A Posteriori Error Bounds for Two Point Boundary Value Problems: A Green's Function Approach
Provides rigorous error bounds for BVPs, including those with unstable initial value problems, benefiting researchers needing computer-assisted proofs in dynamical systems.
The paper presents a computer-assisted method for generating existence proofs and rigorous a posteriori error bounds for solutions to two-point boundary value problems, applicable to n-dimensional systems without special form requirements. The method is demonstrated on singularly perturbed BVPs and used to rigorously prove a periodic orbit in the Lorenz system.
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.