Problem $(2)$: A semi$-$infinite solid domain is shown that extends to $$ in the $+x$ direction. Heat conduction occurs only in the $x$ direction. Originally, the temperature in the solid is uniform at $T_0.$ At time $t=0,$ the solid is suddenly exposed to or immersed in a large mass of ambient fluid at temperature $T_1,$ which is constant. The convection coefficient $h$ is present and is constant; that is a surface resistant is present. Hence, the temperature $T_S$ at the surface is not the same as $T_1.$ Show that the temperature distribution within the solid domain is given by

$\frac{T-T_0}{T_1-T_0}=$erfc$(\frac{x}{2\sqrt[\phantom{2}]{t}})-$exp$[\frac{h\sqrt[\phantom{2}]{t}}{k}(\frac{x}{\sqrt[\phantom{2}]{t}}+\frac{h\sqrt[\phantom{2}]{t}}{k})]erfc(\frac{x}{2\sqrt[\phantom{2}]{t}}+\frac{h\sqrt[\phantom{2}]{t}}{k})$

where $k$ is the thermal conductivity of solid domain, $=\frac{k}{}{c}_p,$ and erfc is the complementary error function. Hint: Use Laplace Transform.