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Exact results for a fractional derivative of elementary functions

by Gavriil Shchedrin, Nathanael C. Smith, Anastasia Gladkina, Lincoln D. Carr

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Submission summary

As Contributors: Gavriil Shchedrin
Arxiv Link: http://arxiv.org/abs/1711.07126v4 (pdf)
Date accepted: 2018-05-10
Date submitted: 2018-04-03 02:00
Submitted by: Shchedrin, Gavriil
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical

Abstract

We present exact analytical results for the Caputo fractional derivative of a wide class of elementary functions, including trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian, and Lorentzian functions. These results are especially important for multi-scale physical systems, such as porous materials, disordered media, and turbulent fluids, in which transport is described by fractional partial differential equations. The exact results for the Caputo fractional derivative are obtained from a single generalized Euler's integral transform of the generalized hyper-geometric function with a power-law argument. We present a proof of the generalized Euler's integral transform and directly apply it to the exact evaluation of the Caputo fractional derivative of a broad spectrum of functions, provided that these functions can be expressed in terms of a generalized hyper-geometric function with a power-law argument. We determine that the Caputo fractional derivative of elementary functions is given by the generalized hyper-geometric function. Moreover, we show that in the most general case the final result cannot be reduced to elementary functions, in contrast to both the Liouville-Caputo and Fourier fractional derivatives. However, we establish that in the infinite limit of the argument of elementary functions, all three definitions of a fractional derivative - the Caputo, Liouville-Caputo, and Fourier- converge to the same result given by the elementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourier fractional derivatives.

Ontology / Topics

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Caputo fractional derivative Disordered media Porous materials Transport Turbulent fluids

Published as SciPost Phys. 4, 029 (2018)



Submission & Refereeing History

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Resubmission 1711.07126v4 on 3 April 2018

Reports on this Submission

Anonymous Report 2 on 2018-4-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1711.07126v4, delivered 2018-04-06, doi: 10.21468/SciPost.Report.410

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