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Ordinary Differential Equations 183
Theoretical considerations of the problems proposed above are presented in many
books on ordinary differential equations. We refer for instance to golomb and shanks
(1965), brauer and nohel(1967), simmons(1972), etc.. Once we gain enough experi-
ence with the numerical treatment of these elementary equations, we will turn to more
realistic problems. Several non trivial examples are discussed in chapter twelve.
9.2 Approximation of linear equations
A technique to approximate the solution of problem (9.1.1) is the so called tau method.
Early applications were considered in lanczos(1956). Theoretical developments are
given for instance in gottlieb and orszag(1977), for Chebyshev and Legendre ex-
pansions. The unknown function U is approximated by a sequence of polynomials
pn ∈ Pn, n ≥ 1, which are expressed in the frequency space relative to some Jacobi
weight function w. For any n ≥ 1, these are required to be solutions of the following
problem:
(9.2.1)
⎧
⎨
p ′ n = Πw,n−1f in ] − 1,1],
⎩
pn(−1) = σ.
The operator Πw,n, n ∈ N, is the same operator introduced in section 2.4. In this
scheme the boundary condition is the same as in (9.1.1), but the function f is replaced
by a truncation of the series (6.2.11). In practice, we project the differential equation
in the space Pn−1. Following the suggestions provided in section 7.3, where now
q := Πw,n−1f, we can reduce (9.2.1) to a linear system. The vector on the right-hand
side contains the first n Fourier coefficients of the function f and the datum σ. The
unknown vector consists of the Fourier coefficients of the polynomial pn. Note that
these do not coincide with the first n + 1 Fourier coefficients of the function U.
The question is to check whether limn→+∞ pn = U, and how to interpret this limit.
By (9.1.2), we easily get