Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

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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

Abstract
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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 and Continuous Dynamical Systems, 2015, 35 (10) : 4765-4789. doi: 10.3934/dcds.2015.35.4765

References:

[1]	A. W. Baker, M. Dellnitz and O. Junge, A topological method for rigorously computing periodic orbits using Fourier modes, Discrete Contin. Dyn. Syst., 13 (2005), 901-920. doi: 10.3934/dcds.2005.13.901.
[2]	J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications Inc., Mineola, NY, 2001.
[3]	M. Breden, J.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: A 3-component reaction-diffusion system, Acta Appl. Math., 128 (2013), 113-152. doi: 10.1007/s10440-013-9823-6.
[4]	M. Breden, L. Desvillettes and J.-P. Lessard, MATLAB codes to perform the proofs,, , ().
[5]	R. Castelli and J.-P. Lessard, Rigorous numerics in Floquet theory: Computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst., 12 (2013), 204-245. doi: 10.1137/120873960.
[6]	P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation, 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.
[7]	S. Day, O. Junge and K. Mischaikow, A rigorous numerical method for the global analysis of infinite-dimensional discrete dynamical systems, SIAM J. Appl. Dyn. Syst., 3 (2004), 117-160 (electronic). doi: 10.1137/030600210.
[8]	M. Gameiro and J.-P. Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, J. Differential Equations, 249 (2010), 2237-2268. doi: 10.1016/j.jde.2010.07.002.
[9]	M. Gameiro and J.-P. Lessard, Efficient Rigorous Numerics for Higher-Dimensional PDEs via One-Dimensional Estimates, SIAM J. Numer. Anal., 51 (2013), 2063-2087. doi: 10.1137/110836651.
[10]	Y. Hiraoka and T. Ogawa, Rigorous numerics for localized patterns to the quintic Swift-Hohenberg equation, Japan J. Indust. Appl. Math., 22 (2005), 57-75. doi: 10.1007/BF03167476.
[11]	A. Hungria, J.-P. Lessard and J. D. Mireles-James, Radii polynomial approach for analytic solutions of differential equations: Theory, examples, and comparisons, To appear in Math. Comp., 2015.
[12]	G. Kiss and J.-P. Lessard, Computational fixed-point theory for differential delay equations with multiple time lags, J. Differential Equations, 252 (2012), 3093-3115. doi: 10.1016/j.jde.2011.11.020.
[13]	D. E. Knuth, The Art of Computer Programming, Vol. 2. Seminumerical Algorithms, Second edition, Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981.
[14]	V. R. Korostyshevskiy and T. Wanner, A Hermite spectral method for the computation of homoclinic orbits and associated functionals, J. Comput. Appl. Math., 206 (2007), 986-1006. doi: 10.1016/j.cam.2006.09.016.
[15]	V. R. Korostyshevskiy, A Hermite Spectral Approach to Homoclinic Solutions of Ordinary Differential Equations, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)-University of Maryland, Baltimore County.
[16]	J.-P. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach,, in preparation., ().
[17]	S. M. Rump, INTLAB - INTerval LABoratory, 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.
[18]	P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Found. Comput. Math., 1 (2001), 255-288. doi: 10.1007/s002080010010.

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References:

[1]	A. W. Baker, M. Dellnitz and O. Junge, A topological method for rigorously computing periodic orbits using Fourier modes, Discrete Contin. Dyn. Syst., 13 (2005), 901-920. doi: 10.3934/dcds.2005.13.901.
[2]	J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications Inc., Mineola, NY, 2001.
[3]	M. Breden, J.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: A 3-component reaction-diffusion system, Acta Appl. Math., 128 (2013), 113-152. doi: 10.1007/s10440-013-9823-6.
[4]	M. Breden, L. Desvillettes and J.-P. Lessard, MATLAB codes to perform the proofs,, , ().
[5]	R. Castelli and J.-P. Lessard, Rigorous numerics in Floquet theory: Computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst., 12 (2013), 204-245. doi: 10.1137/120873960.
[6]	P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation, 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.
[7]	S. Day, O. Junge and K. Mischaikow, A rigorous numerical method for the global analysis of infinite-dimensional discrete dynamical systems, SIAM J. Appl. Dyn. Syst., 3 (2004), 117-160 (electronic). doi: 10.1137/030600210.
[8]	M. Gameiro and J.-P. Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, J. Differential Equations, 249 (2010), 2237-2268. doi: 10.1016/j.jde.2010.07.002.
[9]	M. Gameiro and J.-P. Lessard, Efficient Rigorous Numerics for Higher-Dimensional PDEs via One-Dimensional Estimates, SIAM J. Numer. Anal., 51 (2013), 2063-2087. doi: 10.1137/110836651.
[10]	Y. Hiraoka and T. Ogawa, Rigorous numerics for localized patterns to the quintic Swift-Hohenberg equation, Japan J. Indust. Appl. Math., 22 (2005), 57-75. doi: 10.1007/BF03167476.
[11]	A. Hungria, J.-P. Lessard and J. D. Mireles-James, Radii polynomial approach for analytic solutions of differential equations: Theory, examples, and comparisons, To appear in Math. Comp., 2015.
[12]	G. Kiss and J.-P. Lessard, Computational fixed-point theory for differential delay equations with multiple time lags, J. Differential Equations, 252 (2012), 3093-3115. doi: 10.1016/j.jde.2011.11.020.
[13]	D. E. Knuth, The Art of Computer Programming, Vol. 2. Seminumerical Algorithms, Second edition, Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981.
[14]	V. R. Korostyshevskiy and T. Wanner, A Hermite spectral method for the computation of homoclinic orbits and associated functionals, J. Comput. Appl. Math., 206 (2007), 986-1006. doi: 10.1016/j.cam.2006.09.016.
[15]	V. R. Korostyshevskiy, A Hermite Spectral Approach to Homoclinic Solutions of Ordinary Differential Equations, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)-University of Maryland, Baltimore County.
[16]	J.-P. Lessard, J. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach,, in preparation., ().
[17]	S. M. Rump, INTLAB - INTerval LABoratory, 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.
[18]	P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Found. Comput. Math., 1 (2001), 255-288. doi: 10.1007/s002080010010.

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