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Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach

Authors: Allan Hungria, Jean-Philippe Lessard and J. D. Mireles James
Journal: Math. Comp. 85 (2016), 1427-1459
MSC (2010): Primary 65L60, 65G40, 34C25, 35K57
Published electronically: September 28, 2015
MathSciNet review: 3454370
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Abstract: Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to effective strategies for computer assisted analysis of nonlinear operator equations. The method of radii polynomials is an efficient tool for bounding the smallest and largest neighborhoods on which a Newton-like operator associated with a nonlinear equation is a contraction mapping. The method has been used to study solutions of ordinary, partial, and delay differential equations such as equilibria, periodic orbits, solutions of initial value problems, heteroclinic and homoclinic connecting orbits in the $ C^{\mathbf {k}}$ category of functions. In the present work we adapt the method of radii polynomials to the analytic category. For ease of exposition we focus on studying periodic solutions in Cartesian products of infinite sequence spaces. We derive the radii polynomials for some specific application problems and give a number of computer assisted proofs in the analytic framework.

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  • [1] Alan R. Champneys and Björn Sandstede, Numerical computation of coherent structures, Numerical continuation methods for dynamical systems, Underst. Complex Syst., Springer, Dordrecht, 2007, pp. 331-358. MR 2359337 (2008m:37139),
  • [2] Oscar E. Lanford III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 427-434. MR 648529 (83g:58051),
  • [3] Hans Koch, Alain Schenkel, and Peter Wittwer, Computer-assisted proofs in analysis and programming in logic: a case study, SIAM Rev. 38 (1996), no. 4, 565-604. MR 1420838 (97k:39001),
  • [4] 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), no. 2, 117-160. MR 2067140 (2005c:37165),
  • [5] J. D. Mireles-James and K. Mischaikow, Computational proofs in dynamics, Encyclopedia of Applied Computational Mathematics, 2016. To appear.
  • [6] Siegfried M. Rump, Verification methods: rigorous results using floating-point arithmetic, Acta Numer. 19 (2010), 287-449. MR 2652784 (2011j:65093),
  • [7] Mitsuhiro T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, International Workshops on Numerical Methods and Verification of Solutions, and on Numerical Function Analysis (Ehime/Shimane, 1999), Numer. Funct. Anal. Optim. 22 (2001), no. 3-4, 321-356. MR 1849323 (2002g:65055),
  • [8] Marian Gidea and Piotr Zgliczyński, Covering relations for multidimensional dynamical systems. II, J. Differential Equations 202 (2004), no. 1, 59-80. MR 2060532 (2005c:37019b),
  • [9] Sarah Day, Jean-Philippe Lessard, and Konstantin Mischaikow, Validated continuation for equilibria of PDEs, SIAM J. Numer. Anal. 45 (2007), no. 4, 1398-1424. MR 2338393 (2008k:37169),
  • [10] Marcio Gameiro and Jean-Philippe Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, J. Differential Equations 249 (2010), no. 9, 2237-2268. MR 2718657 (2011g:35420),
  • [11] Jan Bouwe van den Berg and Jean-Philippe Lessard, Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst. 7 (2008), no. 3, 988-1031. MR 2443030 (2009g:37030),
  • [12] Gábor Kiss and Jean-Philippe Lessard, Computational fixed-point theory for differential delay equations with multiple time lags, J. Differential Equations 252 (2012), no. 4, 3093-3115. MR 2871794,
  • [13] Marcio Gameiro and Jean-Philippe Lessard, Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates, SIAM J. Numer. Anal. 51 (2013), no. 4, 2063-2087. MR 3077902,
  • [14] Marcio Gameiro and Jean-Philippe Lessard, Existence of secondary bifurcations or isolas for PDEs, Nonlinear Anal. 74 (2011), no. 12, 4131-4137. MR 2802992 (2012f:35225),
  • [15] Maxime Breden, Jean-Philippe Lessard, and Matthieu 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. MR 3125637,
  • [16] Jean-Philippe Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Differential Equations 248 (2010), no. 5, 992-1016. MR 2592879 (2011b:34201),
  • [17] Marcio Gameiro, Jean-Philippe Lessard, and Konstantin Mischaikow, Validated continuation over large parameter ranges for equilibria of PDEs, Math. Comput. Simulation 79 (2008), no. 4, 1368-1382. MR 2487806 (2010a:65201),
  • [18] Jean-Philippe Lessard, Jason D. Mireles James, and Christian Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields, J. Dynam. Differential Equations 26 (2014), no. 2, 267-313. MR 3207723,
