Heat conduction: a telegraph-type model with self-similar behavior of solutions

Abstract

For heat flux $q$ and temperature $T$ we introduce a modified Fourier--Cattaneo law $q_t+l\frac{q}{t}=-k{T}_x.$ The consequence of it is a non-autonomous telegraph-type equation. % $\epsilon S_{tt}+\frac{a}{t}{S}_t=S_{xx}$ . This model already has a typical self-similar solution which may be written as product of two travelling waves modulo a time-dependent factor and might play a role of intermediate asymptotics.

5 Pages, 2 Figures, it will be submitted to Physical Review