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

2022/09/30 by Stefano Pozza, Pozza, Stefano, Niel Van Buggenhout +1
Computer Science · Mathematics · Physics and Astronomy · #34A30 #41A10 #65L05 #FOS: Mathematics #Image and Signal Denoising Methods #Numerical Analysis (math.NA) #Numerical methods for differential equations #Scientific Research and Discoveries

paper · pdf · doi:10.48550/arxiv.2209.15533

openalex publication_date 2022/09/30 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

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 ⋆-algebra. This algebra is equipped with a product, the ⋆-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 ⋆-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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