Environmental Engineering Reference
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Also
+
∞
+
∞
+
∞
1
√
π
2
e
−
η
V
|
t
=
0
d
ξ
=
lim
0
V
d
ξ
=
d
η
=
1
.
t
→
+
−
∞
−
∞
−
∞
Therefore,
V
|
t
=
0
=
δ
(
x
−
ξ
)
.
Remark 4.
The structure of
W
ϕ
(
x
,
t
)
in Eq. (3.27) shows the following properties of
the temperature distribution due to
ϕ
(
x
)
.
1.
u
(
x
,
t
)
is odd or even in
x
if
ϕ
(
x
)
is an odd or even function of
x
;
2.
u
is a periodic function of
x
;
3. The effect of a local, initial disturbance can propagate very quickly to a very long
distance.
(
x
,
t
)
is periodic in
x
if
ϕ
(
x
)
To demonstrate this, consider
k
>
0
, |
x
−
x
0
|≤
δ
,
ϕ
(
x
)=
,
|
−
| >
δ
,
0
x
x
0
where
δ
is an arbitrary small constant. For a very large
|
x
1
|
and a very small
t
1
,
+
∞
k
x
0
+
δ
x
0
−
δ
(
,
)=
(
,
ξ
,
)
ϕ
(
ξ
)
ξ
=
(
,
ξ
,
)
ξ
>
.
u
x
1
t
1
V
x
1
t
1
d
V
x
1
t
1
d
0
−
∞
This property is the signature of the Fourier law of heat conduction (see Section 1.3).
3.4.3 Problems in Semi-Infinite Domain and the Method
of Continuation
Boundary Condition of the First Kind
Example 1.
Find the solution of
⎧
⎨
a
2
u
xx
+
u
t
=
f
(
x
,
t
)
,
0
<
x
<
+
∞
,
0
<
t
,
u
(
0
,
t
)=
0
,
(3.36)
⎩
u
(
x
,
0
)=
ϕ
(
x
)
.
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