QUESTION 2 Consider the temperature u (r, t) at time t at each point a of a rod length 5, if one of its end is in contact with ice at 0 and the other end is exposed to warm air at 35". Assume that initially the rod is submerged in boiling water at 1009. a. Derive the initial-boundary value problem of the problem described above. Explain all the steps. (5 Marks) b. Show that the solution of the initial-boundary value problem obtained in part (a) has a unique solution. [Hint: apply the maximum principle to the heat equation.] (10 Marks) c. Deduce that the unique solution u (r, () obtained in part (b.) is such that u (x, t) 2 0, Vr e [0, 5], Vt 2 0. (5 Marks) [20 Marks]