Abstract

The Unified Transform Method (UTM) – a method for analysing boundary value problems for linear and integrable nonlinear PDEs – was pioneered in the late ’90s by A.S. Fokas. From the very beginning, the UTM has attracted a great deal of interest in the applied mathematics community. A multitude of versions of the original method have since been developed, each dealing with a specific family of equations. In this talk, we focus on the result of A.S. Fokas and A.A. Kapaev (2003) pertaining to the study of boundary value problems for the Laplacian in convex polygons. Their original approach relied on a variety of tools (spectral analysis of a parameter-dependent ODE ; Riemann-Hilbert techniques, etc.) but it was observed by D.G. Crowdy (2015) that the method can be recast within a complex function-theoretic framework which, in turn, extends the applicability to so-called circular domains (domains bounded by arcs of circles, with line segments being a special case).Our aim is to extend the original approach of Fokas and Kapaev for convex polygons, to arbitrary convex domains. It turns out that ellipses (which are not circular in the sense of Crowdy) are of particular relevance in applications to engineering because the most popular heat exchangers (namely the shell-and-tube exchangers) have elliptical cross section. In this talk I will describe a complex function-theory based new algorithm for convex domains, and will highlight the numerical challenges that arise when implementing it.This is joint work with J. Hulse (Syracuse University), S.G. Llewellyn Smith (UCSD & Scripps Institute of Oceanography) and E. Luca (The Cyprus Institute).