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Ordinary Differential Equations 215
these situations as well, although a lot of robust algorithms are already available in the
framework of finite-difference computations. The success of a certain approximation
technique is strictly dependent on the type of nonlinearity, so that it is difficult to
provide general recipes. Some examples are discussed in chapter twelve. As far as the
theoretical analysis is concerned, references are very few, both for the difficulty and the
modernity of the subject. We give some hints for the implementation. Consider the
problem
(9.8.1)
⎧
⎨U
′ (x) = F(x,U(x)) ∀x ∈] − 1,1],
⎩
U(−1) = σ,
where F : [−1,1] × R → R is a given function and σ ∈ R. We recover the linear
equation (9.1.3) by setting F(x,y) := f(x) − A(x)y.
Appropriate conditions on F insure existence, uniqueness and regularity of the solu-
tion U (see for instance golomb and shanks (1965), brauer and nohel(1967),
simmons(1972), etc.). According to the remarks of section 3.3, approximations in the
physical space are preferable to approximations in the frequency space, although one
can note a little deterioration in the numerical results, due to the effects of the aliasing
errors (see section 4.2 and orszag(1972)). It is common opinion to attribute this phe-
nomenon to the spacing of the collocation grid, when it is not adequate to resolve high
frequency oscillations. De-aliasing procedures have been proposed by various authors.
The reader is addressed to boyd(1989) for a collection of references.
The collocation method applied to problem (9.8.1) consists in finding pn ∈ Pn, n ≥ 1,
such that
(9.8.2)
⎧
⎨
⎩
p ′ n(η (n)
i ) = F(η (n)
i ,pn(η (n)
i )) 1 ≤ i ≤ n,
pn(η (n)
0 ) = σ.
In the Legendre case, a convergence result is obtained by reviewing the proof of theorem
9.2.1. This time, we assume that there exists a constant ǫ > 0 such that