A posteriori error bounds for two point boundary value problems: A green's function approach

Jeremiah  Birrell

doi:10.3934/jcd.2015001

Article Contents

2015, Volume 2, Issue 2: 143-164. Doi: 10.3934/jcd.2015001

This issue Previous Article Next Article Compressed sensing and dynamic mode decomposition

A posteriori error bounds for two point boundary value problems: A green's function approach

Jeremiah Birrell^1,

1.
Program in Applied Mathematics, The University of Arizona, Tucson, Arizona 85721

Received: January 2015

Revised: August 2015

Published: June 2015

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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.

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Mathematics Subject Classification: Primary: 37C27, 65G20, 34B15; Secondary: 34B27, 65L10, 65L11.

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