Many problems in applied mathematics can be solved using the Wiener-Hopf technique. Such problems arise in numerous application areas, including electromagnetism, solid mechanics, water wave theory, acoustics, financial mathematics, theoretical physics and probability theory; many are not amenable to any known alternative approaches. In particular, Wiener-Hopf equations can be used to model abrupt changes that are very difficult to capture using computer simulations, since these often work under the assumption that physical quantities can be accurately approximated using smooth functions. This has proved essential in developing theory that underpins non-destructive evaluation techniques, in which cracks and other defects are detected and sized by transmitting ultrasonic waves into a structure (such as an aeroplane wing, submarine hull or nuclear power plant component) and studying the scattered response.
To visualise an application of a Wiener-Hopf equation, consider a sound wave travelling through a cylindrical pipe, towards an open end. If an observer stands directly facing the end of the pipe and moves to one side, the sound will gradually become inaudible; it will not suddenly disappear as the observer moves out of line with the pipe. The continuous nature of the acoustic field is intricately linked to the diffraction effect that occurs as the wave propagates from the pipe into the open air, and this can be accurately modelled by solving an appropriate Wiener-Hopf equation [1, Section 3.4].
This project is concerned with a new approach to simultaneously solving coupled Wiener-Hopf equations. These so-called matrix problems are amongst the most difficult, but they arise frequently in modern applied mathematics, due to the need to model complicated structures and materials. Within certain constraints, solutions are known to exist. However, the proof of this is nonconstructive, meaning it does not provide any indication as to how the solutions can actually be obtained. Even the construction of approximate solutions is difficult, because Wiener-Hopf equations are extremely delicate, and a small change to the coefficients can drastically affect the solution, causing it to violate physical laws, and rendering it unusable.
The project is motivated by the solution to a very complicated matrix Wiener-Hopf equation which appeared in a recent paper [2]. This does not appear to be amenable to earlier approaches based on making simplifying approximations in the equation itself. Instead, it was solved by an 'Implicit Quadrature Scheme' which works by representing the unknown terms using Cauchy's integral formula. Roughly, this shows that a function which possesses certain attributes is wholly determined by its values along a single path. A set of nodes is distributed along this path, and function values at the nodes are constructed using the matrix Wiener-Hopf equation. Finally, the full solution is approximated using its values at the nodes. Increasing the number of nodes improves the accuracy of the solution. No approximations are applied in the Wiener-Hopf equation itself, so there is no risk of generating an invalid solution. Our principal objective is to further explore the Implicit Quadrature Scheme, widening the range of problems to which it can be applied and optimising its performance. We will also compare it with other, earlier methods where these are available. We will then begin development of a numerical library to efficiently implement the Implicit Quadrature Scheme. This will enable physicists, engineers and other mathematicians to quickly apply the method to important practical problems which are intractable using existing approaches.
References
[1] B. Noble "Methods Based on the Wiener-Hopf Technique". Chelsea, 1988.
[2] I. Thompson "Wave diffraction by a rigid strip in a plate modelled by Mindlin theory". Proceedings of the Royal Society A 476(2243), 2020.
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