Related papers: A spectral approach to a constrained optimization …
The calculation of a three-dimensional underwater acoustic field has always been a key problem in computational ocean acoustics. Traditionally, this solution is usually obtained by directly solving the acoustic Helmholtz equation using a…
We study the Cauchy problem for Fokker--Planck--Kolmogorov equations with unbounded and degenerate coefficients. Sufficient conditions for the existence and uniqueness of solutions are indicated.
This paper introduces a new method for discretizing and solving integral equation formulations of Maxwell's equations which achieves spectral accuracy for smooth surfaces. The approach is based on a hybrid Nystr\"om-collocation method using…
This paper addresses the numerical implementation of the transparent boundary condition (TBC) and its various approximations for the free Schr\"odinger equation on a rectangular computational domain. In particular, we consider the exact TBC…
We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value…
This paper presents a robust numerical solution to the electromagnetic scattering problem involving multiple multi-layered cavities in both transverse magnetic and electric polarizations. A transparent boundary condition is introduced at…
We consider a homogenization of elliptic spectral problem stated in a perforated domain, Fourier boundary conditions being imposed on the boundary of perforation. The presence of a locally periodic coefficient in the boundary operator gives…
We explain how to use smooth bivariate splines of arbitrary degree to solve the exterior Helmholtz equation based on a Perfectly Matched Layer (PML) technique. In a previous study (cf. [26]), it was shown that bivariate spline functions of…
This paper considers efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. The equation we consider contains an integral operator that typically has a singularity at the…
A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a…
We consider the problem of analyzing and designing gradient-based discrete-time optimization algorithms for a class of unconstrained optimization problems having strongly convex objective functions with Lipschitz continuous gradient. By…
The radiation condition is the key question in the mathematical modelling for scattering problems in unbounded domains. Mathematically, it plays the role as the "boundary condition" at the infinity, which guarantees the well-posedness of…
This work develops a functional analytic framework for making computer assisted arguments involving transverse heteroclinic connecting orbits between hyperbolic periodic solutions of ordinary differential equations. We exploit a…
Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an…
A variety of problems in acoustic and electromagnetic scattering require the evaluation of impedance or layered media Green's functions. Given a point source located in an unbounded half-space or an infinitely extended layer, Sommerfeld and…
This paper studies an inverse boundary value problem for a semilinear Helmholtz equation with Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n\ge2$). The objective is to recover the unknown linear and…
Fourier transform-based methods enable accurate, dispersion-free simulations of time-domain scattering problems by evaluating solutions to the Helmholtz equation at a discrete set of frequencies sufficient to approximate the inverse Fourier…
We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an $N$-point configuration that maximizes the minimum value of its potential over a set $A$ in $p$-dimensional Euclidean space. This problem…
Chebyshev pseudospectral (PS) methods are reported to provide highly accurate solution using polynomial approximation. Use of polynomial basis functions in PS algorithms limits the formulation to univariate systems constraining it to tensor…
Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate…