The tau method with perturbation term depending on the differential operator

摘要

The tau method approximates the solution of a differential equation with a polynomial, which is the exact solution of a differential equation obtained by adding to the right hand side of the given equation a perturbation term, consisting of a suitably chosen linear combination of polynomials.Until now the Chebyshev and Legendre polynomials have been used to this purpose, but the determination of the best perturbation term is, still, an open problem.In this paper a perturbation term depending on the differential equation is chosen. For this formulation of the tau method, the existence of an infinite sequence of tau approximants and the convergence of the error to zero is proved. An estimate of the local truncation error is also given, and the stability properties are discussed.Numerical results are also reported.