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

TL;DR

This paper introduces a spectral method for solving non-autonomous ODEs using a novel $igstar$-algebra framework, enabling high-accuracy solutions through discretized $igstar$-products and linear algebra techniques.

Contribution

It presents a new spectral solver based on $igstar$-algebra for non-autonomous ODEs, achieving spectral accuracy and efficient computation.

Findings

Achieves spectral accuracy in solving non-autonomous ODEs.

Uses Legendre polynomial expansion for discretization.

Demonstrates effectiveness through numerical experiments.

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.