
806 SECOND-ORDER ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES
Solution:
w(r, ϕ) =−R
Z
β
0
f
1
(η)
∂
∂ξ
G(r, ϕ, ξ, η)
ξ=R
dη+
Z
R
0
f
2
(ξ)
1
ξ
∂
∂η
G(r, ϕ, ξ, η)
η=0
dξ
−
Z
R
0
f
3
(ξ)
1
ξ
∂
∂η
G(r, ϕ, ξ, η)
η=β
dξ+
Z
β
0
Z
R
0
Φ(ξ, η)G(r, ϕ, ξ, η)ξ dξ dη.
1
◦
. For β = π/n, where n is a positive integer, the Green’s function is expressed as
G(r, ϕ, ξ, η) =
n−1
X
k=0
G
1
(r, ϕ, ξ, 2kβ + η) − G
1
(r, ϕ, ξ, 2kβ − η)
,
G
1
(r, ϕ, ξ, η) =
1
4π
ln
r
2
ξ
2
− 2R
2
rξ cos(ϕ − η) + R
4
R
2
[r
2
− 2rξ cos(ϕ − η) + ξ
2
]
.
⊙ Literature: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).
2
◦
. For arbitrary β, the Green’s function is given by
G(r, ϕ, ξ, η) =
1
2π
ln
z
π/β
−
¯
ζ
π/β
R
2π/β
− (
¯
ζz)
π/β
z
π/β
− ζ
π/β
R
2π/β
− (ζz)
π/β ...