The temperature u(x, t) of a long, thin rod of constant cross section and homogeneous conducting material is governed by the one-dimensional heat equation. If heat is generated in the material, for example, by resistance to current or nuclear reaction, the heat equation becomes
∂2u / ∂x2 + Kr ρC = K ∂u / ∂t, 0< x < l, 0< t,
where l is the length, ρ is the density, C is the specific heat, and K is the thermal diffusivity of the rod. The function r = r(x, t, u) represents the heat generated per unit volume. Suppose that
l = 1.5 cm, K = 1.04 cal/cm · deg · s, ρ = 10.6 g/cm3, C = 0.056 cal/g · deg,
and
r(x, t, u) = 5.0 cal/cm3 · s.
If the ends of the rod are kept at 0◦C, then
u(0, t) = u(l, t) = 0, t > 0.
Suppose the initial temperature distribution is given by
u(x, 0) = sin πx / l , 0≤ x ≤ l.
Use the results of Exercise 15 to approximate the temperature distribution with h = 0.15 and k = 0.0225.