
104 Applied calculus of variations for engineers
L(G(P, Q)) = δ(P − Q),
where δ is the Dirac function. Let us work with the two-dimensional Poisson’s
equation of the form
Δu(x, y)=f(x, y).
Here L(x, y)=Δ=∇
2
and its Green’s function is
G(P, Q)=
1
2π
ln(r),
where
r =
(x
p
− x
q
)
2
+(y
p
− y
q
)
2
.
Green’s theorem’s generic form (a consequence of Gauss’ divergence theorem)
may be written as
Ω
(u∇
2
v − v∇
2
u)dΩ=
Γ
(u
∂v
∂n
− v
∂u
∂n
)dΓ.
Using Green’s function in place of v we obtain
Ω
(u∇
2
G − G∇
2
u)dΩ=
Γ
(u
∂G
∂n
− G
∂u
∂n
)dΓ.
By definition
L(G(P, Q)) = ∇
2
G(P, Q)=δ(P − Q),
and due to the characteristics of the Dirac function the first term on the
left-hand side reduces to u(x, y). Substituting ...