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2015 | 13 | 1 |
Article title

Fractional thermal diffusion and the heat equation

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EN
Abstracts
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.
Publisher

Journal
Year
Volume
13
Issue
1
Physical description
Dates
online
17 - 2 - 2015
accepted
2 - 10 - 2014
received
9 - 9 - 2014
Contributors
  • Centro Nacional de
    Investigación y Desarrollo Tecnológico. Tecnológico Nacional de
    México. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490,
    Cuernavaca, Morelos, México
author
  • 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
References
  • [1] D. Ben-Avraham, S. Havlin, Diffusion and reactions in fractalsand disordered systems (Cambridge University Press, UnitedKingdom, 2000)
  • [2] R. Metzler, A.V. Chechkin, J. Klafter, Encyclopedia of complexityand systems science (Springer, New York, 2009)
  • [3] H. Scher, E.W. Montroll, Phys. Rev. B. 12, 2455 (1975)[Crossref]
  • [4] K.B. Oldham, J. Spanier, The fractional calculus (AcademicPress, New York, 1974)
  • [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)
  • [19] R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)
  • [20] J. Bisquert, A. Compte, J. Electroanal. Chem. 499, 1 (2001)
  • [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)
  • [27] Y.Z. Povstenko, J. Mol. Liq. 137, 1 (2008)
  • [28] J. Xiaoyun, X. Mingyu, Physica A. 389, 17 (2010)
  • [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)
  • [30] J.F. Gómez Aguilar, D. Baleanu, Proc. Rom. Acad. A 1, 15 (2014)
  • [31] H.J. Haubold, A.M. Mathai, R.K. Saxena, J. Appl. Math. 2011,298628 (2011)
Document Type
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.-psjd-doi-10_1515_phys-2015-0023
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