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

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