of Prabhakar type within Mikusinski's theory, by defining them on appropriate function spaces and using judiciously their fundamental properties such as semigroup and series formulae. This theoretical framework enables explicit solutions to be found, for the first time, for a general class of linear Prabhakar fractional differential equations, with the solutions expressed as double or multiple infinite series.