The *-product approach for linear ODEs: a numerical study of the scalar case

TL;DR

This paper investigates a novel numerical approach based on the *-product for solving scalar non-autonomous ODEs, demonstrating its effectiveness through numerical examples and laying the groundwork for extension to systems.

Contribution

It introduces and analyzes a *-product based method for scalar ODEs, providing insights into its properties and potential for extension to systems of ODEs.

Findings

The method effectively solves scalar ODEs with promising accuracy.

Numerical examples confirm the approach's efficacy.

The approach has potential for extension to systems of ODEs.

Abstract

Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the main ideas at the core of this approach in the simpler setting of a scalar ODE. Understanding the scalar case is fundamental since the method can be easily extended to the more challenging problem of systems of ODEs. Numerical examples illustrate the method's efficacy and its properties in the scalar case.