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Ordinary Differential Equations 209
(9.5.6)
+∞
(−p
0
′′ n + 2p ′ +∞
n − pn)φw dx = (
0
Ĩw,nf)φw dx, ∀φ ∈ P 0 n−1.
In (9.5.6), w(x) := xαe−x , x ∈ I, −1 < α < 1, is the Laguerre weight function. At this
point, we use the quadrature formula (3.6.1). Taking as test functions the Lagrange
polynomials ˜l (n)
i , 1 ≤ i ≤ n − 1, we obtain the collocation scheme
⎧
⎨
(9.5.7)
⎩
−p ′′ n(η (n)
i ) + 2p ′ n(η (n)
i ) + (µ − 1)pn(η (n)
i ) = f(η (n)
i ), 1 ≤ i ≤ n − 1,
pn(η (n)
0 ) = 0,
which is the same one presented in (7.4.26) with σ = 0, q := Ia,n−1f, and a(x) :=
xw(x), x ∈ I. For the numerical implementation we suggest taking into account the
results of sections 3.10 and 7.5.
Other differential equations are approximated in a similar way. Examples are found
in maday, pernaud-thomas and vandeven(1985), and mavriplis(1989). For the
domain I ≡ R, Hermite functions are a suitable basis for computations when the
solution decays like e −x2
kavian(1988).
at infinity. Results in this direction are given in funaro and
The literature offers several other numerical techniques for the treatment of prob-
lems in unbounded domains by spectral methods. For instance, one can map the do-
main I in the interval ] −1,1[ and approximate the new problem by Jacobi polynomial
expansions. The main difficulty is to find coordinate transformations preserving the
smoothness of the problem to avoid singularities and retain spectral accuracy. Early
results were given in grosch and orszag(1977). Further indications and comparisons
are provided in boyd(1982) and boyd(1989), chapter 13. Using the mapping approach,
the solution of the original problem is not required to decay exponentially at infinity as
in the case of Laguerre or Hermite approximations.
The idea of truncating the domain is often used in applications. Here, it is not
easy in general to detect the appropriate size of the computational domain. In addition,
one introduces artificial boundaries, where a non correct specification of the behavior of
the approximating polynomials may cause loss in accuracy. We describe an example in
section 12.2.