Fractional calculus via Laplace transform and its application in relaxation processes

The fractional calculus has been receiving considerable interest in recent decades, mainly due to its several interesting applications. In this paper we provide a very intuitive approach to the fractional calculus based on the Laplace transform and ideas from the theory of distributions. Our approach reveals the deep relationship between the Riemann-Liouville and the Caputo definitions of fractional derivative, and opens the way for the formulation of other definitions, which we explore accordingly. As an example of its different many applications, we use it to formulate some generalizations of a relaxation function model, and discuss some limitations that these models impose on possible definitions of fractional derivatives, with focus on two recently proposed definitions of fractional derivatives.