Environmental Engineering Reference
In-Depth Information
Consider a function
g
that satisfies
⎧
⎨
Δ
g
=
0
,
M
∈
Ω
,
1
⎩
g
|
∂Ω
=
r
,
4
π
∂Ω
and removes
∂
u
in Eq. (7.95). Eq. (7.93) reduces to
∂
n
u
∂
G
u
(
M
0
)=
−
n
d
S
,
(7.96)
∂
∂Ω
where
1
G
(
M
,
M
0
)=
r
−
g
(
M
,
M
0
)
(7.97)
4
π
is called the
Green function of the Laplace operator in
Ω
for the first kind of
boundary-value problems
.
Let
so
u
is the solution of Poisson equations. Subtracting
Eq. (7.94a) from Eq. (7.92) yields
Δ
u
=
F
(
M
)(
M
∈
Ω
)
u
∂
G
u
(
M
0
)=
−
n
d
S
−
G
(
M
,
M
0
)
F
(
M
)
d
Ω
.
(7.98)
∂
Ω
∂Ω
Once the Green function
G
(
M
,
M
0
)
is available, the solutions of
Δ
u
=
0
,
M
∈
Ω
,
(7.99)
u
|
∂Ω
=
f
(
M
)
,
Δ
u
=
F
(
M
)
,
M
∈
Ω
,
(7.100)
u
|
∂Ω
=
0
and
Δ
u
=
F
(
M
)
,
M
∈
Ω
,
(7.101)
u
|
∂Ω
=
f
(
M
)
are thus
)
∂
G
u
(
M
0
)=
−
f
(
M
n
d
S
,
u
(
M
0
)=
−
G
(
M
,
M
0
)
F
(
M
)
d
Ω
,
∂
Ω
∂Ω
and
)
∂
G
(
)=
−
(
−
(
,
)
(
)
Ω
,
u
M
0
f
M
n
d
S
G
M
M
0
F
M
d
∂
Ω
∂Ω
Search WWH ::
Custom Search