Environmental Engineering Reference
In-Depth Information
1.4 Conditions and Problems for Determining Solutions
Equations of mathematical physics are drawn from physical problems. In applica-
tions, their solutions always refer to particular solutions subjected to certain physical
conditions. Such physical conditions are called the
conditions for determining so-
lutions
,the
CDS
for short. The CDS is normally divided into initial conditions and
boundary conditions. Finding solutions of equations of mathematical physics sub-
jected to the CDS is called the
problem for determining solutions
,orthe
PDS
for
short.
1.4.1 Initial Conditions
For wave equations containing
u
tt
, the initial conditions refer to the initial values of
u
and
u
t
. The initial instant is normally
t
=
0, but can also be
t
=
t
0
for a known time
instant
t
0
.Let
M
be a point
x
,
(
x
,
y
)
or
(
x
,
y
,
z
)
in one-, two- or three-dimensional
space; the initial conditions are thus
u
(
M
,
0
)=
ϕ
(
M
)
,
u
t
(
M
,
0
)=
ψ
(
M
)
,
(1.77)
where
are known functions of one, two or three variables. For the transverse
displacement of a vibrating string, the initial conditions are the initial displacement
and velocity of the string. The highest-order temporal derivative in the classical heat-
conduction equation is
u
t
. The initial conditions for the classical heat-conduction
equation thus give
ϕ
and
ψ
u
(
M
,
0
)=
ϕ
(
M
)
,
which represents the initial temperature distribution. Both the hyperbolic and the
dual-phase-lagging heat-conduction equations contain the term
u
tt
. The required
initial conditions are the same as those in Eq. (1.77). Here
are the
initial temperature distribution and the initial rate of temperature changes in time,
respectively. Note that the physical meaning of the initial conditions depends on
physics and the nature of the dependent variable
u
.
In potential equations, the dependent variables are functions of position, but not
of time; thus they require no initial conditions.
ϕ
(
M
)
and
ψ
(
M
)
Remark 1.
Initial conditions represent the initial state of a whole system, not just
some parts or points of the system. Consider the vibration of a string of length
l
,
fixed at the end points and subjected to an initial displacement
h
at the middle. The
initial conditions are
0
⎧
⎨
2
h
l
l
2
x
,
x
∈
,
,
u
(
x
,
0
)=
l
2
,
l
and
u
t
(
x
,
0
)=
0
.
⎩
2
h
l
(
l
−
x
)
,
x
∈
,
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