
3.2. Heat Equation with Axial or Central Symmetry and Related Equations 291
Here,
G(r, ξ, t) =
2
R
2
ξ +
2
R
2
∞
X
n=1
ξ
J
2
0
(µ
n
)
J
0
µ
n
r
R
J
0
µ
n
ξ
R
exp
−
aµ
2
n
t
R
2
,
where the µ
n
are positive zeros of the first-order Bessel function, J
1
(µ) = 0. Below are the
numerical values of the first ten roots:
µ
1
= 3.8317, µ
2
= 7.0156, µ
3
= 10.1735, µ
4
= 13.3237, µ
5
= 16.4706,
µ
6
= 19.6159, µ
7
= 22.7601, µ
8
= 25.9037, µ
9
= 29.0468, µ
10
= 32.1897.
As n → ∞, we have µ
n+1
− µ
n
→ π.
Example 3.12. The initial temperature of the cylinder is uniform, f(r) = w
0
. The lateral surface
is maintained at constant thermal flux, g(t) = g
R
.
Solution:
w(r, t) = w
0
+ g
R
R
2
at
R
2
−
1
4
+
r
2
2R
2
−
∞
X
n=1
2
µ
2
n
J
0
(µ
n
)
exp
−µ
2
n
at
R
2
J
0