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8.1.10
Uniqueness Revisited
In Sect. 8.1.3 we used the energy bound (8.7) to prove that the model problem ( 8.1 )-
( 8.3 ) can have at most one smooth solution. We will now see that it is also possible
to utilize (8.41)-( 8.43 ) to prove this.
Suppose that both u and v are smooth functions satisfying ( 8.16 )-( 8.18 ). Let
e.x; t/ D u .x; t / v .x; t /
for x 2 Œ0; 1; t 0;
and note that
e t
D . u v / t
D u t v t
D u xx v xx D . u v / xx D e xx :
Thus, e satisfies the diffusion equation. Furthermore,
e.0; t/ D u .0; t / v .0; t / D g 1 .t / g 1 .t / D 0
for t>0;
e.1; t/ D u .1; t / v .1; t / D g 2 .t / g 2 .t / D 0
for t>0;
e.x; 0/ D u .x; 0/ v .x; 0/ D f.x/ f.x/ D 0
for x 2 .0; 1/;
and consequently (8.41)-( 8.43 ) imply that
0 e.x; t/ 0
for all x 2 Œ0; 1; t 0;
or
u .x; t / v .x; t / D 0
for all x 2 Œ0; 1; t 0:
The problem ( 8.16 )-( 8.18 ) can have at most one smooth solution.
8.2
Separation of Variables and Fourier Analysis
The great French mathematician Jean Baptiste Joseph Fourier (1768-1830)did
important mathematical work on the theory of heat. He invented 14 a new framework
for studying the diffusion equation. Fourier's work has had an enormous impact, not
only on the theory of PDEs, but on several branches of modern mathematics. If you
decide to continue to study mathematics you will most certainly run into theories
that somehow can be tracked back to Fourier's original work on heat conduction.
14 Are new mathematical theories inventions or discoveries? It seems like neither mathematicians
nor philosophers are able to agree upon this. The interested reader can consult the topics [12]and
[11] for a general discussion of the “nature of mathematics”.
 
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