
5.3. Other Equations with Three Space Variables 543
The problems presented in this subsection are assumed to refer to a simply connected
bounded domain V with smooth boundary S. It is also assumed that ρ(r) > 0, a(r) > 0,
and q(r) ≥ 0.
◮ First boundary value problem.
The following conditions are prescribed:
w = f (r) at t = 0 (initial condition),
w = g(r, t) for r ∈ S (boundary condition).
Solution:
w(r, t) =
Z
t
0
Z
V
Φ(ξ, τ)G(r, ξ, t−τ ) dV
ξ
dτ +
Z
V
f(ξ)ρ(ξ)G(r, ξ, t) dV
ξ
−
Z
t
0
Z
S
g(ξ, τ)a(ξ)
∂
∂N
ξ
G(r, ξ, t−τ )
dS
ξ
dτ. (1)
Here, the modified Green’s function is given by
G(r, ξ, t) =
∞
X
n=1
u
n
(r)u
n
(ξ)
ku
n
k
2
exp(−λ
n
t), ku
n
k
2
=
Z
V
ρ(r)u
2
n
(r) dV, ξ = {ξ
1
, ξ
2
, ξ
3
},
(2)
where the