Fractional Scale Calculus: Hadamard vs. Liouville

Manuel D. Ortigueira; Gary W. Bohannan

doi:10.3390/fractalfract7040296

Fractional Scale Calculus: Hadamard vs. Liouville

Manuel D. Ortigueira, Gary W. Bohannan

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Abstract

A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives is established and reformulated. A new derivative similar to the Grünwald–Letnikov’s is deduced. Tempered versions are also introduced. Scale-invariant systems are described and exemplified. For solving the corresponding differential equations, a new logarithmic Mittag-Leffler series is proposed.

Original language	English
Article number	296
Number of pages	19
Journal	Fractal and Fractional
Volume	7
Issue number	4
DOIs	https://doi.org/10.3390/fractalfract7040296
Publication status	Published - Apr 2023

Keywords

Hadamard derivative
Liouville derivative
logarithmic series
scale derivative
tempered derivative

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10.3390/fractalfract7040296Licence: CC BY

Fractional Scale Calculus Hadamard vs. LiouvilleFinal published version, 362 KBLicence: CC BY

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keywords = "Hadamard derivative, Liouville derivative, logarithmic series, scale derivative, tempered derivative",

author = "Ortigueira, {Manuel D.} and Bohannan, {Gary W.}",

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N2 - A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives is established and reformulated. A new derivative similar to the Grünwald–Letnikov’s is deduced. Tempered versions are also introduced. Scale-invariant systems are described and exemplified. For solving the corresponding differential equations, a new logarithmic Mittag-Leffler series is proposed.

AB - A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives is established and reformulated. A new derivative similar to the Grünwald–Letnikov’s is deduced. Tempered versions are also introduced. Scale-invariant systems are described and exemplified. For solving the corresponding differential equations, a new logarithmic Mittag-Leffler series is proposed.

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KW - Liouville derivative

KW - logarithmic series

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KW - tempered derivative

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Fractional Scale Calculus: Hadamard vs. Liouville

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Keywords

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