Multivariate Canonical Polynomials in the Tau Method with Applications to Optimal Control Problems

Abstract

The Tau method is a highly accurate technique that approximates differential equations efficiently. This paper discusses two approaches of the Tau Method: recursive and spectral. In the recursive Tau, the approximate solution of the differential equation is obtained in terms of a special polynomial basis called canonical polynomials. The present paper extends this concept to the multivariate canonical polynomial vectors and proposes a self starting algorithm to generate those vectors. In the spectral Tau, the approximate solution is obtained as a truncated series expansions in terms of a set of orthogonal polynomials where the coefficients of the expansions are obtained by forcing the defect of the differential equation to vanish at the some selected points. In this paper we use the spectral Tau to solve a class of optimal control problems associated with a nonlinear system of differential equations. Some numerical examples that confirm our method are given.