Rigorous numerics for nonlinear differential equations using Chebyshev series

Authors: Lessard, Jean-Philippe; Reinhardt, Christian
Abstract: A computational method based on Chebyshev series to rigorously compute solutions of initial and boundary value problems of analytic nonlinear vector fields is proposed. The idea is to recast solutions as fixed points of an operator defined on a Banach space of rapidly decaying Chebyshev coefficients and to use the so-called radii polynomials to show the existence of a unique fixed point near an approximate solution. As applications, solutions of initial value problems in the Lorenz equations and symmetric connecting orbits in the Gray--Scott equation are rigorously computed. The symmetric connecting orbits are obtained by solving a boundary value problem with one of the boundary values in the stable manifold.
Issue Date: 2 January 2014
Open Access Date: 16 May 2016
Document version: VoR
Permalink: http://hdl.handle.net/20.500.11794/1261
This document was published in: SIAM Journal on Numerical Analysis, Vol. 52 (1), 1–22 (2014)
http://dx.doi.org/10.1137/13090883X
Society for Industrial and Applied Mathematics
Alternative version: 10.1137/13090883X
Collection:Articles publiés dans des revues avec comité de lecture

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