Călin Gheorghiu: Analytic Continuation of Solutions to Some Nonlinear BVPs. Applications in Fluid and Rational Mechanics (Seminar Talk)

On Wednesday 8th of April 2026 starting at 10am, Călin Gheorghiu will give a talk at the Institute Seminar.

Title: Analytic Continuation of Solutions to Some Nonlinear BVPs. Applications in Fluid and Rational Mechanics

Abstract: We revisit a classical second-order singularly perturbed nonlinear boundary value problem (BVP) and solve it directly as a boundary value problem–avoiding the traditional, now out dated shooting approach. This avoids instability, missed branches, and the sensitivity issues inherent in shooting. A key feature of the model is that the diffusive term is scaled by a small parameter, causing the convective term to dominate and generating multiple boundary layers. Using Chebyshev collocation, implemented through Chebfun, we accurately resolve the steep interior layer as well as the two endpoint layers on the integration interval. All three boundary-layer solutions are captured with high fidelity, and their thicknesses are quantified. To investigate the analytic structure of these solutions, we employ the adaptive Antoulas–Anderson (AAA) rational approximation algorithm to continue them into the complex plane and identify the poles and zeros of the resulting rational representations. The presence of complex poles–weak singularities that lie extremely close to the real axis–explains the emergence of the interior boundary layer. In contrast, the comparatively sparse real poles correspond to effective singularities. Finally, we examine how the complex poles migrate toward the real axis as the perturbation parameter decreases, revealing an additional and unexpected aspect of the singular perturbation structure of the problem. Joint work with Eduard Grigoriciuc.