A WEIGHTED NORM INEQUALITY FOR THETA(t)-TYPE OSCILLATORY SINGULAR INTEGRALS

The theta (t) -type oscillatory singular integral operators has been discussed.With the non- negative locally integrable weighted function , the weighted norm inequality of theta (t)-type oscillatory singular integral operators is proved, and the weighted function hasreplaced by action of Hardy-Littlewood maximal operators several times .     

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.     

Numerical solution of singular perturbation problems for the fourth-order elliptic differential equations

In this paper we construct the finite-difference scheme for the singularly perturbedboundary value problem for the fourth-order elliptic differential equation on the basis ofpaper[1],and prove the uniform convergence of this scheme with respect to the smallparameterεin the discrete energy norm,Finally,we give a numerical example.     

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.     

Coupling of high order multiplication perturbation method and reduction method for variable coefcient singular perturbation problems

Based on the precise integration method(PIM), a coupling technique of the high order multiplication perturbation method(HOMPM) and the reduction method is proposed to solve variable coefcient singularly perturbed two-point boundary value problems(TPBVPs) with one boundary layer. First, the inhomogeneous ordinary diferential equations(ODEs) are transformed into the homogeneous ODEs by variable coefcient dimensional expansion. Then, the whole interval is divided evenly, and the transfer matrix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efcient.     

Coupling of high order multiplication perturbation method and reduction method for variable coefficient singular perturbation problems

Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.     

REGULARIZATION OF NEARLY SINGULAR INTEGRALS IN THE BOUNDARY ELEMENT METHOD OF POTENTIAL PROBLEMS

IntroductionAsanimportantnumericalmethod ,BoundaryElementMethod (BEM)hasbeenappliedinmanyareas[1].However,theBEMhasthedifficultiesofcalculatingsingularintegralsatnodesonboundaryoratinteriorpointsveryclosetotheboundary .TheaccuracyoftheBEMdependsontheprecisionofthecalculatedvaluesofthesingularintegrals,toagreatdegree.Manyresearchersdevotethemselvestothetreatmentofthesingularintegrals[2～3],whicharereviewedindetailbyRef.[4] .Ageneralregularizationalgorithmofevaluatingthephysicalquantitiesa…     

各向异性位势问题边界元法中几乎奇异积分的解析算法

导出了一种解析积分算法,精确计算了二维各向异性位势问题边界元法中近边界点的几乎奇异积分。对线性单元,几乎奇异积分可用解析公式直接计算。对二次单元,可将其细分为几个线性单元,采用该解析公式间接近似计算。当内点离积分单元较远时,仍然保持常规高斯数值积分模式;而当内点离其较近时,高斯积分结果失效,采用该解析积分取代高斯数值积分。数值算例证明了该算法的有效性和精确性。二次元比线性元计算结果更精确。     

Direct perturbation method for reanalysis of matrix singular value decomposition

I.IntroductionMatrixsingularvaluedecomposition(SVD)isoneofthemostimportantandfundamentalcomputationalanalysistoolsformodernnumericallinearalgebra.ThematrixSVDhasverygoodnumericalstability,andsoitisthemostreliableandbeautifulnumericalanalysismethodinmanyth…     

A CLASS OF NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, first, the outer solution of the original problem was obtained. Secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer was constructed. Finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems was studied, and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation were discussed.     

Uniquenes of solution of field point of singular source outside-region-distribution method

The uniqueness of solution of field point, inside a convex region due to singular source(s) with kernel function decreasing with distance increasing, outsidergeion-distribution(s) such that the boundary condition expressed by the response of the source(s) is satisfied, is proved by using the condition of kernel function decreasing with distance increasing and an integral inequality. Examples of part ofthese singular sources such as Kelvin’s point force, Point-Ring-Couple (PRC) etc.are given. The proof of uniqueness of solution of field point in a twisted shaft of revolution due to PRC distribution is given as an example of application.     

On homoclinic bifurcation system in the presence of stochastic perturbation

By virture of the singular point theory for one-dimension diffusion process and the stochastic averaging approach of energy envelop, the bifurcation behavior of a homoclinic bifurcation system, which is in the presence of parametric white noise and is concealed behind a codimension two bifurcation point, is investigated in this paper. Supported by the National Science Foundation of China under Grant No. 19602016.     

