Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions

Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions

Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit inf...

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Bibliographic Details
Main Author: Jocelyn Sabatier
Format: Article
Language:English
Published: MDPI AG 2021-06-01
Series:Symmetry
Subjects:
fractional models
infinite memory
initial conditions
Online Access:https://www.mdpi.com/2073-8994/13/6/1099
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spelling doaj-e43ebf97bd9044feba94908d6feeab352021-07-01T00:46:11ZengMDPI AGSymmetry2073-89942021-06-01131099109910.3390/sym13061099Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some SolutionsJocelyn Sabatier0IMS Laboratory, Bordeaux University, UMR 5218 CNRS, 351 Cours de la Libération, 33405 Talence, FranceUsing a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed.https://www.mdpi.com/2073-8994/13/6/1099fractional modelsinfinite memoryinitial conditions
collection DOAJ
language English
format Article
sources DOAJ
author Jocelyn Sabatier
spellingShingle Jocelyn Sabatier
Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions
Symmetry
fractional models
infinite memory
initial conditions
author_facet Jocelyn Sabatier
author_sort Jocelyn Sabatier
title Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions
title_short Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions
title_full Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions
title_fullStr Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions
title_full_unstemmed Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions
title_sort fractional order models are doubly infinite dimensional models and thus of infinite memory: consequences on initialization and some solutions
publisher MDPI AG
series Symmetry
issn 2073-8994
publishDate 2021-06-01
description Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed.
topic fractional models
infinite memory
initial conditions
url https://www.mdpi.com/2073-8994/13/6/1099
work_keys_str_mv AT jocelynsabatier fractionalordermodelsaredoublyinfinitedimensionalmodelsandthusofinfinitememoryconsequencesoninitializationandsomesolutions
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