  • [19] Jean-Philippe Lessard and Christian Reinhardt, Rigorous numerics for nonlinear differential equations using Chebyshev series, SIAM J. Numer. Anal. 52 (2014), no. 1, 1-22. MR 3148084,
  • [20] R. Castelli and H. Teismann, Rigorous numerics for NLS: bound states, spectra, and controllability. Preprint, 2013.
  • [21] Jan Bouwe van den Berg, Jason D. Mireles-James, Jean-Philippe Lessard, and Konstantin Mischaikow, Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation, SIAM J. Math. Anal. 43 (2011), no. 4, 1557-1594. MR 2821596 (2012g:34109),
  • [22] J. B. van den Berg, C. M. Groothedde, and J. F. Williams, Rigorous computation of a radially symmetric localized solution in a Ginzburg-Landau problem, SIAM J. Appl. Dyn. Syst. 14 (2015), no. 1, 423-447. MR 3323206,
  • [23] Anaïs Correc and Jean-Philippe Lessard, Coexistence of nontrivial solutions of the one-dimensional Ginzburg-Landau equation: A computer-assisted proof, European J. Appl. Math. 26 (2015), no. 1, 33-60. MR 3294286,
  • [24] Roberto Castelli and Jean-Philippe Lessard, Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst. 12 (2013), no. 1, 204-245. MR 3032858,
  • [25] Roberto Castelli, Jean-Philippe Lessard, and J. D. Mireles James, Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form, SIAM J. Appl. Dyn. Syst. 14 (2015), no. 1, 132-167. MR 3304254,
  • [26] Jan Bouwe van den Berg, Jean-Philippe Lessard, and Konstantin Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp. 79 (2010), no. 271, 1565-1584. MR 2630003 (2011i:65246),
  • [27] M. Gameiro, J.-P. Lessard, and A. Pugliese, Computation of smooth manifolds of solutions of PDEs via rigorous multi-parameter continuation. To appear in Foundations of Computational Mathematics, 2015.
  • [28] Marcio Gameiro and Jean-Philippe Lessard, Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation, Numer. Math. 117 (2011), no. 4, 753-778. MR 2776917 (2012d:35148),
  • [29] J.-P. Eckmann, H. Koch, and P. Wittwer, A computer-assisted proof of universality for area-preserving maps, Mem. Amer. Math. Soc. 47 (1984), no. 289, vi+122. MR 727816 (85e:58119),
  • [30] William Arveson, A Short Course on Spectral Theory, Graduate Texts in Mathematics, vol. 209, Springer-Verlag, New York, 2002. MR 1865513 (2002j:47001)
  • [31] Jan Bouwe van den Berg, Andréa Deschênes, Jean-Philippe Lessard, and Jason D. Mireles James, Stationary coexistence of hexagons and rolls via rigorous computations, SIAM J. Appl. Dyn. Syst. 14 (2015), no. 2, 942-979. MR 3353132,
  • [32] M. Gameiro R. de la Llave, J.-L. Figueras and J.-P. Lessard, Theoretical results on the numerical computation and a-posteriori verification of invariant objects of evolution equations, in preparation, 2014.
  • [33] Donald E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. MR 633878 (83i:68003)
  • [34] Àngel Jorba and Maorong Zou, A software package for the numerical integration of ODEs by means of high-order Taylor methods, Experiment. Math. 14 (2005), no. 1, 99-117. MR 2146523 (2006d:65068)
  • [35] Konstantin Mischaikow and Marian Mrozek, Chaos in the Lorenz equations: a computer-assisted proof, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 66-72. MR 1276767 (95e:58121),
  • [36] Zin Arai and Konstantin Mischaikow, Rigorous computations of homoclinic tangencies, SIAM J. Appl. Dyn. Syst. 5 (2006), no. 2, 280-292 (electronic). MR 2237148 (2008e:37011),
  • [37] J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15(1) (1977).
  • [38] S. M. Rump, INTLAB - INTerval LABoratory, in Tibor Csendes, editor, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, pp. 77-104.

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Additional Information

Allan Hungria
Affiliation: Department of Mathematical Sciences, Ewing Hall, University of Delaware, Newark, Delaware 19716

Jean-Philippe Lessard
Affiliation: Département de Mathématiques et de Statistique, Université Laval, 1045 avenue de la Médecine, Québec, QC, G1V0A6, Canada

J. D. Mireles James
Affiliation: Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Rd., Boca Raton, Florida 33431

Received by editor(s): March 16, 2014
Received by editor(s) in revised form: November 22, 2014
Published electronically: September 28, 2015
Additional Notes: The second author was supported by NSERC and the FRQNT program Établissement de nouveaux chercheurs.
The third author was partially supported by NSF grant DSM 1318172.
Article copyright: © Copyright 2015 American Mathematical Society

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