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Let us consider a problem with a modified initial condition,
v t
D v xx
for x 2 .0; 1/; t > 0;
(8.12)
v .0; t / D v .1; t / D 0
for t>0;
(8.13)
v .x; 0/ D g.x/
for x 2 .0; 1/:
(8.14)
If g is close to f , will v be approximately equal to u ? To answer this question we
will study the difference e between u and v , that is, we will analyze the function
e.x; t/ D u .x; t / v .x; t /
for x 2 Œ0; 1; t 0:
From ( 8.1 )-( 8.3 )and( 8.12 )-( 8.14 ), we find that
e t
D . u v / t
D u t v t
D u xx v xx D . u v / xx D e xx ;
and furthermore
e.0; t/ D u .0; t / v .0; t / D 0 0 D 0 for t>0;
e.1; t/ D u .1; t / v .1; t / D 0 0 D 0 for t>0;
e.x; 0/ D u .x; 0/ v .x; 0/ D f.x/ g.x/
for x 2 .0; 1/:
So, interestingly, e solves the diffusion equation with homogeneous Dirichlet bound-
ary conditions at x D 0; 1 and with initial condition
h D f g;
that is, e satisfies
e t D e xx for x 2 .0; 1/; t > 0;
e.0; t/ D e.1; t/ D 0
for t>0;
e.x; 0/ D f.x/ g.x/
for x 2 .0; 1/:
In particular, this means that e must satisfy the properties derived above for the
model problem ( 8.1 )-( 8.3 ). Hence, by inequality (8.7), we conclude that
Z 1
e 2 .x; t / dx Z 1
0
h 2 .x/ dx
for t>0;
0
or
Z 1
. u .x; t / v .x; t // 2 dx Z 1
0
.f .x/ g.x// 2 dx
for t>0:
(8.15)
0
 
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