Related papers: Representation of solutions to wave equations with…
The solution of the wave equation in a polyhedral domain in $\mathbb{R}^3$ admits an asymptotic singular expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation…
In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that…
This paper presents a new approach for the computation of eigenvalues of the generalized spheroidal wave equations. The novelty of the present method is in the use of the analytical derivatives of the eigenvalues to minimize losses in…
We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on some domain $\Omega$ from their values at scattered points $x_1,\dots,x_n\subset \Omega$. This problem typically arises when sampling acoustic…
The free-particle propagator, a key operator in various algorithms for simulating the time evolution of the Schr\"odinger equation, is studied. A multiscale approximation of this propagator is constructed, representing the semigroup…
We discuss a new numerical schema for solving the initial value problem for the Korteweg-de Vries equation for large times. Our approach is based upon the Inverse Scattering Transform that reduces the problem to calculating the reflection…
In this article, we consider the reconstruction of $\rho(t)$ in the (time-fractional) diffusion equation $(\partial_t^\alpha-\triangle)u(x,t)=\rho(t)g(x)$ ($0<\alpha \le 1$) by the observation at a single point $x_0$. We are mainly…
The problem of immersing a simply connected surface with a prescribed shape operator is discussed. From classical and more recent work, it is known that, aside from some special degenerate cases, such as when the shape operator can be…
This paper aims at developing new shape functions adapted to smooth vanishing coefficients for scalar wave equation. It proposes the numerical analysis of their interpolation properties. The interpolation is local but high order convergence…
In this paper, we develop and numerically implement a novel approach for solving the inverse source problem of the acoustic wave equation in three dimensions. By injecting a small high-contrast droplet into the medium, we exploit the…
In the paper we derive two formulas representing solutions of Cauchy problem for two Schr\"{o}dinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally…
This paper presents an efficient approach to image segmentation that approximates the piecewise-smooth (PS) functional in [12] with explicit solutions. By rendering some rational constraints on the initial conditions and the final solutions…
We propose a scheme for imaging periodic surfaces using a superlens. By employing an inverse scattering model and the transformed field expansion method, we derive an approximate reconstruction formula for the surface profile, assuming…
Wavelets are a useful basis for constructing solutions of the integral and differential equations of scattering theory. Wavelet bases efficiently represent functions with smooth structures on different scales, and the matrix representation…
In this paper, we seek to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. We first present a conditional result on the construction of nontrivial global solutions to a general system of…
The goal for this paper is twofold. Our first main objective is to develop Bahouri-Gerard type profile decompositions for waves on hyperbolic space. Recently, such profile decompositions have proved to be a versatile tool in the study of…
We discuss the time evolution of the wave function which is solution of a stochastic Schroedinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and…
The dynamical system under consideration is \begin{align*} & u_{tt}-u_{xx}+Vu=0,\qquad x>0,\,\,\,t>0;\\ & u|_{t=0}=u_t|_{t=0}=0,\,\,x\geqslant 0;\quad u|_{x=0}=f,\,\,t\geqslant 0, \end{align*} where $V=V(x)$ is a matrix-valued function…
For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation…
In this article we study the reconstruction problem in TAT/PAT on an attenuating media. Namely, we prove a reconstruction procedure of the initial condition for the damped wave equation via Neumann series that works for arbitrary large…