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so that the initial temperature is the distribution
u
(
x,0
) =
δ
(
x
). The exact
final temperature and approximate solution of the inverse problem (1) ob-
tained by Eq. (9) at the time
t
= 700 s for this case, is given in Figures
(3.1)
and (3.2).
FIGURE 3.1
Numerical approximation of
u
(
x
, 0) =
δ
(
x
1
, x
2
) obtained by Eq. (9) for the
final function (12) at
t
= 700 s with
N
= 65. The values of
u
(
x
1
,
x
2
, 700) and the corresponding
approximated values
u
q
(0) are given on the vertical axes of the left-hand and right-hand
side figures, respectively.
FIGURE 3.2
Numerical approximation of
u
(
x,
0)
= δ
(
x
1
,
x
2
) obtained by Eq. (9) for
the final function (12) at
t
= 250 s with
N
= 191. The values of
u
(
x
1
, x
2
,
250) and the
corresponding approximated values
u
q
(0) are given on the vertical axes of the left-hand and
right-hand side figures, respectively.
In order to compute a solution for a long time, the value of
N
should
be relatively small; for example, for
N
= 65, the attainable backward time
is about 700 s. Beyond this value, the method is unable to render good
results. Note that in this example, the number of nodes is odd. This is
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