Environmental Engineering Reference
In-Depth Information
into
⎧
⎨
a
2
v
xx
a
2
w
xx
v
tt
=
+
−
w
tt
,
(
0
,
l
)
×
(
0
,
+
∞
)
,
v
(
0
,
t
)=
v
(
l
,
t
)=
0
,
⎩
v
(
x
,
0
)=
ϕ
(
x
)
−
w
(
x
,
0
)
,
v
t
(
x
,
0
)=
−
w
t
(
x
,
0
)
.
Therefore, it is of crucial importance to have a function
w
satisfying the nonho-
mogeneous boundary conditions. Such kinds of auxiliary functions depend on the
type of boundary conditions. Here we list such auxiliary functions for three typical
kinds of boundary conditions
(
x
,
t
)
1.
w
(
x
,
t
)=
xf
2
(
t
)+
f
1
(
t
)
for
u
(
0
,
t
)=
f
1
(
t
)
,
u
x
(
l
,
t
)=
f
2
(
t
)
;
2.
w
(
x
,
t
)=(
x
−
l
)
f
1
(
t
)+
f
2
(
t
)
for
u
x
(
0
,
t
)=
f
1
(
t
)
,
u
(
l
,
t
)=
f
2
(
t
)
;
f
2
(
t
)
−
f
1
(
t
)
x
2
3.
w
(
x
,
t
)=
+
f
1
(
t
)
x
for
u
x
(
0
,
t
)=
f
1
(
t
)
,
u
x
(
l
,
t
)=
f
2
(
t
)
.
2
l
For the case of
u
(
0
,
t
)=
0,
u
(
l
,
t
)=
A
sin
ω
t
, both
sin
ω
x
a
sin
ω
Ax
l
w
1
(
x
,
t
)=
sin
ω
t
and
w
2
(
x
,
t
)=
A
sin
ω
t
l
a
can serve as the auxiliary function of homogenization. As
a
2
w
xx
−
w
tt
=
0, however,
(
,
)
the function transformation using
w
2
as the auxiliary function will not change
the source term of the equation and is thus more desirable. Such a transformation
can preserve the homogenenity of the original equation.
Remark 2.
To understand the Green function, we must discuss the Dirac function,
or
x
t
-function can be
found in Appendix B. Here, we introduce it from the point of view of physics and
engineering.
The
δ
-function for short. A precise mathematical discussion of the
δ
δ
-function is often called the
unit impulse function
and is defined by
0
+
∞
,
=
,
x
x
0
δ
(
x
−
x
0
)=
δ
(
x
−
x
0
)
d
x
=
1
.
∞
,
=
,
x
x
0
−
∞
Its introduction comes from the desire to describe some localized phenomena. For
example, consider an infinite wire (
) with all the charges of unit quan-
tity of electricity located at point
x
0
. The charge density
−
∞
<
x
<
+
∞
ρ
(
x
−
x
0
)
satisfies
0
+
∞
,
x
=
x
0
,
ρ
(
x
−
x
0
)=
ρ
(
x
−
x
0
)
d
x
=
1
.
∞
,
x
=
x
0
,
−
∞
Search WWH ::
Custom Search