Environmental Engineering Reference
In-Depth Information
n
2
n
2
Note that
α
n
<
0,
α
n
=
0
(
)
and
β
n
=
0
(
)
as
n
→
+
∞
. For any natural number
N
and
t
>
0, we thus have
O
1
n
N
e
α
n
t
=
(
n
→
+
∞
)
.
2
u
2
u
3
u
∂
(
x
,
t
)
∂
(
x
,
t
)
and
∂
(
x
,
t
)
Since e
α
n
t
occurs in all
u
(
x
,
t
)
,
,
, the demand for the
∂
x
2
∂
t
2
∂
t
∂
x
2
smoothness of
ϕ
(
x
)
and
ψ
(
x
)
is very weak. Provided that their Fourier coefficients
exist, we always have
x
2
≤
O
1
n
2
O
1
n
2
2
u
n
∂
∂
|
u
n
(
x
,
t
)
|≤
,
,
≤
≤
O
1
n
2
O
1
n
2
2
u
n
∂
3
u
n
∂
∂
,
.
t
2
x
2
∂
t
∂
2
u
2
u
1
n
2
∂
(
x
,
t
)
∂
(
x
,
t
)
∑
Note that the series
is convergent. Thus, the series
u
(
x
,
t
)
,
,
∂
x
2
∂
t
2
3
u
and
∂
(
x
,
t
)
x
2
are all uniformly convergent so that the
u
in Eq. (6.33) is indeed the
solution of PDS (6.22).
∂
t
∂
Remark.
Variation of constants for second-order nonhomogeneous ordinary differ-
ential equations
: Consider a nonhomogeneous ODE
y
+
p
(
x
)
y
+
q
(
x
)
y
=
f
(
x
)
.
(6.34)
Let the solution of its corresponding homogeneous ODE be
y
=
c
1
y
1
(
x
)+
c
2
y
2
(
x
)
,
where
c
1
and
c
2
are constants.
Consider the solution of Eq. (6.34) of type
y
=
c
1
(
x
)
y
1
(
x
)+
c
2
(
x
)
y
2
(
x
)
,
(6.35)
where
c
1
(
x
)
and
c
2
(
x
)
are functions to be determined. We have
y
=
c
1
(
y
1
+
c
2
(
y
2
.
x
)
y
1
+
c
1
(
x
)
x
)
y
1
+
c
1
(
x
)
(6.36)
Let
c
1
(
c
2
(
x
)
y
1
+
x
)
y
2
=
0
.
(6.37)
By substituting Eq. (6.37) into Eq. (6.36), we can obtain
y
and consequently
y
.By
substituting
y
,
y
and
y
into Eq. (6.34) and using
y
1
+
y
1
+
y
2
+
y
2
+
p
(
x
)
q
(
x
)
y
1
=
0
,
p
(
x
)
q
(
x
)
y
2
=
0
,
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