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which is transformed to
k
@
u
@x
D
xq.x/
k
x
v
:
(7.51)
This condition is similar to a cooling law in heat conduction. The bottom line is
that programs for the standard one-dimensional diffusion equation can also handle
a class of spherically symmetric diffusion problems.
7.3.5
Scaling
Scaling is a very useful but difficult topic in mathematical modeling. Readers who
are eager to see how a diffusion problem is solved on a computer can jump to
Sect.
7.4
. There we work with a scaled problem, but knowing the details of scal-
ing is not a prerequisite for Sect.
7.4
; one can just imagine that the input data in
the problem have been given specific values (typically values of unity). The forth-
coming material on scaling gives a gentle introduction to the scaling of PDEs and
must be viewed as a collection of a few examples, and not a complete exposition of
scaling as a mathematical technique. In a sense, scaling is, mathematically, a very
simple mechanical procedure, but applying scaling correctly to a problem is often
very demanding. The reader is hereby warned about the inherent difficulties in the
material on the forthcoming pages.
A Heat Conduction Model Problem
Let us look at a heat conduction problem without heat sources and with the
temperature controlled at constant values at the boundaries:
@t
D
k
@
2
u
%c
v
@
u
;
x
2
.a; b/; t > 0;
(7.52)
@x
2
u
.a; t /
D
U
a
;
t > ;
(7.53)
u
.b; t /
D
U
b
;
t > ;
(7.54)
u
.x; 0/
D
I.x/;
x
2
Œa; b :
(7.55)
The heat conduction takes place in a homogeneous material such that %, c
v
,and
k are constants. To be specific, we assume that I.x/ is a step function (relevant
for the introductory example of bringing together two metal pieces at different
temperatures):
I.x/
D
U
a
;a
x<.b
a/=2;
U
b
;.b
a/=2
x
b:
(7.56)
If you wonder how the solution
u
.x; t
`
/ behaves, you can take a look at Fig.
7.10
on
page
313
. Here we have plotted
u
.x; t
`
/ at some time points t
`
. You should notice
