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then its Fourier sine series will converge quickly toward f.x/for all x in the closed
interval Œ0; 1. Finally note that, even if (8.82) does not hold, (8.76) provides the
correct formal solution of the problem (
8.77
)-(
8.79
). We will not dwell upon this
issue. Further details on this advanced topic can, e.g., be found in Weinberger [29].
Example 8.6.
Above we mentioned that we were not in a position to solve our model
problem for even the simplest possible initial condition, i.e., we could not solve
(
8.77
)-(
8.79
)iff were a constant (
¤
0). This is no longer the case. According to
(8.76)and(8.81),
X
40
.2k
1/
e
.2k
1/
2
2
t
sin..2k
1/x/
u
.x; t /
D
(8.83)
k
D
1
is the formal solution of
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
u
.0; t /
D
u
.1; t /
D
0
for t>0;
u
.x; 0/
D
10
for x
2
.0; 1/:
In Figs.
8.2
and
8.3
we have graphed the function given by the 100th partial sum
of the series (8.83) at time t
D
0:5 and t
D
1, respectively. Note that the solution
decays rapidly. This is in accordance with the physics of the underlying heat transfer
problem: Recall that the temperature at the boundary of the solution domain is kept
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 8.2
A plot of the function given by the sum of the first 100 terms of the series defining the
formal solution of the problem studied in Example
8.6
. The figure shows a snapshot of this function
at time t
D
0:5