Environmental Engineering Reference
In-Depth Information
m
+
1
as time increases since
q
(
increases toward
is odd, how-
ever, the last term depends not only on
t
but also on a function of first degree of
x
.
τ
t
)
>
0. When
P
N
(
x
)
0
Remark 5.
Solution (6.188) of PDS (6.181) is a function of
x
and
t
expressed by
the polynomial of initial value
P
N
(
x
)
.Itisthe
W
P
N
(
x
,
t
)
by the solution structure
theorem. The solution of
u
t
/
τ
0
+
A
2
u
xx
+
ε
R
1
u
tt
=
u
txx
+
P
m
(
x
)
,
×
(
0
,
+
∞
)
,
(6.196)
u
(
x
,
0
)=
P
l
(
x
)
,
u
t
(
x
,
0
)=
P
N
(
x
)
is thus
1
x
2
W
P
l
(
t
2
τ
0
+
∂
∂
u
=
t
−
ε
x
,
t
)+
W
P
N
(
x
,
t
)+
W
P
m
(
x
,
t
−
τ
)
d
τ
,
(6.197)
∂
∂
0
where
P
N
(
x
)
,
P
l
(
x
)
and
P
m
(
x
)
are the
N
-th,
l
-th and
m
-th polynomials respectively.
are separable regarding
x
and
t
. This is the same as that for
hyperbolic heat-conduction equations and is very useful for examining features of
All terms in
W
P
N
(
x
,
t
)
m
n
=
0
a
n
(
t
)
x
n
, the third term in Eq. (6.197) reads
heat conduction. When
P
m
(
x
)=
t
W
P
m
τ
(
x
,
t
−
τ
)
d
τ
,
P
m
τ
=
P
m
(
x
,
τ
)
.
0
6.9.4 Solutions for Initial-Value of Lower-Order Polynomials
The lower-order polynomials find their applications in many fields. For
n
≤
5, we
discuss
W
P
N
(
of
u
t
/
τ
0
+
x
,
t
)
A
2
u
xx
+
ε
R
1
u
tt
=
u
txx
,
×
(
0
,
+
∞
)
,
(6.198)
u
(
x
,
0
)=
0
,
u
t
(
x
,
0
)=
P
n
(
x
)
,
n
=
0
,
1
,
2
,
3
,
4
,
5
.
Define
I
0
b
A
2
u
2
u
2
k
d
u
A
(
t
−
τ
)
G
2
k
(
)=
(
−
τ
)
2
−
,
=
,
,
,
t
t
k
1
2
3
(6.199)
−
A
(
t
−
τ
)
where
2
x
2
2
m
d
t
G
2
k
(
τ
0
∞
m
=
0
1
1
2
A
d
t
e
−
(
)=
,
=
τ
0
,
g
2
k
(
)=
)
.
I
0
x
b
t
t
2
(6.200)
(
m
!
)
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