Environmental Engineering Reference
In-Depth Information
Existence Theorem.
Both series (2.20) and series
u
x
,
u
xx
,
u
t
,
u
tt
, constructed by
taking derivatives of series (2.20) term by term, converge uniformly in
[
0
,
l
]
×
[
0
,
T
]
,
t
∈
[
0
,
T
]
with
T
as an arbitrary positive number if
C
4
C
3
1.
ϕ
(
x
)
∈
[
0
,
l
]
,
ψ
(
x
)
∈
[
0
,
l
]
(constraints on
ϕ
and
ψ
);
ϕ
(
ψ
(
2.
ϕ
(
x
)
,
x
)
,
ψ
(
x
)
and
x
)
are all vanished at
x
=
0and
x
=
l
(consistency
conditions at two ends).
Proof.
An application of integration by parts and the consistency conditions for
ϕ
yields
l
0
ϕ
(
l
0
ϕ
(
2
l
sin
k
π
x
2
k
cos
k
π
x
a
k
=
x
)
d
x
=
x
)
d
x
l
π
l
l
0
ϕ
(
2
l
sin
k
π
x
=
−
x
)
d
x
=
···
2
(
k
π
)
l
O
1
k
4
l
0
ϕ
(
4
)
(
2
l
3
sin
k
π
x
=
x
)
d
x
=
,
(
π
)
4
k
l
where
O
1
k
4
stands for the infinitesimal of higher or the same order of
1
k
4
;
0
ϕ
(
4
)
(
sin
k
π
x
(
k
→
∞
)
x
)
d
x
→
0as
k
→
∞
by Riemann lemma. Similarly,
l
k
4
.
Thus, every term
u
k
(
O
1
b
k
=
x
,
t
)
in the series (2.20) satisfies
O
1
k
4
;
|
u
k
(
x
,
t
)
|≤|
a
k
|
+
|
b
k
|
=
≤
O
1
k
3
;
∂
u
k
(
x
,
t
)
k
l
(
|
a
k
|
+
|
b
k
|
)=
∂
x
≤
k
2
O
1
k
2
2
u
k
(
∂
x
,
t
)
l
(
|
a
k
|
+
|
b
k
|
)=
.
∂
x
2
Similarly,
≤
O
1
k
3
≤
O
1
k
2
2
u
k
(
∂
u
k
(
x
,
t
)
∂
x
,
t
)
,
.
∂
t
∂
t
2
1
k
p
converges for
p
∞
k
=
1
1. Therefore, all series
u
,
u
x
,
u
xx
,
u
t
and
u
tt
in the theorem converge uniformly, by the Weierstrass test, so that series (2.20)
is the solution of PDS (2.21).
Also, the
p
-series
>
Remark 1.
The conditions in the existence theorem are sufficient conditions for
existence, but they are not necessary conditions.
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