
4.2. Heat Equation with a Source
∂w
∂t
= a∆
2
w + Φ(x, y, t) 451
The solution w(r, z, t) is determined by the formula in the previous paragraph (for the
second boundary value problem) where
G(r, z, ξ, η, t) = G
1
(r, ξ, t) G
2
(z, η, t),
G
1
(r, ξ, t) =
1
πR
2
∞
X
n=1
µ
2
n
(k
2
1
R
2
+µ
2
n
)J
2
0
(µ
n
)
J
0
µ
n
r
R
J
0
µ
n
ξ
R
exp
−
aµ
2
n
t
R
2
,
G
2
(z, η, t) =
∞
X
m=1
ϕ
m
(z)ϕ
m
(η)
kϕ
m
k
2
exp(−aλ
2
m
t), ϕ
m
(z) = cos(λ
m
z)+
k
2
λ
m
sin(λ
m
z),
kϕ
m
k
2
=
k
3
2λ
2
m
λ
2
m
+k
2
2
λ
2
m
+k
2
3
+
k
2
2λ
2
m
+
l
2
1+
k
2
2
λ
2
m
,
and the µ
n
and λ
m
are positive roots of the transcendental equations
µJ
1
(µ) − k
1
RJ
0
(µ) = 0,
tan(λl)
λ
=
k
2
+ k
3
λ
2
− k
2
k
3
.
◮ Domain: 0 ≤ r ≤ R, 0 ≤ z ≤ l. Mixed boundary value problems.
1
◦
. A circular cylinder of finite length is considere