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explicit (lower), exact, implicit (upper), N =100, T=0.9
14
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Fig. 2.2 The figure shows numerical approximations computed by an explicit and an implicit
scheme. The upper curve is the solution generated by the implicit scheme and the lower curve is
computed by the explicit scheme; the analytical solution is in between
1 p 1 4t z n
:
1
2t
z nC1
D
(2.37)
In Fig. 2.2 we have plotted the two numerical solutions and the exact solution for t
ranging from 0 to 0:9: In the numerical computations we used N D 100: We note
that both numerical solutions behave well.
Example 2.5. We consider the problem above but with a negative initial condition:
y 0 .t / D y 2 .t /;
(2.38)
y.0/ D 10:
The solution of this problem is given by 17
10
1 C 10t :
y.t/ D
 
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