31
where -7r < x < 7r, and -5 < a < -1. If a # -1, the graph of f(x) is shown in
Figure 4-1. We can find the maximum of f(x) is -a/(a2 1)3/2. When a tends
to -1, the maximum of f(x) approaches to infinity. If a = -1, the graph of f(x) is
shown in Figure 4-2. We can see the supremum of f(x) is infinity. Hence,
300s-
250
20J
y15:
10
I \
-3 -2 -1 0
1 2 3
x
Figure 4-2: Graph of
S sin2 x 1
f () = (,, ) when a = -1.
(a + cos )2
sin2 03
sup = +00.
-7/h d h2(1 2p K(At)j?) sin23 03
(E(cos0j 1))2
In other words, the scheme is unstable when 0 K< p < 1/2. U
4.2 Error Analysis
In this section, the error generated from the discretizing process is analyzed.
Lemma 3.4.1 and Lemma 4.2.1 are applied in the error analysis. Lemma 3.4.1 gives