
426 SECOND-ORDER PARABOLIC EQUATIONS WITH TWO SPACE VARIABLES
◮ Domain: 0 ≤ r ≤ R, 0 ≤ z < ∞. Third boundary value problem.
A semiinfinite circular cylinder is considered. The following conditions are prescribed:
w = f (r, z) at t = 0 (initial condition),
∂
r
w + k
1
w = g
1
(z, t) at r = R (boundary condition),
∂
z
w − k
2
w = g
2
(r, t) at z = 0 (boundary condition).
The solution w(r, z, t) is determined by the formula in the previous paragraph (for the
second boundary value problem) where
G(r, z, ξ, η, t) = G
1
(r, ξ, t) G
2
(z, η, t),
G
1
(r, ξ, t) =
1
πR
2
∞
X
n=1
µ
2
n
(k
2
1
R
2
+ µ
2
n
)J
2
0
(µ
n
)
J
0
µ
n
r
R
J
0
µ
n
ξ
R
exp
−
aµ
2
n
t
R
2
,
G
2
(z, η, t) =
1
2
√
πat
exp
−
(z − η)
2
4at
+ exp
−
(z + η)
2
4at ...