Multifrequency Analysis for the Helmholtz Equation

This paper is a brief review of a new numerical method for the multifrequency analysis of the three‐dimensional Helmholtz equation. We describe the principles of this method which is based on the identity of the Fourier transform with respect to the wave number. Some numerical examples for the solution were presented at the oral session.     

On the Completeness of Sets of Solutions to the Helmholtz Equation

Completeness in L²(D) is established for sets of functions formedfrom solutions to the two-dimensional Helmholtz equation ina domain D. Each function is a linear combination of a solution(found by separation of variables) and its normal derivativeon D, so the sets may be used to solve impedance-type boundaryvalue problems. Sets that contain either regular Bessel functionsor singular Hankel functions are considered. Methods of proofare employed that provide alternatives to the conventional potential-theoreticapproaches. In the majority of cases, the domain of interestis bounded and simply connected. One completeness result fora bounded, doubly-connected domain is proved. In some circumstances,one of the methods leads to a mild but inessential eigenvaluerestriction.     

Parallel Solution of the Helmholtz Equation in a Multilayer Domain

We study time-harmonic wave propagation in layered, heterogeneous media. Solving this relatively complex application problem numerically is a challenging task. The full potential of algorithms, parallel programming models and computer architectures must be exploited. Our aim is to give a broad perspective on the various considerations that come into play. The basic parts of our algorithms consist of finite difference discretizations, domain decomposition and preconditioned iterative methods. We present two serial algorithms with different properties. Then, we discuss parallelization strategies using a local memory model, a shared memory model, or a combination of the two. The numerical experiments highlight the differences between the approaches and show results for three different combinations of algorithm and computer architecture that lead to viable solution methods.     

Oblique Derivative Problem for the Helmholtz Equation in a Disk

We show that the oblique derivative problem for the Helmholtz equation in a disk is uniquely solvable under certain restrictions on the parameter.     

Singularities and the Rayleigh Hypothesis for Solutions to the Helmholtz Equation

Possible and actual singularities in the analytic continuationof the solution to the Dirichlet problem for the two-dimensionalHelmholtz equation are studied in order to investigate the Rayleighhypothesis of scattering theory. The procedure uses a representationfor the solution to the analytic Cauchy problem and relatesone type of singularity in the Schwarz function associated withthe boundary curve to singularities in the solution. The resultsprovide confiration of criteria for the validity of the Rayleighhypothesis that have been given by van den Berg and Fokkema.     

Radiation Conditions for the Magnetic Helmholtz Equation

Mathematical Notes - It is proved that, to select a uniqueness class for the magnetic Helmholtz equation, it suffices to impose radiation conditions weaker than the Ikebe–Saito conditions....     

On Vector Helmholtz Equation with a Coupling Boundary Condition

The Helmholtz equation is sometimes supplemented by conditions that include the specification of the boundary value of the divergence of the unknown.In this paper, we study the vector Helmholtz problem in domains of both C~(1,1)and Lipschitz.We es- tablish a rigorous variational analysis such as equivalence,existence and uniqueness. And we propose finite element approximations based on the uncoupled solutions.Fi- nally we present a convergence analysis and error estimates.     

Legendre Gauss Spectral Collocation for the Helmholtz Equation on a Rectangle

A spectral collocation method with collocation at the Legendre Gauss points is discussed for solving the Helmholtz equation –u+(x,y)u=f(x,y) on a rectangle with the solution u subject to inhomogeneous Robin boundary conditions. The convergence analysis of the method is given in the case of u satisfying Dirichlet boundary conditions. A matrix decomposition algorithm is developed for the solution of the collocation problem in the case the coefficient (x,y) is a constant. This algorithm is then used in conjunction with the preconditioned conjugate gradient method for the solution of the spectral collocation problem with the variable coefficient (x,y).     

Mixed Oblique Derivative Problem for the Helmholtz Equation in a Half-Disk

A mixed oblique derivative boundary value problem is considered for the Helmholtz equation in a half-disk. We prove the unique solvability of this problem for sufficiently large values of the parameter occurring in the equation, the leading part of the inverse operator being constructed explicitly.     

Mixed Boundary Value Problems for the Helmholtz Equation in a Quadrant

The main objective is the study of a class of boundary value problems in weak formulation where two boundary conditions are given on the half-lines bordering the first quadrant that contain impedance data and oblique derivatives. The associated operators are reduced by matricial coupling relations to certain boundary pseudodifferential operators which can be analyzed in detail. Results are: Fredholm criteria, explicit construction of generalized inverses in Bessel potential spaces, eventually after normalization, and regularity results. Particular interest is devoted to the impedance problem and to the oblique derivative problem, as well.     

Optimization of a Finite-Difference Scheme for Numerical Solution of the Helmholtz Equation

Computational Mathematics and Mathematical Physics - In this article, we propose an optimization method for a difference scheme for the numerical solution of the Helmholtz equation, applicable for...     

一个新的与Hadamard不等式相关的映射

对于一个最近发表的凸函数的Hadamard不等式的推广形式的不等式,本文引进了—个与这个不等式相关的的映射,从而给出了该不等式连续的加细.同时提及了它的某些应用.     

Inverse Scattering for the 1-D Helmholtz Equation

We prove a uniqueness result for Nevanlinna functions. and this result is then used to give an elementary proof of the uniqueness in the inverse scattering problem for the equation \( u'' + \frac{k^2}{c^2}u=0 \) on \({\mathbb R}\). Here \(c\) is a real positive measurable function that is bounded from below by a positive constant, and is close to \(1\) at \(\pm \infty \).     

Limiting Absorption Principle for the Dissipative Helmholtz Equation

Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the uniform resolvent estimate for the high-frequency Helmholtz equation when trapped classical trajectories meet the region where the absorption coefficient is non-zero. We also give the resolvent estimate in Besov spaces.     

Mean Value Integral Equations and the Helmholtz Equation

This paper considers the mean-value functional equation $$ f(x)={1 \over \mu({B_{R}(x))}} \int_{B_{R}(x)}{f(t)dt} $$ where B_r(x) is a ball of radius R centred at x ∈ R ⁿ, and μ represents Lebesgue measure. By using the spectral synthesis, we find that the space of continuous functions satisfying this equation (in the case n = 1,2) is characterized in terms of solutions of the Helmholtz equation.     

Properties of a Semi-discrete Approximation to the Beam Equation

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx = 0, 0 < x < L, t > 0, where it is assumed that w and p are positive on the interval[O, L], is approximated by using the method of straight lines.The resulting approximation is a linear system of differentialequations with coefficient matrix S. The matrix S is studiedunder very general boundary conditions which result in a conservativesystem. In all cases the matrix S is either an oscillation matrixor possesses nearly all the properties of an oscillation matrix.     

The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L_∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5]. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.