Variable-order fractional calculus: a change of perspective

Several approaches to the formulation of a fractional theory of calculus of "variable order" have appeared in the recent literature to describe physical systems showing special memory effects with features changing over time. Unfortunately, most of these proposals lack a rigorous mathematical framework. Here, we consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the Laplace domain representation of standard kernels functions involved in (constant-order) fractional calculus. We discuss how the variable-order Scarpi derivative can be framed within Kochubei's General Fractional Calculus, thus pin pointing it is possible to frame variable-order Scarpi's derivatives and integrals in the context of a more general and robust mathematical theory. Then, taking advantage of powerful numerical methods for the inversion of Laplace transforms, we discuss some practical applications of the variable-order Scarpi integral and derivative to describe some kinds of transitions in the features of the system memory.