
498 SECOND-ORDER PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
◮ Domain: R
1
≤ r ≤ R
2
, 0 ≤ ϕ ≤ 2π, −∞ < z < ∞. Th ird bound ary value
problem.
An infinite hollow circular cylinder is considered. The following conditions are prescribed:
w = f (r, ϕ, z) at t = 0 (initial condition),
∂
r
w − k
1
w = g
1
(ϕ, z, t) at r = R
1
(boundary condition),
∂
r
w + k
2
w = g
2
(ϕ, z, t) at r = R
2
(boundary condition).
The solution w(r, ϕ, z, t) is determined by the formula in the previous paragraph (for
the second boundary value problem) where
G(r, ϕ, z, ξ, η, ζ, t)=
1
2
√
πat
exp
−
(z−ζ)
2
4at
G
1
(r, ϕ, ξ, η, t),
G
1
(r, ϕ, ξ, η, t)=
1
π
∞
X
n=0
∞
X
m=1
A
n
µ
2
nm
Z
n
(µ
nm
r)Z
n
(µ
nm
ξ) cos[n(ϕ−η)] e x