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SUMMARY:Integral equations for wave scattering by fractals - David Hewett 
 (UCL)
DTSTART:20230608T140000Z
DTEND:20230608T150000Z
UID:TALK198055@talks.cam.ac.uk
CONTACT:Matthew Colbrook
DESCRIPTION:Integral equations are a powerful and popular tool for the num
 erical solution of linear PDEs for which a fundamental solution is availab
 le. They are of particular importance in the study of acoustic\, electroma
 gnetic and elastic wave propagation\, where wave scattering problems posed
  in unbounded domains can often be formulated as an integral equation over
  the (typically bounded) scatterer or its boundary. For scatterers with sm
 ooth boundaries this is classical\, but many real-life scatterers (e.g. tr
 ees/vegetation\, snowflakes/ice crystal aggregates) are highly irregular. 
 The case where the scatterer (or its boundary) is fractal poses particular
 ly interesting challenges\, and our recent investigations into this topic 
 have led to new results in function spaces\, variational problems\, numeri
 cal quadrature and integral equations\, which I will survey in this talk. 
 Computationally\, we have studied two main approaches: (1) approximate the
  fractal by a smoother "prefractal" shape\, and (2) work with integral equ
 ations formulated directly on the fractal\, with respect to the appropriat
 e fractal (Hausdorff) measure. The latter approach seems to provide a clea
 rer pathway for rigorous convergence analysis\, but for numerical implemen
 tation requires accurate quadrature rules for evaluating singular integral
 s with respect to fractal measures.
LOCATION:Centre for Mathematical Sciences\, MR14
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