Environmental Engineering Reference
In-Depth Information
3.3.1 Existence
We have obtained the nominal solution of PDS (3.4) in Section 3.2 by using the
method of separation of variables or the Fourier method,
⎧
⎨
+
∞
k
=
1
b
k
e
−
(
2
t
sin
k
π
x
k
π
a
l
)
u
=
,
l
(3.18)
l
0
ϕ
(
⎩
2
l
sin
k
π
x
b
k
=
x
)
d
x
.
l
To prove the series solution (3.18) satisfies PDS (3.17) so that it is the classical
solution of PDS (3.17), we should show the convergence of series (3.18) and the
uniform convergence of the resulting series after taking the derivative with respect
to
t
and the second derivative with respect to
x
, term by term. Let
D
be the domain:
0
≤
x
≤
l
,
0
≤
t
≤
T
(
T
is an arbitary positive constant) and
u
k
(
x
,
t
)
is the general
term of series (3.18). We have
k
2
+
∞
k
=
1
∂
u
k
+
∞
k
=
1
−
π
a
2
t
sin
k
π
x
b
k
e
−
(
k
π
a
)
t
=
,
(3.19)
l
∂
l
l
k
2
+
∞
k
=
1
∂
+
∞
k
=
1
−
2
u
k
∂
l
2
t
sin
k
π
x
b
k
e
−
(
k
π
a
)
x
2
=
.
(3.20)
l
l
For any natural number
k
,
≤
l
0
|
ϕ
(
sin
k
π
x
2
l
1
,
|
b
k
|≤
x
)
|
d
x
=
constant
,
l
k
N
e
k
2
=
and for any fixed natural number
N
,
lim
0, there exists a constant
C
such
k
→
+
∞
that
≤
≤
2
u
k
∂
C
k
2
,
∂
u
k
∂
C
k
2
,
∂
C
k
2
.
|
u
k
(
x
,
t
)
|≤
x
2
t
+
∞
k
=
1
C
k
2
is also convergent. Therefore, series (3.18), (3.19) and (3.20) are all uni-
formly convergent so we can take derivatives of series (3.18) term by term. Thus the
solution (3.18) can satisfy PDS (3.17).
Remark 1.
The uniform convergence of a series with function terms is not sufficient
for the term by term differentiation of the series. It is, however, ensured by the
uniform convergence of the series itself and all the resulting series from term by
term differentiation.
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