Environmental Engineering Reference
In-Depth Information
where
T
m
(
t
)
is a function to be determined later and
μ
m
are the positive zero-points
x
lh
. Substituting Eq. (4.9) into the equation of (4.7) with
f
of
f
(
x
)=
tan
x
+
=
0
yields
μ
m
a
l
2
τ
0
T
m
(
T
m
(
t
)+
t
)+
T
m
(
t
)=
0
.
(4.10)
Its general solution can be easily found as
t
e
−
T
m
(
t
)=
2
τ
0
(
a
m
cos
γ
m
t
+
b
m
sin
γ
m
t
)
,
(4.11)
where
a
m
and
b
m
are undetermined constants,
sin
4
τ
0
μ
m
a
l
2
1
2
γ
m
t
,
γ
=
0
,
m
γ
=
−
1and
sin
γ
m
t
=
(4.12)
m
,
=
.
τ
t
γ
0
m
0
We thus have
+
∞
m
=
1
e
−
t
sin
μ
m
x
l
u
(
x
,
t
)=
2
τ
0
(
a
m
cos
γ
m
t
+
b
m
sin
γ
m
t
)
.
Applying the initial condition
u
(
x
,
0
)=
0 leads to
a
m
=
0. Applying the initial con-
(
,
)=
ψ
(
)
dition
u
t
x
0
x
yields
γ
m
,
γ
m
=
+
∞
m
=
1
b
m
γ
m
sin
μ
m
x
0
,
l
=
ψ
(
x
)
,
γ
m
=
1
,
γ
m
=
0
.
which requires
l
0
ψ
(
1
γ
m
M
m
sin
μ
m
x
l
b
m
=
x
)
d
x
.
Here
M
m
is the normal square of the eigenfunction set
sin
μ
m
x
l
.
Finally, we have
⎧
⎨
τ
0
+
∞
m
=
1
b
m
sin
γ
m
t
sin
μ
m
x
t
e
−
u
(
x
,
t
)=
W
ψ
(
x
,
t
)=
2
,
l
(4.13)
l
0
ψ
(
⎩
1
γ
m
M
m
sin
μ
m
x
l
b
m
=
x
)
d
x
.
Therefore the solution of PDS (4.7) under the boundary conditions (4.8) is, by the
solution structure theorem,
1
W
ϕ
(
t
τ
0
+
∂
u
=
x
,
t
)+
W
ψ
(
x
,
t
)+
W
f
τ
(
x
,
t
−
τ
)
d
τ
.
(4.14)
∂
t
0
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