Environmental Engineering Reference
In-Depth Information
Finally, we obtain Eq. (7.133) by letting
is a har-
monic function in
D
, in particular, we obtain the
fundamental integral formula of
harmonic functions in unbounded plane domains
ε
→
0and
R
→
∞
.When
u
(
M
)
u
∂
∂
ln
1
r
ln
1
r
∂
d
s
1
2
u
u
(
M
0
)=
−
,
(7.134)
π
n
∂
n
∂
D
=
where
r
r
M
0
M
is the distance between
M
0
and
M
. Therefore, we may express the
value of harmonic functions at any point in unbounded domains by using a line
integral over the domain boundary of the function and its normal derivative.
Let
v
be a harmonic function of two variables. The second Green
formula in a plane leads to
=
g
(
M
,
M
0
)
g
∂
d
s
u
u
∂
g
n
−
−
g
Δ
u
d
σ
=
0
.
(7.135)
∂
∂
n
D
∂
D
Subtracting Eq. (7.135) from Eq. (7.133) yields
u
g
1
2
g
∂
d
s
)
∂
∂
1
2
ln
1
r
ln
1
u
u
(
M
0
)=
(
M
−
+
r
−
n
π
π
∂
n
D
∂
g
1
2
ln
1
r
+
−
σ
.
Δ
u
d
π
D
Similar to the case of a three-dimensional unbounded domain, let
g
(
M
,
M
0
)
be the
solution of
⎧
⎨
M
0
∈
Ω
,
Δ
g
=
0
,
(7.136)
1
2
ln
1
⎩
g
|
∂Ω
=
r
,
r
=
r
MM
0
is distance between
M
and
M
0
.
π
The solution of the external plane Dirichlet problem
D
,
=
(
)
,
∈
Δ
u
F
M
M
(7.137)
u
|
∂
D
=
f
(
M
)
.
is thus
)
∂
G
(
)=
−
(
−
(
,
)
(
)
σ
,
u
M
0
f
M
n
d
s
G
M
M
0
F
M
d
(7.138)
∂
∂
D
D
where
n
is the external normal of the domain boundary and
1
2
ln
1
G
(
M
,
M
0
)=
r
−
g
(
M
,
M
0
)
.
(7.139)
π
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