Environmental Engineering Reference
In-Depth Information
5.3.2 Verify the Solution for
u
(
x
,
0
)=
1
and
u
t
(
x
,
0
)=
0
Verify that the solution of
⎧
⎨
u
t
τ
0
+
A
2
u
xx
, −
∞
<
u
tt
=
x
<
+
∞
,
0
<
t
,
(5.48)
⎩
u
(
x
,
0
)=
1
,
u
t
(
x
,
0
)=
0
is
⎧
⎨
ϕ
(
x
−
At
)+
ϕ
(
x
+
At
)
t
e
−
u
=
2
τ
0
⎩
2
⎡
I
0
b
2
x
+
At
1
2
A
1
2
t
2
⎣
0
b
+
(
At
)
−
(
x
−
ξ
)
+
τ
x
−
At
0
2
2
4
τ
(
At
)
−
(
x
−
ξ
)
2
⎤
⎫
⎬
⎭
.
I
1
b
2
⎦
ϕ
(
ξ
)
·
(
At
)
−
(
x
−
ξ
)
d
ξ
(5.49)
The Initial Conditions
By Eq. (5.49), it is clear that
u
(
x
,
0
)=
1. Substituting
ϕ
(
x
)=
1 into Eq. (5.49) and
taking the derivative of Eq. (5.49) with respect to
t
yields
u
t
(
x
,
0
)=
0
,
wherewehaveused
e
−
1
t
=
0
=
−
1
2
t
2
τ
0
·
0
,
τ
e
−
τ
0
I
0
b
2
d
t
=
0
=
x
+
At
1
2
A
1
1
2
t
2τ
0
2
(
At
)
−
(
x
−
ξ
)
ξ
τ
0
,
2
x
−
At
⎡
⎤
t
=
0
2
d
b
x
+
At
1
2
A
t
⎣
t
2τ
0
⎦
e
−
2
0
b
I
1
(
(
At
)
−
(
x
−
ξ
)
ξ
x
−
At
2
2
(
)
−
(
−
ξ
)
4
τ
At
x
e
−
t
=
0
=
τ
0
t
4
t
2τ
0
=
0
.
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