A posterior error analysis for the nonconforming discretization of Stokes eigenvalue problem

In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.     

FINITE ELEMENT AND DISCONTINUOUS GALERKIN METHOD FOR STOCHASTIC HELMHOLTZ EQUATION IN TWO- AND THREE-DIMENSIONS

In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d （d = 2, 3）. Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.     

On Generalized Nonlinear Burgers＇ Equation

In this work, we study the similarity analysis of the generalized nonlinear Burgers＇ equation having diffusivity and viscosity as a general functions of the particle velocity. We perform the symmetry analysis and exhibit exact solutions for various forms of the diffusivity and viscosity.     

BIFURCATION OF CUBIC INTEGRABLE SYSTEM UNDER CUBIC PERTURBATION

In this paper, we consider the bifurcation for a class of cubic integrable system under cubic perturbation. Using bifurcation theory and qualitative analysis, we obtain a complete bifurcation diagram of the system in a neighbourhood of the origin for parameter plane.     

QUALITATIVE ANALYSIS ON A CLASS OF ECOSYSTEM

In this paper, we give complete qualitative analysis on a class of biological system and prove the nonexistence, existence and uniqueness of a limit cycle for the system.     

Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation

This paper discusses convergence and complexity of arbitrary,but fixed,order adaptive mixed element methods for the Poisson equation in two and three dimensions.The two main ingredients in the analysis,namely the quasi-orthogonality and the discrete reliability,are achieved by use of a discrete Helmholtz decomposition and a discrete inf-sup condition.The adaptive algorithms are shown to be contractive for the sum of the error of flux in L2-norm and the scaled error estimator after each step of mesh refinement and to be quasi-optimal with respect to the number of elements of underlying partitions.The methods do not require a separate treatment for the data oscillation.     

Convergence analysis of Oja’s iteration for solving online PCA with nonzero-mean samples

Principal component analysis(PCA)is one of the most popular multivariate data analysis techniques for dimension reduction and data mining,and is widely used in many fields ranging from industry and biology to finance and social development.When working on big data,it is of great necessity to consider the online version of PCA,in which only a small subset of samples could be stored.To handle the online PCA problem,Oja(1982)presented the stochastic power method under the assumption of zero-mean samples,and there have been lots of theoretical analysis and modified versions of this method in recent years.However,a common circumstance where the samples have nonzero mean is seldom studied.In this paper,we derive the convergence rate of a nonzero-mean version of Oja’s algorithm with diminishing stepsizes.In the analysis,we succeed in handling the dependency between each iteration,which is caused by the updated mean term for data centering.Furthermore,we verify the theoretical results by several numerical tests on both artificial and real datasets.Our work offers a way to deal with the top-1 online PCA when the mean of the given data is unknown.     

A Sampling Method for Solving Inverse Scattering Problems with a Locally Perturbed Half Plane

In this paper,we will consider the scattering of time—harmonic electromagnetic waves by aninfinitely long cylinder with a perfectly conducting boundary.The problem is modelled by aDirichlet problem for the Helmholtz equation.Adopting Cartesion coordinates(Xl,X2,X3)in础we assume that the boundary is the XlX3一plane with the perturbation independent of z3and local with respect to X1.Furthermore,we assume that the elect:tic and magnetic fieldsare invariant with respect to X3 and we restrict …     

A New Sparse Recovery Method for the Inverse Acoustic Scattering Problem

Based on sparse information recovery,we develop a new method for locating multiple multiscale acoustic scatterers.Firstly,with the prior information of the scatterers’shape,we reformulate the location identification problem into a sparse information recovery model which brought the power of sparse recovery method into this type of inverse scattering problems.Specifically,the new model can advance the judgment of the existence of alternative scatterers and,in the meantime,conclude the number and locating of each existing scatterers.Secondly,as well known,the core model(l0-minimization)in sparse information recovery is an NP-hard problem.According to the characteristics of the proposed sparse model,we present a new substitute method and give a detailed theoretical analysis of the new substitute model.Relying on the properties of the new model,we construct a basic algorithm and an improved one.Finally,we verify the validity of the proposed method through two numerical experiments.     

Instability of Solitary Waves in Nonlinear Composite Media

In this paper,we investigate a class of Hamiltonian systems arising in nonlinear composite media.By detailed analysis and computation we obtain a decaying estimates on the semigroup and prove the orbitalinstability of two families of explicit solitary wave solutions (slow family in anisotropic case and solitary wavesin isotropic case),which theoretically verify the related guess and numerical results.     

Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions

In this paper, we propose complete radiation boundary conditions (CRBCs) for solutions of the convected Helmholtz equation with a uniform mean flow in a waveguide. We first study CRBCs for the Helmholtz equation in a waveguide. Noting that the convected Helmholtz equation is associated with the Helmholtz equation via the Prandtl–Glauert transformation, CRBCs for the convected Helmholtz equation is derived from CRBCs for the Helmholtz equation. We analyse well-posedness and convergence of approximate solutions satisfying CRBCs for the convected Helmholtz equation. In addition, simple numerical experiments will be presented to confirm the theoretical results.     

Integral representations in quaternionic analysis related to the helmholtz operator

As the Laplacian also the Helmholtz operator can be factorized in Clifford analysis. Using the fundamental solution of the Helmholtz equation, a fundamental solution can be constructed for the factors of the Helmholtz operator. These are used in the case of quaternions to prove Cauchy–Pompeiu type representation formulas in terms of powers of the factors as well as in terms of powers of the Helmholtz operator.     

Two-Dimensional Helmholtz Spaces

We study two-dimensional manifolds whose infinitesimally small neighborhoods have the structures of Helmholtz planes. We deal with the main objects related to Helmholtz spaces: in particular, we determine the metric function f and introduce the concept of quasimetric connection and geodesic. For some Helmholtz spaces we prove the existence of isothermal coordinates.     

Some analytical and numerical investigation of a family of fractional-order Helmholtz equations in two space dimensions

In this article, we aim at solving a family of two-dimensional fractional-order Helmholtz equations by using the Laplace-Adomian Decomposition Method (LADM). The fractional-order derivatives, which we use in this investigation, follows the Liouville-Caputo definition. Our results based upon the LADM are obtained in series form that helps us in analyzing the analytical solutions of the fractional-order Helmholtz equations considered here. For illustration and verification of the analytical procedure using the LADM, several numerical examples and graphical representations are presented for the analytical solution of the fractional-order Helmholtz equations. The mathematical analytic procedure, which we have used here, has shown that the LADM is a fairly accurate and computable method for the solution of problems involving fractional-order Helmholtz equations in two dimensions. In an analogous manner, one can apply the LADM for finding the analytical solution of other classes of fractional-order partial differential equations.     

The uniqueness and existence of solutions for the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation

In this paper, we study the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation, allowing the presence of guided waves. Our assumptions on the perturbed and source terms are too few. On the basis of the Green's function for the 3D homogeneous Helmholtz equation in a step‐index waveguide without perturbation, we introduce a generalized (out‐going) Sommerfeld–Rellich radiation condition, and then we prove the uniqueness and existence of solutions for the studied 3D Helmholtz equation satisfying our radiation condition. Copyright © 2012 John Wiley & Sons, Ltd.     

关于Helmholtz外问题的边界积分方程解的唯一性问题

本文用能量分析的观点探讨了用边界积分方程描述Helmholtz外问题时,解的唯一性不能保持的原因.文中证明了,当利用积分方程来描述问题时,实际上将无穷远处的Sommerfeld条件改成了既适合于外向波(辐射波),又适合于内向波(吸收波),即整个系统的能量保持守恒.并根据此观点解释了保持唯一性的算法.     

半圆域内的二维线性椭圆偏微分方程

本文研究了半圆域内的二维线性椭圆偏微分方程.利用Fokas提出的求解凸多边形区域内的线性椭圆偏微分方程的变换方法,我们改进了这个方法来研究半圆域内Laplace方程,修改Helmholtz方程和Helmholtz方程的解,并且导出了这些方程解的积分表达式,讨论了Helmholtz方程的广义Dirichlet到Neumann映射.     

Multi‐periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n‐torus

In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus, the solutions can be expressed as multi‐periodic eigensolutions to the Dirac operator associated with a complex parameter λ∈?. Physically, these can be interpreted as the solutions to the time‐harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green‐type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n‐torus. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical evaluation of the two-dimensional partition of unity boundary integrals for Helmholtz problems

There has been considerable attention given in recent years to the problem of extending finite and boundary element-based analysis of Helmholtz problems to higher frequencies. One approach is the Partition of Unity Method, which has been applied successfully to boundary integral solutions of Helmholtz problems, providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy. These benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions. In this paper we adapt the numerical steepest descent method to evaluate these integrals for two-dimensional problems. The approach is successful in reducing the computational effort for most integrals encountered. The paper includes some numerical features that are important for successful practical implementation of the algorithm.     

Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green’s formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.