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Integral equations on fractal domains: analysis and computation

分形域上的积分方程：分析与计算

基本信息

批准号：
EP/S01375X/1
负责人：
David Peter Hewett
金额：
$ 31.64万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FS01375X%2F1
关键词：
Integral equations fractal domains analysis

项目摘要

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.

积分方程是泛函分析数学领域的基本对象。它们也是分析和求解由偏微分方程描述的许多物理过程的数学模型的有力工具。特别地，它们广泛用于声学、电磁和弹性波散射应用，例如噪声控制、雷达/声纳/医学/地震成像、移动的通信和气候科学。基于积分方程的计算方法的重要例子包括边界元法和离散偶极子近似，现有的用于波散射中的积分方程分析和计算的数学工具仅适用于散射体相对简单的情况，具有一定程度的数学“光滑性”。然而，在许多应用中，散射体可以是高度复杂和极其粗糙的，具有多个长度尺度上的微观结构。例子包括树木和植被，建筑物外墙，海洋表面，电气工程中的某些天线设计，以及大气颗粒，如雪/冰晶和灰尘聚集体。这种散射体通常被建模为“分形”，非光滑的数学对象，在所有长度尺度上表现出自相似性。这个项目的目的是推广的积分方程理论，能够处理这样的分形散射体。这需要数学分析和数值近似的进步。（A）在函数空间和积分算子理论方面取得新的数学成果，从而能够对超出现有理论范围的分形散射问题进行严格分析;（B）精确有效地求解分形域上积分方程的新数值方法，支持（不同于目前在文献中）通过系统的数学分析。该项目的理论结果将被应用到实际相关的应用，包括（i）电子工程中的分形天线设计，以及（ii）气象学和气候科学中的分形大气颗粒（雪/冰/灰尘）的光/雷达散射。