Environmental Engineering Reference
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a
mn
and
c
mn
such that
∞
∑
n
=
0
,
m
=
1
(
a
mn
cos
n
θ
+
c
mn
sin
n
θ
)
J
n
(
k
mn
r
)=
0
=
=
and hence
a
mn
c
mn
0, by using (1) the completeness and the orthogonality of
{
,
θ
,
θ
, ··· ,
θ
,
θ
, ···}
(
=
,
, ···
)
[
−
π
,
π
]
1
cos
sin
cos
n
sin
n
n
1
2
in
, (2) the complete-
ness of
{
J
n
(
k
mn
r
)
}
in
[
0
,
a
0
]
for every fixed value of
n
, and (3) the orthogonality of
{
J
n
(
k
mn
r
)
}
in
[
0
,
a
0
]
with respect to the weight function
r
for every fixed value of
n
,
i.e.
a
0
J
n
(
k
mn
r
)
J
n
(
k
ln
r
)
r
d
r
=
0
,
m
=
l
.
0
To satisfy the initial condition
u
t
(
r
,
θ
,
0
)=
Ψ
(
r
,
θ
)
,
b
mn
and
d
mn
must be determined
such that
∞
∑
1
(
b
mn
ω
mn
cos
n
θ
+
d
mn
ω
mn
sin
n
θ
)
J
n
(
k
mn
r
)=
Ψ
(
r
,
θ
)
.
n
=
0
,
m
=
Thus we obtain the solution of PDS (2.37) for the case
f
=
ϕ
=
0
⎧
⎨
∞
∑
u
=
W
Ψ
(
r
,
θ
,
t
)=
1
(
b
mn
cos
n
θ
+
d
mn
sin
n
θ
)
J
n
(
k
mn
r
)
sin
ω
mn
t
,
n
=
0
,
m
=
π
a
0
1
=
Ψ
(
,
θ
)
(
)
,
b
m
0
d
θ
r
J
0
k
m
0
r
r
d
r
2
πω
m
0
M
m
0
−
π
0
(2.44)
π
a
0
⎩
1
πω
mn
M
mn
b
mn
=
d
θ
Ψ
(
r
,
θ
)
J
n
(
k
mn
r
)
r
cos
n
θ
d
r
,
−
π
0
π
a
0
1
πω
mn
M
mn
d
mn
=
d
θ
Ψ
(
r
,
θ
)
J
n
(
k
mn
r
)
r
sin
n
θ
d
r
,
−
π
0
a
0
(
n
m
a
0
,
M
mn
J
n
(
=
μ
=
)
where
k
mn
k
mn
r
r
d
r
is the normal square of eigenfunc-
0
tion set
.
Finally, the solution of PDS (2.37) follows from the solution structure theorem,
{
J
n
(
k
mn
r
)
}
t
=
∂
∂
u
t
W
Φ
+
W
Ψ
(
r
,
θ
,
t
)+
W
f
τ
(
r
,
θ
,
t
−
τ
)
d
τ
.
(2.45)
0
Remark 1
. The solution of PDS (2.37) for the other two kinds of boundary condi-
tions can also be written in the form of Eqs. (2.44) and (2.45). However, the
μ
(
n
)
m
1
a
0
xJ
n
(
,
J
n
(
are different and are the positive zero-points of
J
n
(
x
)
x
)
and
x
)+
hJ
n
(
x
)
|
r
=
a
0
=
|
r
=
a
0
=
+
|
r
=
a
0
=
for the first (
u
0), the second (
u
r
0) and the third ((
u
r
hu
)
0)
μ
(
0
)
1
kind of boundary conditions, respectively. The only exception is
=
0forthe
case of a boundary condition of the second kind.
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