A ⋆-product solver with spectral accuracy for non-autonomous ordinary differential equations

Abstract

A new method for solving non-autonomous ordinary differential equations is proposed, the method achieves spectral accuracy. It is based on a new result which expresses the solution of such ODEs as an element in the so called STAR OPERATOR-algebra.

This algebra is equipped with a product, the STAR OPERATOR-product, which is the integral over the usual product of two bivariate distributions. Expanding the bivariate distributions in bases of Legendre polynomials leads to a discretization of the STAR OPERATOR-product and this allows for the solution to be approximated by a vector that is obtained by solving a linear system of equations.

The effectiveness of this approach is illustrated with numerical experiments.

Keywords

• non-autonomous linear system of ordinary differential equations
• spectral accuracy
• Legendre polynomials