Environmental Engineering Reference
In-Depth Information
The solution of PDS (3.39) follows directly from the solution structure theorem as
t
u
=
W
f
τ
(
x
,
t
−
τ
)
d
τ
0
V
d
t
+
∞
2
a
2
−
τ
)+
ξ
−
(
t
−
τ
)
h
=
d
τ
f
(
ξ
,
τ
)
(
x
,
ξ
,
t
h
V
(
x
,−
ξ
,
t
−
τ
)
ξ
.
2
a
2
ξ
+
(
t
−
τ
)
0
0
(3.40)
Remark 3.
Solutions of PDS (3.37) and (3.39) that are subjected to boundary con-
dition of the second kind can be obtained by letting
h
=
0 in Eqs. (3.38) and (3.40).
3.4.4 PDS with Variable Thermal Conductivity
, the task of seeking PDS
solutions of heat-conduction equations becomes difficult. There exists no well-
developed method for developing analytical solutions. Numerical method are nor-
mally appealed for numerical solutions. Here we detail the procedure of developing
analytic solutions of heat-conduction equations with variable thermal conductivity
of type
k
When thermal conductivity is variable so that
k
=
k
(
x
,
t
)
=
k
(
x
)
by using the example
⎧
⎨
cu
t
=
∂
∂
ρ
x
[
k
(
x
)
u
x
]
,
(
0
,
l
)
×
(
0
,
+
∞
)
,
(3.41)
⎩
u
(
0
,
t
)=
u
(
l
,
t
)=
0
,
u
(
x
,
0
)=
f
(
x
)
,
where
and
c
are the density and the specific heat of the material. The thermal
conductivity
k
ρ
(
x
)
>
0 is a differentiable function of
x
.
Transform into an Eigenvalue Problem of an Integral Equation
Consider a solution of type
u
(
x
,
t
)=
X
(
x
)
T
(
t
)
. Substitute it into the equation of
PDS (3.41) to obtain eigenvalue problems
d
d
x
[
k
(
x
)
X
x
]+
λ
X
=
0
,
X
(
0
)=
X
(
l
)=
0
(3.42)
and
T
t
+
λ
ρ
c
T
=
0
.
(3.43)
Here
λ
is the separation constant.
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