Fractional heat equation and the second law of thermodynamics. (English) Zbl 1273.80002

Summary: 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 fractional calculus.


80A10 Classical and relativistic thermodynamics
35Q79 PDEs in connection with classical thermodynamics and heat transfer
35R11 Fractional partial differential equations
Full Text: DOI


[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. http://dx.doi.org/10.1088/0305-4470/37/9/015; · Zbl 1059.35030
[2] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, Elsevier (2006).; · Zbl 1092.45003
[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.; · Zbl 1189.37095
[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; · Zbl 1054.35156
[5] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000), 1-77. http://dx.doi.org/10.1016/S0370-1573(00)00070-3; · Zbl 0984.82032
[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. http://dx.doi.org/10.1016/S0301-0104(02)00537-2;
[7] J.W. Nunziato, On heat conduction in materials with memory. Quart. Applied Mathematics 29 (1971), 187-204.; · Zbl 0227.73011
[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. http://dx.doi.org/10.1016/S0378-4371(00)00491-X; · Zbl 0978.82080
[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. http://dx.doi.org/10.1063/1.2121167; · Zbl 1111.35049
[10] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego et al. (1999).; · Zbl 0924.34008
[11] M.B. Rubin, Hyperbolic heat conduction and the second law. Int. J. Engng. Sci. 30, No 11 (1992), 1665-1676. http://dx.doi.org/10.1016/0020-7225(92)90134-3; · Zbl 0764.73006
[12] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Switzerland (1993).; · Zbl 0818.26003
[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 http://dx.doi.org/10.1016/j.cnsns.2010.05.027; · Zbl 1221.26002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.