
9.2. Poisson Equation ∆
2
w = −Φ(x) 801
◮ Domain: 0 ≤ x ≤ a, 0 ≤ y ≤ b. First bound ary value problem.
A rectangle is considered. Boundary conditions are prescribed:
w = f
1
(y) at x = 0, w = f
2
(y) at x = a,
w = f
3
(x) at y = 0, w = f
4
(x) at y = b.
Solution:
w(x, y) =
Z
a
0
Z
b
0
Φ(ξ, η)G(x, y, ξ, η) dη dξ
+
Z
b
0
f
1
(η)
∂
∂ξ
G(x, y, ξ, η)
ξ=0
dη −
Z
b
0
f
2
(η)
∂
∂ξ
G(x, y, ξ, η)
ξ=a
dη
+
Z
a
0
f
3
(ξ)
∂
∂η
G(x, y, ξ, η)
η=0
dξ −
Z
a
0
f
4
(ξ)
∂
∂η
G(x, y, ξ, η)
η=b
dξ.
Two forms of representation of the Green’s function:
G(x, y, ξ, η) =
2
a
∞
X
n=1
sin(p
n
x) sin(p
n
ξ)
p
n
sinh(p
n
b)
H
n
(y, η) =
2
b
∞
X
m=1
sin(q
m
y) sin(q
m
η)
q
m
sinh(q
m
a)
Q
m
(x, ξ),
where
p
n
=
πn
a
, H
n
(y, η) =
(
sinh(p
n
η) sinh[p
n
(b − y)] for b ≥ y > η ≥ 0,
sinh ...