Environmental Engineering Reference
In-Depth Information
From this special case, we obtain the integral
t
ch
A
A
2
2
2
2
(
t
−
τ
)
−
(
x
−
ξ
)
−
(
y
−
η
)
t
−
τ
2
e
−
τ
0
d
τ
A
2
d
ξ
d
η
(
t
−
τ
)
2
−
(
x
−
ξ
)
2
−
(
y
−
η
)
2
0
D
A
(
t
−
τ
)
τ
0
t
+
τ
0
e
−
1
t
τ
0
=
2
π
A
−
.
T
−
2
in PDS (5.75), we have
T
−
2
in Eq. (5.76).
Remark 1.
Since
[
1
]=
Θ
[
u
]=
Θ
y
and
z
.This
can be verified physically. The three PDS are thus equivalent to the following three
initial-value problems of ordinary differential equations.
⎧
⎨
Remark 2.
The
u
(
x
,
y
,
z
,
t
)
in last three examples is independent of
x
,
d
2
u
d
t
2
+
1
τ
0
du
dt
=
=
τ
0
1
τ
0
0
,
t
e
−
Solution:
u
−
.
⎩
u
(
u
(
0
)=
0
,
0
)=
1
.
⎧
⎨
d
2
u
d
t
2
+
1
τ
0
d
u
d
t
=
0
,
=
.
Solution:
u
1
⎩
u
(
u
(
0
)=
1
,
0
)=
0
.
⎧
⎨
d
2
u
d
t
2
+
1
τ
0
d
u
d
t
=
0
e
−
1
1
,
t
τ
0
Solution:
u
=
τ
0
t
+
τ
−
.
⎩
u
(
u
(
0
)=
0
)=
0
.
5.5 Domains of Dependence and Influence, Measuring
τ
0
by Characteristic Cones
The hyperbolic heat-conduction equation shares features of wave equations includ-
ing the domain of dependence and the domain of influence.
5.5.1 Domain of Dependence
Equations (5.67), (5.69) and (5.70) in Section 5.4 show that the effect of initial val-
ues
ϕ
,
ψ
and source term
f
is similar to that for wave equations. The solution due to
in
D
At
, but not on those
ϕ
(
)
ψ
(
,
)
xy
and
x
y
depends only on the initial values
ϕ
and
ψ
outside
D
At
.The
u
(
x
0
,
y
0
,
t
0
)
in Eq. (5.67) is, for example, determined completely
in
D
M
0
At
0
. In the three-dimensional space
Oxyt
,
D
M
0
by
ψ
(
x
,
y
)
is the intersecting area
At
0
between plane
t
=
0 and the cone of top point
P
0
(
x
0
,
y
0
,
t
0
)
(Fig. 5.6), i.e.
D
M
0
At
0
=
(
x
,
y
,
t
)
|
(
x
−
x
0
)
2
+(
y
−
y
0
)
2
≤
A
(
t
0
−
t
)
∩{
t
|
t
=
0
} .
The region
D
M
0
At
0
is called the
domain of dependence
of point
P
0
on the initial value.
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