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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 |
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. |
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), 901-920.
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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