Information Technology Reference
In-Depth Information
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
 
Search WWH ::




Custom Search