A Posteriori Error Bounds for Two Point Boundary Value Problems: A Green's Function Approach

TL;DR

This paper introduces a computer-assisted, rigorous method for deriving a posteriori error bounds for solutions to two-point boundary value problems, applicable to high-dimensional systems and stable BVPs with unstable initial value problems.

Contribution

It develops a general Green's function-based approach that accounts for all errors and does not require special vector field forms, demonstrated on singularly perturbed BVPs and the Lorenz system.

Findings

Successfully applied to singularly perturbed BVPs

Rigorously proved the existence of a Lorenz system periodic orbit

Method accounts for truncation and rounding errors

Abstract

We present a computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs). All truncation errors are accounted for and, if combined with interval arithmetic to bound the rounding errors, the computer generated results are mathematically rigorous. The method is formulated for $n$-dimensional systems and does not require any special form for the vector field of the differential equation. It utilizes a numerically generated approximation to the BVP fundamental solution and Green's function and thus can be applied to stable BVPs whose initial value problem is unstable. The utility of the method is demonstrated on a pair of singularly perturbed model BVPs and by using it to rigorously show the existence of a periodic orbit in the Lorenz system.