-
Over the past five years, I have focused on the spectral collocation method based on Chebyshev polynomials (ChC) in its conventional form, and subsequently utilised the Chebfun platform. We have addressed, in turn, problems of singular eigenvalues (Schrödinger) and nonlinear and singular boundary problems with applications in fluid mechanics and physico-chemical hydrodynamics. The singularity refers to the fact that the domain of integration is unbounded. We study the numerical analytic continuation using the best or near-best rational approximation (interpolation), the AAA algorithm (adaptive Antoulas-Anderson), to extend the real solution of the BVP into the complex plane. In this way, we estimate the locations of real or complex singularities of the solution (poles, branch points, etc.). Accordingly, we assert that achieving a comprehensive understanding of the dynamics inherent in certain challenging problems necessitates an examination of complex singularities.