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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