Generalized heat conduction equation

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Abstract
The linear constitutive relation between heat flux and temperature gradient known as Fourier’s law is modified to be nonlocal in time; i.e., the heat flux depends on the past history of the temperature gradient. It is shown that this leads to a generalized heat conduction equation which can be hyperbolic and thus possess a wavelike solution with finite velocity of propagation of a heat pulse. In two extreme limits of very short and very long memory the equation reduces, respectively, to the usual parabolic heat conduction equation and to a hyperbolic wave equation. For in between memory the equation has the properties of a telegraph equation characterized by a relaxation time. Such equations have arisen in treatments of second sound, exciton transport, and quasiparticle transport in 3He.

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