Arod of length l coincides with the interval [0, l] on the x-axis. set up the boundary-value problem problem for the temperature u(x, t) in the following scenarios:

(a) the left end is held at temperature u0, and the right end is held at temperature u1. the initial temperature is zero throughout.

(b) there is heat transfer from the left end into a surrounding medium at temperature 20◦ , and the right end is insulated. the initial temperature is f(x) throughout.

(c) the ends are insulated, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at 50◦ . the initial temperature is 100◦ throughout.