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Ordinary Differential Equations 219
Problem (9.10.1) admits solutions satisfying appropriate integrability conditions. The-
oretical results have been given in tricomi(1957), kantorovich and krylov(1964),
pogorzelski(1966), hochstadt(1973).
Galerkin and collocation type approximations of the solution U have been studied by
many authors. In the latter approach, the integral is replaced by suitable quadrature
sums based on Chebyshev polynomials. Consider the case A = 0 and B = 1. A
classical scheme is obtained by defining V (s) := U(s) √ 1 − s 2 , s ∈] − 1,1[. For n ≥ 1,
recalling (3.4.1) and (3.4.6), we approximate the integral by the formula
(9.10.3)
1
−1
V (s)
s − x
ds
√ 1 − s 2
≈ π
n
n
j=1
V (ξ (n)
j )
ξ (n)
j
x = ξ(n) j , 1 ≤ j ≤ n,
− x,
where the ξ (n)
j ’s are the zeroes of Tn (see (3.1.4)).
Finally, for n ≥ 2, V is approximated by the polynomial pn ∈ Pn−2 satisfying the set
of equations
(9.10.4)
1
n
n
j=1
pn(ξ (n)
j )
ξ (n)
j
− η(n)
i
= f(η (n)
i ), 1 ≤ i ≤ n − 1,
where the η (n)
j ’s are given in (3.1.11).
It is easy to set up the corresponding linear system. Instead, the theoretical analysis of
convergence is not a trivial matter. Hints are provided in krenk(1975). For other results
and generalizations, we quote the following papers: delves(1968), erdogan, gupta
and cook(1973), krenk(1978), elliott(1981), ioakimidis(1981), monegato(1982),
elliott(1982).
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