
10.2. Poisson Equation ∆
3
w + Φ(x) = 0 869
◮ First boundary value problem.
The solution of the first boundary value problem for the Poisson equation
∆
3
w + Φ(r) = 0 (1)
in a domain V with the nonhomogeneous boundary condition
w = f (r) for r ∈ S
can be represented in the form
w(r) =
Z
V
Φ(ρ)G(r, ρ) dV
ρ
−
Z
S
f(ρ)
∂G
∂N
ρ
dS
ρ
. (2)
Here, G(r, ρ) is the Green’s function of the first boundary value problem,
∂G
∂N
ρ
is the deriva-
tive of the Green’s function with respect to ξ, η, ζ along the outward normal N to the bound-
ary S of the domain V . The integration is everywhere with respect to ξ, η, ζ.
The volume elements in solution (2) for basic coordinate systems are presented ...