Variable-Order Fractional Calculus: from Old to New Approaches

Different approaches to introduce fractional derivatives and integrals of variable order have been proposed. The majority of these operators are obtained by replacing the constant order with some function in the time-domain formulation of the usual fractional-order operators. Here we present a completely different approach relying on the generalization of fractional integrals and derivatives in the Laplace transform domain, according to some pioneering ideas by Scarpi. By framing these ideas in the modern mathematical theory of General Fractional Derivatives and Integrals (based on the well-known Sonine condition) we are able to overcome the main issues of other variable-order operators. Numerical strategies for handling these operators are also discussed.