Advanced Search
Article Contents
Article Contents

Computer-assisted proofs for radially symmetric solutions of PDEs

  • * Corresponding author: Jean-Philippe Lessard

    * Corresponding author: Jean-Philippe Lessard 
The first, fourth and sixth authors were supported by the Hungarian Scientific Research Fund (NKFIH-OTKA), Grant No. K109782. The second author was supported in part by NWO-Vici grant 639.033.109. The fifth and the seventh authors were supported by NSERC.
Abstract Full Text(HTML) Figure(6) Related Papers Cited by
  • We obtain radially symmetric solutions of some nonlinear (geometric) partial differential equations via a rigorous computer-assisted method. We introduce all main ideas through examples, accessible to non-experts. The proofs are obtained by solving for the coefficients of the Taylor series of the solutions in a Banach space of geometrically decaying sequences. The tool that allows us to advance from numerical simulations to mathematical proofs is the Banach contraction theorem.

    Mathematics Subject Classification: 58J05, 58C40, 65G40, 41A58, 58J20, 65M99, 65G40, 35K57.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  (Left) Ten relative equilibria of (CR4BP) with equal masses. (Right) Eight relative equilibria of (CR4BP) with masses $m_1 = 0.9987451087$, $m_2 = 0.0010170039$ and $m_3 = 0.0002378873$. In both plots, some level sets of the effective potential $\Omega$ are depicted.

    Figure 2.  (Left) The first solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11). Since $r_{\min}<10^{-8}$, the true solution lies with the line-width by Theorem 2.1.

    Figure 3.  The second solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11).

    Figure 4.  (Left) The third solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11).

    Figure 5.  Six solutions of (22) for $\lambda \in \{118.2,120,250,350,450,500\}$.

    Figure 6.  (Left) A stationary solution of the Swift-Hohenberg equation (20) on the unit ball in $\mathbb{R}^3$ at $\lambda = 500$. (Right) The corresponding graph of $u(s) = u(\sqrt{x^2+y^2+z^2})$.

