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of our model problem (
8.1
)-(
8.3
). Under what circumstances will this function
define a “proper” solution of this problem? This issue is beyond the scope of this
text,
21
and thus we will not dwell upon this subject. Instead, we will assume that
these formal solutions are well behaved.
Note that if this series allows for term-wise differentiation, i.e.,
c
k
e
k
2
2
t
sin.kx/
;
X
d
dt
u
t
.x; t /
D
k
D
1
c
k
e
k
2
2
t
sin.kx/
;
X
d
2
dx
2
u
xx
.x; t /
D
k
D
1
then (8.69) defines the unique smooth solution of our model problem. The argument
for this property is completely analogous to that used in the case of an initial condi-
tion given in terms of a finite sine series discussed above, and it is left as an exercise
to the reader.
8.2.6
Computing Fourier Sine Series
So far we have seen that if the initial condition f is given by a sum, finite or infi-
nite, of sine functions that vanish at x
D
0 and at x
D
1, then we can construct,
at least formally, solutions of our model problem (
8.1
)-(
8.3
). But very few func-
tions are given in terms of sine series. What about fairly simple initial temperature
distributions? For example,
f.x/
D
x.1
x/
for x
2
.0; 1/;
or
f.x/
D
10
for x
2
.0; 1/:
So, the state of affairs is really bad. We cannot even handle what, from a physi-
cal point of view, most people would characterize as the simplest case, an initially
uniform temperature distribution! How can we extend the present theory to handle
such cases? Fourier himself asked this question and pursued the subject with great
interest.
22
Our goal now is to develop a more general approach to this problem.
Faced with the fact that very few functions are given in terms of sine series, what
can we do? Well, we can sort of take the “opposite view” to handle this problem.
More precisely, for a given initial condition f we might ask if we can find constants
c
1
;c
2
;c
3
;::: such that
21
It is an important topic in more advanced texts on Fourier analysis, see, e.g., Weinberger [29]
22
As is so often the case with breakthroughs in science, Fourier's work on this subject caused a
lot of controversy. Fourier's results were basically correct but his arguments contained errors. He
even had problems with getting his work accepted for publication!
