A Factory of Fractional Derivatives

Author:

Ortigueira Manuel D.1ORCID

Affiliation:

1. CTS-UNINOVA and LASI, NOVA School of Science and Technology, NOVA University of Lisbon, 2829-516 Caparica, Portugal

Abstract

This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn.

Funder

Fundação para a Ciência e Tecnologia

Publisher

MDPI AG

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