Environmental Engineering Reference
In-Depth Information
Definition 2 comes from a similar argument to that used in Definition 1, by elim-
inating u
in Eq. (7.92). However, PDS (7.105) has no solution. In heat conduc-
tion, the boundary condition G
| ∂Ω
n ∂Ω =
0 means that the boundary is well-insulated.
The heat generated by the internal source
cannot flow through the bound-
ary. Thus we cannot have a steady temperature field so the G
δ (
M
,
M 0 )
(
M
,
M 0 )
satisfying
PDS (7.105) does not exist.
Definition 3. Let
Ω
be a bounded domain with a piecewise smooth boundary
∂Ω
,and g
(
M
,
M 0 )
satisfies
Δ
g
=
0
,
M
Ω ,
g ∂Ω =
n r + σ
g
1
4
1
r
n + σ
,
σ >
0
.
π
(
,
)
Then G
M
M 0
is defined by
1
G
(
M
,
M 0 )=
r
g
(
M
,
M 0 )
4
π
or
Δ
G
(
M
,
M 0 )= δ (
M
M 0 ) ,
M
Ω ,
G ∂Ω =
(7.106)
G
n + σ
0
.
and is called the Green function of the three-dimensional Laplace operator in
Ω
for
the third kind of boundary-value problems .
The Green functions defined by Eq. (7.104) can be obtained for some simple
regular domain
, and can be used to solve Dirichlet problems of potential equa-
tions. However, the Green functions defined by Eq. (7.105) and (7.106) are either
non-existent or very difficult to find. Therefore, we use the Green functions only for
solving the Dirichlet problems in some simple regular domains, and we use the other
methods such as the transformation of potential equations into integral equations for
the Neumann and Robin problems.
Ω
7.6.2 Properties of Green Functions of the Dirichlet Problems
with the same order of 1
r
1. G
(
M
,
M 0 )
tends to
+
as M
M 0 .
Proof. Note that r
0as M
M 0 .Since g
(
M
,
M 0 )
is a harmonic function in
Ω
1
and lim
M M 0
=+
, g
(
M
,
M 0 )
is continuous at M 0 . Thus
4
π
r
1
4
rg
1
4
lim
M
G
/ (
1
/
r
)=
lim
M
π
=
π .
M 0
M 0
 
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