Mikusiński Algebraic Calculus and Multivariate Fractional Operators for Representing Solutions of Linear Fractional Systems and Functionals

ABSTRACT The aim of this paper is to advance in the field of fractional calculus, in the topics of Mikusiński algebraic calculus and multivariate operators. On the one hand, we show how the Mikusiński method can be used for finding closed‐form solutions to multi‐order systems of linear fractional equations. We focus on the three‐dimensional situation, where multivariate Mittag–Leffler functions of large size emerge and symbolic computation is utilized. Each addend of the system has a distinct fractional order, in terms of Riemann–Liouville operators, for which incommensurate Caputo equations are a particular case. As a case study, we adapt the operational method to distinct functional memory orders of a benchmark model in nuclear physics, the Bateman equations. We employ the recently developed fractional calculus with respect to functions and multivariate Mittag–Leffler series. In general, continuous responses and forcing terms can be assumed, highlighting the improvement over the Laplace transform. On the other hand, we develop the role of fractional calculus in the representation of continuous linear functionals in terms of multidimensional integro‐differential operators, to extend the classical Peano–Sard or Peano kernel theorem.