Untitled - Cdm.unimo.it

Examples 279
By imposing the condition U(1) = 0, it is possible to show that the eigenvalue problem
(12.4.2) has a countable number of positive real eigenvalues λm, m ≥ 1. Therefore,
it is clear that z (1)
j = λj, j ≥ 1. Our goal is to find an approximation to these
eigenvalues. According to the results of section 8.6, for n ≥ 2, we discretize (12.4.2) by
the eigenvalue problem
⎧
(12.4.3)
⎪⎨
−p ′′ n,m(˜η (n)
i ) − [˜η (n)
i ] −1 p ′ n,m(˜η (n)
i )
+ [˜η (n)
i ] −2 pn,m(˜η (n)
i ) = λn,m pn,m(˜η (n)
i ) 1 ≤ i ≤ n − 1,
⎪⎩
pn,m(0) = λn,m pn,m(0), pn,m(1) = λn,m pn,m(1),
0 ≤ m ≤ n,
where pn,m ∈ Pn, 0 ≤ m ≤ n, and the collocation points are recovered from the
Chebyshev Gauss-Lobatto nodes by setting ˜η (n)
i := 1
2 (η(n) i
+1), 0 ≤ i ≤ n (see (3.1.11)).
The (n + 1) × (n + 1) matrix relative to (12.4.3) can be written in the standard way
(see chapter seven). Two of the corresponding n + 1 eigenvalues λn,m, 0 ≤ m ≤ n,
are equal to 1 (say λn,0 = λn,n = 1). The others approach the exact eigenvalues
λm, m ≥ 1. In table 12.4.1, we report the zeroes z (1)
m and the computed eigenvalues
for 1 ≤ m ≤ 5 (the first five zeroes of J1 are tabulated in luke(1969), Vol.2, p.233).
In figure 12.4.1, we plot J1 in the window [0,18] × [−1,1]. An approximation of the
zeroes of Jk, k ≥ 2, is obtained with the same arguments.
m z (1)
m
λ8,m
λ12,m
λ16,m
1 3.831705970207 3.831705650608 3.831705970200 3.831705970207
2 7.015586669815 7.015773798574 7.015586712768 7.015586669820
3 10.17346813506 10.16416616370 10.17347111164 10.17346813668
4 13.32369193631 13.50667201292 13.32368681006 13.32369172635
5 16.47063005087 15.87375296304 16.46311800444 16.47062464105
Table 12.4.1 - Approximation of the zeroes of J1.