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

<p style='text-indent:20px;'>In this paper, we introduce a rigorous computational approach to prove existence of rotation invariant patterns for nonlinear Laplace-Beltrami equation posed on the 2-sphere. After changing spherical coordinates, problem becomes singular second order boundary value (BVP) interval <inline-formula><tex-math id="M1">\begin{document}$ (0,\frac{\pi}{2}] $\end{document}</tex-math></inline-formula> with <i>removable</i> singularity at zero. The is removed by solving Taylor series id="M2">\begin{document}$ (0,\delta] (with id="M3">\begin{document}$ \delta small) while Chebyshev expansion used solve id="M4">\begin{document}$ [\delta,\frac{\pi}{2}] $\end{document}</tex-math></inline-formula>. two setups are incorporated in larger zero-finding form id="M5">\begin{document}$ F(a) = 0 id="M6">\begin{document}$ containing coefficients and series. id="M7">\begin{document}$ F solved rigorously using Newton-Kantorovich argument.</p>