Environmental Engineering Reference
In-Depth Information
Therefore,
τ
0
can also be obtained by measuring
u
(
x
0
,
t
0
)
.
We have discussed several methods of measuring
τ
0
. For better accuracy, we may
measure
τ
0
by using different methods and take their mean value as its value.
5.4 Method of Descent for Two-Dimensional Problems
and Discussion Of Solutions
In this section we first transform Cauchy problems of two-dimensional hyperbolic
heat-conduction equations to Cauchy problems of three-dimensional wave equa-
tions by using a function transformation.We then apply the method of descent to ob-
tain solutions of Cauchy problems of two-dimensional hyperbolic heat-conduction
equations. Finally, we analyze the solutions for some special cases in order to obtain
a better understanding of solutions and of
τ
0
.
5.4.1 Transform to Three-Dimensional Wave Equations
In this section, we attempt to solve
⎧
⎨
⎩
u
t
τ
0
+
A
2
R
2
u
tt
=
Δ
u
+
f
(
x
,
y
,
t
)
,
×
(
0
,
+
∞
)
,
(5.61)
u
(
x
,
y
,
0
)=
ϕ
(
x
,
y
)
,
u
t
(
x
,
y
,
0
)=
ψ
(
x
,
y
)
.
This can be achieved, by the solution structure theorem, if we can solve
⎧
⎨
u
t
τ
A
2
R
2
0
+
u
tt
=
Δ
u
,
×
(
0
,
+
∞
)
,
(5.62)
⎩
u
(
x
,
y
,
0
)=
0
,
u
t
(
x
,
y
,
0
)=
ψ
(
x
,
y
)
.
t
2τ
0
of eliminating the term of first
e
−
By a function transformation
u
=
v
(
x
,
y
,
t
)
derivative, we transform PDS (5.62) into
v
tt
A
2
c
2
v
1
2
R
2
=
+
,
=
τ
0
,
×
(
,
+
∞
)
,
Δ
v
c
0
(5.63)
(
,
,
)=
,
(
,
,
)=
ψ
(
,
)
.
v
x
y
0
0
v
t
x
y
0
x
y
Consider another function transformation of eliminating term
c
2
v
in PDS (5.63),
c
A
z
e
−
(
,
,
)=
(
,
,
,
)
,
v
x
y
t
V
x
y
z
t
(5.64)
where
A
z
=
1,
[
V
]=[
v
]=[
u
]
. The PDS (5.63) is thus transformed to
Search WWH ::
Custom Search