The left at $x=0$ and right $($at $x=L$ surfaces of an "infinite" slab of thickness $L$ are kept at

known temperatures $T_0$ and $T_L(T_0>T_L),$ respectively, until steady heat flow is established. The

temperature of the left surface only is then suddenly changed to $T_L$ and maintained at this

temperature.

a$)$ Write the initial condition, $T(x,0)$ for this problem.

b$)$ Write the boundary conditions for this problem.

c$)$ Solve the PDE using the separation of variables technique to develop an expression for the

temperature distribution $T(x,t)$ in the slab following the aforementioned change.

The slab properties $($density$,$ heat capacity, and thermal conductivity$)$ may be assumed

constant.