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SUMMARY:Cauchy and log transforms on intervals\, curves\, wedges and polyg
 ons - Sheehan Olver (Imperial College London)
DTSTART:20230208T094500Z
DTEND:20230208T103000Z
UID:TALK194776@talks.cam.ac.uk
DESCRIPTION:Green's integral representations for two-dimensional PDEs aris
 ing in scattering can be expressed as smooth modifications of logarithmic 
 and Cauchy kernels. An efficient way to numerically solve these is to find
  exact expressions for the logarithmic and Cauchy kernels on the underlyin
 g geometries. This talk explores how this can be accomplished using orthog
 onal polynomials on an interval\, and by mapping the interval to a curve o
 r wedge by a polynomial\, rational\, or algebraic map. The Cauchy transfor
 m over the mapped domain can be expressed exactly in terms of the unmapped
  Cauchy transform using new change-of-variable formulae\, and multiple wed
 ges can be combined to solve PDEs on polygons\, that is\, by breaking the 
 polygon into elements but where the elements have a corner in the middle. 
 These maps exactly capture the singularities that arise in corners for sol
 utions ot PDEs leading to spectral convergence.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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