Environmental Engineering Reference
In-Depth Information
For a force of unit impulse at time instant
t
0
, the force
F
(
t
−
t
0
)
can also be
expressed by
0
+
∞
,
t
=
t
0
,
(
−
)=
(
−
)
=
.
F
t
t
0
F
t
t
0
d
t
1
∞
,
t
=
t
0
,
−
∞
In applications, the
δ
-function is often viewed as the limit of unit impulsive func-
tions. For example,
δ
(
x
−
x
0
)
can be regarded as the limit as
h
→
0of
⎧
⎨
h
2
< |
0
,
x
−
x
0
| ,
δ
h
(
x
−
x
0
)=
1
h
,
h
2
.
⎩
|
x
−
x
0
|≤
i.e.
δ
(
x
−
x
0
)=
h
→
0
δ
h
(
lim
x
−
x
0
)
.
Similarly,
2
4μ
t
λ
2
−
|
x
|
1
)
−
δ
(
x
)=
lim
λ
→
+
∞
and
δ
(
x
)=
lim
t
0
(
4
πμ
t
,
(
μ
,
t
>
0
)
.
π
(
1
+
λ
2
x
2
)
→
For any continuous function
ϕ
(
x
)
,the
δ
-function can also be defined by
0
+
∞
,
x
=
x
0
,
δ
(
x
−
x
0
)=
ϕ
(
x
)
δ
(
x
−
x
0
)
d
x
=
ϕ
(
x
0
)
.
∞
,
x
=
x
0
.
−
∞
This can also be extended to the
δ
-function of multi-variables. For the case of two
variables, for example,
δ
(
x
−
x
0
,
y
−
y
0
)
satisfies
0
,
(
x
,
y
)
=(
x
0
,
y
0
)
,
δ
(
−
,
−
)=
x
x
0
y
y
0
∞
,
(
x
,
y
)=(
x
0
,
y
0
)
,
+
∞
+
∞
δ
(
x
−
x
0
,
y
−
y
0
)
ϕ
(
x
,
y
)
d
x
d
y
=
ϕ
(
x
0
,
y
0
)
,
−
∞
−
∞
where
ϕ
(
x
,
y
)
is a continuous function of
x
and
y
.
Letting
f
(
x
,
t
)=
δ
(
x
−
x
0
,
t
−
t
0
)
in Eq. (2.11), we have
u
(
x
,
t
)=
G
(
x
,
x
0
,
t
−
t
0
)
.
Therefore,
u
(
x
,
t
)=
G
(
x
,
ξ
,
t
−
τ
)
is the solution of
⎧
⎨
a
2
G
xx
+
δ
(
G
tt
=
x
−
ξ
,
t
−
τ
)
,
0
<
x
<
l
,
0
<
τ
<
t
<
+
∞
G
|
x
=
0
=
G
|
x
=
l
=
0
,
⎩
G
|
t
=
τ
=
G
t
|
t
=
τ
=
0
,
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