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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
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0.7
0.8
0.9
1
Fig. 7.18
Numerical results in a problem with a sine series solution given in (8.84). The
solid
line
represents the first 100 terms in the sine series (8.84). The
dotted
,
dash-dotted
,and
dashed
curves
are the numerical results generated in the cases of ˛
D
D
D
0:3, ˛
0:45,and˛
0:4971,
respectively
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. 7.19
Same problem as in Fig.
7.18
, but now the numerical solution corresponds to ˛
D
0:5828
Let us fix the number of grid points used in the space dimension to 13, and
study the performance of the method as the time step is reduced. Will we observe
the same type of behavior as that in Figs.
7.16
and
7.17
?Yes,inFigs.
7.19
and
7.20
we used ˛
D
0:5828 and ˛
D
0:6760, respectively. Clearly, oscillations and
instabilities occur. The numerical method we have presented so far to solve the
