On a Generic Fractional Derivative Associated with the Riemann-Liouville Fractional Integral

Yuri Luchko

doi:10.20944/preprints202407.2116.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 25 July 2024 / Approved: 25 July 2024 / Online: 26 July 2024 (10:28:12 CEST)

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann-Liouville fractional integral. Then the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied for derivation of its properties. In particular, we characterize its domain, null-space, and projector operator, establish the interrelations between its different realisations, and present a generalized fractional Taylor formula involving the generic fractional derivative. Then we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed form formula for its unique solution, and study its complete monotonicity.

Riemann-Liouville fractional integral; left-inverse operator; projector operator; generalized fractional Taylor formula; fractional differential equations; complete monotonicity

Computer Science and Mathematics, Analysis

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On a Generic Fractional Derivative Associated with the Riemann-Liouville Fractional Integral

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