
10.2. Poisson Equation ∆
3
w + Φ(x) = 0 879
where
r
(1)
nm
=
p
(x −ξ − 2na)
2
+ (y − η −2ma)
2
+ (z − ζ)
2
,
r
(2)
nm
=
p
(x + ξ − 2na)
2
+ (y − η −2ma)
2
+ (z − ζ)
2
,
r
(3)
nm
=
p
(x −ξ − 2na)
2
+ (y + η −2ma)
2
+ (z − ζ)
2
,
r
(4)
nm
=
p
(x + ξ − 2na)
2
+ (y + η −2ma)
2
+ (z − ζ)
2
.
⊙ Literature: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
◮ Domain: 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z < ∞. First boundary value problem.
A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary
conditions are prescribed:
w = f
1
(y, z) at x = 0, w = f
2
(y, z) at x = a,
w = f
3
(x, z) at y = 0, w = f
4
(x, z) at y = b,
w = f
5
(x, y) at z = 0.
Green’s function:
G(x, y, z, ξ, η, ζ) =
4
ab
∞
X
n