Environmental Engineering Reference
In-Depth Information
cannot satisfy
p
(
x
)
>
0in
[
−
1
,
1
]
. The associated Legendre equation
m
2
d
d
x
d
y
d
x
x
2
(
1
−
)
−
x
2
y
+
λ
y
=
0
,
(D.14)
1
−
m
2
x
2
p
(
x
)=
1
−
,
q
(
x
)=
x
2
,
ρ
(
x
)=
1
1
−
also cannot satisfy
p
(
x
)
>
0 or the continuous requirement for
q
(
x
)
in
[
−
1
,
1
]
.The
Bessel equation
r
d
R
d
r
2
r
R
d
d
r
−
γ
+
λ
rR
=
0
,
p
(
r
)=
r
,
(D.15)
2
r
q
(
r
)=
γ
,
ρ
(
r
)=
r
does not satisfy conditions (D.3) and (D.4) either because, in
[
0
,
a
0
]
,
p
(
0
)=
0,
r
→
0
q
lim
(
r
)=
∞
and
ρ
(
0
)=
0.
Equation (D.2) is called the
singular S-L equation
if one of following three con-
ditions is valid:
1. the domain is semi-infinite or infinite,
2.
p
(
x
)
or
q
(
x
)
is vanished at one or two ends of the finite domain
[
a
,
b
]
,
3. the pole of
q
(
x
)
or
ρ
(
x
)
appears at the end point of finite domain
[
a
,
b
]
;and
C
1
a.
p
(
x
)
∈
[
a
,
b
]
,
q
(
x
)
,
ρ
(
x
)
∈
C
(
a
,
b
)
,
b.
in
(
a
,
b
)
,
p
(
x
)
>
0,
q
(
x
)
≥
0and
ρ
(
x
)
>
0.
Equation (D.13) satisfies condition 2. Equation (D.14) satisfies both conditions 2
and 3. Equation (D.15) satisfies conditions 2 and 3 when
γ
=
0. Therefore, they are
all singular S-L equations.
A singular S-L equation with a boundary condition that satisfies the self-conju-
gate relation is called a
singular S-L problem
.
Boundary conditions that satisfies the self-conjugate relation are singularity-
dependent. Here we briefly discuss three commonly-used singular S-L equations.
In Eq. (D.13),
p
1. The Lagrange equality (D.7)
shows that the self-conjugate relation (D.8) is valid in
(
x
)=
0 at the end points
x
=
±
[
−
1
,
1
]
if
u
,
v
and their deriva-
and
y
(
±
tives at
x
=
±
1 are bounded. Therefore Eq. (D.12) with bounded
y
(
±
1
)
1
)
forms a singular S-L problem.
In Eq.(D.14),
p
(
)=
=
±
=
±
x
0 at the end points
x
1.
x
1 are also the poles of
(
)
q
x
. The Lagrange equality (D.7) reads, in generalized integrals,
b
−
ε
2
−
)
u
(
)
1
−
ε
2
−
v
(
lim
ε
+
ε
1
(
vLu
−
uLv
)
d
x
=
lim
p
(
x
x
)
v
(
x
)
−
u
(
x
)
x
+
ε
1
.
1
a
→
0
ε
→
0
1
1
ε
→
0
ε
→
0
2
2
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