# American Institute of Mathematical Sciences

June & December  2018, 5(1&2): 61-80. doi: 10.3934/jcd.2018003

## 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

Fund Project: 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.

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.

Citation: István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003
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(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.
(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.
The second solution of (9) on the unit sphere $S^2 \subset \mathbb{R}^3$. (Right) The corresponding (numerical) solution of the BVP (11).
(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).
Six solutions of (22) for $\lambda \in \{118.2,120,250,350,450,500\}$.
(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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