Environmental Engineering Reference
In-Depth Information
T
−
1
in (4.15), so that
Remark 4.
Let
f
(
x
,
t
)=
δ
(
x
−
x
0
,
t
−
t
0
)
,
[
δ
]=
Θ
t
l
u
=
d
τ
G
(
x
,
ξ
,
t
−
τ
)
δ
(
ξ
−
x
0
,
τ
−
t
0
)
d
ξ
=
G
(
x
,
x
0
,
t
−
t
0
)
,
[
u
]=
Θ
0
0
(
,
,
−
)
G
is thus the temperature distribution in a rod of heat conduction due to
a source term of the unit
x
x
0
t
t
0
, or the temperature distribution
caused by a point source with a unit changing rate of temperature at time instant
t
0
and at spatial point
x
0
. Therefore,
G
δ
-function
δ
(
x
−
ξ
,
t
−
τ
)
(
x
,
ξ
,
t
−
τ
)
satisfies
⎧
⎨
a
2
G
xx
+
δ
(
G
t
+
τ
0
G
tt
=
x
−
ξ
,
t
−
τ
)
,
0
<
x
<
l
,
0
<
τ
<
t
<
+
∞
,
G
|
x
=
0
=(
G
x
+
hG
)
|
x
=
l
=
0
,
(4.18)
⎩
G
|
t
=
τ
=
G
t
|
t
=
τ
=
0
or
⎧
⎨
a
2
G
xx
G
t
+
τ
0
G
tt
=
,
0
<
x
<
l
,
0
<
τ
<
t
<
+
∞
,
G
|
x
=
0
=(
G
x
+
hG
)
|
x
=
l
=
0
,
(4.19)
⎩
G
t
|
t
=
τ
=
δ
(
x
−
ξ
)
τ
0
G
|
t
=
τ
=
0
,
.
Remark 5.
We can obtain the Green function
G
(
x
,
ξ
,
t
−
τ
)
by using different meth-
ods.
1. We may obtain
G
by applying the generalized Fourier method of ex-
pansion to solve PDS (4.15) and expanding
f
(
x
,
ξ
,
t
−
τ
)
(
x
,
t
)
using the eigenfunction set
.
2. We may apply the solution structure theorem to obtain
G
sin
μ
m
x
l
(
x
,
ξ
,
t
−
τ
)
as in Re-
mark 2.
3. We may obtain
G
by applying the generalized Fourier method of
expansion to solve PDS (4.19). The
G
(
x
,
ξ
,
t
−
τ
)
(
x
,
ξ
,
t
−
τ
)
in Eq. (4.17) follows from
)=
δ
(
x
−
ξ
)
τ
0
Eq. (4.13) by substituting
ψ
(
x
and replacing
t
by
t
−
τ
.
4.2.2 Mixed Boundary Conditions of the Second
andtheThirdKind
Here we propose another way of using the solution structure theorem following
a procedure of
u
3
⇒
u
2
⇒
u
1
instead of the normal procedure
u
2
⇒
u
1
⇒
u
3
.Find
the solution of
⎧
⎨
a
2
u
xx
+
u
t
+
τ
0
u
tt
=
f
(
x
,
t
)
,
(
0
,
l
)
×
(
0
,
+
∞
)
,
u
x
(
0
,
t
)=
u
x
(
l
,
t
)+
hu
(
l
,
t
)=
0
,
(4.20)
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
.
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