On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function of four variables. In addition, the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found.     

Nonreflecting boundary condition for the Helmholtz equation

To solve the Helmholtz equation in an infinite three-dimensional domain a spherical artificial boundary is introduced to restrict the computational domain Ω. To determine the nonreflecting boundary condition on ∂Ω, we start with a finite number of spherical harmonics for the Helmholtz equation. With a precise choice of (primary) nodes on the sphere, the theorem on Gauss-Jordan quadrature establishes the discrete orthogonality of the spherical harmonics when summed over these nodes. An approximate nonreflecting boundary condition for the Helmholtz equation follows readily upon solving the exterior Dirichlet problem. The accuracy of the boundary condition is determined using a point source, and the computational results are presented for the scattering of a wave from a sphere.     

Suppression of electroosmotic flow by polyampholyte brush

Molecular dynamics simulations are conducted to investigate the suppression of electroosmotic flow by grafting polyampholyte brushes onto two parallel channel walls. The effects of grafting density and charge distribution of polyampholyte brushes on the electroosmotic flow velocity, salt ion distribution, and conformational characteristics of grafted brushes are studied in detail. Simulation results indicate that increasing the grafting density induces a stronger suppression of electroosmotic flow. The flow velocity is significantly influenced by the different charge distributions of polyampholyte brushes. In addition, an important flow phenomenon we have found is that the flow velocity profile shows a valley at the center of the channel. These results reveal that the flow velocity is dependent not only on the conformation of the polyampholyte brushes but also on the anion and cation distributions. The hierarchical distribution of salt ions is caused by the special properties of polyampholyte brushes.     

A compression technique for the boundary integral equation reduced from Helmholtz equation

Applying the trigonometric wavelets and the multiscale Galerkin method, we investigate the numerical solution of the boundary integral equation reduced from the exterior Dirichlet problem of Helmholtz equation by the potential theory. Consequently, we obtain a matrix compression strategy, which leads us to a fast algorithm. Our truncated treatment is simple, the computational complexity and the condition number of the truncated coefficient matrix are bounded by a constant. Furthermore, the entries of the stiffness matrix can be evaluated from the Fourier coefficients of the kernel of the boundary integral equation. Examples given for demonstrating our numerical method shorten the runtime obviously.     

A numerical method for transmission problems for the Helmholtz equation

This method consists in decoupling the transmission problem into two boundary value problems, which can be solved separately by well known procedures. A convergence proof is given with the help of the integral equation method and convergence results on projection methods.     

A mollification method for a Cauchy problem for the Helmholtz equation

The Cauchy problem for the Helmholtz equation is considered. This problem is severely ill-posed, that is, the solution does not depend continuously on the data. To solve the problem numerically a mollification method is proposed. Convergence on error estimates between the exact solution and its approximation are obtained. Some numerical examples are given to show that the method works effectively.     

基于Poisson-Nernst-Planck模型的微通道电渗流数值模拟

采用由电解质溶液离子输运Nernst-Planck方程、流体运动Navier-Stokes方程和电场Possion方程建立的Possion-Nernst-Planck模型,应用有限元分析方法研究二维光滑微通道电渗流输运特性和离子分布。对比分别基于Possion-Nernst-Planck模型和Poisson-Boltzmann模型数值模拟结果,结果表明:Possion-Nernst-Planck模型能更准确地模拟计算微通道中的电渗流输运特性和离子分布。     

Two efficient methods for a multifrequency solution of the Helmholtz equation

Abstact In this paper, two efficient methods for the multifrequency analysis for the Helmholtz equation are compared. The first approach is based on the direct collocation method with a subsequent approximation of the matrix by the Adaptive Cross Approximation method. Using the so-called Fourier scheme, the elements of the matrices for a series of frequencies can be computed efficiently. In the second approach an indirect Galerkin type method with piecewise linear ansatz and test functions is combined with a special source simulation technique. This combination allows a rather accurate and systematic approximation of acoustical results in major parts of the considered frequency range, leading to a significant reduction of the computer time needed to calculate complete frequency spectra. A representative example demonstrates how the two proposed approaches can be used. Both procedures turn out to be very promising steps towards a more efficient calculation of complex sound radiation problems. Communicated by: O. Steinbach     

Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation

In this paper, the Cauchy problem for the modified Helmholtz equation in a rectangular domain is investigated. We use a quasi-reversibility method and a truncation method to solve it and present convergence estimates under two different a priori boundedness assumptions for the exact solution. The numerical results show that our proposed numerical methods work effectively.     

