Eur. Phys. J. Special Topics 193, 173–184 (2011) c  EDP Sciences, Springer-Verlag 2011 DOI: 10.1140/epjst/e2011-01389-y T HE EUROPEAN P HYSICAL JOURNAL SPECIAL TOPICS Regular Article A non-local model of fractional heat conduction in rigid bodies G. Borino, M. Di Paola, and M. Zingales 1, a Dipartimento di Ingegneria Civile, Aerospaziale e Ambientale, viale delle Scienze Ed.8, 90128 Palermo, Italy Received 01 December 2010 / Received in final form 27 January 2011 Published online 4 April 2011 Abstract. In recent years several applications of fractional differential calculus have been proposed in physics, chemistry as well as in en- gineering fields. Fractional order integrals and derivatives extend the well-known definitions of integer-order primitives and derivatives of the ordinary differential calculus to real-order operators. Engineering applications of fractional operators spread from viscoelastic models, stochastic dynamics as well as with thermoelasticity. In this latter field one of the main actractives of fractional operators is their capability to interpolate between the heat flux and its time-rate of change, that is related to the well-known second sound effect. In other recent stud- ies a fractional, non-local thermoelastic model has been proposed as a particular case of the non-local, integral, thermoelasticity introduced at the mid of the seventies. In this study the autors aim to introduce a different non-local model of extended irreverible thermodynamics to account for second sound effect. Long-range heat flux is defined and it involves the integral part of the spatial Marchaud fractional derivatives of the temperature field whereas the second-sound effect is accounted for introducing time-derivative of the heat flux in the transport equa- tion. It is shown that the proposed model does not suffer of the patho- logical problems of non-homogenoeus boundary conditions. Moreover the proposed model coalesces with the Povstenko fractional models in unbounded domains. 1 Introduction The nowadays concepts in technology, material sciences and nano-technology require a generalization of the heat transport equations beyhond the classical relation widely used in thermodynamics. For an istance, as far as the dimensions of the studied sys- tem become comparable with the mean-free path of molecules, then, the classical equilibrium conditions assumed in clasical thermodynamics do not account for some important phenomenon. Some of these latter observations have already been discussed at the beginning of the last century such as the well-known second-sound effect in Helium II (He II). In fact experimental evidences show that in steady-state fluxes of a e-mail: massimiliano.zingales@unipa.it