Environmental Engineering Reference
In-Depth Information
tions can be written as
μ
m
lh
1
2
sin
μ
m
x
l
+
ϕ
m
)=
μ
m
lh
1
cos
μ
m
x
sin
μ
m
x
X
m
(
x
l
+
l
=
1
+
,
ϕ
m
=
μ
m
where tan
lh
1
.
Finally, we have eigenvalues
λ
m
and eigenfunctions
X
m
(
x
)
μ
m
l
2
Eigenvalues
λ
m
=
,
μ
m
are the positive zero points of
f
(
x
)
in Eq. (6.44).
sin
μ
m
x
l
+
ϕ
m
ϕ
m
=
μ
m
Eigenfunctions
X
m
(
x
)=
,
tan
lh
1
. Normal square of
eigenfunction set
l
sin
2
μ
m
x
l
m
d
x
2
X
m
(
x
)
=(
X
m
(
x
)
,
X
m
(
x
)) =
+
ϕ
0
l
1
β
m
sin
2
=
(
β
m
x
+
ϕ
m
)
d
(
β
m
x
+
ϕ
m
)
0
l
1
β
m
1
−
cos2
(
β
m
x
+
ϕ
m
)
=
d
(
β
m
x
+
ϕ
m
)
2
0
1
2
(
β
l
1
β
1
4
sin2
=
m
x
+
ϕ
)
−
(
β
m
x
+
ϕ
)
m
m
m
0
1
2
l
ϕ
m
1
β
m
1
4
sin2
1
4
sin2
=
β
m
−
(
l
β
m
+
ϕ
m
)+
1
l
2
1
=
−
μ
m
[
sin2
(
μ
m
+
ϕ
m
)
−
sin2
ϕ
m
]
2
1
l
2
sin
μ
m
=
−
cos
(
μ
m
+
2
ϕ
m
)
.
(6.45)
μ
m
Remark 1
. Eigenvalues and eigenfunctions of the other eight combinations can also
be obtained using a similar approach. The results for all nine combinations are listed
in Table 2.1. Therefore we can use Table 2.1 to solve the mixed problems of dual-
phase-lagging heat-conduction equations subject to different boundary conditions.
Remark 2
. We can also apply integration by parts to prove that the eigenvalues
of Eq. (6.43) cannot be negative. Integrating the equation in Eq. (6.43) from
x
=
0
yields
l
0
)
X
(
)
d
x
=
(
)
to
x
l
after multiplying
X
x
X
(
x
x
)+
λ
X
(
x
=
0, which be-
comes, by making use of integration by parts, to
l
0
X
(
X
(
x
)
x
)
d
x
,
l
l
)
0
X
(
)
2
d
x
l
0
+
X
2
X
(
λ
(
x
)
d
x
=
−
X
(
x
)
x
x
.
0
It becomes, after applying the boundary conditions in Eq. (6.43),
l
l
X
(
)
2
d
x
X
2
h
1
X
2
h
2
X
2
λ
(
x
)
d
x
=
(
0
)+
(
l
)+
x
,
0
0
Search WWH ::
Custom Search