3. Given the heat conduction equation

256uxx = ut, 0 < x < 10, t > 0,

and the initial temperature distribution u(x, 0) = f (x) = 60. Also consider the following 5 pairs of

boundary conditions.

(1) u(0,t) = 0, u(10,t) = 0.

(2) ux(0,t) = 0, ux(10,t) = 0.

(3) u(0,t) = 100, u(10,t) = 0.

(4) u(0,t) = 0, u(10,t) = 100.

(5) u(0,t) = 100, u(10,t) = 100.

(a) Find the steady-solution corresponding to each pair of boundary conditions.

(b) For each pair of boundary conditions, find the limit of u(8,t) as t approaches .

(c) Suppose the initial condition is changed to u(x, 0) = g(x) = 20x2 - 2x3. Repeat part (b). Is any of

your answers different from its counterpart in (b)?