ANZIAM J.
48 (2007), 361386

Efficient spectralGalerkin algorithms for direct solution of the integrated forms of secondorder equations using ultraspherical polynomials

E. H. Doha
Department of Mathematics
Faculty of Science
Cairo University
Giza
Egypt
eiddoha@frcu.eun.eg



A. H. Bhrawy
Department of Mathematics
Faculty of Science
BeniSuef University
BeniSuef
Egypt
alibhrawy@yahoo.co.uk



Abstract

It is well known that spectral methods (tau,
Galerkin, collocation) have a condition number of
where N
is the number of retained modes of polynomial
approximations. This paper presents some
efficient spectral algorithms, which have a
condition number of
, based on the ultrasphericalGalerkin methods
for the integrated forms of secondorder elliptic
equations in one and two space variables. The key
to the efficiency of these algorithms is to
construct appropriate base functions, which lead
to systems with specially structured matrices
that can be efficiently inverted. The
complexities of the algorithms are a small
multiple of
operations for a
ddimensional domain with
unknowns, while the convergence rates of the
algorithms are exponentials with smooth
solutions.

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