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1.5 x 10 −4
1
0.5
0
−0.5
−1
−1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 7.16
The dashed line represents numerical results generated by Algorithm 7.1 with ˛
D
D
0:5112 (x
1=59). The solid line is the graph of the exact solution (7.106)
5
4
3
2
1
0
−1
−2
−3
−4
−5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 7.17
D
1=59). Observe that for the chosen x and t values, the method fails to solve the problem under
consideration!
A plot of the numbers generated by Algorithm 7.1 when ˛
D
0:5157 > 0:5 (x
In Fig. 7.18 we have plotted the function given by the sum of the first 100 terms
of the sine series of the solution (8.84), along with the results from three different
numerical experiments. Quick convergence is obtained. Even by using “modest”
discretization parameters (x D 1=13 and t D 1=34), the explicit scheme seems
to produce accurate approximations. Note that ˛ 0:5 for all the parameters used
in these experiments.
 
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