Environmental Engineering Reference
In-Depth Information
is
1
2
1
G
(
x
,
y
;
x
0
,
y
0
)=
ln
z
0
)
|
.
π
|
w
(
z
,
G
(
x
,
y
;
x
0
,
y
0
)
is the required Green function.
Remark 5
. Two-dimensional problems can also sometimes be solved directly by
using a conformal transformation. For example, consider
⎧
⎨
2
T
∂
x
2
2
T
∂
y
2
∂
+
∂
=
0
,
T
0
, |
x
| <
a
,
⎩
T
|
y
=
0
=
0
, |
x
| >
a
.
ln
z
−
a
z
=
=
ξ
+
Using the conformal transformation
w
, it is transformed into a
boundary-value problem of an ordinary differential equation
⎧
⎨
i
η
+
a
d
2
T
dη
=
0
2
⎩
T
|
η
=
0
=
0
,
T
|
η
=
π
=
T
0
.
Its solution is
Im
ln
z
T
0
π
T
0
π
−
a
T
(
x
,
y
)=
η
=
z
+
a
T
0
π
T
0
π
2
ay
=
Im
[
ln
(
z
−
a
)
−
ln
(
z
+
a
)] =
arctan
a
2
.
x
2
+
y
2
−
Search WWH ::
Custom Search