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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 ε, (where ε is any number greater than zero) are calculated. It is proved that for integer values of ε 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. Moreover, it was shown that integral went into the required results in special case of the whole ε obtained in n-dimensional case, where n=1,2..., etc.

About the authors

S. O. Gladkov

Moscow Aviation Institute (National Research University)

Author for correspondence.
Email: morenov@ssau.ru

S. B. Bogdanova

Moscow Aviation Institute (National Research University)

Email: morenov@ssau.ru


  1. Gladkov S.O. K teorii gidrodinamicheskikh iavlenii v kvaziodnomernykh sistemakh . ZhTF , 2001, Vol. 71, no 11, pp. 130–132. Available at: https://journals.ioffe.ru/articles/38956 .
  2. Gladkov S.O. K teorii odnomernoi i kvaziodnomernoi teploprovodnosti . ZhTF , 1997, Vol. 67, no 7, pp. 8–12. Available at: https://journals.ioffe.ru/articles/33167 .
  3. Mandelbrot B. Fraktal’naia geometriia prirody . Izhevsk: RKhD, 2002, 665 p. .
  4. Feder J. Fraktaly . M.: Mir, 1991, 524 p. .
  5. Ivanova V.S. Sinergetika i fraktaly v materialovedenii . М.: Nauka, 1994, 383 p. .
  6. Schroeder M. Fraktaly, khaos, stepennye zakony . Izhevsk: RKhD, 2001, 528 p. .
  7. Gladkov S.O., Bogdanova S.B. K teorii prodol’noi magnitnoi vospriimchivosti kvazitrekhmernykh ferromagnitnykh dielektrikov . FTT , 2012, Vol. 54, no 1, pp. 70–73. Available at: https://journals.ioffe.ru/articles/482 .
  8. Gladkov S.O., Bogdanova S.B. K voprosu o magnitnoi vospriimchivosti fraktal’nykh ferromagnitnykh provolok . Izvestiya vuzov. Fizika, 2014, Vol. 57, no 4, pp. 44–47. Available at: http://i.uran.ru/webcab/system/files/journalspdf/izvestiya-vuzov.ser.fizika/izvestiya-vuzov.ser.fizika-2014-t.57-n-4/ 42014.pdf .
  9. Bagley R.L., Torvik P.J. A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity. Journal of Rheology, 1983, Vol. 27 (201), pp. 201–210. doi: 10.1122/1.549724 .
  10. Bagley R.L., Torvik P.J. On the fractional calculus model of viscoelastic behavior. Journal of Rheology, 1986, Vol. 30 (1), pp. 133–155. doi: 10.1122/1.549887 .
  11. Kochubey A.N. Diffuziia drobnogo poriadka . Differentsial’nye uravneniia , 1990, Vol. 26, no 4, pp. 660–670. Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=de&paperid=7144&option_lang=rus .
  12. Nigmatullin R.R. Drobnyi integral i ego fizicheskaia interpretatsiia . Teoreticheskaia i matematicheskaia fizika , 1992, Vol. 90, no 3, pp. 354–368. Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=5547&option_lang=rus .
  13. Nakhushev A.M. Strukturnye i kachestvennye svoistva operatora, obratnogo operatoru drobnogo integro–differentsirovaniia s fiksirovannym nachalom i kontsom . Differentsial’nye uravneniia , 2000, Vol. 36, no 8, pp. 1093–1100. DOI: https://doi.org/10.1007/BF02754189 .
  14. Samko S.G., Kilbas A.A, Marichev O.I. Integraly i proizvodnye drobnogo poriadka i nekotorye ikh prilozheniia . Minsk: Nauka i tekhnika, 1987, 688 p. .
  15. Landau L.D., Lifshits Ye.M. Gidrodinamika . M.: Nauka, 1986, 736 p. .
  16. Gladkov S.O., Bogdanova S.B. The heat-transfer theory for quasi-n-dimensional system. Physica B: Condensed Matter, 2010, Vol. 405, pp. 1973–1975 .

Copyright (c) 2019 С. О. Гладков, С. Б. Богданова

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