Fractional Derivatives: Fourier, Elephants, Memory Effects, Viscoelastic Materials, and Anomalous Diffusions

DOI10.1090/NOTI2663zbMATH Open1522.35564arXiv2212.02279WikidataQ125473410 ScholiaQ125473410MaRDI QIDQ6073076

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Publication date: 16 October 2023

Published in: Notices of the American Mathematical Society (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/2212.02279

Mathematics Subject Classification ID

Viscoelastic fluids (76A10) Fractional partial differential equations (35R11) Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials) (74Dxx)