Integral equations on fractal domains: analysis and computation

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Integral equations are fundamental objects in the mathematical area of functional analysis. They are also a powerful tool for analysing and solving mathematical models of many physical processes described by partial differential equations. In particular they are widely used in acoustic, electromagnetic and elastic wave scattering applications such as noise control, radar/sonar/medical/seismic imaging, mobile communications and climate science. Important examples of computational methods based on integral equations include the boundary element method and the discrete dipole approximation.Existing mathematical tools for analysis and computation with integral equations in wave scattering apply only to situations in which the scatterer is relatively simple, possessing a certain degree of mathematical "smoothness". However, in many applications scatterers can be highly complex and extremely rough, with microstructure on multiple lengthscales. Examples include trees and vegetation, building facades, surface of the ocean, certain antenna designs in electrical engineering, and atmospheric particles such as snow/ice crystals and dust aggregates. Such scatterers are often modelled as "fractals", non-smooth mathematical objects exhibiting self-similarity on all lengthscales. This project aims to generalise the theory of integral equations to be able to handle such fractal scatterers. This requires advances in mathematical analysis and numerical approximation. The project will lead to:(A) New mathematical results in the theory of function spaces and integral operators, permitting the rigorous analysis of fractal scattering problems that are beyond the scope of existing theory;(B) New numerical methods for accurately and efficiently solving integral equations on fractal domains, supported (unlike those currently available in the literature) by a systematic mathematical analysis.The theoretical results of the project will be applied to practically relevant applications including (i) fractal antenna design in electrical engineering, and (ii) light/radar scattering by fractal atmospheric particles (snow/ice/dust) in meteorology and climate science.