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

Hungria, Allan; Lessard, Jean-Philippe; Mireles James, J.

doi:10.1090/mcom/3046

Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach
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by Allan Hungria, Jean-Philippe Lessard and J. D. Mireles James PDF

Math. Comp. 85 (2016), 1427-1459 Request permission

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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