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202 Polynomial Approximation of Differential Equations
(9.4.14) Bw(pn,φ) = Fw(φ), ∀φ ∈ P 0 n,
where Bw and Fw are respectively given in (9.3.2) and (9.3.3). The convergence
analysis is straightforward, since now the linear operator on the right-hand side does
not depend on n. In addition, (9.4.14) is equivalent to the problem
(9.4.15)
⎧
⎨
−p ′′ n = Πa,n−2f in I,
⎩
pn(±1) = 0,
where a(x) := (1 − x 2 )w(x), x ∈ I. The solution in this particular case has the
form pn = ˆ Π 1 0,w,nU, ∀n ≥ 2 (see (6.4.13)). However, the linear system associated to
(9.4.15) is not easy to recover when solving the problem in the frequency space relative
to the weight function w. Usually, this system is obtained by testing (9.4.14) on the set
of polynomials: φ(x) := P (α,β)
k
(x) − 1
2
(1 − x)P (α,β)
k
(−1) − 1
2
(α,β)
(1 + x)P k (1), x ∈ Ī,
2 ≤ k ≤ n. By the orthogonality of Jacobi polynomials, we obtain the corresponding
(n − 1) × (n − 1) matrix which does not explicitly contain the rows relative to the
boundary conditions as in the tau method (clarifying remarks about this point are
given in boyd(1989), chapter six).
Let us consider now the approximation of other differential equations, such as
(9.1.5). For X ≡ H 1 w(I), a variational formulation is given by (9.3.4), where Bw and
Fw are defined in (9.3.18) and (9.3.19) respectively. Polynomial approximations of the
solution U can be determined in several ways. We leave the analysis of tau and Galerkin
methods to the reader. We examine the collocation method.
We require pn ∈ Pn, n ≥ 2, to be solution of
(9.4.16) Bw,n(pn,φ) = Fw,n(φ), ∀φ ∈ Pn.
Here, Bw,n : Pn × Pn → R and Fw,n : Pn → R are respectively defined by