A Tau method with perturbed boundary conditions for certain ordinary differential equations

Ortiz' recursive formulation of the Lanczos Tau method (TM) is a powerful and efficient technique for producing polynomial approximations for initial or boundary value problems. The method consists in obtaining a polynomial which satisfies (i) aperturbed version of the given differential equati...

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Bibliographic Details
Published in:Numerical algorithms 2005-03, Vol.38 (1-3), p.31-45
Main Author: El-Daou, Mohamed K.
Format: Article
Language:English
Subjects:
Approximation
Boundary conditions
Boundary value problems
Chebyshev approximation
Differential equations
Mathematical analysis
Minimax technique
Ordinary differential equations
Polynomials
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Summary:Ortiz' recursive formulation of the Lanczos Tau method (TM) is a powerful and efficient technique for producing polynomial approximations for initial or boundary value problems. The method consists in obtaining a polynomial which satisfies (i) aperturbed version of the given differential equation, and (ii) the imposed supplementary conditionsexactly. This paper introduces a new form of the TM, (denoted by PTM), for a restricted class of differential equations, in which the differential equations as well as the supplementary conditions areperturbed simultaneously. PTM is compared to the classical TM from the point of view of their errors: it is found that the PTM error is smaller and more oscillatory than that of the TM; we further find that approximations nearly as accurate as minimax polynomial approximations can be constructed by means of the PTM. Detailed formulae are derived for the polynomial approximations in TM and PTM, based on Canonical Polynomials. Moreover, various limiting properties of Tau coefficients are established and it is shown that the perturbation in PTM behaves asymptotically proprtional to a Chebyshev polynomial.
ISSN:1017-1398
1572-9265
DOI:10.1007/BF02810614