Time-fractional derivatives in relaxation processes: a tutorial survey

TL;DR

This tutorial survey reviews the theory of relaxation processes modeled by fractional differential equations, comparing Riemann-Liouville and Caputo derivatives, and discusses their implications in viscoelasticity.

Contribution

It provides a comprehensive overview of fractional derivatives in relaxation processes, highlighting differences between Riemann-Liouville and Caputo derivatives and their roles in viscoelastic models.

Findings

Contrast between Riemann-Liouville and Caputo derivatives in handling initial conditions

Historical context of fractional calculus in viscoelasticity

Clarification of fractional derivatives' role in relaxation processes

Abstract

The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in their ability to take into account initial conditions in the constitutive equations of fractional order. We also provide historical notes on the origins of the Caputo derivative and on the use of fractional calculus in viscoelasticity.