An $$infinite$$ slab of thickness L is initially at a uniform temperature, T$0.$ The temperature of the left $($x $=0)$ and right $($x $=$ L$)$ surface is suddenly increased to TL and maintained at this temperature.

$(3$ pt$)$ Sketch what the temperature profile will look like at different time points.

$(2$ pt$)$ State the relevant boundary conditions.

$(2$ pt$)$ Write the initial condition.

$(12$ pt$)$ Solve the PDE using the separation of variables technique to develop an expression for the temperature profile, T$($x$,$t$)$ in the slab.

$(2$ pt$)$ Should n $=0$ be included in your solution? If yes, determine the constant associated with n $=0.$ If no$,$ explain why not.

$(1$ pt$)$ Determine the steady state temperature profile $($i$.$e$.$ determine T$($x$)$ as t$->\backslash$infty $)$ in the slab.

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