Mikusinski's operational calculus for general conjugated fractional derivatives

Abstract

Algebraic conjugations (transmutations) can be useful in constructing modified versions of fractional calculus. In a general model of fractional calculus given by conjugating the operators of standard fractional calculus by an arbitrary linear bijection, it is possible to construct a version of Mikusinski's operational calculus. This process involves combining different algebraic ideas, from the conjugation itself to the ring and field of fractions that are part of Mikusinski's theory. As applications of the general theory thus obtained, we indicate how it can be used to solve fractional differential equations in various settings, including left-sided and right-sided fractional calculus on arbitrary intervals and fractional calculus with respect to functions.