
3.2. Heat Equation with Axial or Central Symmetry and Related Equations 301
Example 3.15. The initial temperature is unifor m, f(r) = w
0
, and the surface of the sphere is
maintained at constant temperature, g(t) = w
R
.
Solution:
w(r, t) − w
R
w
0
− w
R
=
2R
πr
∞
X
n=1
(−1)
n+1
n
sin
πnr
R
exp
−
aπ
2
n
2
t
R
2
.
The average temperature w depends on time t as follows:
w −w
R
w
0
− w
R
=
6
π
2
∞
X
n=1
1
n
2
exp
−
aπ
2
n
2
t
R
2
, w =
1
V
Z
v
w dv,
where V is the volume of the sphere of radiu s R.
⊙ Literature: H. S. Carslaw and J. C. Jaeger (1984).
◮ Domain: 0 ≤ r ≤ R. Second boundary value problem.
The following conditions are prescribed:
w = f (r) at t = 0 (initial condition),
∂
r
w = g(t) at r = R (bound