Untitled - Cdm.unimo.it

Ordinary Differential Equations 203
(9.4.17) Bw,n(ψ,φ) :=
(ψ −
I
˜ ψ) ′ [(φ − ˜ φ)w] ′ dx +
+ µ (Ia,n−1ψ)(φ −
I
˜
φ)w dx + µ
(9.4.18) Fw,n(φ) :=
(Ia,n−1f)(φ −
I
˜ φ)w dx +
I
˜ψ ′ ˜ φ ′ dx
(Ia,n−1ψ)
I
˜ φ dx, ∀ψ,φ ∈ Pn,
+σ2φ(1) − σ1φ(−1), ∀φ ∈ Pn,
(Ia,n−1f)
I
˜ φ dx
where a(x) := (1 − x 2 )w(x), x ∈ I, and the interpolation operator Ia,n−1, n ≥ 2, is
defined in section 3.3.
Our aim is to show that (9.4.16) leads to a collocation type approximation. First,
assume φ ∈ P 0 n so that ˜ φ ≡ 0. With the help of integration by parts and quadrature
formula (3.5.1), we obtain
(9.4.19) −
n
j=0
=
p ′′ n(η (n)
j )φ(η (n)
j ) ˜w (n)
j
n
j=0
+ µ
n
j=0
(Ia,n−1pn)(η (n)
j )φ(η (n)
j ) ˜w (n)
j
(Ia,n−1f)(η (n)
j )φ(η (n)
j ) ˜w (n)
j , ∀φ ∈ P 0 n.
Choosing φ := ˜ l (n)
i , 1 ≤ i ≤ n − 1, (9.4.19) implies
(9.4.20) −p ′′ n(x) + µ(Ia,n−1pn)(x) = (Ia,n−1f)(x) at x = η (n)
i , 1 ≤ i ≤ n − 1.
Note that, for any g ∈ C0 ( Ī), one has
(n)
(n)
(Ia,n−1g)(η i ) = ( Ĩw,ng)(η i ) = g(η (n)
i ),
1 ≤ i ≤ n − 1. This gives −p ′′ n(η (n)
i ) + µpn(η (n)
i ) = f(η (n)
i ), 1 ≤ i ≤ n − 1. Moreover,
for a general φ ∈ Pn, we have a relation like (9.3.21). Both −p ′′ n + µIa,n−1pn and
Ia,n−1f are polynomials of degree n −2. Since they coincide at n −1 points, (9.4.20)
holds for any x ∈ Ī. This enables us to remove the integrals and recover the boundary
conditions −p ′ n(−1) = −σ1 and p ′ n(1) = σ2. To obtain the corresponding linear
system we follow the suggestions of section 7.4 (see (7.4.13)).