Convergence analysis of Schwarz methods without overlap for the Helmholtz equation

In this paper, the continuous and discrete optimal transmission conditions for the Schwarz algorithm without overlap for the Helmholtz equation are studied. Since such transmission conditions lead to non-local operators, they are approximated through two different approaches. The first approach, called optimized, consists of an approximation of the optimal continuous transmission conditions with partial differential operators, which are then optimized for efficiency. The second approach, called approximated, is based on pure algebraic operations performed on the optimal discrete transmission conditions. After demonstrating the optimal convergence properties of the Schwarz algorithm new numerical investigations are performed on a wide range of unstructured meshes and arbitrary mesh partitioning with cross points. Numerical results illustrate for the first time the effectiveness, robustness and comparative performance of the optimized and approximated Schwarz methods on a model problem and on industrial problems.     

A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation

In this paper, a Cauchy problem for the Helmholtz equation is considered. It is known that such a problem is severely ill-posed, i.e. the solution does not depend continuously on the given Cauchy data. We propose a quasi-reversibility regularization method to solve it. Convergence estimates are established under two different a priori assumptions for an exact solution. Numerical results obtained by two different schemes show that our proposed methods work well.     

Ultrafast measurement of transient electroosmotic flow in microfluidics

We present a non-intrusive molecular dye based method, i.e., laser-induced fluorescence photobleaching anemometer (LIFPA), to significantly increase temporal resolution (TR) for velocity measurement of fast transient electrokinetic flows. To our knowledge, the TR has been for the first time achieved to 5–10 μs, about 100 times better than that published from state-of-the-art micro particle image velocimetry (μPIV), which is currently the most widely used velocimetry in the microfluidics community. The new method provides us with new opportunities to study experimentally the fundamental phenomena of unsteady electrokinetics (EK) and to validate relevant theoretical models. One application of the new method is demonstrated by measuring the rise time of DC electroosmotic flows (EOFs) in a microcapillary of 10 μm in diameter.     

An optimal compact sixth-order finite difference scheme for the Helmholtz equation

In this paper, we present an optimal compact finite difference scheme for solving the 2D Helmholtz equation. A convergence analysis is given to show that the scheme is sixth-order in accuracy. Based on minimizing the numerical dispersion, a refined optimization rule for choosing the scheme’s weight parameters is proposed. Numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters.     

Three-dimensional eigenvalue analysis of the Helmholtz equation by multiple reciprocity boundary element method

The eigenvalue of the three-dimensional Helmholtz equation is determined efficiently by extending the previously developed method for the two-dimensional problem. Boundary integral equation is formulated in the realm of the multiple reciprocity method, using higher order fundamental solutions for the Laplace equation; yielding polynomial coefficient matrices in terms of unknown wavenumber (eigenvalue). The Newton iteration method with the help of LU decomposition is employed to search eigenvalue, which can reduce the computational task significantly.     

A coordinate transformation approach for efficient repeated solution of Helmholtz equation pertaining to obstacle scattering by shape deformations

A computational model is developed for efficient solutions of electromagnetic scattering from obstacles having random surface deformations or irregularities (such as roughness or randomly-positioned bump on the surface), by combining the Monte Carlo method with the principles of transformation electromagnetics in the context of finite element method. In conventional implementation of the Monte Carlo technique in such problems, a set of random rough surfaces is defined from a given probability distribution; a mesh is generated anew for each surface realization; and the problem is solved for each surface. Hence, this repeated mesh generation process places a heavy burden on CPU time. In the proposed approach, a single mesh is created assuming smooth surface, and a transformation medium is designed on the smooth surface of the object. Constitutive parameters of the medium are obtained by the coordinate transformation technique combined with the form-invariance property of Maxwell’s equations. At each surface realization, only the material parameters are modified according to the geometry of the deformed surface, thereby avoiding repeated mesh generation process. In this way, a simple, single and uniform mesh is employed; and CPU time is reduced to a great extent. The technique is demonstrated via various finite element simulations for the solution of two-dimensional, Helmholtz-type and transverse magnetic scattering problems.     

Nonexistence of solutions to the Riccati equation

The Riccati equation does not necessarily give a positive definite solution when the plant has finite escape time in the time interval considered.     

Almost analytic representation for the solution of the differentialmatrix Riccati equation

An almost analytic representation is presented for the solution of the nonhomogeneous and homogeneous, time-invariant and time-variant differential matrix Riccati equation. This representation gives a new insight into the behavior of the optimal estimator on a finite time interval. For the time-invariant case this solution is a basis for an `exact/true' iterative algorithm. For the time-invariant case an approximate iterative algorithm is derived     

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.