# American Institute of Mathematical Sciences

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

## Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

 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.
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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