Rigorous numerics for nonlinear operators with tridiagonal dominant linear parts

Authors: Breden, Maxime; Desvillettes, Laurent; Lessard, Jean-Philippe
Abstract: We present a method designed for computing solutions of infinite dimensional nonlinear operators f(x) = 0 with a tridiagonal dominant linear part. We recast the operator equation into an equivalent Newton-like equation x = T(x) = x - Af(x), where A is an approximate inverse of the derivative Df(¯x) at an approximate solution ¯x. We present rigorous computer-assisted calculations showing that T is a contraction near ¯x, thus yielding the existence of a solution. Since Df(¯x) does not have an asymptotically diagonal dominant structure, the computation of A is not straightforward. This paper provides ideas for computing A, and proposes a new rigorous method for proving existence of solutions of nonlinear operators with tridiagonal dominant linear part.
Document Type: Article de recherche
Issue Date: 1 April 2015
Open Access Date: 16 May 2016
Document version: AM
Permalink: http://hdl.handle.net/20.500.11794/1301
This document was published in: Discrete and Continuous Dynamical Systems, Vol. 35 (10), 4765–4789 (2015)
http://dx.doi.org/10.3934/dcds.2015.35.4765
Dept. of Mathematics, Southwest Missouri State University
Alternative version: 10.3934/dcds.2015.35.4765
1503.06315
Collection:Articles publiés dans des revues avec comité de lecture

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