On calculus of local fractional derivatives

Babakhani, Abolfazl; Daftardar-Gejji, Varsha

doi:10.1016/s0022-247x(02)00048-3

Cited by 131 publications

(97 citation statements)

References 2 publications

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“…It should be noted that statement [26,27] that the Leibniz rule in the unviolated form (17) holds for non-differentiable functions is incorrect [8]. Let us give some explanations below.…”

Section: Proofmentioning

confidence: 99%

“…(2) It is easy to see that nowhere in the proofs proposed in [26,27], the requirement that the functions f (x) and g(x) are not classically differentiable is not used. Therefore, using the same proofs, we can get the statement that the Leibniz rule (17) holds for fractional-differentiable functions f (x), g(x) without the useless assumption that these functions are not classically differentiable.…”

Section: Proofmentioning

confidence: 99%

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

Tarasov

2015

Int. J. Appl. Comput. Math

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In this paper we prove that local fractional derivatives of differentiable functions are integer-order derivative or zero operator. We demonstrate that the local fractional derivatives are limits of the left-sided Caputo fractional derivatives. The Caputo derivative of fractional order α of function f (x) is defined as a fractional integration of order n − α of the derivative f (n) (x) of integer order n. The requirement of the existence of integer-order derivatives allows us to conclude that the local fractional derivative cannot be considered as the best method to describe nowhere differentiable functions and fractal objects. We also prove that unviolated Leibniz rule cannot hold for derivatives of orders α = 1.MSC: 26A33 Fractional derivatives and integrals PACS: 45.10.Hjj Perturbation and fractional calculus methods

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“…It should be noted that statement [26,27] that the Leibniz rule in the unviolated form (17) holds for non-differentiable functions is incorrect [8]. Let us give some explanations below.…”

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

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

Tarasov

2015

Int. J. Appl. Comput. Math

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“…For further works, see [4], [8], [27], [33], [36], [38], [41] and [42]. This section is dedicated to constructing a time scale analogue of the expression obtained in [21].…”

Section: Taylor Theoremsmentioning

confidence: 99%

Fractional calculus on time scales with Taylor’s theorem

Williams

2012

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We present a definition of the Riemann-Liouville fractional calculus for arbitrary time scales through the use of time scales power functions, unifying a number of theories including continuum, discrete and fractional calculus. Basic properties of the theory are introduced including integrability conditions and index laws. Special emphasis is given to extending Taylor's theorem to incorporate our theory.MSC 2010 : Primary 26A33, Secondary 39A12

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“…Recently, the local fractional calculus was successfully applied to non-differentiable problems arising in the areas of solid mechanics [33], heat transfer and wave propagation [34], diffusion [35], hydrodynamics [36], vehicular traffic flow [37] and other topics [38][39][40][41][42] (see also references therein).…”

Section: Introductionmentioning

confidence: 99%