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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