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SUMMARY:Approximate solutions to Wiener-Hopf equations via the implicit qu
 adrature scheme - Ian Thompson (University of Liverpool)
DTSTART:20240704T094500Z
DTEND:20240704T101500Z
UID:TALK214957@talks.cam.ac.uk
DESCRIPTION:Obtaining approximate solutions to Wiener-Hopf equations is ma
 de difficult by the intricate coupling of near and far field effects. Repl
 acing a term with a numerical approximation that is accurate in a region o
 f the complex plane can have unexpected consequences due to inaccuracies t
 hat occur elsewhere.\nIn this presentation\, we will discuss the 'implicit
  quadrature scheme'\, a numerical method that can accurately solve matrix 
 Wiener-Hopf equations. &nbsp\;The scheme does not rely on obtaining an app
 roximate equation that can itself be solved by some exact procedure\, but 
 instead solves the Wiener-Hopf equation directly\, in a single step. &nbsp
 \;Unknown functions are represented by Cauchy integrals\, and the only app
 roximation occurs when these are eventually evaluated using quadrature for
 mulae. &nbsp\;In this respect\, the method is similar to the standard proc
 edure for solving scalar Wiener-Hopf equations\, since this typically requ
 ires certain functions to be represented as Cauchy integrals\, and these m
 ust be computed numerically in most cases. The main difference is that the
  implicit quadrature scheme requires the construction and inversion of a l
 inear system of algebraic equations. The size of this system might once ha
 ve been considered 'large' but on modern computers it can usually be solve
 d in a few seconds at most.\nThe implicit quadrature scheme was introduced
  in [1]\, as a means of solving a matrix Wiener-Hopf equation that contain
 s very complicated terms. This work was presented at the INI in 2019. &nbs
 p\;A numerical library that can be used to easily obtain solutions to Wien
 er-Hopf equations via the implicit quadrature scheme then began developmen
 t\, funded by an EPSRC Mathematical Sciences small grant (EP/W000504/1). W
 e will discuss the capabilities and limitations of this library\, and show
  results obtained for problems for which accurate solutions are available 
 by other means.\n[1] Thompson\, I. ''Wave diffraction by a rigid strip in 
 a plate modelled by Mindlin theory.'' &nbsp\;Proceedings of the Royal Soci
 ety of London A 476(2243)\, 2020.
LOCATION:Seminar Room 1\, Newton Institute
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