Environmental Engineering Reference
In-Depth Information
Finally, we have
⎧
⎨
l
f
(
p
n
)=
f
(
x
)
cos
p
n
x
d
x
,
0
p
n
+
h
2
(
)
⎩
n
f
f
(
x
)=
2
(
p
n
)
cos
p
n
x
.
(
p
n
+
h
2
)+
l
h
If the conditions are
v
|
x
=
0
=
0and
(
v
x
+
hv
)
|
x
=
l
=
0
,
(
h
>
0
)
, we consider
)=
n
b
n
sin
q
n
x
,
f
(
x
where the
q
n
and the
b
n
can be determined by following a similar approach.
Remark 3.
The Fourier sine/cosine transformation can also be used to solve un-
steady problems. We discuss this by considering the following PDS arising from the
heat transfer enhancement in energy, chemical engineering and aerospace engineer-
ing
⎧
⎨
⎩
a
2
u
t
=
Δ
u
,
0
<
x
<
+
∞
,
0
<
y
<
b
,
t
>
0
,
y
=
0
=
∂
u
y
=
b
=
∂
u
u
0
,
y
+
σ
0
,
∂
y
∂
⎧
⎨
(7.40)
x
=
0
=
A
sin
t
,
t
∈
(
2
n
π
,
(
2
n
+
1
)
π
)
,
∂
u
f
(
t
)=
0
,
otherwise
,
∂
x
⎩
u
(
x
,
y
,
0
)=
v
0
.
where
v
0
is a constant.
Consider a Fourier cosine transformation in
(
0
,
+
∞
)
with respect to
x
so that
+
∞
(
,
,
)=
(
,
,
)
u
x
y
t
u
x
y
t
cos
ω
x
d
x
0
Since, with
δ
(
ω
)
as the
δ
-function,
+
∞
2
u
2
u
u
xx
cos
ω
x
d
x
=
−
u
x
(
0
,
y
,
t
)
−
ω
=
−
f
(
t
)
−
ω
,
0
+
∞
+
∞
v
0
2
v
0
cos
ω
x
d
x
=
cos
ω
x
d
x
=
v
0
πδ
(
ω
)
.
0
−
∞
PDS (7.40) is transformed into a mixed problem in
[
0
,
b
]
⎧
⎨
a
2
u
yy
+(
ω
2
u
a
2
f
u
t
−
a
)
=
−
(
t
)
,
(
0
,
b
)
×
(
0
,
+
∞
)
,
y
=
0
=
u
y
=
b
=
∂
u
∂
∂
u
0
,
y
+
σ
0
,
(7.41)
⎩
∂
y
u
(
x
,
y
,
0
)=
v
0
πδ
(
ω
)
.
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