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.
Highlights
The Tau method is a highly accurate technique that approximates differential equations without requiring the discretization of the given differential operator
In the recursive Tau, the approximate solution of the differential equation is obtained in terms of a special polynomial basis called canonical polynomials
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
Summary
The Tau method is a highly accurate technique that approximates differential equations without requiring the discretization of the given differential operator. The recursive approach, proposed in (Ortiz, 1969), permits to obtain an approximate polynomial solution expressed in terms of a special polynomials basis called canonical polynomials This technique has been thoroughly investigated in a series of papers (see for example (Crisci & Russo, 1983), (Freilich & Ortiz, 1982) and (El-Daou & Ortiz, 1994-1998)). We will show that the spectral Tau method is highly effective in tackling a class of optimal control problems (see (Flores Tlacuahuac, Terrazas Moreno, & Biegler, 2008) and (Jaddu & Majdalawi, 2014)) Numerical examples illustrating the efficiency of our method are provided throughout the paper
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