Environmental Engineering Reference
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tion (7.164) such that
⎛
⎞
1
4
F
(
P
)
⎝
−
⎠
=
Δ
d
Ω
F
(
M
)
.
π
r
PM
Ω
Therefore, the particular solution is
1
4
F
(
P
)
u
∗
=
−
d
Ω
.
(7.168)
π
r
PM
Ω
Note that the
u
∗
(
M
)
is obtained on the assumption that the solution
u
(
M
)
of the
Poisson equation exists and has continuous second derivatives on
S
. A verifica-
tion by substituting into the equation can serve as the justification that Eq. (7.168)
is indeed the particular solution of the Poisson equation.
Therefore boundary-value problems of Poisson equations can be transformed
into boundary-value problems of Laplace equations, which are then reduced into
the four integrals equations (7.160')-(7.163'). These integral equations have been
studied extensively in literature.
Ω
+
7.9.4 Two-Dimensional Potential Equations
Dirichlet internal and external problems of Laplace equations in two-dimensional
space read
D
,
Δ
u
=
0
,
M
∈
D
,
Δ
u
=
0
,
M
∈
and
(7.169)
|
C
=
(
)
|
C
=
(
)
,
u
f
M
u
f
M
respectively. Here
D
and
D
are the plane domains inside of closed boundary curve
C
and outside of
C
, respectively. The Neumann internal and external problems
of Laplace equations can also be written out simply by replacing
u
|
C
=
f
(
M
)
in
C
=
Eq. (7.169) by
∂
u
.
Following a similar approach for three-dimensional cases, we can transform
these four problems into four integral equations regarding
f
(
M
)
∂
n
τ
(
M
)
or
μ
(
M
)
)=
C
K
1
π
τ
(
M
(
M
,
P
)
τ
(
P
)
d
s
−
f
(
M
)
,
)=
−
C
K
1
π
τ
(
M
(
M
,
P
)
τ
(
P
)
d
s
+
f
(
M
)
,
(7.170)
)=
−
C
K
1
π
μ
(
M
(
P
,
M
)
μ
(
P
)
d
s
+
f
(
M
)
,
)=
C
K
1
π
μ
(
M
(
P
,
M
)
μ
(
P
)
d
s
+
f
(
M
)
,
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