Environmental Engineering Reference
In-Depth Information
Once the limits of
u
,
v
and their derivatives at
x
=
±
1 are finite or bounded we have
b
−
ε
2
0or
b
lim
ε
1
→
+
ε
1
(
vLu
−
uLv
)
d
x
=
a
(
vLu
−
uLv
)
d
x
=
0
,
a
0
ε
2
→
0
so that the self-conjugate relation (D.8) is valid. Therefore, Eq. (D.14) with bounded
y
and
y
(
±
(
±
1
)
1
)
forms a singular S-L problem.
0, it is similar to
Eq. (D.14). Here the singular point is the left end of the interval
When
γ
=
0, Eq. (D.15) is similar to Eq. (D.12). When
γ
=
[
0
,
a
0
]
. Thus the
Lagrange equality reads
a
0
(
ε
)
u
(
ε
)
v
(
ε
)
,
lim
ε
→
(
vLu
−
uLv
)
d
r
=
lim
ε
→
0
p
v
(
ε
)
−
u
(
ε
)
0
ε
where
u
and
v
are assumed to satisfy boundary condition (D.6) at
r
=
a
0
so that
a
0
)+
β
2
u
(
a
0
)+
β
2
v
(
β
1
u
(
a
0
)=
0
,
β
1
v
(
a
0
)=
0
.
Note that
lim
ε
→
0
p
(
ε
)=
0
.
Therefore if
u
,
v
and their derivatives are bounded at
r
=
0, then the self-conjugate
relation holds, i.e.
a
0
(
vLu
−
uLv
)
d
r
=
0
.
0
Thus, under the boundary condition
R
(
<
∞
,
a
0
)+
β
2
R
(
|
R
(
0
)
| <
∞
,
0
)
β
1
R
(
a
0
)=
0
,
Eq.(D.15) forms a singular S-L problem.
The self-conjugate relation can be used to show properties of eigenvalues of sin-
gular S-L problems. Examples of such properties are: (1) all eigenvalues are real,
and (2) eigenfunction sets are orthogonal with respect to the weight function
ρ
(
x
)
.
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