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Probabilistic Interpretations of Fractional Operators and Fractional Behaviours: Extensions, Applications and Tribute to Prof. José Tenreiro Machado’s Ideas

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  • Jocelyn Sabatier

    (IMS Laboratory, Bordeaux University, UMR 5218 CNRS, 351 Cours de la Libération, 33405 Talence, France)

Abstract

This paper extends and illustrates a probabilistic interpretation of the fractional derivative operator proposed by Pr. José Tenreiro Machado. While his interpretation concerned the probability of finding samples of the derivate signal in the expression of the fractional derivative, the present paper proposes interpretations for other fractional models and more generally fractional behaviours (without using a model). It also proposes probabilistic interpretations in terms of time constants and time delay distributions. It shows that these probabilistic interpretations in terms of time delay distributions can be connected to the physical behaviour of real systems governed by adsorption or diffusion phenomena.

Suggested Citation

  • Jocelyn Sabatier, 2022. "Probabilistic Interpretations of Fractional Operators and Fractional Behaviours: Extensions, Applications and Tribute to Prof. José Tenreiro Machado’s Ideas," Mathematics, MDPI, vol. 10(22), pages 1-17, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4184-:d:967325
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    References listed on IDEAS

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    1. Jocelyn Sabatier & Christophe Farges & Vincent Tartaglione, 2020. "Some Alternative Solutions to Fractional Models for Modelling Power Law Type Long Memory Behaviours," Mathematics, MDPI, vol. 8(2), pages 1-16, February.
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