
478 SECOND-ORDER PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
where
G(x, y, z, ξ, η, ζ, t)=
2
l
1
l
2
√
πat
exp
−
(z−ζ)
2
4at
∞
X
n=1
sin
πnx
l
1
sin
πnξ
l
1
exp
−
π
2
n
2
at
l
2
1
×
1
2
+
∞
X
m=1
cos
πmx
l
2
cos
πmξ
l
2
exp
−
π
2
m
2
at
l
2
2
.
◮ Domain: 0 ≤ x ≤ l
1
, 0 ≤ y ≤ l
2
, 0 ≤ z < ∞. First boundary value problem.
A semiinfinite cylindrical domain of a rectangular cross-section is considered. The follow-
ing conditions are prescribed:
w = f (x, y, z) at t = 0 (initial condition),
w = g
1
(y, z, t) at x = 0 (boundary condition),
w = g
2
(y, z, t) at x = l
1
(boundary condition ),
w = g
3
(x, z, t) at y = 0 (boundary condition),
w = g
4
(x, z, t) at y = l
2
(boundary condition ),
w = g
5
(x,