Related papers: Modified representations for the close evaluation …
In this work, we present a new solution representation for the Helmholtz transmission problem in a bounded domain in $\mathbb{R}^2$ with a thin and periodic layer of finite length. The layer may consists of a periodic pertubation of the…
Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer…
We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The…
In this paper, we review the construction of periodic fundamental solutions and periodic layer potentials for various differential operators. Specifically, we focus on the Laplace equation, the Helmholtz equation, the Lam\'e system, and the…
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…
A classical problem in acoustic (and electromagnetic) scattering concerns the evaluation of the Green's function for the Helmholtz equation subject to impedance boundary conditions on a half-space. The two principal approaches used for…
A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equations that arise in mathematical models for the electrostatics of molecules in solvent. The proposed method used an implicit…
The recently developed quadrature by expansion (QBX) technique accurately evaluates the layer potentials with singular, weakly or nearly singular, or even hyper singular kernels in the integral equation reformulations of partial…
This article develops a solution for an inverse problem through the generalized method of lines. We consider a Laplace equation on a domain with internal and external boundaries with standard Dirichlet boundary conditions. Also, we specify…
In the present paper, we solve three boundary value problems related to the temperature field in oil strata -- the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional…
A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated…
Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient…
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…
This paper presents a mixed basis approach for Laplace eigenvalue problems, which treats the boundary as a perturbation of the free Laplace operator. The method separates the boundary from the volume via a generic function that can be…
We introduce a two-phase approximation method designed to resolve singularities in three-dimensional harmonic Dirichlet problems. The approach utilizes the classical Green's function representation, decomposing the function into its…
In this paper, using the approximate particular solutions of Helmholtz equations, we solve the boundary value problems of Helmholtz equations by combining the methods of fundamental solutions (MFS) with the methods of particular solutions…
The solution of constrained linear partial-differential equations can be described via parametric representations of linear relations. To study these representations, we provide a novel definition of boundary triplets for linear relations…
In this paper a generalized fundamental solution using the boundary element method to solve the Helmholtz equation is proposed. It is observed that the commonly used fundamental solution is only valid for good conductors since the…
This paper presents a quadrature method for evaluating layer potentials in two dimensions close to periodic boundaries, discretized using the trapezoidal rule. It is an extension of the method of singularity swap quadrature, which recently…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…