Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Feng, Jonathan

Managing Editor: Lesna-Szreter, Paulina

1 Issue per year


IMPACT FACTOR 2016 (Open Physics): 0.745
IMPACT FACTOR 2016 (Central European Journal of Physics): 0.765

CiteScore 2016: 0.82

SCImago Journal Rank (SJR) 2015: 0.458
Source Normalized Impact per Paper (SNIP) 2015: 1.142

Open Access
Online
ISSN
2391-5471
See all formats and pricing
More options …
Volume 13, Issue 1 (Feb 2015)

Issues

Fractional thermal diffusion and the heat equation

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
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ 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
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ 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
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ 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
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ 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
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-02-17 | DOI: https://doi.org/10.1515/phys-2015-0023

Abstract

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.

Keywords: anomalous diffusion; Caputo derivative; fractional differential equations; Mittag-Leffler function

References

  • [1] D. Ben-Avraham, S. Havlin, Diffusion and reactions in fractals and disordered systems (Cambridge University Press, United Kingdom, 2000) Google Scholar

  • [2] R. Metzler, A.V. Chechkin, J. Klafter, Encyclopedia of complexity and systems science (Springer, New York, 2009) Google Scholar

  • [3] H. Scher, E.W. Montroll, Phys. Rev. B. 12, 2455 (1975) CrossrefGoogle Scholar

  • [4] K.B. Oldham, J. Spanier, The fractional calculus (Academic Press, New York, 1974) Google Scholar

  • [5] M. Duarte Ortiguera, Fractional calculus for scientists and engineers (Springer, New York, 2011) Google Scholar

  • [6] I. Podlubny, Fractional differential equations (Academic Press, New York, 1999) Google Scholar

  • [7] H. Nasrolahpour, Commun. Nonlinear Sci. Numer. Simul. 18, 9 (2013) CrossrefGoogle Scholar

  • [8] J.F. Gómez Aguilar, D. Baleanu, Z. Naturforsch. 69a, 539 (2014) Google Scholar

  • [9] R.P. Agarwal, B.D. Angrade, G. Siracusa. Compt.Math. Appl. 62, 1143 (2011) CrossrefGoogle Scholar

  • [10] F. Gómez, J. Bernal, J. Rosales, T. Córdova, J. Electr. Bioimp. 3, 1 (2012) Google Scholar

  • [11] R. Gorenflo, F.Mainardi, Eur. Phys. J. Special Topics 193, 1 (2011) Google Scholar

  • [12] F.Mainardi, Fractional calculus and waves in linear viscoelasticity (Imperial College Press, London, 2010) Google Scholar

  • [13] J.F. Gómez-Aguilar, R. Razo-Hernández, D. Granados- Lieberman. Rev. Mex. Fís. 60, 1 (2014) Google Scholar

  • [14] D. Baleanu, A.K. Golmankhaneh, A.K. Golmankhaneh, M.C. Baleanu, Int. J. Theor. Phys. 48, 11 (2009) Google Scholar

  • [15] D. Baleanu, A.K. Golmankhaneh, R. Nigmatullin, A.K. Golmankhaneh, Centr. Eur. J. Phys. 8, 1 (2010) Google Scholar

  • [16] J.F. Gómez Aguilar, J.R. Razo Hernández, Revista Investigación y Ciencia de la Universidad Autónoma de Aguascalientes 22, 61 (2014) Google Scholar

  • [17] Mohamed A.E. Herzallah, I. Muslih Sami, D. Baleanu, M. Rabei Eqab, Nonlinear Dynam. 66, 4 (2011) Google Scholar

  • [18] F. Mainardi, Y. Luchko, G. Pagnini, Fract. Calc. Appl. Anal. 4, 2 (2001) Google Scholar

  • [19] R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000) Google Scholar

  • [20] J. Bisquert, A. Compte, J. Electroanal. Chem. 499, 1 (2001) Google Scholar

  • [21] J.F. Gómez Aguilar, M.M. Hernández, Abstr. Appl. Anal. 2014, 283019 (2014) Google Scholar

  • [22] Mohamed A.E. Herzallah, Ahmed M.A. El-Sayed, D. Baleanu, Rom. J. Phys. 55, 3 (2010) Google Scholar

  • [23] M.A. Ezzat, AA. El-Bary, M.A. Fayik, Mech. Adv.Mater. Struc. 20, 1 (2013) Google Scholar

  • [24] Y.Z. Povstenko, J. Ther. Stresses 28, 1 (2004) Google Scholar

  • [25] O. Narayan, S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002) CrossrefGoogle Scholar

  • [26] X.-J. Yang, D. Baleanu, Therm. Sci. 17, 2 (2013) Google Scholar

  • [27] Y.Z. Povstenko, J. Mol. Liq. 137, 1 (2008) Google Scholar

  • [28] J. Xiaoyun, X. Mingyu, Physica A. 389, 17 (2010) Google Scholar

  • [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) Google Scholar

  • [30] J.F. Gómez Aguilar, D. Baleanu, Proc. Rom. Acad. A 1, 15 (2014) Google Scholar

  • [31] H.J. Haubold, A.M. Mathai, R.K. Saxena, J. Appl. Math. 2011, 298628 (2011) Google Scholar

About the article

Received: 2014-09-09

Accepted: 2014-10-02

Published Online: 2015-02-17


Citation Information: Open Physics, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2015-0023.

Export Citation

©2015 F. Gómez et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in