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QUESTION 1 Suppose that f and s are integrable. Show that fy is also integrable. [20 Marks] QUESTION 2 Consider the heat distribution (I. t) in a semi-infinite rod. Assume that the finite end is kept in contact with ice at rc and if initially the rod is submerged in hot water at 100C (a) Use the appropriate Fourier transform to show that (explain the choice of the Fourier transform) 4 (0:1)+ ¢a® (a,0) = 1022" with All (0,0) = 0, where is the appropriate Fourier transform of and is the specific heat of the rod. (15 Marks) (b) Establish that (t) = (1-2) Then derive that solution 200 (t) = 100 - (15 Marks) [30 Marks) 2002 *** sinondo 19 QUESTION 3 Write down the initial value problem that models the heat transfer in an infinitely long rod. As- sume that the rod is exchanging heat with the surrounding medium (convection, the coefficient of convection is t, the heat specific of the rod is 1, and the initial temperature is given by f(s). [20 Marks] QUESTION 4 Find the steady state temperature in the wedge region in the figure below (explain In details) [30 Marks)