Environmental Engineering Reference
In-Depth Information
7.6 Green Functions
In this section we introduce Green functions of the Laplace operator and discuss
their properties.
7.6.1 Green Function
The third Green formula reads (see Eq. (7.65))
u
1
r
d
S
1
4
∂
∂
1
r
∂
u
(
M
)
1
4
Δ
u
r
u
(
M
0
)=
−
(
M
)
−
−
d
Ω
,
(7.92)
π
n
∂
n
π
∂Ω
Ω
where
M
0
and
M
are two points in
Ω
, and their distance
r
is
2
2
2
r
=
(
x
−
x
0
)
+(
y
−
y
0
)
+(
z
−
z
0
)
.
If
u
(
M
)
is a harmonic function, Eq. (7.92) reduces to
u
1
r
d
S
1
4
)
∂
∂
1
r
∂
u
(
M
)
(
)=
−
(
−
.
u
M
0
M
(7.93)
π
n
∂
n
∂Ω
∂Ω
and
∂
u
(
M
)
Note that the right-hand side of Eq. (7.93) contains both
u
.
In order to use Eq. (7.93) for expressing solutions of Dirichlet or Neumann problems
of potential equations, we must eliminate
∂
(
M
)
|
∂Ω
∂
n
∂Ω
u
or
u
|
∂Ω
in Eq. (7.93).
∂
n
Consider a harmonic function
g
(
M
,
M
0
)
in
Ω
with
M
0
∈
Ω
as the parameter and
M
∈
Ω
as the variable. By letting
v
=
g
, the second Green formula yields
g
∂
d
S
u
u
∂
g
n
−
=
g
Δ
u
d
Ω
.
(7.94a)
∂
∂
n
∂Ω
Ω
For a harmonic function
u
(
M
, Eq. (7.94a) reduces to
)
g
∂
d
S
u
u
∂
g
n
−
=
0
.
(7.94b)
∂
∂
n
∂Ω
Subtracting Eq. (7.94b) from Eq. (7.93) yields
u
∂
∂
g
1
4
g
∂
d
S
1
u
u
(
M
0
)=
−
+
r
−
.
(7.95)
n
4
π
r
π
∂
n
∂Ω
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