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SUMMARY:THE UNIFIED TRANSFORM METHOD FOR THE LAPLACIAN ON CONVEX PLANAR DO
 MAINS - Loredana Lanzani (Syracuse University)
DTSTART:20230728T090000Z
DTEND:20230728T100000Z
UID:TALK202825@talks.cam.ac.uk
DESCRIPTION:The Unified Transform Method (UTM) - a method for analysing bo
 undary value problems for linear and integrable nonlinear PDEs - was pione
 ered in the late &rsquo\;90s by A.S. Fokas. From the very beginning\, the 
 UTM has attracted a great deal of interest in the applied mathematics comm
 unity. A multitude of versions of the original method have since been deve
 loped\, each dealing with a specific family of equations.&nbsp\;In this ta
 lk\, we focus on the result of A.S. Fokas and A.A. Kapaev (2003) pertainin
 g to the study of boundary value problems for the Laplacian in convex poly
 gons. Their original approach relied on a variety of tools (spectral analy
 sis 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 appl
 icability to so-called circular domains (domains bounded by arcs of circle
 s\, with line segments being a special case).Our aim is to extend the orig
 inal approach of Fokas and Kapaev for convex polygons\, to arbitrary conve
 x domains. It turns out that ellipses (which are not circular in the sense
  of Crowdy) are of particular relevance in applications to engineering bec
 ause the most popular heat exchangers (namely the shell-and-tube exchanger
 s) have elliptical cross section. In this talk I will describe a complex f
 unction-theory based new algorithm for convex domains\, and will highlight
  the numerical challenges that arise when implementing it.This is joint wo
 rk with J. Hulse (Syracuse University)\, S.G. Llewellyn Smith (UCSD & Scri
 pps Institute of Oceanography) and E. Luca (The Cyprus Institute).
LOCATION:Seminar Room 1\, Newton Institute
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