Redo Exercise 2.3.8 if α < 0. [Be especially careful if −α/k = (nπ/L)2.] Reference...

Redo Exercise 2.3.8 if α < 0. [Be especially careful if −α/k = (nπ/L)².]

Reference Exercise 2.3.8:

Consider

This corresponds to a one-dimensional rod either with heat loss through the lateral sides with outside temperature 0° (α > 0, see Exercise 1.2.4) or with insulated lateral sides with a heat sink proportional to the temperature. Suppose that the boundary conditions are

u(0, t) = 0 and u(L, t) = 0.

(a) What are the possible equilibrium temperature distributions if α > 0?

(b) Solve the time-dependent problem [u(x, 0) = f(x)] if α > 0. Analyze the temperature for large time (t→∞) and compare to part (a).

Reference Exercise 1.2.4:

Derive the diffusion equation for a chemical pollutant.

(a) Consider the total amount of the chemical in a thin region between x and x + Δx.

(b) Consider the total amount of the chemical between x = a and x = b.