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In-Depth Information
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 8.4
The first (
dashed line
), the first seven (
dash-dotted line
), and the first 100 (
solid line
)
terms of the Fourier sine series of the function f.x/
x
2
. It is impossible to distinguish
between the figures representing the first seven and the first 100 terms. They are both accurate
approximations of x
D
x
x
2
u
t
D
u
xx
for x
2
.0; 1/; t > 0;
u
.0; t /
D
u
.1; t /
D
0
for t>0;
sin..2k
1/x/
X
1
8
..2k
1//
3
u
.x; 0/
D
for x
2
.0; 1/;
k
D
1
and then it follows by (8.76)that
e
.2k
1/
2
2
t
sin..2k
1/x/
X
8
..2k
1//
3
u
.x; t /
D
(8.84)
k
D
1
defines a formal solution of it. As in Example
8.6
we observe that this solution
decays as time t increases, see Figs.
8.5
and
8.6
. This is in agreement with our
physical intuition of the problem under consideration.
8.2.9
Analysis of an Explicit Finite Difference Scheme
In Sect. 7.4.5 we found that the ˛ parameter in the explicit finite difference scheme
from Algorithms 7.1-7.3 must be chosen to be less than one-half. In the present
section we will give an alternative theoretical explanation why the explicit scheme
