Environmental Engineering Reference
In-Depth Information
which shows
u
t
τ
0
+
u
tt
=
0
.
t
τ
0
and
t
τ
0
from the
e
−
e
−
This can also be obtained by noting that
u
t
=
τ
0
u
tt
=
−
theory of series. Note that the temperature distribution
u
(
x
,
t
)
comes only from
u
t
(
x
,
0
)=
1. Since the initial condition
u
t
(
x
,
0
)=
1is
x
-independent,
u
(
x
,
t
)
can-
not depend on
x
either so
u
xx
=
0; consequently, Eq. (5.43) follows. From this point
of view, the solution of PDS (5.40) should be that of an initial-value problem of an
ordinary differential equation
t
=
0
=
τ
0
d
2
u
d
u
d
t
=
d
u
d
t
d
t
2
+
0
,
u
(
0
)=
0
,
1
.
Its solution can be readily obtained
)=
τ
0
1
τ
0
t
e
−
u
(
x
,
t
−
,
(5.46)
which agrees with Eq. (5.44). This, with Eq. (5.41), leads to
x
+
At
I
0
b
2
d
τ
0
1
τ
0
t
t
e
−
2
(
At
)
−
(
x
−
ξ
)
ξ
=
2
A
τ
0
e
2
−
,
(5.47)
x
−
At
1
2
A
=
where
b
τ
0
.
Note.
By Eq. (5.44), we have
t
τ
0
2
t
τ
0
3
u
τ
0
=
t
τ
0
−
1
2!
1
3!
+
−··· .
Also, from series theory,
t
τ
0
t
τ
0
2
t
τ
0
3
e
−
t
τ
0
+
1
2!
1
3!
=
1
−
−
+
··· .
Adding these two equations together yields
0
1
τ
0
t
e
−
=
τ
−
,
u
which is the same as Eq. (5.46).
Consider a charging process of a RC-circuit. By replacing
τ
0
in Eq. (5.46) by the
time constant RC, the
u
in Eq. (5.46) will reflect how the electrical voltage varies.
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