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140 Polynomial Approximation of Differential Equations
(7.4.5) p ′ + γ[p(η (n)
0 ) − σ] ˜l (n)
0
= q.
This kind of approach was proposed in funaro and gottlieb(1988) for Chebyshev
nodes. In terms of matrices, this is equivalent to (we take for instance n = 3)
(7.4.6)
⎡
⎢
⎣
˜d (1)
00 + γ d ˜(1) 01
˜d (1)
10
˜d (1)
20
˜d (1)
30
˜d (1)
11
˜d (1)
21
˜d (1)
31
˜d (1)
02
˜d (1)
12
˜d (1)
22
˜d (1)
32
˜d (1)
03
˜d (1)
13
˜d (1)
23
˜d (1)
33
⎤⎡
p(η
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎦⎣
(n)
0 )
p(η (n)
1 )
p(η (n)
⎤
⎥
2 ) ⎥
⎦
p(η (n)
3 )
=
⎡
q(η
⎢
⎣
(n)
0 ) + γσ
q(η (n)
1 )
q(η (n)
⎤
⎥
⎥.
2 ) ⎥
⎦
q(η (n)
3 )
If γ = 0, (7.4.4) is not a well-posed problem, because the determinant of the corre-
sponding matrix vanishes. Also note that, if q ∈ Pn−1, problems (7.4.1) and (7.4.4) are
equivalent. Indeed, in this case, the n conditions p ′ (η (n)
i ) = q(η (n)
i ), 1 ≤ i ≤ n, imply
that p ′ ≡ q in Ī, hence p′ (η (n)
0 ) = q(η (n)
0 ). Since γ = 0, we obtain the equivalence.
Let A : Ī → R be a continuous function. A generalization of problem (7.4.1) is
obtained by setting
(7.4.7)
⎧
⎨
⎩
p ′ (η (n)
i ) + (Ap)(η (n)
i ) = q(η (n)
i ) 1 ≤ i ≤ n,
p(η (n)
0 ) = σ,
where q ∈ Pn−1 and σ ∈ R.
The corresponding system is obtained by adding to the matrix of system (7.4.1), the
(n + 1) × (n + 1) diagonal matrix diag{0, A(η (n)
1 ), · · · , A(η (n)
n )}.
Problem (7.4.4) is generalized in a similar way. In particular, (7.4.5) becomes
(7.4.8) p ′ + Iw,n(Ap) + γ[p(η (n)
0 ) − σ] ˜l (n)
0
= q,
where q ∈ Pn, σ ∈ R, and γ ∈ R, γ = 0. Now, we must add to the matrix relative to
(7.4.4), the (n + 1) × (n + 1) diagonal matrix diag{A(η (n)
0 ), A(η (n)
1 ), · · · , A(η (n)
n )}.