Environmental Engineering Reference
In-Depth Information
Using a similar approach to that in Section 2.4.2, we can readily show
d
E
d
t
=
0or
E
(
t
)=
E
(
0
)
.
Multiplying the above inequality by e
−
t
and using
E
(
)=
(
)
t
E
0
leads to
d
t
E
0
e
−
t
≤
d
e
−
t
(
)
(
)
.
t
2
E
0
Or, by integration with respect to
t
over
[
0
,
t
]
e
t
E
0
(
e
t
E
0
(
t
)
≤
0
)+
2
E
(
0
)(
−
1
)
.
(2.23)
For stability, we substitute
ϕ
(
x
)
and
ψ
(
x
)
into the right hand side of (2.23) to
obtain
2
e
T
1
2
2
2
e
T
a
2
ϕ
u
≤
ϕ
+
ψ
+
−
e
T
2
2
2
a
2
ϕ
≤
ϕ
+
ψ
+
,
b
f
2
where
f
=
(
x
)
d
x
is the normal of function
f
(
x
)
in
[
a
,
b
]
.
a
ϕ
<
ε
ϕ
<
ε
ψ
<
ε
For
,
and
, therefore,
e
T
2
a
2
2
2
u
≤
+
ε
or
u
≤
C
ε
.
Here,
is a small positive constant, and
C
is a nonnegative constant. This shows that
the solution (2.20) is stable with respect to the initial conditions.
ε
2.4.4 Generalized Solution
A
classical solution
refers to a solution that satisfies the conditions in the exis-
tence theorem in Section 2.4.1. Such solutions must have continuous second-order
derivatives and over-restricted initial functions
, thus their applica-
tion is limited. Consider the string of two fixed ends discussed in Chapter 1, for
example. If we raise the string at point
x
0
and release it without initial velocity, the
string will have a free vibration. The PDS governing the string displacement should
have a unique solution. However,
ϕ
(
x
)
and
ψ
(
x
)
is not differentiable at
x
0
. Due to the reflec-
tion at the two ends, the discontinuity of
ϕ
(
x
)
.
There exists no solution with continuous second-order derivatives in the whole do-
main
ϕ
(
x
)
at
x
0
propagates in
(
0
,
l
)
×
(
0
,
+
∞
)
(
,
)
×
(
,
+
∞
)
, thus no classical solution exists. By the D'Alembert formula,
however, the solution is
0
l
0
)=
ϕ
(
x
−
at
)+
ϕ
(
x
+
at
)
u
(
x
,
t
,
2
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