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October  2015, 35(10): 4765-4789. doi: 10.3934/dcds.2015.35.4765

Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

Maxime Breden 1, , Laurent Desvillettes 1, and  Jean-Philippe Lessard 2,

1. 

CMLA, ENS Cachan & CNRS, 61 avenue du Président Wilson, 94230 Cachan, France, France
2. 

Département de Mathématiques et de Statistique, Université Laval, 1045 avenue de la Médecine, Québec, QC, G1V0A6, Canada

Received  June 2014 Revised  January 2015 Published  April 2015

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 $Df(\overline{x})$ at an approximate solution $\overline{x}$. We present rigorous computer-assisted calculations showing that $T$ is a contraction near $\overline{x}$, thus yielding the existence of a solution. Since $Df(\overline{x})$ 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.
Keywords: contraction mapping, Fourier series., rigorous numerics, Tridiagonal operator.
Mathematics Subject Classification: Primary: 47H10, 97N20; Secondary: 42A10, 34B08, 65L1.
Citation: Maxime Breden, Laurent Desvillettes, Jean-Philippe Lessard. Rigorous numerics for nonlinear operators with tridiagonal dominant linear part. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4765-4789. doi: 10.3934/dcds.2015.35.4765
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show all references

References:
[1]

Discrete Contin. Dyn. Syst., 13 (2005), 901-920. doi: 10.3934/dcds.2005.13.901.  Google Scholar

[2]

Second edition, Dover Publications Inc., Mineola, NY, 2001.  Google Scholar

[3]

Acta Appl. Math., 128 (2013), 113-152. doi: 10.1007/s10440-013-9823-6.  Google Scholar

[4]

M. Breden, L. Desvillettes and J.-P. Lessard, MATLAB codes to perform the proofs,, , ().   Google Scholar

[5]

SIAM J. Appl. Dyn. Syst., 12 (2013), 204-245. doi: 10.1137/120873960.  Google Scholar

[6]

With the assistance of Bernadette Miara and Jean-Marie Thomas, Translated from the French by A. Buttigieg, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989.  Google Scholar

[7]

SIAM J. Appl. Dyn. Syst., 3 (2004), 117-160 (electronic). doi: 10.1137/030600210.  Google Scholar

[8]

J. Differential Equations, 249 (2010), 2237-2268. doi: 10.1016/j.jde.2010.07.002.  Google Scholar

[9]

SIAM J. Numer. Anal., 51 (2013), 2063-2087. doi: 10.1137/110836651.  Google Scholar

[10]

Japan J. Indust. Appl. Math., 22 (2005), 57-75. doi: 10.1007/BF03167476.  Google Scholar

[11]

To appear in Math. Comp., 2015. Google Scholar

[12]

J. Differential Equations, 252 (2012), 3093-3115. doi: 10.1016/j.jde.2011.11.020.  Google Scholar

[13]

Second edition, Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981.  Google Scholar

[14]

J. Comput. Appl. Math., 206 (2007), 986-1006. doi: 10.1016/j.cam.2006.09.016.  Google Scholar

[15]

ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)-University of Maryland, Baltimore County.  Google Scholar

[16]

J.-P. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach,, in preparation., ().   Google Scholar

[17]

in Developments in Reliable Computing (ed. Tibor Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77-104. Available from: http://www.ti3.tu-harburg.de/rump/. doi: 10.1007/978-94-017-1247-7_7.  Google Scholar

[18]

Found. Comput. Math., 1 (2001), 255-288. doi: 10.1007/s002080010010.  Google Scholar

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