Solving ordinary differential equations using Schur decomposition

TL;DR

This paper presents a novel approach to solving ordinary differential equations by leveraging Schur decomposition to transform systems into upper triangular form, enabling sequential solution generation, and extends this to perturbed non-linear systems using operator theory and polynomial representations.

Contribution

Introduces a new methodology combining Schur decomposition and operator theory for solving linear and perturbed non-linear differential equations, with algorithms for automation.

Findings

Effective transformation of linear systems into upper triangular form.

Approximate solutions for perturbed non-linear systems using polynomial and Legendre representations.

Algorithms provided for automated implementation of the methods.

Abstract

This work introduces a methodology to solve ordinary differential equations using the Schur decomposition of the linear representation of the differential equation. This is done by first transforming the system into an upper triangular system using the Schur decomposition, and second, by generating the solution sequentially following the upper triangular structure. In addition, and when dealing with non-linear perturbed systems, this work proposes a methodology based on operator theory to find an approximate linear representation of perturbed non-linear systems. Particularly, we focus on polynomial differential equations and the use of Legendre polynomials to represent the solution. Based on these results, a perturbation technique is also proposed to study these problems. A set of algorithms to automate these methodologies are included.