Environmental Engineering Reference
In-Depth Information
Appendix D
Eigenvalue Problems
In this appendix, we discuss eigenvalue problems of second-order equations
y
(
y
+
a
(
x
)
x
)+
b
(
x
)
c
(
x
)
y
(
x
)+
λ
y
(
x
)=
0
.
(D.1)
Here
λ
is a parameter,
a
(
x
)
,
b
(
x
)
and
c
(
x
)
are functions of
x
.
D.1 Regular Sturm-Liouville Problems
Let
b
(
x
)
a
(
x
)
d
x
c
(
x
)
p
(
x
)=
e
,
q
(
x
)=
−
p
(
x
)
,
a
(
x
)
p
p
(
x
)
d
d
x
d
d
x
ρ
(
x
)=
)
,
L
=
−
(
x
)
+
q
(
x
)
.
a
(
x
Equation (D.1) is thus reduced to
Ly
−
λρ
(
x
)
y
=
0
.
(D.2)
It is called the
Sturm-Liouville equation
,the
S-L
equation for short.
In order for it to have nontrivial solutions of Eq. (D.2) in interval
[
a
,
b
]
,
p
(
x
)
,
q
(
x
)
and
ρ
(
x
)
should satisfy
p
(
(1)
p
(
x
)
,
x
)
,
q
(
x
)
ρ
(
x
)
∈
C
[
a
,
b
]
.
(D.3)
(2)
p
(
x
)
>
0
,
q
(
x
)
≥
0
,
ρ
(
x
)
>
0in
[
a
,
b
]
.
(D.4)
The boundary conditions are homogeneous and separable at both ends, i.e.,
)+
α
2
y
(
1
2
α
1
y
(
a
a
)=
0
,
α
+
α
=
0
.
(D.5)
)+
β
2
y
(
1
2
β
1
y
(
b
b
)=
0
,
β
+
β
=
0
.
(D.6)
where
α
i
and
β
i
(
i
=
1
,
2
)
are known constants.
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