IJPAM: Volume 1, No. 1 (2002)
APPROXIMATION METHODS FOR
DIFFERENTIAL EQUATIONS



College of Technological Studies, KUWAIT
e-mail: modaou@paaet.edu.kw

Imperial College, London, U.K.
e-mail: e.ortiz@ic.ac.uk
Abstract.We introduce sequences of linear functionals uniquely
associated with each of several methods for the approximate solution of
differential equations and discuss the classification of the latter in
terms of these functionals. We use Davis's [#!dav!#] notion of
permanence to discuss the structure of these methods further.
We show that the Tau method, formulated in terms of sequences of
canonical polynomials (see Ortiz [#!ort69!#]), is a permanent technique
which makes no direct reference to a Hilbert structure. We also show
that, for a given operator, the linear functionals associated with it
are biorthogonal to the canonical polynomials. Furthermore, we make
precise here a duality which exists between the Tau method and
collocation, using the linear functionals associated with each of
these methods. We relate them to spectral or Fourier expansion
techniques.
Received: January 6, 2002
AMS Subject Classification: 65L60, 65J10
Key Words and Phrases: interpolation, collocation, Tau method, canonical polynomials, orthogonal polynomials
Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2002
Volume: 1
Issue: 1