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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