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

被引：1

作者：

van den Berg, Jan Bouwe ^{[1
]}

Duchesne, Gabriel William ^{[2
]}

Lessard, Jean-Philippe ^{[2
]}

机构：

[1] Vrije Univ Amsterdam, Dept Math, De Boelelaan 1081, NL-1081 HV Amsterdam, Netherlands

[2] McGill Univ, Dept Math & Stat, 805 Sherbrooke St West, Montreal, PQ H3A 0B9, Canada

来源：

JOURNAL OF COMPUTATIONAL DYNAMICS | 2022年 / 9卷 / 02期

关键词：

Rotation invariant patterns; elliptic PDEs on manifolds; computer-assisted proofs; Chebyshev series; Taylor series; contraction mapping theorem; RATIONAL INTERPOLATION; VERIFICATION METHODS; PROOF;

D O I：

10.3934/jcd.2022005

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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 (0,pi/2] with a removable singularity at zero. The singularity is removed by solving the equation with Taylor series on (0,delta] (with delta small) while a Chebyshev series expansion is used to solve the problem on [delta,pi/2]. 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.

引用

页码：253 / 278

页数：26