- 标题
- 摘要
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Radial multipliers in solutions of the Helmholtz equations
摘要： We define radial multipliers using solutions of the Helmholtz equation, which depend on the radial coordinate, and we find the recurrence relations between them in the space of any dimension m > 1, in which the Helmholtz operator is defined. It is shown that the procedure of differentiation of these multipliers leads to a system of solutions of the Helmholtz equation, represented as products of the radial multipliers and harmonic polynomials. Theorems about the properties of radial multipliers and the structure of harmonic polynomials in the solutions of Helmholtz equation are given. These solutions constructed using radial multipliers and harmonic polynomials are proposed to be used in gradient elasticity for multi-layered domains with spherical and cylindrical boundaries, since they allow to present boundary conditions in explicit algebraic form.
关键词： Gradient elasticity,generalized Papkovich–Neuber representation,Helmholtz equation,radial multipliers,harmonic polynomials
更新于2025-09-23 15:23:52
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A limiting absorption principle for the Helmholtz equation with variable coefficients
摘要： We prove a limiting absorption principle for a generalized Helmholtz equation on an exterior domain with Dirichlet boundary conditions under a Sommerfeld radiation condition at infinity. The operator L is a second order elliptic operator with variable coefficients; the principal part is a small, long range perturbation of the Laplacian, while lower order terms can be singular and large. The main tool is a sharp uniform resolvent estimate, which has independent applications to the problem of embedded eigenvalues and to smoothing estimates for dispersive equations.
关键词： Smoothing estimates,limiting absorption principle,variable coefficients,Helmholtz equation
更新于2025-09-23 15:22:29
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A regularized approach evaluating origin intensity factor of singular boundary method for Helmholtz equation with high wavenumbers
摘要： Evaluation of the origin intensity factor of the singular boundary method for Helmholtz equation with high wavenumbers has been a difficult task for a long time. In this study, a regularized approach is provided to bypass this limitation. The core idea of the subtraction and adding-back technique is to substitute an artificially constructed general solution of the Helmholtz equation into the boundary integral equation or the hyper boundary integral equation to evaluate the non-singular expressions of the fundamental solutions at origin. The core difficulty is to derive the appropriate artificially constructed general solution. The regularized approach avoids the unstable inverse interpolation and has strict mathematical derivation process. Therefore, it is easy-to-program and free of mesh dependency. Numerical experiments show that the proposed technique can be used successfully to avoid singularity and hyper singularity difficulties encountered in the boundary element method and the singular boundary method.
关键词： Three-dimensional Helmholtz equation,Singularity and hyper singularity,Boundary element method,Origin intensity factor,Singular boundary method
更新于2025-09-23 15:22:29
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FEM and CIP-FEM for Helmholtz Equation with High Wave Number and Perfectly Matched Layer Truncation
摘要： The Helmholtz scattering problem with high wave number is truncated by the perfectly matched layer (PML) technique and then discretized by the linear continuous interior penalty–finite element method (CIP-FEM). It is proved that the truncated PML problem satisfies the inf-sup condition with inf-sup constant of order O(k?1). Stability and convergence of the truncated PML problem are discussed. In particular, the convergence rate is twice the previous result. The preasymptotic error estimates in the energy norm of the linear CIP-FEM as well as FEM are proved to be C1kh + C2k3h2 under the mesh condition that k3h2 is sufficiently small. Numerical tests are provided to illustrate the preasymptotic error estimates and show that the penalty parameter in the CIP-FEM may be tuned to reduce greatly the pollution error.
关键词： perfectly matched layer,FEM,CIP-FEM,Helmholtz equation with high wave number,wave-number-explicit estimates
更新于2025-09-23 15:22:29
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Comments on iterative schemes for high order compact discretizations to the exterior Helmholtz equation
摘要： We consider various formulations of higher order absorbing boundary conditions for the Helmholtz equation.
关键词： absorbing boundary conditions,Helmholtz equation
更新于2025-09-23 15:21:21
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Mixed Oblique Derivative Problem for the Helmholtz Equation in a Half-Disk
摘要： 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.
关键词： unique solvability,mixed oblique derivative problem,half-disk,Helmholtz equation
更新于2025-09-23 15:21:01
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Error analysis of PML-FEM approximations for the Helmholtz equation in waveguides
摘要： In this paper, we study ?nite element approximate solutions to the Helmholtz equation in waveguides by using a perfectly matched layer (PML). The PML is de?ned in terms of a piecewise linear coordinate stretching function with two parameters for absorbing propagating and evanescent components respectively, and truncated with a Neumann condition on an arti?cial boundary rather than a Dirichlet condition for cuto? modes that waveguides may allow. In the ?nite element analysis for the PML problem, we have to deal with two di?culties arising from the lack of full regularity of PML solutions and the anisotropic nature of the PML problem with, in particular, large PML damping parameters. Anisotropic ?nite element meshes in the PML regions depending on the damping parameters are used to handle anisotropy of the PML problem. As a main goal, we establish quasi-optimal a priori error estimates, that does not depend on anisotropy of the PML problem (when no cuto? mode is involved), including the exponentially convergent PML error with respect to the width and the strength of PML. The numerical experiments that con?rm the convergence analysis will be presented.
关键词： PML,Helmholtz equation,waveguide,?nite element method
更新于2025-09-23 15:19:57
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Dirichlet-to-Neumann boundary conditions for multiple scattering in waveguides
摘要： In this paper, we study a multiple Dirichlet-to-Neumann (MDtN) boundary condition for solving a time-harmonic multiple scattering problem governed by the Helmholtz equation in waveguides that include multiple obstacles, cavities or inhomogeneities with straight waveguides placed between them. The MDtN condition is derived by analyzing analytic solutions represented by Fourier series in the straight waveguides between obstacles, cavities or inhomogeneities. The proposed method is then to remove the straight waveguides between scatterers and impose the MDtN condition on artificial boundaries resulting from domain truncation. This numerical technique can allow a great reduction of computational efforts. The well-posedness of the reduced problem with the full MDtN condition and the reduced problem with truncated MDtN conditions are established. Also the exponential convergence of approximate solutions satisfying truncated MDtN conditions will be proved.
关键词： Helmholtz equation,Multiple scattering,Multiple Dirichlet-to-Neumann condition,Waveguide
更新于2025-09-19 17:13:59
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Mathematical Analysis of Surface Plasmon Resonance by a Nano-Gap in the Plasmonic Metal
摘要： We develop a mathematical theory for the excitation of surface plasmon resonance on an infinitely thick metallic slab with a nano-gap defect. Using layer potential techniques, we establish the well-posedness of the underlying scattering problem. We further obtain the asymptotic expansion of the scattering solution in order to characterize the leading-order term of the surface plasmonic waves, and derive sharp estimates for both the plasmonic and nonplasmonic parts of the solution. The explicit dependence of the surface plasmon resonance on the size of the nano-gap, and the real and imaginary parts of the metal dielectric constant, are given.
关键词： surface plasmon resonance,Helmholtz equation,subwavelength structure,integral equation
更新于2025-09-16 10:30:52
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A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets
摘要： In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is a combination of the local fractional Laplace transform (LFLT) and the homotopy perturbation method (HPM). The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.
关键词： local fractional homotopy perturbation method,local fractional Laplace transform,local fractional derivative operator,Helmholtz equation
更新于2025-09-16 10:30:52