Consider heat conduction in a semiinfinite slab, where the slab is initially \((t=0)\) at a constant temperature \(u_{m}\) and the end of the slab (at \(x=0\) ) is maintained at the constant temperature \(u_{w}\). The partial differential equation for the temperature distribution \(u(x, t)\) for thermal diffusivity \(\kappa\) is

\[
\frac{\partial u}{\partial t}=\kappa \frac{\partial^{2} u}{\partial x^{2}}, \quad x>0, t>0
\]

with \(u(x, 0)=u_{m}, x>0, u(0, t)=u_{w}, t>0,|u(x, t)|0)\). Using a Laplace transform method:

(a) Show that the transformed solution is \(U(x, s)=\left(u_{w}-u_{m}ight) \frac{e^{-\sqrt{\frac{s}{\kappa}} x}}{s}+\frac{u_{m}}{s}\).

(b) Using the inverse Laplace transform \(\mathscr{L}^{-1}\left\{\frac{e^{-a \sqrt{s}}}{s}ight\}=\operatorname{erfc}\left(\frac{a}{2 \sqrt{t}}ight)\), show that the solution is

\[
u(x, t)=\mathscr{L}^{-1}\{U(x, s)\}=\left(u_{w}-u_{m}ight) \operatorname{erfc}\left(\frac{x}{2 \sqrt{\kappa t}}ight)+u_{m}
\]