
10.2. Poisson Equation ∆
3
w + Φ(x) = 0 889
where
r = {x, y, z}, x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ
r
0
= {x
0
, y
0
, z
0
}, x
0
= ξ s in η cos ζ, y
0
= ξ s in η sin ζ, z
0
= ξ cos η.
⊙ Literature: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin et al. (1964), B. M. Budak, A. A. Samarskii,
and A. N. Tikhonov (1980).
◮ Domain: 0 ≤ r ≤ R, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π. Second boundary value problem.
A spherical domain is considered. A boundary condition is prescribed:
∂
r
w = f (ϕ, θ) at r = R.
Green’s function:
G(r, θ, ϕ, ξ, η, ζ) =
1
4π
1
|r −r
0
|
+
R
|r
0
||r
1
|
+
1
R
ln
2R
2
R
2
+ |r
0
||r
1
| − (r · r
0
)
,
where
|r −r
0
| =
p
r
2
− 2rξ cos γ + ξ
2
, |r
0
||r
1
| =
p
r
2
ξ
2
− 2R
2
rξ cos γ + R
4
,
|r
0
| = ξ