
508 SECOND-ORDER PARABOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
◮ Domain: 0 ≤ r < ∞, 0 ≤ ϕ ≤ ϕ
0
, 0 ≤ z < ∞. First boun dary value problem.
The upper half of a dihedral angle is considered. The following conditions are prescribed:
w = f (r, ϕ, z) at t = 0 (initial condition),
w = g
1
(r, z, t) at ϕ = 0 (boundary condition),
w = g
2
(r, z, t) at ϕ = ϕ
0
(boundary condition ),
w = g
3
(r, ϕ, t) at z = 0 (boundary condition).
Solution:
w(r, ϕ, z, t) =
Z
∞
0
Z
ϕ
0
0
Z
∞
0
f(ξ, η, ζ) G(r, ϕ, z, ξ, η, ζ, t)ξ dξ dη dζ
+ a
Z
t
0
Z
∞
0
Z
∞
0
g
1
(ξ, ζ, τ )
1
ξ
∂
∂η
G(r, ϕ, z, ξ, η, ζ, t − τ)
η=0
dξ dζ dτ
− a
Z
t
0
Z
∞
0
Z
∞
0
g
2
(ξ, ζ, τ )
1
ξ
∂
∂η
G(r, ϕ, z, ξ, η, ζ, t − τ)
η=ϕ
0
dξ dζ dτ
+ a
Z
t
0
Z
ϕ
0
0
Z
∞
0
g