Environmental Engineering Reference
In-Depth Information
Eq. (2.50) (
λ
m
=
λ
l
). The right-hand side of Eq. (2.50) reduces to
r
J
n
(
a
0
J
n
(
k
mn
r
)
J
n
(
k
ln
r
)
−
J
n
(
k
mn
r
)
k
ln
r
)
a
0
k
mn
J
n
J
n
J
n
k
ln
J
n
0
μ
(
n
)
μ
(
n
)
l
μ
(
n
)
μ
(
n
)
l
=
−
,
m
m
which is zero for the boundary condition of the first
[
R
(
a
0
)=
0
]
, the second
R
(
[
a
0
)=
0
]
and the third
[
R
(
a
0
)=
cR
r
(
a
0
)
with
c
as a constant
]
kinds. In arriv-
ing at the last case, we have used
J
n
ck
mn
J
n
J
n
ck
ln
J
n
)
r
=
a
0
=
μ
(
n
)
μ
(
n
)
μ
(
n
)
l
μ
(
n
)
l
cJ
n
(
=
k
mn
r
,
=
.
m
m
This, together with Eq. (2.49), shows that the Bessel function set is orthogonal in
[
0
,
with respect to the weight function
r
.
An operator satisfying
b
a
0
]
b
C
2
L
2
u
(
x
)
L
[
v
(
x
)]
d
x
−
v
(
x
)
L
[
u
(
x
)]
d
x
=
0
,
∀
u
(
x
)
,
v
(
x
)
∈
(
a
,
b
)
∩
[
a
,
b
]
a
a
is called a
self-conjugate operator
.
A corresponding eigenvalue problem of
L
0 that is subject to cer-
tain boundary conditions is called a
self-conjugate eigenvalue problem
.
Remark 4
. Here we prove the solution structure theorem in a polar coordinate sys-
tem, which was used in obtaining Eq. (2.45).
[
y
]
−
λρ
(
x
)
y
=
Theorem
The solution of
⎧
⎨
a
2
u
tt
=
Δ
u
(
r
,
θ
,
t
)+
F
(
r
,
θ
,
t
)
,
0
<
r
<
R
,
0
<
t
L
(
u
,
u
r
)
|
r
=
R
=
0
,
lim
r
→
0
|
u
(
r
,
θ
,
t
)
| <
∞
,
u
(
r
,
θ
+
2
π
,
t
)=
u
(
r
,
θ
,
t
)
,
(2.51)
⎩
u
(
r
,
θ
,
0
)=
Φ
(
r
,
θ
)
,
u
t
(
r
,
θ
,
0
)=
Ψ
(
r
,
θ
)
,
0
<
r
<
R
is
t
=
∂
∂
u
t
W
Φ
+
W
Ψ
(
r
,
θ
,
t
)+
W
F
τ
(
r
,
θ
,
t
−
τ
)
d
τ
,
0
where
u
2
=
W
Ψ
(
r
,
θ
,
t
)
is the solution for the case of
Φ
=
F
=
0
,
F
τ
=
F
(
r
,
θ
,
τ
)
,
and
2
2
Δ
=
∂
1
∂
1
∂
r
2
+
r
+
2
.
r
2
∂
r
∂
∂θ
Also note that
L
(
u
,
u
r
)
|
r
=
R
=
0 covers all three cases of boundary conditions.
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