Variable-order fractional calculus: a change of perspective

TL;DR

This paper introduces a new perspective on variable-order fractional calculus by reformulating Scarpi's ideas within the modern General Fractional Derivatives framework, supported by numerical methods and practical applications.

Contribution

It provides a rigorous mathematical foundation for variable-order fractional calculus based on Scarpi's approach and recent theory, enhancing understanding and applicability.

Findings

Established a connection between Scarpi's approach and General Fractional Derivatives.

Analyzed properties of the new variable-order operators.

Demonstrated practical applications using numerical Laplace inversion methods.

Abstract

Several approaches to the formulation of a fractional theory of calculus of "variable order" have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpi's ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative.