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) a perturbed version of the given differential equation, and (ii) the imposed supplementary conditions exactly. 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 are perturbed 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 proportional to a Chebyshev polynomial.