Environmental Engineering Reference
In-Depth Information
Again, a linear combination of the dependent variable
u
and its normal deriva-
tive is given on the boundary
S
. Such a boundary condition is called a
convective
boundary condition
in heat conduction. In particular, if the body is a rod of length
l
(
0
≤
x
≤
l
)
, then we have at the two endpoints
x
=
l
=
∂
x
=
l
x
=
0
=
−
∂
x
=
0
.
∂
u
u
∂
u
u
and
∂
n
∂
x
∂
n
∂
x
The above convective boundary condition reduces to
∂
hu
x
=
0
=
ϕ
1
(
∂
hu
x
=
l
=
ϕ
2
(
u
u
x
−
t
)
,
x
+
t
)
,
∂
∂
where
are known functions.
When a linear combination of the dependent variable
u
and its normal derivative
ϕ
(
t
)
and
ϕ
(
t
)
1
2
∂
u
n
is known on the boundary, in general, the boundary condition reads
∂
∂
u
S
=
ϕ
(
∂
u
S
=
ϕ
(
u
u
n
+
σ
M
,
t
)
or
n
+
σ
M
)
(1.81)
∂
∂
ϕ
(
,
)
ϕ
(
)
σ
>
where
M
t
and
M
are known and the constant
0. Such a boundary con-
dition is called a
boundary condition of the third kind
.
All three kinds of boundary conditions discussed above are linear in the depen-
dent variable
u
and its normal derivative. They are thus called
linear boundary
conditions
. If a boundary condition is not linear in the dependent variable and/or
its normal derivative, it is called a
nonlinear boundary condition
. If the free term
(the right-hand side) in the three kinds of boundary conditions is zero, then we call
them
homogeneous boundary conditions
. Otherwise, we call them
nonhomogeneous
boundary conditions
.
Remark 1.
Initial conditions describe the physical state of the whole system at an
initial time instant, not just the initial state of one point. Boundary conditions rep-
resent the system state at the boundary over the whole process, not just the instant
state at the boundary. Potential equations are independent of time; no initial condi-
tions are required. While boundary conditions describe physical states on the system
boundary, external forces or sources may act inside the system.
Remark 2.
The effects of boundary conditions can sometimes be neglected. Con-
sider, for example, the slow unidirectional diffusion of some molecules from
x
=
0
to
x
. If our attention is on diffusion in a sufficiently
short time, we may neglect the effect of boundary conditions at
x
=
l
in a rod of length
l
(
0
≤
x
≤
l
)
=
l
, replace the
=
→
+
∞
real boundary condition at
x
l
by an infinite boundary
x
, and study the
≤
<
+
∞
diffusion in a semi-infinite rod 0
.
Remark 3.
Boundary conditions can be different on different parts of the boundary.
For the longitudinal vibration of a rod of length
l
x
(
0
≤
x
≤
l
)
, we have boundary
conditions of the first kind and the second kind at
x
=
0and
x
=
l
for the fixed end
x
=
0 and the free end
x
=
l
such that
u
(
0
,
t
)=
0and
u
x
(
l
,
t
)=
0. If the end
x
=
0
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