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A $\star$-product solver with spectral accuracy for non-autonomous ordinary differential equations (2209.15533v1)

Published 30 Sep 2022 in math.NA and cs.NA

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 $\star$-algebra. This algebra is equipped with a product, the $\star$-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 $\star$-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.

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