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u(x,t=0)
u(x,t=0.001)
u(x,t=0.025)
u(x,t=0.25)
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Fig. 7.8
Curve plots of the temperatures along the line y
D
0:5 for the same physical problem as
in Fig.
7.7
Mathematical Splitting of Dimensions
Reduction of the number of space dimensions in a model can also result from purely
mathematical techniques. For example, one can think of a situation where physical
and/or mathematical insight leads to a split of the unknown function's dependence
on space coordinates: typically,
u
.x; y;
z
;t/
D
F.x;t/
C
G.y;
z
;t/
u
.x; y;
z
;t/
D
F.x;t/G.y;
z
;t/:
or
If one can easily deduce what the G function is, one is left with a one-dimensional
problem for finding F.x;t/.
Neglecting Variations in Time
When discussing the reduction of the dimensions of a model, we should also men-
tion that the time parameter can sometimes be discarded, if the changes of the
unknown function in time are sufficiently small. Such problems are usually referred
to as
stationary
or
time-independent problems
. One example is the physical problem
corresponding to Figs.
7.4
and
7.5
, where two metal pieces at different temperatures
are brought into contact. Our everyday experience tells us that the temperature jump
will be smoothed out in time. As time increases, the solution approaches a constant
state. Mathematically, the solution and thus also the model do not depend on the
time parameter in the limit t
!1
. In real computations we reach an approximately
stationary (time-independent) problem after a finite time.
