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October  2010, 14(3): 793-816. doi: 10.3934/dcdsb.2010.14.793

Minimum free energy in the frequency domain for a heat conductor with memory

Giovambattista Amendola 1, , Mauro Fabrizio 2, and  John Murrough Golden 3,

1. 

Department of Applied Mathematics "U. Dini”, via Diotisalvi 2, 56126-Pisa, Italy
2. 

Department of Mathematics, Piazza di Porta S. Donato 5, 40127-Bologna, Italy
3. 

School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Received  February 2009 Revised  January 2010 Published  July 2010

An explicit expression is given in the frequency domain for the minimum free energy related to a particular state of a linear rigid heat conductor with memory effects in the constitutive equations, using the fact that this quantity coincides with the maximum recoverable work obtainable from that state. The constitutive equations for the internal energy and the heat flux are expressed as linear functionals of the histories of temperature and its gradient, respectively, together with the present value of the latter quantity. Another equivalent expression for the minimum free energy is also deduced and used to derive explicit formulae for a discrete spectrum model.
Keywords: heat conduction., Memory effects.
Mathematics Subject Classification: 80A20, 74F0.
Citation: Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden. Minimum free energy in the frequency domain for a heat conductor with memory. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 793-816. doi: 10.3934/dcdsb.2010.14.793
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