  • [1] G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7.
    [2] G. Arioli and H. Koch, Integration of dissipative partial differential equations: A case study, SIAM J. Appl. Dyn. Syst., 9 (2010), 1119-1133.  doi: 10.1137/10078298X.
    [3] G. Arioli and H. Koch, Validated numerical solutions for some semilinear elliptic equations on the disk, 2017. Preprint.
    [4] I. Balázs, J.B. van den Berg, J. Courtois, J. Dudás, J.-P. Lessard, A. Vörös-Kiss, J. F. Williams and X. Y. Yin, MATLAB code for "Computer-assisted proofs for radially symmetric solutions of PDEs", 2017, http://www.math.vu.nl/~janbouwe/code/radialpdes/.
    [5] A. N. Baltagiannis and K. E. Papadakis, Periodic solutions in the Sun—Jupiter—Trojan Asteroid—Spacecraft system, Planetary and Space Science, 75 (2013), 148-157. 
    [6] J. F. Barros and E. S. G. Leandro, The set of degenerate central configurations in the planar restricted four-body problem, SIAM J. Math. Anal., 43 (2011), 634-661.  doi: 10.1137/100789701.
    [7] J. F. Barros and E. S. G. Leandro, Bifurcations and enumeration of classes of relative equilibria in the planar restricted four-body problem, SIAM J. Math. Anal., 46 (2014), 1185-1203.  doi: 10.1137/130911342.
    [8] B. BreuerJ. HorákP. J. McKenna and M. Plum, A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, J. Diff. Eq., 224 (2006), 60-97.  doi: 10.1016/j.jde.2005.07.016.
    [9] J. Burgos-García and M. Gidea, Hill's approximation in a restricted four-body problem, Celestial Mech. Dynam. Astronom., 122 (2015), 117-141.  doi: 10.1007/s10569-015-9612-9.
    [10] CAPD: Computer assisted proofs in dynamics, a package for rigorous numerics, http://capd.ii.uj.edu.pl/.
    [11] A. Castro and E. M. Fischer, Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres, Canad. Math. Bull., 58 (2015), 723-729.  doi: 10.4153/CMB-2015-056-7.
    [12] L. Cesari, Functional analysis and periodic solutions of nonlinear differential equations, Contributions to Differential Equations, 1 (1963), 149-187. 
    [13] L. Cesari, Functional analysis and Galerkin's method, Michigan Math. J., 11 (1964), 385-414.  doi: 10.1307/mmj/1028999194.
    [14] J.-L. FiguerasM. GameiroJ.-P. Lessard and R. de la Llave, A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations, SIAM J. Appl. Dyn. Syst., 16 (2017), 1070-1088.  doi: 10.1137/16M1073777.
    [15] A. HungriaJ.-P. Lessard and J. D. Mireles James, Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, Math. Comp., 85 (2016), 1427-1459.  doi: 10.1090/mcom/3046.
    [16] H. KochA. Schenkel and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: A case study, SIAM Rev., 38 (1996), 565-604.  doi: 10.1137/S0036144595284180.
    [17] O. E. Lanford, Ⅲ. A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X.
    [18] E. S. G. Leandro, On the central configurations of the planar restricted four-body problem, J. Diff. Eq., 226 (2006), 323-351.  doi: 10.1016/j.jde.2005.10.015.
    [19] P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.
    [20] S. McCalla and B. Sandstede, Snaking of radial solutions of the multi-dimensional Swift-Hohenberg equation: A numerical study, Phys. D, 239 (2010), 1581-1592.  doi: 10.1016/j.physd.2010.04.004.
    [21] R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898717716.
    [22] M. T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim., 22 (2001), 321-356.  doi: 10.1081/NFA-100105107.
    [23] S. M. Rump, INTLAB - INTerval LABoratory, In Tibor Csendes, editor, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, 77-104. http://www.ti3.tu-harburg.de/rump/.
    [24] S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.  doi: 10.1017/S096249291000005X.
    [25] A. Scheel, Radially symmetric patterns of reaction-diffusion systems, Mem. Amer. Math. Soc., 165 (2003), ⅷ+86 pp. doi: 10.1090/memo/0786.
    [26] C. Simó, Relative equilibrium solutions in the four-body problem, Celestial Mech., 18 (1978), 165-184.  doi: 10.1007/BF01228714.
    [27] J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 1977.
    [28] W. Tucker, A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2 (2002), 53-117.  doi: 10.1007/s002080010018.
    [29] W. Tucker, Validated Numerics, Princeton University Press, Princeton, NJ, 2011. A short introduction to rigorous computations.
    [30] J. B. van den BergA. DeschênesJ.-P. Lessard and J. D. Mireles James, Stationary coexistence of hexagons and rolls via rigorous computations, SIAM J. Appl. Dyn. Syst., 14 (2015), 942-979.  doi: 10.1137/140984506.
    [31] J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015), 1057-1061.  doi: 10.1090/noti1276.
    [32] J. B. van den BergJ.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.  doi: 10.1090/S0025-5718-10-02325-2.
    [33] M. J. Ward, Spots, traps, and patches: Asymptotic analysis of localized solutions to some linear and nonlinear diffusive systems, Nonlinearity, 31 (2018), R189-R239.  doi: 10.1088/1361-6544/aabe4b.
    [34] N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal., 35 (1998), 2004-2013.  doi: 10.1137/S0036142996304498.
    [35] P. Zgliczyński, Rigorous numerics for dissipative partial differential equations. Ⅱ. Periodic orbit for the Kuramoto-Sivashinsky PDE—a computer-assisted proof, Found. Comput. Math., 4 (2004), 157-185.  doi: 10.1007/s10208-002-0080-8.
    [36] 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.
  • 加载中



Article Metrics

HTML views(2890) PDF downloads(389) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint