A new matrix equation expression for the solution of non-autonomous linear systems of ODEs

TL;DR

This paper introduces a novel matrix equation approach for solving non-autonomous linear ODE systems, extending a spectral accuracy method from scalar to systems, with demonstrated efficiency and low-rank solutions.

Contribution

It extends a spectral accuracy method based on a generalized convolution product to systems of non-autonomous linear ODEs, expressing solutions via a matrix equation.

Findings

Method achieves spectral accuracy.

Matrix solutions exhibit low-rank properties.

Numerical examples confirm efficacy.

Abstract

The solution of systems of non-autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in the scalar case. The method is based on a product that generalizes the convolution. In this work, we show that it is possible to extend the method to solve systems of non-autonomous linear ordinary differential equations (ODEs). In this new approach, the ODE solution can be expressed through a linear system that can be equivalently rewritten as a matrix equation. Numerical examples illustrate the method's efficacy and the low-rank property of the matrix equation solution.