= Error-function isobaric cooling front
{title2=$T=T_0\operatorname{erf}\!\left(m/(2\sqrt{\lambda_0\tau})\right)$}
For conductivity $\lambda=\lambda_0T$, no cooling in the hot phase and a zero-<temperature> boundary at $m=0$, the mass-coordinate equation is linear diffusion. Its similarity profile solves $f''+\xi f'/2=0$, $f(0)=0$, $f(\infty)=1$, giving the displayed <error function>. The thermal layer broadens as the square root of time. Conducted energy enters the cold radiating reservoir at rate per area $L=\lambda_0A T_m(0)=A T_0\sqrt{\lambda_0}/\sqrt{\pi\tau}$, hence $L\propto t^{-1/2}$.
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