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Integral equations for wave scattering by fractals

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Integral equations are a powerful and popular tool for the numerical solution of linear PDEs for which a fundamental solution is available. They are of particular importance in the study of acoustic, electromagnetic 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 smooth boundaries this is classical, but many real-life scatterers (e.g. trees/vegetation, snowflakes/ice crystal aggregates) are highly irregular. The case where the scatterer (or its boundary) is fractal poses particularly interesting challenges, and our recent investigations into this topic have led to new results in function spaces, variational problems, numerical 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 equations formulated directly on the fractal, with respect to the appropriate fractal (Hausdorff) measure. The latter approach seems to provide a clearer pathway for rigorous convergence analysis, but for numerical implementation requires accurate quadrature rules for evaluating singular integrals with respect to fractal measures.

This talk is part of the Applied and Computational Analysis series.

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