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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 - A, 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.  doi: 10.3934/dcds.2005.13.901.  Google Scholar

[2]

J. P. Boyd, Chebyshev and Fourier Spectral Methods,, Second edition, (2001).   Google Scholar

[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.  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]

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.  doi: 10.1137/120873960.  Google Scholar

[6]

P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation,, With the assistance of Bernadette Miara and Jean-Marie Thomas, (1989).   Google Scholar

[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.  doi: 10.1137/030600210.  Google Scholar

[8]

M. Gameiro and J.-P. Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs,, J. Differential Equations, 249 (2010), 2237.  doi: 10.1016/j.jde.2010.07.002.  Google Scholar

[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.  doi: 10.1137/110836651.  Google Scholar

[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.  doi: 10.1007/BF03167476.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1016/j.jde.2011.11.020.  Google Scholar

[13]

D. E. Knuth, The Art of Computer Programming, Vol. 2. Seminumerical Algorithms,, Second edition, (1981).   Google Scholar

[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.  doi: 10.1016/j.cam.2006.09.016.  Google Scholar

[15]

V. R. Korostyshevskiy, A Hermite Spectral Approach to Homoclinic Solutions of Ordinary Differential Equations,, ProQuest LLC, (2005).   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]

S. M. Rump, INTLAB - INTerval LABoratory,, in Developments in Reliable Computing (ed. Tibor Csendes), (1999), 77.  doi: 10.1007/978-94-017-1247-7_7.  Google Scholar

[18]

P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation,, Found. Comput. Math., 1 (2001), 255.  doi: 10.1007/s002080010010.  Google Scholar

show all references

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.  doi: 10.3934/dcds.2005.13.901.  Google Scholar

[2]

J. P. Boyd, Chebyshev and Fourier Spectral Methods,, Second edition, (2001).   Google Scholar

[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.  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]

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.  doi: 10.1137/120873960.  Google Scholar

[6]

P. G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation,, With the assistance of Bernadette Miara and Jean-Marie Thomas, (1989).   Google Scholar

[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.  doi: 10.1137/030600210.  Google Scholar

[8]

M. Gameiro and J.-P. Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs,, J. Differential Equations, 249 (2010), 2237.  doi: 10.1016/j.jde.2010.07.002.  Google Scholar

[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.  doi: 10.1137/110836651.  Google Scholar

[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.  doi: 10.1007/BF03167476.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1016/j.jde.2011.11.020.  Google Scholar

[13]

D. E. Knuth, The Art of Computer Programming, Vol. 2. Seminumerical Algorithms,, Second edition, (1981).   Google Scholar

[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.  doi: 10.1016/j.cam.2006.09.016.  Google Scholar

[15]

V. R. Korostyshevskiy, A Hermite Spectral Approach to Homoclinic Solutions of Ordinary Differential Equations,, ProQuest LLC, (2005).   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]

S. M. Rump, INTLAB - INTerval LABoratory,, in Developments in Reliable Computing (ed. Tibor Csendes), (1999), 77.  doi: 10.1007/978-94-017-1247-7_7.  Google Scholar

[18]

P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation,, Found. Comput. Math., 1 (2001), 255.  doi: 10.1007/s002080010010.  Google Scholar

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