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This study proposes a novel coupled-mode theory for two-dimensional exterior Helmholtz problems. The proposed approach is based on the separation of the entire space R2 into a fictitious disk and its exterior. The disk is allocated in such a way that it comprises all the inhomogeneity; therefore, the exterior supports cylindrical waves with a continuous spectrum. For the interior, we expand an unknown wave field using normal modes that satisfy some auxiliary boundary conditions on the surface of the disk. For the interior expansion, we propose combining the Neumann and Dirichlet normal modes. We show that the proposed expansion sacrifices L2 orthogonality but significantly improve the convergence. Finally, we present some numerical verifications of the proposed coupled-mode theory., Comment: 26 pages, 7 figures

A mesh-free approach for modelling beam-wall interactions in particle accelerators is proposed. The key idea of our method is to use a deep neural network as a surrogate for the solution to a set of partial differential equations involving the particle beam, and the surface impedance concept. The proposed approach is applied to the coupling impedance of an accelerator vacuum chamber with thin conductive coating, and also verified in comparison with the existing analytical formula., Comment: 3 pages, 3 figures, submitted for IET possible publications

We consider a two-phase heat conductor in two dimensions consisting of a core and a shell with different constant conductivities. When the medium outside the two-phase conductor has a possibly different conductivity, we consider the Cauchy problem in two dimensions where initially the conductor has temperature 0 and the outside medium has temperature 1. It is shown that, if there is a stationary isothermic surface in the shell near the boundary, then the structure of the conductor must be circular. Moreover, as by-products of the method of the proof, we mention other proofs of all the previous results of [S] in N(≥ 2) dimensions and two theorems on their related two-phase elliptic overdetermined problems.

This paper applies the Modified separation of variables method (MSV) suggested by Pishkoo and Darus towards obtaining a solution for fractional Laplace's equation. The closed form expression for potential function is formulated in terms of Meijer's G-functions (MGFs). Moreover, the relationship between fractional dimension and parameters of Meijer’s G-function for spherical Laplacian in R, for radial functions is derived.

This paper applies the Modified separation of variables method (MSV) suggested by Pishkoo and Darus towards obtaining a solution for fractional Laplace's equation. The closed form expression for potential function is formulated in terms of Meijer's G-functions (MGFs). Moreover, the relationship between fractional dimension and parameters of Meijer’s G-function for spherical Laplacian in R, for radial functions is derived.

We study the problem of finding functions, defined within and on an ellipse, whose Laplacian is -1 and which satisfy a homogeneous Robin boundary condition on the ellipse. The parameter in the Robin condition is denoted by beta. The general solution and various asymptotic approximations are obtained. To find the general solution, the boundary value problem is formulated in elliptic cylindrical coordinates. A Fourier series solution is then derived. The Fourier coefficients satisfy a 3-term recurrence relation which can be solved. The integral of the solution over the ellipse, denoted by Q, is a quantity of interest in some physical applications. The dependence of Q on beta and the ellipse geometry is found. Finding asymptotics directly from the pde formulations is easier than from our series solution. We use the asymptotic approximations to Q as checks on the series solution. Several other inequalities are also used to check the solution. It is intended that this arXiv preprint will be referenced by the journal version, which will be submitted soon, as the arXiv contains material, e.g. codes for calculating Q, not in the much shorter journal version. Maple codes used in deriving or checking results in this paper are in the process of being tidied prior to being made available via links given at the URL given in the pdf version.