Heat equation: Arbitrary temperatures at ends. If the ends x = 0 and x = L of the bar in the text are kept at constant temperatures U1 and U2, respectively. The initial temp distribution is given by u(x, 0) = f (x).

(a) What is the temperature u1(x) in the bar after a long time (theoretically, as t)? First guess, then calculate.

(b) What is the temperature at any t. Use the heat equation given by ut = c^2uxx