Related papers: The lightning method for the heat equation
This paper shows how the enclosure method which was originally introduced for elliptic equations can be applied to inverse initial boundary value problems for parabolic equations. For the purpose a prototype of inverse initial boundary…
This paper builds rigorous analysis on the root-exponential convergence for the lightning schemes via rational functions in approximating corner (branch) singularity problems with uniform exponentially clustered poles proposed by Gopal and…
A nonlinear Helmholtz equation (NLH) with high wave number and Sommerfeld radiation condition is approximated by the perfectly matched layer (PML) technique and then discretized by the linear finite element method (FEM).…
The Helmholtz equation arises in the study of electromagnetic radiation, optics, acoustics, etc. In spherical coordinates, its general solution can be written as a spherical harmonic series which satisfies the radiation condition at…
Building on introducing exponentially clustered poles, Trefethen and his collaborators introduced lightning algorithms for approximating functions of singularities. These schemes may achieve root-exponential convergence rates. In…
Consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper is concerned with the numerical solutions of the transverse…
Boundary integral equations are an efficient and accurate tool for the numerical solution of elliptic boundary value problems. The solution is expressed as a layer potential; however, the error in its evaluation grows large near the…
We present a method for computing the spectrum of black hole radiation of a scalar field satisfying a wave equation with high frequency dispersion. The method involves a combination of Laplace transform and WKB techniques for finding…
A finite element approach for approximating the solution of a mathematical model for the response of a penetrable, bounded object (obstacle) to the excitation by an external electromagnetic field is presented and investigated. The model…
This paper is about a method for solving infinite series in closed form by using inverse and forward Laplace transforms. The resulting integral is to be solved instead. The method is extended by parametrizing the series. A further Laplace…
An inverse source problem for the heat equation is considered. Extraction formulae for information about the time and location when and where the unknown source of the equation firstly appeared are given from a single lateral boundary…
This paper presents an integral formulation for Helmholtz problems with mixed boundary conditions. Unlike most integral equation techniques for mixed boundary value problems, the proposed method uses a global boundary charge density. As a…
This work introduces novel numerical algorithms for computational quantum mechanics, grounded in a representation of the Laplace operator -- frequently used to model kinetic energy in quantum systems -- via the heat semigroup. The key…
The LIT approach is reviewed both for inclusive and exclusive reactions. It is shown that the method reduces a continuum state problem to a bound-state-like problem, which then can be solved with typical bound-state techniques. The LIT…
New numerical algorithms based on rational functions are introduced that can solve certain Laplace and Helmholtz problems on two-dimensional domains with corners faster and more accurately than the standard methods of finite elements and…
This paper investigates the numerical modeling of a time-dependent heat transmission problem by the convolution quadrature boundary element method. It introduces the latest theoretical development into the error analysis of the numerical…
We investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the…
New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the…
We investigate the Helmholtz equation with suitable boundary conditions and uncertainties in the wavenumber. Thus the wavenumber is modeled as a random variable or a random field. We discretize the Helmholtz equation using finite…
A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a…