Environmental Engineering Reference
In-Depth Information
same essence as the homogenization of equations and the impulsive method. It can
also be obtained by using other methods such as integral transformation. To demon-
strate some mathematical techniques, we re-develop it here by using the method of
characteristics.
Example 4.
Using the method of characteristics, find the solution of
u
tt
a
2
u
xx
=
+
f
(
x
,
t
)
, −
∞
<
x
<
+
∞
,
0
<
t
,
(2.99)
u
(
x
,
0
)=
u
t
(
x
,
0
)=
0
.
Solution.
Introduce new variables
ξ
and
η
such that
x
ξ
=
=
ξ
+
η
2
x
+
at
,
,
or
η
=
x
−
at
=
ξ
−
η
2
a
t
,
ξ
>
η
.
The equation reduces to (Section 1.1.5)
4
a
2
f
ξ
+
η
1
,
ξ
−
η
2
a
u
ξη
=
g
(
ξ
,
η
)
,
g
(
ξ
,
η
)=
−
.
2
Applying the initial condition
u
(
x
,
0
)=
0 yields
u
(
ξ
,
η
)
|
ξ
=
η
=
0. By the initial
condition
u
t
(
x
,
0
)=
0, we have
u
η
ξ
=
η
=
u
ξ
−
u
t
=
u
ξ
·
a
+
u
η
·
(
−
a
)
or
0
.
(2.100)
We thus arrive at a Goursat problem
|
ξ
=
η
=
u
u
η
ξ
=
η
=
u
ξη
=
g
(
ξ
,
η
)
,
u
ξ
−
0
.
(2.101)
i.e.
u
ξ
ξ
=
η
=
u
η
ξ
=
η
=
u
ξη
=
g
(
ξ
,
η
)
,
0
.
(2.102)
Integrating the equation in (2.102) with respect to
ξ
from
ξ
to
η
and applying
u
η
ξ
=
η
=
0 yields
η
(
ξ
,
η
)
ξ
.
u
η
=
−
g
d
ξ
Integrating it with respect to
η
from
ξ
to
η
and using
u
|
η
=
ξ
=
0 leads to
η
ξ
d
η
(
ξ
,
η
)
η
,
u
(
ξ
,
η
)=
−
g
d
(2.103)
ξ
ξ
which is the solution of PDS (2.101). In Eq. (2.103),
ξ
and
η
are the parametric
ξ
and
η
are the variables of integration. Introduce new variables
x
variables and
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