Fractional derivatives: Fourier, elephants, memory effects, viscoelastic materials and anomalous diffusions

TL;DR

This paper reviews the history, theory, and recent advances in fractional derivatives, highlighting their applications in population dynamics, viscoelasticity, and anomalous diffusion phenomena.

Contribution

It provides a comprehensive overview of fractional calculus, introduces modern methods for defining fractional derivatives, and discusses recent theoretical developments and practical applications.

Findings

Recent advances in fractional derivative theory

Applications to population growth with memory

Modeling viscoelastic materials and anomalous diffusion

Abstract

This paper, that will appear in the Notices of the AMS, begins with a brief historical account of the beginnings of fractional calculus and the crucial roles played by Leibniz and Fourier. Fourier's definition of fractional derivative is introduced and unpacked by using the modern technique known as the method of semigroups. Recent advances on the theory of fractional derivatives are presented. Furthermore, we address some questions that have been raised by some in the scientific community. Finally, we present three different applications: population growth with memory, viscoleastic materials and anomalous diffusions.