ROTATION INVARIANT PATTERNS FOR A NONLINEAR LAPLACE-BELTRAMI EQUATION: A TAYLOR-CHEBYSHEV SERIES APPROACH

Jan Bouwe van den Berg; Gabriel William Duchesne; Jean Philippe Lessard

doi:10.3934/jcd.2022005

ROTATION INVARIANT PATTERNS FOR A NONLINEAR LAPLACE-BELTRAMI EQUATION: A TAYLOR-CHEBYSHEV SERIES APPROACH

Jan Bouwe van den Berg, Gabriel William Duchesne, Jean Philippe Lessard^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

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Abstract

In this paper, we introduce a rigorous computational approach to prove existence of rotation invariant patterns for a nonlinear Laplace-Beltrami equation posed on the 2-sphere. After changing to spherical coordinates, the problem becomes a singular second order boundary value problem (BVP) on the interval ((Formula presented)] with a removable singularity at zero. The singularity is removed by solving the equation with Taylor series on (0,δ] (with δ small) while a Chebyshev series expansion is used to solve the problem on [(Formula presented)]. The two setups are incorporated in a larger zero-finding problem of the form F(a) = 0 with a containing the coefficients of the Taylor and Chebyshev series. The problem F = 0 is solved rigorously using a Newton-Kantorovich argument.

Original language	English
Pages (from-to)	253-278
Number of pages	26
Journal	Journal of Computational Dynamics
Volume	9
Issue number	2
DOIs	https://doi.org/10.3934/jcd.2022005
Publication status	Published - Apr 2022

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Keywords

Chebyshev series
computer-assisted proofs
contraction mapping theorem
elliptic PDEs on manifolds
Rotation invariant patterns
Taylor series

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10.3934/jcd.2022005

ROTATION INVARIANT PATTERNS FOR A NONLINEAR LAPLACE-BELTRAMI EQUATION_A TAYLOR-CHEBYSHEV SERIES APPROACHFinal published version, 0.98 MBLicence: Article 25fa Dutch Copyright Act

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title = "ROTATION INVARIANT PATTERNS FOR A NONLINEAR LAPLACE-BELTRAMI EQUATION: A TAYLOR-CHEBYSHEV SERIES APPROACH",

abstract = "In this paper, we introduce a rigorous computational approach to prove existence of rotation invariant patterns for a nonlinear Laplace-Beltrami equation posed on the 2-sphere. After changing to spherical coordinates, the problem becomes a singular second order boundary value problem (BVP) on the interval ((Formula presented)] with a removable singularity at zero. The singularity is removed by solving the equation with Taylor series on (0,δ] (with δ small) while a Chebyshev series expansion is used to solve the problem on [(Formula presented)]. The two setups are incorporated in a larger zero-finding problem of the form F(a) = 0 with a containing the coefficients of the Taylor and Chebyshev series. The problem F = 0 is solved rigorously using a Newton-Kantorovich argument.",

keywords = "Chebyshev series, computer-assisted proofs, contraction mapping theorem, elliptic PDEs on manifolds, Rotation invariant patterns, Taylor series",

author = "{van den Berg}, {Jan Bouwe} and Duchesne, {Gabriel William} and Lessard, {Jean Philippe}",

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AU - Lessard, Jean Philippe

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ROTATION INVARIANT PATTERNS FOR A NONLINEAR LAPLACE-BELTRAMI EQUATION: A TAYLOR-CHEBYSHEV SERIES APPROACH

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Keywords

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