Environmental Engineering Reference
In-Depth Information
τ
(
)
(
,
)
P
cos
PM
n
u
(
M
)=
d
S
−
2
πτ
(
M
)
.
r
PM
S
To obtain a solution
u
that is continuous on
Ω
+
S
and satisfies
u
|
S
=
f
(
M
)
,we
should impose
u
(
M
)=
f
(
M
)
,
so that the boundary-value problem is transformed into the problem of seeking the
solution
τ
(
P
)
of
τ
(
P
)
cos
(
PM
,
n
)
f
(
M
)=
d
S
−
2
πτ
(
M
)
r
PM
S
or
1
2
τ
(
)=
(
,
)
τ
(
)
−
(
)
,
M
K
M
P
P
d
S
f
M
(7.160b)
π
S
cos
(
PM
,
n
)
where
K
(
M
,
P
)=
. This is a Fredholm integral equation of the second
r
PM
2
π
kind regarding
is available from Eq. (7.160b), the solution of the
Dirichlet internal problems can thus readily be obtained from the double-layer po-
tential (7.160a).
τ
(
P
)
.Once
τ
(
P
)
First External Boundary-Value Problems (Dirichlet External Problems)
Similar to the Dirichlet internal problems, we can use the double-layer potential as a
function that satisfies
∈
Ω
and
u
Ω
+
Δ
u
=
0,
M
|
S
=
f
(
M
)
, and is continuous in
S
.
The density function
τ
(
P
)
is determined such that
1
2
τ
(
M
)=
−
K
(
M
,
P
)
τ
(
P
)
d
S
+
f
(
M
)
.
(7.161)
π
S
Equation (7.161) is also a Fredholm integral equation of the second kind.
Second Internal Boundary-Value Problems (Neumann Internal Problems)
Assume that the single-layer potential with the undetermined density function
μ
(
P
)
μ
(
)
r
PM
d
S
P
(
)=
u
M
S
S
=
and
∂
u
is a function that satisfies
Δ
u
=
0,
M
∈
Ω
f
(
M
)
, and is continuous on
∂
n
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