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 Vestnik SamU. Estestvenno-Nauchnaya Ser., 2018, Issue 24, Issue 3, Pages 7–13 (Mi vsgu577)

Mathematics

On fractional differentiation

S. O. Gladkov, S. B. Bogdanova

Department of Applied Software and Mathematical Methods, Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, 125993, Russian Federation

Abstract: Due to the operation of fractional differentiation introduced with the help of Fourier integral, the results of calculating fractional derivatives for certain types of functions are given. Using the numerical method of integration, the values of fractional derivatives for arbitrary dimensionality $\varepsilon$, (where $\varepsilon$ is any number greater than zero) are calculated. It is proved that for integer values of $\varepsilon$ we obtain ordinary derivatives of the first, second and more high orders. As an example it was considered heat conduction equation of Fourier, where spatial derivation was realized with the use of fractional derivatives. Its solution is given by Fourier integral. Mmoreover, it was shown that integral went into the required results in special case of the whole $\varepsilon$ obtained in $n$-dimensional case, where $n = 1, 2…$, etc.

Keywords: fractional differentiation, Fourier integral, Riemann integral, heat conduction, fractal, fractional dimension, Fourier equation, measure.

DOI: https://doi.org/10.18287/2541-7525-2018-24-3-7-13

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Full text: http://www.journals.ssau.ru/.../6447 (published under the terms of the Creative Commons Attribution 4.0 International License)
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Document Type: Article
UDC: 517.9; 544.034

Citation: S. O. Gladkov, S. B. Bogdanova, “On fractional differentiation”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 24:3 (2018), 7–13

Citation in format AMSBIB
\Bibitem{GlaBog18} \by S.~O.~Gladkov, S.~B.~Bogdanova \paper On fractional differentiation \jour Vestnik SamU. Estestvenno-Nauchnaya Ser. \yr 2018 \vol 24 \issue 3 \pages 7--13 \mathnet{http://mi.mathnet.ru/vsgu577} \crossref{https://doi.org/10.18287/2541-7525-2018-24-3-7-13} \elib{http://elibrary.ru/item.asp?id=36731737}