In this talk I will discuss a method to prove the absence of critical points for the Helmholtz equation in 3D. The key element of the approach is the use of multiple frequencies in a fixed range, and the proof is based on the spectral analysis of the associated problem. This question is strictly connected with the Radó-Kneser-Choquet theorem, whose direct extension to the Helmholtz equation or ...

We have developed a three level implicit method for solution of the Helmholtz equation. Using the cubic spline in space and finite difference in time directions. The approach has been modied to drive Numerov type nite difference 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...

In this paper, we consider an unknown source problem for the modified Helmholtz equation. The Tikhonov regularization method in Hilbert scales is extended to deal with ill-posedness of the problem. An a priori strategy and an a posteriori choice rule have been present to obtain the regularization parameter and corresponding error estimates have been obtained. The smoothness parameter and the a ...

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary, in the case of the alternating iterative algorithm of Kozlov, Maz′ya and Fomin (1991) applied to Cauchy problems for the modified Helmholtz equation. A convergence proof of these relaxation methods is given, along with a stopping criterion. The num...

We propose two hybrid convolution quadrature based discretizations of the wave equation on interior domains with broadband Neumann boundary data or source terms. The method transforms time-domain problem into a series Helmholtz problems complex-valued wavenumbers, in which and solutions are connected to those original through

The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. W...

First passage times (FPTs) of two-dimensional Brownian motion have many applications in quantitative finance. However, despite various attempts since the 1960’s, there are few analytical solutions available. By solving a non-homogeneous modified Helmholtz equation in an infinite wedge, we find analytical solutions for the Laplace transforms of FPTs; these Laplace transforms can be inverted nume...

number of iterationsrequired to meet the convergencecriterion. the converged solutions from the previous step. This significantly reduces the interfacial boundaries, the initial estimates for the interfacial flux is given from scheme. Outside of the first time step where zero initial flux is assumed on all between subdomains are satisfied using a Schwarz Neumann-Neumam iteration method which is...

The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral representations are analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation (i.e., for generalized eigenfu...

The Cauchy problem for Helmholtz equation arises from inverse scattering problems. Specific backgrounds can be seen in the existing literature; we can refer to 1–6 and so forth. A number of numerical methods for stabilizing this problem are developed. Several boundary element methods combined with iterative, conjugate gradient, Tikhonov regularization, and singular value decomposition methods a...