The unified transform for mixed boundary condition problems in unbounded domains

Abstract

This paper implements the unified transform to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition problems can be solved without the need for the Wiener–Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geometries involving finite and/or multiple plates. The new approach constructs a global relation that relates known boundary data, such as the scattered normal velocity on a rigid plate, to unknown boundary values, such as the jump in pressure upstream of the plate. By approximating the known data and the unknown boundary values by suitable functions and evaluating the global relation at collocation points, one can accurately obtain the expansion coefficients of the unknown boundary values. The method is illustrated for the modified Helmholtz and Helmholtz equations. In each case, comparisons between the traditional Wiener–Hopf approach, other spectral or boundary methods and the unified transform approach are discussed.

A spectral collocation method for acoustic scattering by multiple elastic plates

“…As a simple demonstration of the method, we now briey consider the case of three rigid plates with γ 1 = [−4, 1], γ 2 = [1.1, 2] and γ 3 = [3,5]. We chose q I = e −ik 0 x cos θ−ik 0 y sin θ for k 0 = 20 and θ = π/4.…”

“…Inspired by the previous numerical method for elastic plates, [3], this paper presents an alternative spectral boundary-based method for obtaining the scattered eld due to the interaction of acoustic sources with nite elastic plates, adapted from a similar approach used for rigid plates [5]. This approach, centred around taking the unied transform [11] (also known as the Fokas transform) of the governing equations both for the structural and acoustic elds, allows us to combine these equations in spectral space and avoids having to calculate singular integrals as is often problematic for BEM codes.…”

“…This approach, centred around taking the unied transform [11] (also known as the Fokas transform) of the governing equations both for the structural and acoustic elds, allows us to combine these equations in spectral space and avoids having to calculate singular integrals as is often problematic for BEM codes. It is also fast, with suitable basis functions leading to rapid convergence (see [5] for comparisons with other spectral and boundary based methods). Our new method can be easily extended to more complicated geometries, such as those consisting of multiple, not necessarily parallel plates, and therefore could provide a valuable tool for a range of aeroacoustic and hydroacoustic applications.…”

Kernel density estimation with linked boundary conditions

“…Our goal in this article is to extend this analysis to linked boundary conditions. A key tool in our analysis is the unified transform (also known as the Fokas method), a novel transform for analyzing boundary value problems for linear (and integrable nonlinear) partial differential equations (PDEs) 7–19 . An excellent pedagogical review of this method can be found in the paper of Deconinck et al 20 …”

The Unified Transform: A Spectral Collocation Method for Acoustic Scattering

“…The method is illustrated first for the simple problem of acoustic scattering by a finite rigid flat plate 7 , for which a fully analytic solution is known in terms of Mathieu functions. 8 Results are then given for acoustic scattering by a finite elastic plate, and quadrupole scattering by rigid plates with finite elastic extensions.…”