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Ordinary Differential Equations 207
The way to write the linear system associated with (9.4.26) is described in section 7.4.
A proof of convergence for ultraspherical weights (ν := α = β, −1 < ν < 1) is provided
in bernardi and maday(1990). In the same paper, another collocation scheme is
proposed and analyzed. In the latter approach the nodes η (n)
i , 1 ≤ i ≤ n − 1, in
(9.4.26) are replaced by the nodes ξ (n−1)
i , 1 ≤ i ≤ n − 1, which are the zeroes of
P (ν+2,ν+2)
n−1 , n ≥ 2, −1 < ν < 1. In both cases, error estimates in the space H2 0,w(I)
(see (9.3.24)) are given. The theory is developed with the help of quadrature formulas
having a double multiplicity at the boundary points (see also gautschi and li(1991)).
Other boundary conditions are analyzed in funaro and heinrichs(1990).
9.5 Approximation in unbounded domains
In this section, we discuss examples where the domain I ⊆ R is not bounded. Let us
assume I :=]0,+∞[. A typical second-order problem consists in finding U : I → R
such that
(9.5.1)
⎧
⎨ −U ′′ + µU = g in I,
⎩
U(0) = σ,
where σ ∈ R, µ > 0, and g : I → R are given.
In order to determine a unique solution of (9.5.1), we need an extra condition. This
is obtained by prescribing the behavior of the unknown U at infinity. For instance, we
can require the convergence of U to a given value. Without loss of generality we can
impose the condition limx→+∞ U(x) = 0. To develop the analysis, we make further
assumptions on the rate of decay of U. This is obtained by assuming that U belongs to
some functional space X. Thus, the best way to argue is to use a suitable variational
formulation of (9.5.1). We remark that it is not restrictive to assume that σ = 0.
Let v(x) := x α e x , x ∈ I, −1 < α < 1, be a weight function. Let us define the Hilbert
space: