Local Fractional Derivatives of Differentiable Functions are Integer-order Derivatives or Zero

Tarasov, Vasily E.

doi:10.1007/s40819-015-0054-6

Cited by 22 publications

(14 citation statements)

References 34 publications

(55 reference statements)

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“…The conditions for applicability of the K-G fractional derivative were not specified in the seminal paper, which leaves space for different interpretations and sometimes confusions. For example, recently Tarasov claimed that local fractional derivatives of fractional order vanish everywhere [36]. In contrast, the results presented here demonstrate that local fractional derivatives vanish only if they are continuous.…”

Section: Discussioncontrasting

confidence: 69%

Fractional Velocity as a Tool for the Study of Non-Linear Problems

Prodanov

2018

Fractal Fract

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Singular functions and, in general, Hölder functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocities as tools to characterize Hölder and singular functions, in particular. Fractional velocities are defined as limits of the difference quotients of a fractional power and they generalize the local notion of a derivative. On the other hand, their properties contrast some of the usual properties of derivatives. One of the most peculiar properties of these operators is that the set of their non trivial values is disconnected. This can be used for example to model instantaneous interactions, for example Langevin dynamics. Examples are given by the De Rham and Neidinger's singular functions, represented by limits of iterative function systems. Finally, the conditions for equivalence with the Kolwankar-Gangal local fractional derivative are investigated.

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Section: Discussioncontrasting

confidence: 69%

Fractional Velocity as a Tool for the Study of Non-Linear Problems

Prodanov

2018

Fractal Fract

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“…250ff), nonconformable local-type fractional derivative [6]. However, after Tarasov's argument for the principle of nonlocality, attempts to found a fractional calculus on local operator now seriously require a theoretical rigor to avoid becoming objectionable [23,24]. In particular, Abdelhakim [1] has shown that conformable fractional derivative is in fact integer-order derivative in fractional disguise, thereby advising the fractional calculus researchers against its use.…”

Section: The Case Of Caputomentioning

confidence: 99%

Principal parts of a vector bundle on projective line and the fractional derivative

Husain¹,

Sultana²

2019

Turk J Math

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This work is an exposition on computational aspects of principal parts of a vector bundle on projective line over the field of characteristic zero. Principal parts help determine the possibility of algebraically formalizing infinitesimal-neighborhoods of subschemes inside some ambient scheme. The purpose of this study is to look for the possibility of formalizing the algebraic geometric interpretation of fractional derivative. For the latter, this study follows the approach proposed by Vasily Tarasov. The difference is that Tarasov proposed a geometric interpretation using finiteorder jet bundles from differential geometry. Present study proposes finite-order principal parts of the structure-sheaf of real projective line as its formal algebraic geometric parallel.

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“…cannot hold for fractional derivatives of order α = 1 for sets of differentiable and non-differentiable functions [19,20,21,22]. Unusual properties of fractional derivatives of non-integer orders that are represented by deformations a of the usual Leibniz rule, and the usual chain rule can be considered as the characteristic properties of fractional-order derivatives.…”

Section: Accepted Manuscriptmentioning

confidence: 99%