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Fractional heat equation and the second law of thermodynamics

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Abstract

In the framework of second law of thermodynamics, we analyze a set of fractional generalized heat equations. The second law ensures that the heat flows from hot to cold regions, and this condition is analyzed in the context of the Fractional Calculus.

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References

  1. A.A. Kilbas, T. Pierantozzi, J.J. Trujillo, L. Vázquez, On the solution of fractional evolution equations. J. Phys. A: Math. Gen. 37 (2004), 3271–3283.

    Article  MATH  Google Scholar 

  2. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, Elsevier (2006).

    MATH  Google Scholar 

  3. Y. Luchko, M. Rivero, J.J. Trujillo, M.P. Velasco, Fractional models, non-locality and complex systems. Computers and Mathematics with Applications 59, No 3 (2010), 1048–1056.

    MATH  MathSciNet  Google Scholar 

  4. F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, No 2 (2001), 153–192; http://www.math.bas.bg/~fcaa/volume4/jlumapa-1.gif

    MATH  MathSciNet  Google Scholar 

  5. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000), 1–77.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Metzler, T.F. Nonnemacher, Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation. Chem. Phys. 284 (2002), 67–90.

    Article  Google Scholar 

  7. J.W. Nunziato, On heat conduction in materials with memory. Quart. Applied Mathematics 29 (1971), 187–204.

    MATH  MathSciNet  Google Scholar 

  8. P. Paradisi, R. Cesari, F. Mainardi, F. Tampieri, A generalized Fick’s law to describe non-local transport processes. Physica A 293 (2001), 130–142.

    Article  MATH  Google Scholar 

  9. T. Pierantozzi, L. Vázquez, An interpolation between the wave and diffusion equations through the fractional evolution equations Dirac like. J. Math. Phys. 46 (2005), 113512.

    Article  MathSciNet  Google Scholar 

  10. I. Podlubny, Fractional Differential Equations. Academic Press, San Diego et al. (1999).

    MATH  Google Scholar 

  11. M.B. Rubin, Hyperbolic heat conduction and the second law. Int. J. Engng. Sci. 30, No 11 (1992), 1665–1676.

    Article  MATH  Google Scholar 

  12. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Switzerland (1993).

    MATH  Google Scholar 

  13. J. Tenreiro Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simulat. 16 (2011), 1140–1153; doi:10.1016/j.cnsns.2010.05.027

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Luis Vázquez & M. Pilar Velasco

  2. Departamento de Análisis Matemático, Universidad de La Laguna, 38271, La Laguna, Tenerife, Spain

    Juan J. Trujillo

Authors
  1. Luis Vázquez
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  2. Juan J. Trujillo
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  3. M. Pilar Velasco
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Correspondence to Luis Vázquez.

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Dedicated to Prof. Stefan Samko on the occasion of his 70th Anniversary

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Vázquez, L., Trujillo, J.J. & Pilar Velasco, M. Fractional heat equation and the second law of thermodynamics. fcaa 14, 334–342 (2011). https://doi.org/10.2478/s13540-011-0021-9

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  • DOI: https://doi.org/10.2478/s13540-011-0021-9

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