Environmental Engineering Reference
In-Depth Information
Proof.
Let
y
(
x
)
be the eigenfunction corresponding to eigenvalue
λ
,i.e.,
py
−
qy
+
λρ
(
x
)
y
=
0
.
Multiplying it by
y
and integrating over
[
a
,
b
]
leads to
b
a
ρ
(
b
b
b
ypy
py
2
qy
2
d
x
b
a
+
y
2
d
x
py
)+
qy
2
d
x
λ
x
)
=
−
y
d
(
=
−
+
a
a
a
b
py
2
qy
2
d
x
=
−
α
1
)+
β
1
y
2
y
2
α
2
p
(
a
)
(
a
β
2
p
(
b
)
(
b
)+
+
≥
0
,
a
in which Eqs. (D.3)-(D.6) have been used. Note that
b
a
ρ
(
y
2
d
x
x
)
>
0
.
Thus
0. By
following a similar approach, we can also show that the eigenvalues of periodic S-L
problems are also positive semi-definite.
λ
≥
0. When
α
2
=
0or
β
2
=
0,
y
(
a
)=
0or
y
(
b
)=
0. We also have
λ
≥
Стклов
(
)
Property 5 (
Expansion Theorem).
If
f
x
has continuous first deriva-
[
,
]
tive and piece-wise continuous second derivative in
and satisfies the boundary
conditions of S-L problems, it can be expanded into an absolutely and uniformly
convergent function series by using the eigenfunction set
a
b
{
y
n
(
x
)
}
,i.e.
d
x
b
a
b
∞
k
=
1
c
k
y
k
(
x
)
,
y
k
(
f
(
x
)=
c
k
=
f
(
x
)
y
k
(
x
)
ρ
(
x
)
x
)
ρ
(
x
)
d
x
.
a
This is called the
generalized Fourier series
.The
c
k
are called the
generalized
Fourier coefficients.
Therefore the
{
y
n
(
x
)
}
form a complete and orthogonal base in
[
a
,
b
]
.
Remark
.If
f
only satisfies the Dirichlet condition, the generalized Fourier
series converges to
f
(
x
)
(
x
)
at a continuous point
x
and it converges to
[
f
(
x
0
−
0
)
)]
2 at a discontinuous point
x
0
.
+
(
+
f
x
0
0
D.4 Singular S-L Problems
Conditions (D.3) and (D.4) are often not satisfied by the coefficients of second-order
linear equations which arise in applications. For example, the Legendre equation in
Chapter 2
d
d
x
d
y
d
x
x
2
x
2
(
1
−
)
+
λ
y
=
0
,
p
(
x
)=
1
−
,
(D.13)
(
)=
,
ρ
(
)=
q
x
0
x
1
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