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

$$\frac{\partial u}{\partial t}=\kappa \frac{{\partial }^{2}u}{\partial x^2},x>0,t>0$$
with $u(x,0)=u_m,x>0,u(0,t)=u_w,t>0,|u(x,t)|0)$. Using a Laplace transform method:

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(a) Show that the transformed solution is U(x, s) = (uw-um) S (b) Using the inverse Laplace transform L-1 a =erfc show that the solution is u(x,t) =LU(x,s)}=(u-um)erfe +um-