相关论文: Coordinate complexification for the Helmholtz equa…
We extend the theoretical framework of non-local optimized Schwarz methods as introduced in [Claeys,2021], considering an Helmholtz equation posed in a bounded cavity supplemented with a variety of conditions modeling material boundaries.…
The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) for a second order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develops the…
In acoustical engineering, analytical methodologies are often restricted to two or three dimensions; however, a general-dimensional approach can enhance learning and implementation efficiency while providing a unified understanding of…
This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network based on the DeepONet framework. We consider a 2D square domain with an inclusion of arbitrary boundary…
The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a…
A simple transformation converts a solution of a partial differential equation with a Dirichlet boundary condition to a function satisfying a Robin (generalized Neumann) condition. In the simplest cases this observation enables the exact…
We study two-dimensional periodic metamaterials in which idealised cylindrical inclusions are modelled by boundary conditions. In the scalar time-harmonic setting, the background field satisfies the Helmholtz equation, and high-contrast…
The orthogonality of Hilbert spaces whose elements can be represented as simple and double layer potentials is determined. Conditions of well-posed solvability of integral equations for the sum of simple and double layer potentials…
We consider the elliptic system of linear elasticity with bounded measurable coefficients in a domain where the second Korn inequality holds. We construct heat kernel of the system subject to Dirichlet, Neumann, or mixed boundary condition…
This paper is concerned with problems of scattering of time-harmonic acoustic waves by a two-layered medium with a non-locally perturbed boundary (called a rough boundary in this paper) in two dimensions, where a Dirichlet or impedance…
Green's function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green's function is a non-trivial exercise, especially for a PDE defined on a…
The purpose of this article is to provide a solution to the $m$-fold Laplace equation in the half space $R_+^d$ under certain Dirichlet conditions. The solutions we present are a series of $m$ boundary layer potentials. We give explicit…
In this paper we consider the setting of a locally compact, non-complete metric measure space $(Z,d,\nu)$ equipped with a doubling measure $\nu$, under the condition that the boundary $\partial Z:=\overline{Z}\setminus Z$ (obtained by…
In the present paper we describe a method for solving inverse problems for the Helmholtz equation in radially-symmetric domains given multi-frequency data. Our approach is based on the construction of suitable trace formulas which relate…
It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundary condition can be used to solve the Helmholtz equation in unbounded domains. In this work, applying such DtN boundary condition and using the finite element method,…
We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality. We propose a notion of \emph{domain with boundary of positive mean curvature} and prove that, for…
We derive and establish a solution concept for the linear mountain wave problem in two dimensions. After linearizing the governing equations and a change of variables, the problem can be stated as a Dirichlet boundary value problem for a…
We present a variational study of employing the trigonometric basis functions satisfying periodic boundary condition for the accurate calculation of eigenvalues and eigenfunctions of quartic double-well oscillators. Contrary to usual…
The present work devoted to the finding explicit solution of a boundary problem with the Dirichlet-Neumann condition for elliptic equation with singular coefficients in a quarter of ball. For this aim the method of Green's function have…
In this paper we consider the Dirichlet form on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-\alpha}\mathcal{B}(x,y)$, where $\mathcal{B}(x,y)$ can be degenerate at the boundary. Unlike our previous works…