Related papers: Integral equation methods for scattering by genera…
We explain how the spectrum of a closed embedded surface $\Sigma \subset \mathbb{R}^3$ relates to the Dirichlet spectrum of the bounded domain $\Omega \subset \mathbb{R}^3$ with $\partial \Omega = \Sigma$. We prove that there exists a…
This paper develops and analyzes some continuous interior penalty finite element methods (CIP-FEMs) using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions.…
In this paper we investigate the boundary value problem ${div(\gamma\nabla u)=0 in \Omega, u=f on \partial\Omega$ where $\gamma$ is a complex valued $L^\infty$ coefficient, satisfying a strong ellipticity condition. In Electrical Impedance…
This paper is devoted to proving the almost global solvability of the Cauchy problem for the Kirchhoff equation in the Gevrey space $\gamma^s_{\eta,L^2}$. Furthermore, similar results are obtained for the initial-boundary value problems in…
The $ k $-configuration space $ B_k\Gamma $ of a topological space $ \Gamma $ is the space of sets of $ k $ distinct points in $ \Gamma $. In this paper, we consider the case where $ \Gamma $ is a graph of circumference at most $1$. We show…
We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form $[0,T] \times \Sigma$, where $\Sigma$ is a compact manifold with smooth boundaries $\partial\Sigma$. By using an appropriate…
The completion of a network of advanced laser-interferometric gravitational-wave observatories around 2001 will make possible the study of the inspiral and coalescence of binary systems of compact objects (neutron stars and black holes),…
This work presents an unfitted boundary algebraic equation (BAE) method for solving three-dimensional elliptic partial differential equations on complex geometries using finite difference on structured meshes. We demonstrate that replacing…
Three types of boundary integral equation (BIE) methods are employed to obtain closed-form solutions of a wave-scattering problem which are compared to the exact, closed-form (reference), solution deriving from the separation-of-variables…
The time-harmonic Maxwell equations at high wavenumber k in domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution…
Given a finite-dimensional, complex simple Lie algebra we exhibit an integral form for the universal enveloping algebra of its map algebra, and an explicit integral basis for this integral form. We also produce explicit commutation formulas…
We refine a remark of Steinerberger (2024), proving that for $\alpha \in \mathbb{R}$, there exists integers $1 \leq b_{1}, \ldots, b_{k} \leq n$ such that \[ \left\| \sum_{j=1}^k \sqrt{b_j} - \alpha \right\| = O(n^{-\gamma_k}), \] where…
We establish Lipschitz stability for both the potential and the initial conditions from a single boundary measurement in the context of a hyperbolic boundary initial value problem. In our setting, the initial conditions are allowed to…
The time-harmonic Maxwell equations at high wavenumber $k$ are discretized by edge elements of degree $p$ on a mesh of width $h$. For the case of a ball and exact, transparent boundary conditions, we show quasi-optimality of the Galerkin…
We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then…
We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the…
An integral formulation for acoustic radiation in moving flows is presented. It is based on a potential formulation for acoustic radiation on weakly non-uniform subsonic mean flows. This work is motivated by the absence of suitable kernels…
Let $\Omega$ be a bounded domain with $C^2$-smooth boundary in an $n$-dimensional oriented Riemannian manifold. It is well-known that for the bi-harmonic equation $\Delta^2 u=0$ in $\Omega$ with the $0$-Dirichlet boundary condition, there…
We prove using an integral criterion the existence and completeness of the wave operators $W_{\pm}(\Delta_h^{(k)}, \Delta_g^{(k)}, I_{g,h}^{(k)})$ corresponding to the Hodge Laplacians $\Delta_\nu^{(k)}$ acting on differential $k$-forms,…
In an attempt to simulate black hole echoes (generated by potential quantum-gravitational structure) in numerical relativity, we recently described how to implement a reflecting boundary outside of the horizon of a black hole in spherical…