On the differential equation for heat conduction

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Abstract
A discussion based on the Boltzmann equation is given of the way in which the heat-conduction equation C(∂T/∂t) = KV2T must be modified when the temperature T changes appreciably within a mean free path. Assuming a temperature-independent relaxation time τ, a hierarchy of linear equations of increasing accuracy is obtained, of which the first member is the modification C[(∂T/∂t) + τ (∂2T/∂t2)] = = KV2T suggested earlier by many authors. Damped-wave solutions for T are shown to exist over a certain frequency range, and the corresponding dispersion relations are obtained. It is shown that if τ is temperature dependent, the above conduction equation takes the nonlinear form