The Tau Method as an Analytic Tool in the Discussion of Equivalence Results Across Numerical Methods

摘要

A Tau Method approximate solution of a given differential equation defined on a compact [a,b] is obtained by adding to the right hand side of the equation a specific minimal polynomal perturbation term H_n(x), which plays the role of a representation of zero in [a,b] by elements of a given subspace of polynomials. Neither discretization nor orthogonality are involved in this process of approxima- tion. However, there are interesting relations between the Tau Method and approximation methods based on the former techniques. In this paper we use equivalence results for collocation and the Tau Method, contributed recently by the authors together with classical results in the literature, to Identify precisely the perturbation term H(x) which would generate a Tau Method approximate Solution, identical to that generated by some specific discrete methods over a given mesh ∏∈[a,b].