Constant surface heating of a variable conductivity half-space

A solution to the problem of constant surface heating of an initially constant-temperature, $T_0^*$, half-space where the material in question has a temperature-dependent thermal conductivity is obtained. The thermal conductivity, ${k^*}$, is specifically given by ${k^*} = k_0^*\exp \left [ {\lambda \left ( {{T^*} - T_0^*} \right )/T_0^*} \right ]$. The solution is valid for both heating and cooling of the material where $\lambda$ and $k_0^*$ are arbitrary in magnitude, and $\lambda$ can be either positive or negative in sign.