Initial value problems should not be associated to fractional model descriptions whatever the derivative definition used

Jocelyn SABATIER; Christophe FARGES; Jocelyn SABATIER; Christophe FARGES

doi:10.3934/math.2021657

AIMS Mathematics

2021, Volume 6, Issue 10: 11318-11329. doi: 10.3934/math.2021657

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Initial value problems should not be associated to fractional model descriptions whatever the derivative definition used

Jocelyn SABATIER ^,,
Christophe FARGES

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

Received: 29 May 2021 Accepted: 15 July 2021 Published: 05 August 2021
MSC : 26A33, 34A12

The paper shows that the Caputo definition of fractional differentiation is problematic if it is used in the definition of a time fractional model and if initial conditions are taken into account. The demonstration is done using simple examples (or counterexamples). The analysis is extended to the Riemann-Liouville and Grünwald-Letnikov definitions. These results thus question the validity of results produced in the field of time fractional model analysis in which initial conditions are involved.
- time fractional derivative,
- Caputo definition,
- Riemann-Liouville definition,
- Grünwald-Letnikov definition,
- initial conditions,
- memory
Citation: Jocelyn SABATIER, Christophe FARGES. Initial value problems should not be associated to fractional model descriptions whatever the derivative definition used[J]. AIMS Mathematics, 2021, 6(10): 11318-11329. doi: 10.3934/math.2021657

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Abstract

The paper shows that the Caputo definition of fractional differentiation is problematic if it is used in the definition of a time fractional model and if initial conditions are taken into account. The demonstration is done using simple examples (or counterexamples). The analysis is extended to the Riemann-Liouville and Grünwald-Letnikov definitions. These results thus question the validity of results produced in the field of time fractional model analysis in which initial conditions are involved.

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