which describes stationary reaction-diffusion phenomena but is often also called Helmholtz equation in certain communities. For example in meteorology, the early seminal papers [48, 56] led an entire community to call equations of the form (2) Helmholtz equations, see for example [14]. Even standard texts in applied mathematics now sometimes use the term Helmholtz equation for both (1) and (2),...

We show how a number of second order time-dependent partial differential equations can be solved numerically by reformulating them in terms of inhomogeneous modified Helmholtz equations. Using recently derived analytic particular solutions for these equations, the resulting boundary value problems can be reduced to solving homogeneous Helmholtz equations which can be done effectively using the ...

Abstract We study the unique solvability of discretized Helmholtz problem with Robin boundary conditions using a conforming Galerkin finite element method. Well-posedness discrete equations is typically investigated by applying compact perturbation argument to continuous so that `sufficiently rich' discretization results in small' and well-posedness inherited via Fredholm’s alternative. The qua...

The simplest parabolic equation of normal type with periodic boundary condition is considered, and the problem of stabilization to zero of its solution with arbitrary initial condition by starting control supported in a prescribed subset is investigated. This problem is reduced to one inequality for starting control, and the proof of this inequality is given. Introduction. This paper is devoted...

‎Radial basis functions (RBFs) are a powerful tool for approximating the solution of high-dimensional problems‎. ‎They are often referred to as a meshfree method and can be spectrally accurate‎. ‎In this paper, we analyze a new stable method for evaluating Gaussian radial basis function interpolants based on the eigenfunction expansion‎. ‎We develop our approach in two-dimensional spaces for so...

We demonstrate the existence of vector Helmholtz-Gauss (vHzG) and vector Laplace-Gauss beams that constitute two general families of localized vector beam solutions of the Maxwell equations in the paraxial approximation. The electromagnetic components are determined starting from the scalar solutions of the two-dimensional Helmholtz and Laplace equations, respectively. Special cases of the vHzG...

CMSC828D notes (adapted from material written with Nail Gumerov). Work in progress 1 Acoustic Waves 1.1 Barotropic Fluids The usual assumptions for acoustic problems are that acoustic waves are perturbations of the medium (fluid) density ρ (r,t), pressure p (r,t), and mass velocity, v (r,t), where t is time. It is also assumed that the medium is inviscid, and that perturbations are small, so th...

This paper1 gives a geometric description of functional spaces related to Domain Decomposition techniques for computing solutions of Laplace and Helmholtz equations. Understanding the geometric structure of these spaces leads to algorithms for solving the equations. It leads also to a new interpretation of classical algorithms, enhancing convergence. The algorithms are given and convergence is ...

we have developed a three level implicit method for solution of the helmholtz equation. using the cubic spline in space and nite dierence in time directions. the approach has been modi ed to drive numerov type nite dierence method. the method yield the tri- diagonal linear system of algebraic equations which can be solved by using a tri-diagonal solver. stability and error estimation of the...

CLASSICAL SCATTERING OF ELECTROMAGNETIC RADIATION Light is an electromagnetic disturbance which is propagated by vector wave equations which are readily derived from Maxwell's equations. The Helmholtz wave equation results from Maxwell's equations. The Helmholtz equation is linear; thus, superposition of solutions is allowed. Huygens' principle is that a point source of light will give rise to ...