Environmental Engineering Reference
In-Depth Information
The
conjugate operator
of
L
is defined by
2
2
=
∂
x
2
−
∂
)
∂
∂
)
∂
∂
M
y
2
−
a
(
x
,
y
x
−
b
(
x
,
y
y
+
c
(
x
,
y
)
.
(5.2)
∂
∂
The conjugate operator of
M
is clearly
L
. Therefore, the
L
and the
M
are
mutually
conjugate operators
.
If
Lu
Mu
, in particular, the
L
(or the
M
) is called the
self-conjugate operator
.
Such self-conjugate operators can be used to define generalized solutions and solve
PDS.
For any twice differentiable functions
u
=
(
x
,
y
)
and
v
(
x
,
y
)
, we have, by the rules of
differentiation,
=
∂
P
x
+
∂
Q
vLu
−
uMv
y
,
∂
∂
(
,
)=(
)
−
(
−
)
(
,
)=
−
(
)
+(
+
)
where
P
x
y
uv
2
v
x
av
u
,
Q
x
y
uv
2
v
y
bv
u
. Thus, for a
x
y
plane domain
D
,
∂
d
P
x
+
∂
Q
(
vLu
−
uMv
)
d
σ
=
σ
∂
∂
y
D
D
C
−
(5.3)
=
[
(
,
)+
(
,
)]
=
+
,
P
cos
n
x
Q
cos
n
y
d
s
Qdx
Pdy
C
where
C
is the positive-directed boundary curve of
D
and
n
is the outer normal of
C
. This is called the
generalized Green formula
.
5.1.2 Cauchy Problems and Riemann Functions
We aim to solve the PDS with its CDS specified on curve
c
:
L
u
=
(
,
)
,
f
x
y
(5.4)
|
c
=
ϕ
(
)
,
|
c
=
ψ
(
)
,
u
x
u
n
x
where
u
n
is the normal derivative of
u
. It reduces to the normal Cauchy problem
when
y
=
t
and
c
is taken as the straight line
t
=
0.
To find the solution of (5.4) at any point
M
0
(
x
0
,
y
0
)
,
u
(
x
0
,
y
0
)
, construct two char-
acteristic curves passing through point
M
0
:
x
y
0
.The
two characteristic curves intersect with curve
c
at points
M
1
and
M
2
. The domain
enclosed by
M
0
M
1
,
+
y
=
x
0
+
y
0
,
x
−
y
=
x
0
−
M
1
M
2
and
M
2
M
0
is denoted by
Δ
M
0
(Fig. 5.1).
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