Initial temperature pulse. Solve the inhomogeneous heat problem with Type I boundary conditions: du Ju t x where t > 0, 0x 1, and = u (0, t) = 0 = u(1, t); Pw (x) = 0 W 0 u(x,0) = Pw (x) if 0 Can you distinguish between a pulse with width w = 1/1000 from one with w = 1/2000, say, by measuring u(,)? (b) Illustrate the solution qualitatively by sketching (i) some typical tempera- ture profiles in the u-t plane (i.e. x = constant) and in the u- x plane (i.e. t = constant), and (ii) some typical level curves u (x, t) = constant in the x-t plane. At what points of the set D = {(x, t): 0x 1, t 0} is u (x, t) discontinuous?