Environmental Engineering Reference
In-Depth Information
Stability
The CDS comes normally from experimental or field measurements with unavoid-
able errors. If the solution of a PDS is not stable with respect to the CDS such that
a small error in obtaining the CDS can cause significant variations of the solution,
then the PDS cannot be used to represent physical reality. A well-posed PDS thus
must have a solution that is stable with respect to the CDS, so that the solution
varies only a little for a small variation of data in the CDS. In other words, the
solution of a well-posed PDS must depend continuously on its CDS. Since a non-
homogeneous boundary condition can normally be homogenized by some function
transformations, stability often refers to the stability of solutions with respect to the
initial conditions.
An analytical definition of stability can be made by introducing the size and norm
of functions in some function space.
If a PDS has a unique and stable solution, it is called a
well-posed PDS
.Other-
wise, it is ill-posed. While a well-posed PDS is normally desirable from the point of
view of applications, some physical problems do lead to an ill-posed PDS. There-
fore the study of ill-posed PDS is also an important branch of partial differential
equations.
A study of well-posedness is normally mathematically-involved. We discuss the
well-posedness only in a few places and focus our attention mainly on how to find
solutions of various PDS in this topic.
1.4.5 Example of Developing PDS
A PDS represents how physical variables distribute and evolve in the form of differ-
ential equations. The generation, distribution and evolution of a physical variable
are the result of and a response to physical causes. The physical causes come nor-
mally from the following three mechanisms:
1. internal sources that are represented by the nonhomogeneous terms in the equa-
tions;
2. initial physical states of systems that are represented by the initial conditions;
3. system states at the boundary that are represented by the boundary conditions.
If all three mechanisms are absent, there would be no generation, distribution or
evolution of physical variables. For example, the mixed problem describing the dis-
placement
u
(
x
,
t
)
of a vibration string
⎧
⎨
a
2
u
xx
u
tt
=
+
f
(
x
,
t
)
,
0
<
x
<
l
,
0
<
t
(
,
)=
μ
(
)
,
(
,
)=
μ
(
)
,
u
0
t
t
u
l
t
t
1
2
⎩
u
(
x
,
0
)=
ϕ
(
x
)
,
u
t
(
x
,
0
)=
ψ
(
x
)
(
,
)
≡
(
,
)=
μ
(
)=
μ
(
)=
ϕ
(
)=
ψ
(
)
≡
0. This is physically-
grounded and can be precisely proven. This simply shows that the string will be at
rest if there is not any cause for vibration.
has
u
x
t
0if
f
x
t
t
t
x
x
1
2
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