
8.3. Equations of the Form
∂
2
w
∂t
2
= a
2
∆
3
w − bw + Φ(x, y, z, t) 735
where
A
n
=
(
1 for n = 0,
2 for n 6= 0,
λ
nmk
= a
2
µ
2
nm
+
a
2
k
2
π
2
l
2
+ b,
Z
nm
(r) = J
n
(µ
nm
R
1
)Y
n
(µ
nm
r) − Y
n
(µ
nm
R
1
)J
n
(µ
nm
r);
the J
n
(r) and Y
n
(r) are Bessel functions, and the µ
nm
are positive roots of the transcen-
dental equation
J
n
(µR
1
)Y
n
(µR
2
) − Y
n
(µR
1
)J
n
(µR
2
) = 0.
2
◦
. A hollow circular cylinder of finite length is considered. The following conditions are
prescribed:
w = f
0
(r, ϕ, z) at t = 0 (initial condition),
∂
t
w = f
1
(r, ϕ, z) at t = 0 (initial condition),
∂
r
w = g
1
(ϕ, z, t) at r = R
1
(boundary condition ),
∂
r
w = g
2
(ϕ, z, t) at r = R
2
(boundary condition ),
w = g
3
(r, ϕ, t) at z = 0 (boundary condition),
w