
7.3. Equations of the Form
∂
2
w
∂t
2
= a
2
∆
2
w − bw + Φ(x, y, t) 673
◮ Domain: 0 ≤ r ≤ R, 0 ≤ z ≤ l. Mixed boundary value problems.
1
◦
. A circular cylinder of finite length is considered. The following conditions are pre-
scribed:
w = f
0
(r, z) at t = 0 (initial condition),
∂
t
w = f
1
(r, z) at t = 0 (initial condition),
w = g
1
(z, t) at r = R (boundary condition),
∂
z
w = g
2
(r, t) at z = 0 (boundary condition),
∂
z
w = g
3
(r, t) at z = l (boundary condition).
Solution:
w(r, z, t) =
∂
∂t
Z
l
0
Z
R
0
f
0
(ξ, η)G(r, z, ξ, η, t) dξ dη
+
Z
l
0
Z
R
0
f
1
(ξ, η)G(r, z, ξ, η, t) dξ dη
− a
2
Z
t
0
Z
l
0
g
1
(η, τ)
∂
∂ξ
G(r, z, ξ, η, t − τ)
ξ=R
dη dτ
− a
2
Z
t
0
Z
R
0
g
2
(ξ, τ )G(r, z, ξ, 0, t − τ) dξ dτ
+ a
2
Z
t
0
Z
R
0
g
3
(ξ,