Computer-assisted proofs for radially symmetric solutions of PDEs

István Balázs; Jan Bouwe van den Berg; Julien Courtois; János Dudás; Jean-Philippe Lessard; Anett Vörös-Kiss; JF Williams; Xi Yuan Yin

doi:10.3934/jcd.2018003

Article Contents

2018, Volume 5: 61-80. Doi: 10.3934/jcd.2018003

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Computer-assisted proofs for radially symmetric solutions of PDEs

1.
MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, Hungary, H-6720
2.
VU Amsterdam, Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
3.
Université de Montréal, Département de Mathématiques et de Statistique, Pavillon André-Aisenstadt, 2920 chemin de la Tour, Montreal, QC, H3T 1J4, Canada
4.
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, Hungary, H-6720
5.
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada
6.
Simon Fraser University, Department of Mathematics, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

* Corresponding author: Jean-Philippe Lessard
* Corresponding author: Jean-Philippe Lessard

Published: November 2018

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.

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Abstract

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.

Keywords:

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

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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.

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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.

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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).

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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).

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

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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 $u(s) = u(\sqrt{x^2+y^2+z^2})$.

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