
4.1. Heat Equation
∂w
∂t
= a∆
2
w 431
Solution:
w(r, z, t) = 2π
Z
l
0
Z
R
0
ξf (ξ, η) G(r, z, ξ, η, t) dξ dη
− 2πaR
Z
t
0
Z
l
0
g
1
(η, τ)
∂
∂ξ
G(r, z, ξ, η, t − τ)
ξ=R
dη dτ
− 2πa
Z
t
0
Z
R
0
ξg
2
(ξ, τ ) G(r, z, ξ, 0, t − τ) dξ dτ
+ 2πa
Z
t
0
Z
R
0
ξg
3
(ξ, τ ) G(r, z, ξ, l, t − τ) dξ dτ.
Here,
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
1
J
2
1
(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) =
1
l
+
2
l
∞
X
n=1
cos
nπz
l
cos
nπη
l
exp
−
an
2
π
2
t
l
2
,
where the µ
n
are positive zeros of the Bessel function, J
0
(µ
n
) = 0. The numerical values
of the first ten µ
n
are specified in Section 3.2.1 (see the first boundary value problem for
0 ≤ r ≤ R).
2
◦
. A circular cylinder of finite length is considered. ...