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Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach

Authors: Allan Hungria, Jean-Philippe Lessard and J. D. Mireles James
Journal: Math. Comp. 85 (2016), 1427-1459
MSC (2010): Primary 65L60, 65G40, 34C25, 35K57
Published electronically: September 28, 2015
MathSciNet review: 3454370
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Abstract: Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to effective strategies for computer assisted analysis of nonlinear operator equations. The method of radii polynomials is an efficient tool for bounding the smallest and largest neighborhoods on which a Newton-like operator associated with a nonlinear equation is a contraction mapping. The method has been used to study solutions of ordinary, partial, and delay differential equations such as equilibria, periodic orbits, solutions of initial value problems, heteroclinic and homoclinic connecting orbits in the $ C^{\mathbf {k}}$ category of functions. In the present work we adapt the method of radii polynomials to the analytic category. For ease of exposition we focus on studying periodic solutions in Cartesian products of infinite sequence spaces. We derive the radii polynomials for some specific application problems and give a number of computer assisted proofs in the analytic framework.

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Additional Information

Allan Hungria
Affiliation: Department of Mathematical Sciences, Ewing Hall, University of Delaware, Newark, Delaware 19716

Jean-Philippe Lessard
Affiliation: Département de Mathématiques et de Statistique, Université Laval, 1045 avenue de la Médecine, Québec, QC, G1V0A6, Canada

J. D. Mireles James
Affiliation: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Rd., Boca Raton, Florida 33431

Received by editor(s): March 16, 2014
Received by editor(s) in revised form: November 22, 2014
Published electronically: September 28, 2015
Additional Notes: The second author was supported by NSERC and the FRQNT program Établissement de nouveaux chercheurs.
The third author was partially supported by NSF grant DSM 1318172.
Article copyright: © Copyright 2015 American Mathematical Society