Abstract

The tau method is a highly accurate technique that approximates differential equations efficiently. It has
three approaches: recursive, spectral and operational. Only the first two approaches concern this paper. In
the recursive Tau method, the approximate solution of the differential equation is obtained in terms of a
special polynomial basis  called {\it canonical polynomials}. The present paper extends this concept to the
{\it multivariate  canonical polynomial vectors} and proposes a self starting algorithm to generate those vectors.
In the spectral Tau method, 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 illustrated how the spectral tau
can be used to solve a class of optimal control problem associated with a nonlinear system of differential equations.
Some numerical examples that confirm our method are given.

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