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206 Polynomial Approximation of Differential Equations
(9.4.25) Fw,n(φ) :=
+ µ
n
j=0
n
j=0
f(η (n)
j ) (φ − ˜ φ)(η (n)
j ) ˜w (n)
j
f(η (n)
j ) ˜ φ(η (n)
j ) χ (n)
j + σ2φ(1) − σ1φ(−1), ∀φ ∈ Pn.
The weights χ (n)
j , 0 ≤ j ≤ n, are defined in section 3.7. When ν = 0 , we obtain
(9.4.21) and (9.4.22) by noting that χ (n)
j
= ˜w(n)
j , 0 ≤ j ≤ n.
Basically, Bw,n and Fw,n are derived from (9.3.18) and (9.3.19) respectively, by
considering quadrature formulas in place of integrals. The errors introduced decay
spectrally, giving the convergence of the method. We remark that, although Clenshaw-
Curtis formula does not have the accuracy of Gaussian formulas, it is only used for
polynomials of low degree (we have ˜ φ ∈ P1 and ψ ˜ φ ∈ Pn+1).
The corresponding variational problem is equivalent to system (7.4.16) where we set
q := Ĩw,nf and γ := [χ (n)
0 ]−1 = [χ (n)
n ] −1 . In general, if γ grows at least as n 2 ,
one gets convergence of pn to the solution of (9.1.5) when n tends to infinity (see
funaro(1988)).
The idea of collocating the equation at the boundary points has been also proposed
in funaro and gottlieb(1988) for first-order equations, such as (9.1.1). This leads
to a collocation scheme like that in (7.4.4), where γ is proportional to n 2 . A proof
of convergence is easily given for the Legendre case following the guideline of theorem
9.2.1.
The techniques introduced and analyzed in this section are also used for the approx-
imation of the solution of fourth-order equations. The collocation method for computing
the solution of problem (9.1.6) is
⎧
(9.4.26)
p IV
n (η (n)
i ) = f(η (n)
i ), 1 ≤ i ≤ n − 1,
⎪⎨
pn(−1) = σ1, pn(1) = σ2,
⎪⎩
p ′ n(−1) = σ3, p ′ n(1) = σ4,
where pn is a polynomial in Pn+2, n ≥ 2.