Environmental Engineering Reference
In-Depth Information
3. Boundary Condition of the Third Kind
Since
R
(
a
0
)=
−
hR
(
a
0
)
and
R
(
0
)=
J
n
(
0
)=
0
(
n
=
1
,
2
, ···
)
, Eq. (2.47) yields
⎡
⎣
⎤
⎦
J
n
2
a
0
2
n
2
+
(
a
0
h
)
−
μ
(
n
)
M
mn
=
1
.
2
m
μ
(
n
)
m
When
n
0,
M
m
0
is also available by using the above formula.
Remark 3
. Here we prove the orthogonality of the Bessel function set in
=
[
0
,
a
0
]
with
respect to the weight function
r
used in developing Eq. (2.44).
Consider the S-L problem under some boundary conditions
L
[
y
]
−
λρ
(
x
)
y
=
0
,
(2.48)
where
λ
is a parameter,
L
is an operator defined by
p
d
d
x
d
d
x
L
=
−
(
x
)
+
q
(
x
)
,
a
<
x
<
b
.
By the definition of eigenvalues and eigenfunctions, we have
L
[
y
m
(
x
)] =
λ
m
ρ
(
x
)
y
m
(
x
)
,
L
[
y
n
(
x
)] =
λ
n
ρ
(
x
)
y
n
(
x
)
,
where
λ
m
and
λ
n
are two distinct eigenvalues of Eq. (2.48)
(
λ
=
λ
)
,
y
m
(
x
)
and
m
n
y
n
are the corresponding eigenfunctions. Subtracting the latter from the former
after multiplying the former by
y
n
(
x
)
(
x
)
and the latter by
y
m
(
x
)
, we obtain
b
b
y
n
(
x
)
L
[
y
m
(
x
)]
d
x
−
y
m
(
x
)
L
[
y
n
(
x
)]
d
x
a
a
(2.49)
b
=(
λ
m
−
λ
n
)
y
m
(
x
)
y
n
(
x
)
ρ
(
x
)
d
x
.
a
C
2
L
2
Also, the Lagrange equality requires, for all
u
(
x
)
,
v
(
x
)
∈
(
a
,
b
)
∩
[
a
,
b
]
,
b
b
u
(
x
)
L
[
v
(
x
)]
d
x
−
v
(
x
)
L
[
u
(
x
)]
d
x
a
a
(2.50)
p
b
a
.
u
(
v
(
=
(
x
)[
x
)
v
(
x
)
−
u
(
x
)
x
)]
Here
L
2
stands for a function group which is both square integrable and square
integrable with respect to a weight function
[
a
,
b
]
. The Lagrange equality comes
directly from an application of integration by parts and can be found in Appendix D.
Equation (2.42) (or Eq. (2.46)) is a special case of Eq. (2.48) at
p
ρ
(
x
)
(
r
)=
ρ
(
r
)=
r
,
(0
<
r
<
a
0
). Take two eigenfunctions
J
n
(
k
mn
r
)
and
J
n
(
k
ln
r
)
as
u
(
x
)
and
v
(
x
)
in
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