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2015 | 13 | 1 |
Tytuł artykułu

Fractional thermal diffusion and the heat equation

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Fractional calculus is the branch of mathematical analysis that deals with operators interpreted as derivatives and integrals of non-integer order. This mathematical representation is used in the description of non-local behaviors and anomalous complex processes. Fourier’s lawfor the conduction of heat exhibit anomalous behaviors when the order of the derivative is considered as 0 < β,ϒ ≤ 1 for the space-time domain respectively. In this paper we proposed an alternative representation of the fractional Fourier’s law equation, three cases are presented; with fractional spatial derivative, fractional temporal derivative and fractional space-time derivative (both derivatives in simultaneous form). In this analysis we introduce fractional dimensional parameters σx and σt with dimensions of meters and seconds respectively. The fractional derivative of Caputo type is considered and the analytical solutions are given in terms of the Mittag-Leffler function. The generalization of the equations in spacetime exhibit different cases of anomalous behavior and Non-Fourier heat conduction processes. An illustrative example is presented.
Wydawca

Czasopismo
Rocznik
Tom
13
Numer
1
Opis fizyczny
Daty
otrzymano
2014-09-09
zaakceptowano
2014-10-02
online
2015-02-17
Twórcy
  • Centro Nacional de
    Investigación y Desarrollo Tecnológico. Tecnológico Nacional de
    México. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490,
    Cuernavaca, Morelos, México
autor
  • Facultad de Ingeniería en Electrónica
    y Comunicaciones. Campus: Poza Rica - Tuxpan. Universidad
    Veracruzana. Av. Venustiano Carranza s/n, Col. Revolución, C.P.
    93390, Poza Rica Veracruz, México
  • Facultad de Ingeniería en Electrónica
    y Comunicaciones. Campus: Poza Rica - Tuxpan. Universidad
    Veracruzana. Av. Venustiano Carranza s/n, Col. Revolución, C.P.
    93390, Poza Rica Veracruz, México
  • Centro Nacional de
    Investigación y Desarrollo Tecnológico. Tecnológico Nacional de
    México. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490,
    Cuernavaca, Morelos, México
  • Centro Nacional de
    Investigación y Desarrollo Tecnológico. Tecnológico Nacional de
    México. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490,
    Cuernavaca, Morelos, México
Bibliografia
  • [1] D. Ben-Avraham, S. Havlin, Diffusion and reactions in fractalsand disordered systems (Cambridge University Press, UnitedKingdom, 2000)
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  • [5] M. Duarte Ortiguera, Fractional calculus for scientists and engineers(Springer, New York, 2011)
  • [6] I. Podlubny, Fractional differential equations (Academic Press,New York, 1999)
  • [7] H. Nasrolahpour, Commun. Nonlinear Sci. Numer. Simul. 18, 9(2013)[Crossref]
  • [8] J.F. Gómez Aguilar, D. Baleanu, Z. Naturforsch. 69a, 539 (2014)
  • [9] R.P. Agarwal, B.D. Angrade, G. Siracusa. Compt.Math. Appl. 62,1143 (2011)[Crossref]
  • [10] F. Gómez, J. Bernal, J. Rosales, T. Córdova, J. Electr. Bioimp. 3, 1(2012)
  • [11] R. Gorenflo, F.Mainardi, Eur. Phys. J. Special Topics 193, 1 (2011)
  • [12] F.Mainardi, Fractional calculus and waves in linear viscoelasticity(Imperial College Press, London, 2010)
  • [13] J.F. Gómez-Aguilar, R. Razo-Hernández, D. Granados-Lieberman. Rev. Mex. Fís. 60, 1 (2014)
  • [14] D. Baleanu, A.K. Golmankhaneh, A.K. Golmankhaneh, M.C.Baleanu, Int. J. Theor. Phys. 48, 11 (2009)
  • [15] D. Baleanu, A.K. Golmankhaneh, R. Nigmatullin, A.K. Golmankhaneh,Centr. Eur. J. Phys. 8, 1 (2010)
  • [16] J.F. Gómez Aguilar, J.R. Razo Hernández, Revista Investigación yCiencia de la Universidad Autónoma de Aguascalientes 22, 61(2014)
  • [17] Mohamed A.E. Herzallah, I. Muslih Sami, D. Baleanu, M. RabeiEqab, Nonlinear Dynam. 66, 4 (2011)
  • [18] F. Mainardi, Y. Luchko, G. Pagnini, Fract. Calc. Appl. Anal. 4, 2(2001)
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  • [21] J.F. Gómez Aguilar, M.M. Hernández, Abstr. Appl. Anal. 2014,283019 (2014)
  • [22] Mohamed A.E. Herzallah, Ahmed M.A. El-Sayed, D. Baleanu,Rom. J. Phys. 55, 3 (2010)
  • [23] M.A. Ezzat, AA. El-Bary, M.A. Fayik, Mech. Adv.Mater. Struc. 20,1 (2013)
  • [24] Y.Z. Povstenko, J. Ther. Stresses 28, 1 (2004)
  • [25] O. Narayan, S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002)[Crossref]
  • [26] X.-J. Yang, D. Baleanu, Therm. Sci. 17, 2 (2013)
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  • [29] J.F. Gómez-Aguilar, J.J. Rosales-García, J.J. Bernal-Alvarado, T.Córdova-Fraga, R. Guzmán-Cabrera, Rev. Mex. Fís. 58, 4 (2012)
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  • [31] H.J. Haubold, A.M. Mathai, R.K. Saxena, J. Appl. Math. 2011,298628 (2011)
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.-psjd-doi-10_1515_phys-2015-0023
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