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Ordinary Differential Equations 205
There are several reasons for preferring conditions (9.4.23) to exact boundary condi-
tions. For instance, as noticed in section 8.6, the eigenvalues of the matrix corresponding
to system (7.4.16) are more appropriate for computation than the eigenvalues of problem
(8.6.8). In addition, for the approximation of the solution of (9.1.5), the new approach
gives better numerical results. We compare the two ways of imposing the boundary
conditions in table 9.4.1. Take µ := 1 and f, σ1, σ2 such that the exact solution of
(9.1.5) is U(x) := cos(e x ), x ∈ Ī. Then, compute the error En := pn − Ĩw,nUw
for various n. In the first column, pn is obtained by the collocation method with the
conditions p ′ n(−1) = σ1, p ′ n(1) = σ2. In the second column, pn is obtained by the
collocation method with the conditions (9.4.23).
n Column 1 Column 2
6 .1090 × 10 −1 .2009 × 10 −3
8 .4130 × 10 −2 .1840 × 10 −4
10 .2723 × 10 −3 .5941 × 10 −6
12 .6896 × 10 −5 .1495 × 10 −7
14 .1404 × 10 −5 .1471 × 10 −8
16 .7122 × 10 −7 .4405 × 10 −10
Table 9.4.1 - Errors corresponding to the Legendre
collocation approximation of problem (9.1.5).
For other Jacobi ultraspherical weights, we generalize (9.4.21) and (9.4.22) by defining
(9.4.24) Bw,n(ψ,φ) :=
+ µ
n
j=0
ψ(η (n)
j ) (φ − ˜ φ)(η (n)
j ) ˜w (n)
j
(ψ −
I
˜ ψ) ′ [(φ − ˜ φ)w] ′ dx +
+ µ
n
j=0
I
˜ψ ′ ˜ φ ′ dx
ψ(η (n)
j ) ˜ φ(η (n)
j ) χ (n)
j , ∀ψ,φ ∈ Pn,