Mathematical Methods in the Physical Sciences · Problem 3.7
FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas
Mathematics — Methods · heat-flow-insulated-bar
Problem
A bar of length l with insulated sides has its ends also insulated from time t=0 on. Initially the temperature is u=x, where x is the distance from one end. Determine the temperature distribution inside the bar at time t. Hints and comments: See the last paragraph of this section, above, and also Problem 2.14. Show that the k=0 solutions are x and constant (time independent). Note that here (unlike Problem 2.15) you do not need the extra solution (namely x) for k=0 since the final steady state is a constant and this is included in the solutions (3.10). Also note that we did need the k=0 solutions in the discussion following (3.15) but were able to simplify the work by observing that these linear solutions simply give the final steady state.