Problem $(1)$: Consider a rectangular $1-$D bar lying in the domain $-x.$ Initially at $t=0$ the first half$-$part of the bar lying in the domain $x>0$ is at a temperature of $T_0$ while the other half is at zero temperature. It is apparent that for $t>0$ heat will start conducting from the first halfpart to the other half thereby leading to development of a temperature profile $T(t,x)$ within the bar. Show that the temperature profile $T(t,x)$ in the entire domain $-x$ is given by

$T(t,x)=\frac{T_0}{2}[1+$erf$(\frac{x}{\sqrt[\phantom{2}]{4t}})]$

where $$ is the thermal diffusivity of the bar material. Hint. Use Laplace transform.