
740 SECOND-ORDER HYPERBOLIC EQUATIONS WITH THREE OR MORE SPA CE VARIABLES
◮ Domain: 0 ≤ r ≤ R, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π. Third boundary value problem.
A spherical domain is considered. The following conditions are prescribed:
w = f
0
(r, θ, ϕ) at t = 0 (initial condition),
∂
t
w = f
1
(r, θ, ϕ) at t = 0 (initial condition),
∂
r
w + kw = g(θ, ϕ, t) at r = R (boundary condition ).
The solution w(r, θ, ϕ, t) is determined by the formula in the previous paragraph (for
the second boundary value problem) where
G(r, θ, ϕ, ξ, η, ζ, t) =
1
2π
√
rξ
∞
X
n=0
∞
X
m=1
n
X
s=0
A
s
B
nms
p
a
2
λ
2
nm
+ b
J
n+1/2
(λ
nm
r)J
n+1/2
(λ
nm
ξ)
× P
s
n
(cos θ)P
s
n
(cos η) cos[s(ϕ − ζ)] sin
t
p
a
2
λ
2
nm
+ b
.
Here,
A
s
=
(
1 for s=