III (25 pts) A plate of thickness L is initially at zero temperature. For times t> 0, the boundary surface at x = L is kept insulated while the surface at x =0 is subjected to a heat flux f(t) varying with time as T x -k- Using the Duhamel's theorem, develop an expression for the temperature distribution T(x,t) in the slab for times (i) t < ti, and (ii) t > ti aT (x, t) x (t = f(t) = {t for 0 0 x = 0 in 0 < x 0 T(x, t) = 0 at x = L, t> 0 fort = 0