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is fortunately a good choice in the present example, since it can be shown mathemat-
ically that
u
in (
7.52
)-(
7.55
) is bounded by the initial condition and the boundary
values.
We are now ready to replace the physical variables x, t ,
u
,andI , i.e. all indepen-
dent and dependent variables plus all function expressions, by their dimensionless
equivalents. From
N
x
D
x
a
b
a
;
N
t
D
t
; I
D
I
U
a
U
b
U
a
;
N
u
D
u
U
a
U
b
U
a
;
t
c
we get
I
D
U
a
C
.U
b
U
a
/ I;
N
t;
x
D
a
C
.b
a/
N
x;
t
D
t
c
u
D
U
a
C
.U
b
U
a
/
N
u
;
which we insert into (
7.52
)-(
7.55
). Noting that
@t
D
@
N
t
@
u
@
N
t
.U
a
C
.U
b
U
a
/
N
u
/
D
1
@
.U
b
U
a
/
@
N
u
@
N
t
;
@t
t
c
with a similar development for the @
u
=@x expression, we arrive at
@
2
N
u
@
N
x
2
%c
v
U
b
U
a
t
c
@
N
u
@
N
t
D
k
U
b
U
a
;
N
x
2
.0; 1/;
N
t>0;
(7.57)
.b
a/
2
N
u
.0;
N
t/
D
0;
N
t>0;
(7.58)
N
u
.1;
N
t/
D
1;
N
t>0;
(7.59)
N
u
.
N
x; 0/
D
0; 0
x
2
;
(7.60)
1;
2
<
N
x
1:
The PDE (
7.57
) can be written in the form
@
N
t
D
ı
@
2
N
u
@
N
u
;
(7.61)
@
N
x
2
with ı being a dimensionless number,
kt
c
%c
v
.b
a/
2
ı
D
:
We have not yet determined t
c
, which is the task of the next paragraph.
Finding a Non-Trivial Scale
The goal of scaling is to have the maximum value of all independent and dependent
variables in the PDE problem of order unity. One can argue that if
N
u
is of order
unity, the coordinates are of order unity, and
N
u
is sufficiently smooth, we expect the