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presents an extension from the previous example: How shall we treat the functions
%.x/, c
v
.x/, k.x/,andf.x;t/? The idea is simple: These functions are known, and
we scale them by their maximum values. For example,
f.x;t/
D
f.x;t/
f
c
D
max
x;t
j
f.x;t/
j
:
;
c
The maximum value of the scaled function is then unity.
Scaling with Neumann Boundary Condition
Letusaddress(
7.52
)-(
7.55
) when the boundary value
u
D
U
b
is replaced by
@
@x
u
.b; t /
D
Q
b
;:
k
(7.72)
i.e., the heat flow is known as
Q
b
on x
D
b. How will this small change affect the
scaling? The detailed arguments become quite lengthy in this case too, but an impor-
tant result is that all scalings, except the one for
u
, are not affected. Our suggested
scaling of
u
in (
7.52
)-(
7.55
) reads
N
u
D
u
U
a
U
b
U
a
:
(7.73)
The question now is whether U
b
U
a
is a characteristic magnitude of
u
U
a
.The
answer depends on the value of the input data in the problem.
Suppose we apply (7.73). The scaled PDE problem becomes (with bars dropped)
@t
D
@
2
u
@
u
;
x
2
.0; 1/; t > 0;
(7.74)
@x
2
u
.0; t /
D
0;
t > 0;
(7.75)
@
@x
u
.1; t /
D
ˇ;
t > 0;
(7.76)
u
.x; 0/
D
0; 0
x
c;
1;
N
c<x
1;
(7.77)
where
ˇ
D
Q
b
.b
a/
k.U
b
U
a
/
(7.78)
is a dimensionless number. If ˇ is too far from unity,
u
has a large derivative at the
boundary, suggesting that
u
itself may have a magnitude much larger than unity. In
that case it would be more natural to base the magnitude of
u
on Q
b
. Estimating
u
as
a straight line based on the knowledge that
u
is U
a
at x
D
a and
u
has a derivative
Q
b
=k at x
D
b (yes, we use the same symbols now for scaled and unscaled
