Consider the following boundary value problem (if necessary, see Section 2.4.1):...

Consider the following boundary value problem (if necessary, see Section 2.4.1):

(a) Give a one-sentence physical interpretation of this problem.

(b) Solve by the method of separation of variables. First show that there are no separated solutions which exponentially grow in time. [Hint: The answer is

What is λn?

(c) Show that the initial condition, u(x, 0) = f(x), is satisfied if

(d) Using Exercise 2.3.6, solve for A0 and An(n ≥ 1).

(e) What happens to the temperature distribution as t → ∞? Show that it approaches the steady-state temperature distribution (see Section 1.4).

Reference Exercise 2.3.6:

Evaluate

Use the trigonometric identity

(Be careful if a − b = 0 or a + b = 0.)