A Tau method with perturbed boundary conditions for certain ordinary differential equations |
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Authors: | Mohamed K El-Daou |
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Institution: | (1) Applied Sciences Department, College of Technological Studies, POB 64287, 70453 Shuwaikh/B, Kuwait |
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Abstract: | 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.
Dedicated to Eduardo L. Ortiz on the occasion of his 70th birthday |
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Keywords: | Tau method polynomial approximation Chebyshev polynomials |
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