Codes for the work "Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: a Taylor-Chebyshev series approach"

Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: a Taylor-Chebyshev series approach

J.B. van den Berg, G.W. Duchesne and J.-P. Lessard

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, \frac{\pi}{2} ]

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

[\delta,\frac{\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.

The paper in its pdf form can be found here.

Here are the MATLAB codes to perform the computer-assisted proofs.
The codes require installing and starting the interval arithmetic package INTLAB.