Environmental Engineering Reference
In-Depth Information
M
M
−
2
v
(
x
,
t
)=
u
(
x
,
t
)+
(
x
−
x
0
)
,
(
x
,
t
)
∈
D
.
4
l
2
Then
M
M
2
l
2
a
2
M
−
a
2
M
−
a
2
v
xx
=
a
2
u
xx
−
v
t
−
u
t
−
=
−
<
0
,
(3.22)
2
l
2
v
(
x
0
,
t
0
)=
u
(
x
0
,
t
0
)=
M
.
(3.23)
Also, on the boundary
Γ
,
M
4
=
3
M
4
=
θ
−
M
M
4
+
M
v
(
x
,
t
)
<
+
M
,
(3.24)
where 0
1.
Eqs. (3.23) and (3.24) show that
v
<
θ
<
(
x
,
t
)
does not take its maximumvalue on
Γ
.Let
(
x
1
,
t
1
)
be the point inside
D
,where
v
(
x
,
t
)
takes its maximumvalue. At point
(
x
1
,
t
1
)
,
a
2
v
xx
≥
we have
v
xx
≤
0,
v
t
≥
0
(
v
t
=
0if
t
1
<
T
,
v
t
≥
0if
t
1
=
T
)
such that
v
t
−
0.
This contradicts Eq. (3.22) so we arrive at Eq. (3.21).
Uniqueness follows from the extremum principle of heat conduction. Let
u
1
and
u
2
be two solutions of PDS (3.17). The function
w
defined by
u
1
−
u
2
satisfies
⎧
⎨
a
2
w
xx
,
w
t
=
(
0
,
l
)
×
(
0
,
+
∞
)
w
(
0
,
t
)=
w
(
l
,
t
)=
0
,
⎩
w
(
x
,
0
)=
0
.
By the extremum principle of heat conduction,
w
(
x
,
t
)
≡
0so
u
1
=
u
2
.
3.3.3 Stability
Consider any small variation of
ϕ
(
x
)
in 0
≤
x
≤
l
from
ϕ
1
(
x
)
to
ϕ
2
(
x
)
,
u
1
(
x
,
0
)
−
u
2
(
x
,
0
)
=
ϕ
1
(
x
)
−
ϕ
2
(
x
)
<
ε
,
where
is the norm in the sense of uniform approxima-
tion. By the extremum principle of heat conduction,
ε
is any small constant,
·
u
1
(
x
,
t
)
−
u
2
(
x
,
t
)
≤
max
0
l
|
ϕ
1
(
x
)
−
ϕ
2
(
x
)
| <
ε
,
≤
x
≤
where
u
1
(
x
,
t
)
and
u
2
(
x
,
t
)
are the two solutions of PDS (3.17) corresponding to
, respectively. Therefore, the solution of PDS (3.17) is stable.
Therefore, PDS (3.17) is well-posed. Well-posedness can also be established for
the other PDS of heat-conduction equations in this topic.
ϕ
1
(
x
)
and
ϕ
2
(
x
)
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