Environmental Engineering Reference
In-Depth Information
Thus
+
∞
m
=
0
e
α
m
t
cos
(
2
m
+
1
)
π
x
u
(
x
,
t
)=
(
A
m
cos
β
m
t
+
B
m
sin
β
m
t
)
,
2
l
+
∞
m
=
0
{
α
m
e
α
m
t
u
t
(
x
,
t
)=
(
A
m
cos
β
m
t
+
B
m
sin
β
m
t
)
cos
(
2
m
+
1
)
π
x
e
α
m
t
+
(
−
A
m
β
m
sin
β
m
t
+
B
m
β
m
cos
β
m
t
)
}
2
l
where
2
4
1
τ
0
+
λ
m
B
2
1
τ
0
+
λ
m
B
2
2
1
2
)
π
1
2
1
2
(
m
+
α
m
=
−
,
β
m
=
λ
m
A
2
−
,
λ
m
=
.
l
Applying the initial condition
u
(
x
,
0
)=
0 yields
A
m
=
0. To satisfy the initial con-
dition
u
t
(
x
,
0
)=
ϕ
(
x
)
,
B
m
must be determined such that
+
∞
m
=
0
B
m
β
m
cos
(
2
m
+
1
)
π
x
=
ψ
(
x
)
.
2
l
Thus
l
0
ψ
(
1
M
m
β
m
cos
(
2
m
+
1
)
π
x
B
m
=
x
)
d
x
,
2
l
is the normal square of
cos
(
.
2
m
+
1
)
π
x
l
2
where
M
m
=
2
l
Finally, we have
⎧
⎨
+
∞
m
=
0
B
m
e
α
m
t
sin
β
m
t
cos
(
2
m
+
1
)
π
x
u
(
x
,
t
)=
W
ψ
(
x
,
t
)=
,
2
l
(6.47)
⎩
cos
(
+
)
π
M
m
β
m
0
ψ
(
1
2
m
1
x
B
m
=
x
)
d
x
2
l
and the solution of PDS (6.46), by the solution structure theorem, is
1
τ
W
ϕ
(
t
0
+
∂
(
,
)=
u
x
t
x
,
t
)+
W
ψ
−
B
2
ϕ
(
x
,
t
)+
W
f
τ
(
x
,
t
−
τ
)
d
τ
.
∂
t
0
Remark 3.
u
actually enjoys a very elegant structure. We may use
this structure and Table 2.1 to write out
W
ψ
(
(
x
,
t
)=
W
ψ
(
x
,
t
)
be the
eigenvalues and eigenfunctions from Table 2.1 based on given boundary conditions.
The structure of
W
ψ
(
x
,
t
)
directly. Let
λ
m
and
X
m
(
x
)
x
,
t
)
is thus
⎧
⎨
+
∞
∑
B
m
e
α
m
t
sin
u
(
x
,
t
)=
W
ψ
(
x
,
t
)=
β
m
t
·
X
m
(
x
)
,
m
=
0or1
(6.48)
l
0
ψ
(
⎩
1
M
m
β
m
=
)
(
)
.
B
m
x
X
m
x
d
x
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