周边嵌入弹性圆薄板非线性自由振动直接摄动解法

采用弹性圆薄板中心无量纲振幅和板厚与半径的比值为参数。将挠度、应力函数对半径的导数以及自由振动频率展开为双参数的幂级数。用直接摄动法获得各级递推线性偏微分方程。应用变分法求得各级递推方程的近似解，从而给出弹性圆薄板非线性自由振动频率的基本公式。     

Prediction of velocity distribution in straight open-channel flow with partial vegetation by singular perturbation method

A numerical analysis model based on two-dimensional shallow water differential equations is presented for straight open-channel flow with partial vegetation across the channel. Both the drag force acting on vegetation and the momentum exchange between the vegetation and non-vegetation zones are considered. The depth-averaged streamwise velocity is solved by the singular perturbation method, while the Reynolds stress is calculated based on the results of the streamwise velocity. Comparisons with the experimental data indicate that the accuracy of prediction is significantly improved by introducing a term for the secondary current in the model. A sensitivity analysis shows that a sound choice of the secondary current intensity coefficient is important for an accurate prediction of the depth-averaged streamwise velocity near the vegetation and non-vegetation interfaces, and the drag force coefficient is crucial for predictions in the vegetation zone.     

A numerical method for Cauchy principal value of singular integral under parametric transformation in BEM

A numerical method for the Cauchy principal value of the singular integral in BEM is developed using the concept of finite part integration under integral variable transformation. It is applied to the numerical integration on isoparametric element successfully, as shown in the examples in this paper.The project supported by National Natural Foundation of China.     

Denoising method based on singular spectrum analysis and its applications in calculation of maximal liapunov exponent

IntroductionSingularSpectrumAnalysis (SSA)asadataanalysismethodhasbeenusedforyearsindigitalsignalprocessing .BroomheadandKing[1]proposedtheapplicationofSSAindynamicalsystemstheories.Vautardetal.[2 ,3]studiedthetheoryandapplicationofSSAindetail.AnalgorithmbasedonSSAisproposedtodenoisechaoticdatainthispaper.Theessenceofthisalgorithmistochooseproperorderofempiricalorthogonalfunctions (EOFs)andprincipalcomponents (PCs)toreconstructthesignal.ThefirstalgorithmtoestimatethemaximalLiapunovex…     

Refined differential equations of deflections in axial symmetrical bending problems of spherical shell and their singular perturbation solutions

This paper deals with the research of accuracy of differential equations of deflections.The basic idea is as follows.Firstly,considering the boundary effect the meridianmidsurface displacement u=0,thus we derive the deflection differential equations;secondly we accurately prove that by use of the deflection differential equations or theoriginal differential equations the same inner forces solutions are obtained;finally,weaccurately prove that considering the boundary effect the meridian surface displacementu=0 is an exact solution.In this paper we give the singular perturbation solution of thedeflection differential equations.Finally we check the equilibrium condition and prove theinner forces solved by perturbation method and the outer load are fully equilibrated.Itshows that perturbation solution is accurate.On the other hand,it shows again that thedeflection differential equation is an exact equation.The features of the new differential equations are as follows:1.The accuracies of the new differentia     

Applications of the extrapolation method to the numerical solution of singular perturbation problems

In this paper we consider applications of extrapolation method to the numerical solution of singular perturbation problem for elliptic—parabolic equation in order to manifesting accuracy of approximations and estimate the order of accuracy. Concerning the uniform convergence in ref. [1], its proof is given in the appendix.     

Numerical solution of singular integral equations in stress concentration problems under longitudinal shear loading

Summary The paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknown functions are densities of body forces distributed in the longitudinal direction of an infinite body. First, four kinds of fundamental density functions are introduced to satisfy completely the boundary conditions for an elliptical boundary in the range 0≤φ _k ≤2π. To explain the idea of the fundamental densities, four kinds of equivalent auxiliary body force densities are defined in the range 0≤φ _k ≤π/2, and necessary conditions that the densities must satisfy are described. Then, four kinds of fundamental density functions are explained as sample functions to satisfy the necessary conditions. Next, the unknown functions of the body force densities are approximated by a linear combination of the fundamental density functions and weight functions, which are unknown. Calculations are carried out for several arrangements of elliptical holes. It is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions. Received 26 May 1998; accepted for publication 27 November 1998     

The perturbation solution of the large elastic curve of buckled bars and the singular perturbation method for its imperfect bifurcation problem

This paper presents the large deflection elastic curve of buckled bars through perturbation method, and the bifurcation diagrams including the influence of the imperfection at the base by using singular perturbation method of imperfect bifurcation theory. The physical meaning of the bifurcation diagrams is discussed.