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

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.

Keywords

• non-autonomous linear system of ordinary differential equations
• matrix equation
• Legendre polynomials