On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains

The Modified Helmholtz Equation on a Regular Hexagon—The Symmetric Dirichlet Problem

“…where F j and A j are given by (38) and (39), respectively; see Figure 5. We also depict the deviation of d(s) from the above expression evaluated at the side of the hexagon, namely at x = √ 3 and y = s ∈ [−1, 1].…”

“…The global relation has had important analytical and numerical implications: first, it has led to novel analytical formulations of a variety of important physical problems from water waves [20][21][22][23][24][25][26] to three-dimensional layer scattering [27]. Second, it has led to the development of new techniques for the Laplace, modified Helmholtz, Helmholtz, biharmonic equations, both analytical [28][29][30][31][32][33][34][35] and numerical [36][37][38][39][40][41][42][43][44][45][46][47].…”

The Unified Transform: A Spectral Collocation Method for Acoustic Scattering

“…where Λ is a domain containing all λ such that the integral converges (which depends on how D behaves at infinity if it is unbounded, and the conditions satisfied by q at infinity). The global relation (9) may be used directly, however to simplify it in the case of a polygonal domain where ∂D consists of M straight sides, we let q j and q j n denote the Dirichlet and Neumann boundary values a Whilst here we discuss convex polygons, for extensions to non-convex polygons one should consult Colbrook et al 9 The unified transform can also be used for circular domains [10][11][12] and non-polygonal domains with general curved edges. 13 on the j th side which connects corners z j and z j+1 .…”

“…For example, for acoustic scattering in an unbounded domain, functions are oscillatory therefore Bessel functions of integer and half-integer order are beneficial. Substituting the function expansions into the global relation for Helmholtz, (9), yields…”

A Mathieu function boundary spectral method for scattering by multiple variable poro-elastic plates, with applications to metamaterials and acoustics

“…See also [32,33] for an expansion scheme of the plate deformation connected to Chebyshev polynomials that tackles the problem of a single elastic plate in a rigid baffle (our numerical scheme can handle this problem with an appropriate modification of the boundary conditions when we separate variables in §3). Another approach for these types of problems is the unified transform [34] (see also [35][36][37][38] for recent developments), a Fourier space boundary spectral collocation method which in certain cases generalizes the Wiener-Hopf method [9,39]. 1 However, using the unified transform in unbounded domains requires the setting up of several global relations by hand, which becomes complicated in complex geometries.…”