Environmental Engineering Reference
In-Depth Information
Similar to its three-dimensional counterpart, the two-dimensional Green function
can also be defined by
1
2
ln
1
G
(
M
,
M
0
)=
r
+
g
(
M
,
M
0
)
.
π
7.7 Method of Green Functions for Boundary-Value Problems of
the First Kind
Once the Green function is available in domain
Ω
, the solution of
Δ
u
=
F
(
M
)
,
M
∈
Ω
,
(7.108)
u
|
∂Ω
=
f
(
M
)
.
is thus
)
∂
G
u
(
M
0
)=
−
f
(
M
n
d
S
−
G
(
M
,
M
0
)
F
(
M
)
d
Ω
.
(7.109)
∂
Ω
∂Ω
In this section we develop the Green functions for some domains
Ω
.
7.7.1 Mirror Image Method for Finding Green Functions
For some domains with certain kinds of symmetry, we can find the electric potential
g
of the electric field generated by induced electric charges. Thus we may
find the Green function, using their physical implications discussed in Section 7.6.
The mirror image method attempts to locate a mirror (symmetric) point
M
1
of
M
0
outside the hollow conductor. Image that there is a point electric charge with ca-
pacity not necessarily equal to
(
M
,
M
0
)
ε
at
M
1
such that its induced electric field completely
1
neutralizes the
π
r
due to the charge at
M
0
, and the electric potential becomes zero
4
on
) of electric field due to the charge at
M
1
is known, we can obtain the Green function
∂Ω
. Once the electric potential
g
(inside
Ω
1
G
(
M
,
M
0
)=
r
−
g
(
M
,
M
0
)
.
(7.110)
4
π
7.7.2 Examples Using the Method of Green Functions
:
x
2
y
2
z
2
R
2
Example 1.
Find the Green function in a spherical domain
Ω
+
+
<
and the solution of the Dirichlet problem in
Ω
.
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