Fractional thermal diffusion and the heat equation

• Autorzy: Francisco Gómez , Luis Morales , Mario González , Victor Alvarado , Guadalupe López
• 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.

Słowa kluczowe

EN

anomalous diffusion; Caputo derivative; fractionaldifferential equations; Mittag-Leffler function

• Czasopismo: Open Physics
• Rocznik: 2015
• Tom: 13
• Numer: 1

Daty

otrzymano

2014-09-09

zaakceptowano

2014-10-02

online

2015-02-17

Twórcy

autor

Francisco Gómez

• 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

Luis Morales

• 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

autor

Mario González

• 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

autor

Victor Alvarado

• 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

Guadalupe López

• 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)
• [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)

Identyfikatory

DOI

10.1515/phys-2015-0023