Fractional line integral

Gabriel Bengochea; Manuel Ortigueira

doi:10.3390/math9101150

Fractional line integral

Gabriel Bengochea, Manuel Ortigueira

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Abstract

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

Original language	English
Article number	1150
Number of pages	11
Journal	Mathematics
Volume	9
Issue number	10
DOIs	https://doi.org/10.3390/math9101150
Publication status	Published - 2 May 2021

Keywords

Fractional integral
Fractional line integral
Grünwald–Letnikov fractional derivative
Liouville fractional derivative

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Fractional Line IntegralFinal published version, 281 KBLicence: CC BY

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title = "Fractional line integral",

abstract = "This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Gr{\"u}nwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.",

keywords = "Fractional integral, Fractional line integral, Gr{\"u}nwald–Letnikov fractional derivative, Liouville fractional derivative",

author = "Gabriel Bengochea and Manuel Ortigueira",

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doi = "10.3390/math9101150",

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AU - Ortigueira, Manuel

N1 - Funding Information: info:eu-repo/grantAgreement/FCT/6817 - DCRRNI ID/UIDB%2F00066%2F2020/PT# Funding: The first author was supported by the Autonomous University of Mexico City (UACM) under the project PI-CCyT-2019-15. The work of the second author was partially funded by Portuguese National Funds through the FCT-Foundation for Science and Technology within the scope of the CTS Research Unit-Center of Technology and Systems/UNINOVA/FCT/NOVA, under the reference UIDB/00066/2020. Publisher Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2021/5/2

Y1 - 2021/5/2

N2 - This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

AB - This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

KW - Fractional integral

KW - Fractional line integral

KW - Grünwald–Letnikov fractional derivative

KW - Liouville fractional derivative

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Fractional line integral

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