Environmental Engineering Reference
In-Depth Information
1.
M
stands for a spatial point in one-, two- and three-dimensional space. Its co-
ordinates are
x
,
is the Laplace operator,
which reduces to
u
xx
for the one-dimensional case. Equation (6.1) or (6.2) is
a third-order linear nonhomogeneous partial differential equation with constant
coefficients.
2. Mixed problems are always specified in certain spatial domains denoted by
(
x
,
y
)
and
(
x
,
y
,
z
)
, respectively. The
Δ
Ω
,
D
and a interval
[
x
1
,
x
2
]
, respectively, for three-, two- and one-dimensional cases.
The boundary of
Ω
and
D
are denoted by
∂Ω
and
∂
D
, respectively. The bound-
ary of
consists of two end points
x
1
and
x
2
. We develop the solution struc-
ture theorem under linear homogeneous boundary conditions of all three kinds
[
x
1
,
x
2
]
(
,
)
|
∂Ω
=
,
L
u
u
n
0
where the
u
n
stands for the normal derivative on the boundary. The solution struc-
ture theorem developed for the three-dimensional case is clearly also valid for the
one- and two-dimensional cases.
3. In Eqs. (6.1) and (6.2),
1
α
∂
[
Δ
L
2
T
,
∂
T
T
∂
=[
Δ
T
]
⇒
[
α
]=
T
]=
∂
t
t
T
T
∂
∂
[
Δ
]=
T
τ
t
Δ
⇒
[
τ
]=
T
,
T
1
α
τ
q
α
2
T
∂
T
∂
=
⇒
[
τ
q
]=
T
,
(
τ
q
(
or
τ
0
)
is also a time constant
)
∂
t
∂
t
2
]=
L
2
,
u
tt
]=
T
2
,
[
Δ
T
]=[
F
(
M
,
t
)]
⇒
[
F
[
f
]=[
α
τ
L
T
,
[
A
]=
=
the speed of thermal waves
,
q
ατ
T
τ
q
=
√
α
=
L
2
T
=[
α
]
.
L
√
T
⇒
[
B
2
[
B
]=
]=
6.1.2 Solution Structure Theorem
Theorem 1.
Let
u
=
W
ψ
(
M
,
t
)
be the solution of well-posed PDS
⎧
⎨
u
t
τ
0
+
B
2
∂
∂
A
2
u
tt
=
Δ
u
+
t
Δ
u
,
Ω
×
(
0
,
+
∞
)
,
(6.3)
⎩
L
(
u
,
u
n
)
|
∂Ω
=
0
,
u
(
M
,
0
)=
0
,
u
t
(
M
,
0
)=
ψ
(
M
)
.
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