THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.     

SINGULAR INTEGRAL EQUATIONS ALONG AN OPEN ARC WITH SOLUTIONS HAVING SINGULARITIES OF HIGHER ORDER

In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.     

THE COUPLING OF NATURAL BOUNDARY ELEMENT AND FINITE ELEMENT METHOD FOR 2D HYPERBOLIC EQUATIONS

In this paper, we investigate the coupling of natural boundary element and finite element methods of exterior initial boundary value problems for hyperbolic equations. The governing equation is first discretized in time, leading to a time-step scheme, where an exterior elliptic problem has to be solved in each time step. Second, a circular artificial boundary FR consisting of a circle of radius R is introduced, the original problem in an unbounded domain is transformed into the nonlocal boundary value problem in abounded subdomain. And the natural integral equation and the Poisson integral formula are obtained in the infinite domainΩ2 outside circle of radius R. The coupled variational formulation is given. Only the function itself, not its normal derivative at artificial boundary ΓR, appears in the variational equation, so that the unknown numbers are reducedand the boundary element stiffness matrix has a few different elements. Such a coupled method is superior to the one based on direct boundary element method. This paper discusses finite element discretization for variational problem and its corresponding numerical technique, and the convergence for the numerical solutions. Finally, the numerical example is presented to illustrate feasibility and efficiency of this method.     

THE HP-VERSION OF BEM - FAST CONVERGENCE,ADAPTIVITY AND EFFICIENT PRECONDITIONING＊

In this survey paper we report on recent developments of the hp-version of the boundary element method （BEM）. As model problems we consider weakly singular and hypersingular integral equations of the first kind on a planar, open surface. We show that the Galerkin solutions （computed with the hp-version on geometric meshes） converge exponentially fast towards the exact solutions of the integral equations. An hp-adaptive algorithm is given and the implementation of the hp-version BEM is discussed together with the choice of efficient preconditioners for the ill-conditioned boundary element stiffness matrices. We also comment on the use of the hp-version BEM for solving Signorini contact problems in linear elasticity where the contact conditions are enforced only on the discrete set of Gauss-Lobatto points. Numerical results are presented which underline the theoretical results.     

Finite Integral Transform Solutions for Free Vibrations of Rectangular Thin Plates With Mixed Boundary Constraints北大核心CSCD

Analytical solutions, with unique research value, can serve as benchmarks for empirical formulas and numerical methods, a tool for rapid parameter analysis and optimization, and a theoretical basis for experimental designs. Conventional analytical methods, e.g., the Lévy solution method, are only applicable to mechanical problems of plates and shells with opposite simply-supported edges, which, however, may fail to obtain analytical solutions for the issues with complex boundary constraints. In recent years, the finite integral transform method for plate and shell problems was developed to deal with non-Lévy-type plates and shells, but it is still infeasible to solve the mixed boundary constrains-induced complex boundary value problems of higher-order partial differential equations. Herein, for the first time, the finite integral transform method was combined with the sub-domain decomposition technique to solve the free vibrations of rectangular thin plates with mixed boundary constraints. The rectangular plate was first divided into 2 sub-domains according to the mixed boundary constraints, and the 2 sub-domains were solved analytically with the finite integral transform method. Finally, the continuity conditions were introduced to obtain the analytical solution of the original problem. Based on the side spot-welded cantilever plates commonly used in engineering, the free vibration problem of a rectangular thin plate with 1 edge subjected to clamped-simply supported constraints and the other 3 edges free, was analyzed. The obtained natural frequencies and mode shapes are in good agreement with those from the finite element method as well as the solutions in literature, thus verifying the accuracy of the proposed method. The solution procedure of the finite integral transform method can be implemented based on the governing equations without any assumption of the solution form. Therefore, this strict analytical method is widely applicable to complex boundary value problems of higher-order partial differential equations for such mechanical problems of plates and shells. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

RANDOM SINGULAR INTEGRAL OF RANDOM PROCESS WITH SECOND ORDER MOMENT

This paper discussses the random singular integral of random process with second order moment, establishes the concepts of the random singular integral and proves that it‘s a linear bounded operator of space H^α(L)(rn, s). Then Plemelj formula and some other properties for random singular integral are proved.     

NONEXISTENCE AND EXISTENCE OF POSITIVE SOLUTIONS FOR 2nTH-ORDER SINGULAR SUPERLINEAR PROBLEMS WITH STRUM-LIOUVILLE BOUNDARY CONDITIONS

This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C 2 n-1 [0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.     

PLANE ELASTIC PROBLEMS OF DIFFERENT MEDIA WITH CRACKS ON THE INTERFACE

The equllibrium problem for the infinite elastic plane consisting of two different media with many cracks on the interface is discussed.It is transferred to a boundary value problem for analytic functions and then further reduced to a singular integral equation,the unique solvability and an effective method of solution for which are eatablished.A practical example in applications is illustrated,the solution of which is obtained in closed form.     

无界区域上Cauchy型奇异积分算子的性质

On the basis of introducing the modified Cauchy kernel, we discuss the Hoelder continuity of the Cauchy-type singular integral operator on unbounded domains for regular functions by dividing into the following three cases： two points are on the boundary of region; one point is on the boundary and another point is in the interior（or exterior） of the region; two points are in the interior （or exterior） of the region.     

ON DIRECT METHOD OF SOLUTION FOR A CLASS OF SINGULAR INTEGRAL EQUATIONS

In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equations with certain analytic inputs. They obtain both the conditions of solvability and the solutions in closed form. It is noteworthy that the method is different from the classical one that is due to Lu.     

Natural Boundary Integral Method and its New Development

In this paper, the natural boundary integral method, and some related methods, including coupling method of the natural boundary elements and finite elements, which is also called DtN method or the method with exact artificial boundary conditions, domain decomposition methods based on the natural boundary reduction, and the adaptive boundary element method with hyper-singular a posteriori error estimates, are discussed.     

ON A CLASS OF SINGULAR INTEGRAL EQUATIONS WITH T ANSLATIONS

In this paper,singular integral equations with upper and lower translations arediscussed and the equivalence for solving them in and are shown.Equations with asingle pair of upper and lower translations are studied in detail.Their solutions as wellas the conditions of solvability are obtained.The method used consists of transferringthem to boundary value problems of analytic functions in upper and lower half-planesand then solving the latter.     

SINGULAR INTEGRAL EQUATIONS WITH COSECANT KERNEL IN SOLUTIONS WITH SINGULARITIES OF ORDER ONE

In this article, periodic Riemann boundary value problem with period 2aπalong closed smooth contours is discussed, and then singular integral equation with kernel csc t-t0/a along closed smooth contours restricted in the strip 0< Rez相似文献   

ERROR ANALYSIS FOR A FAST NUMERICAL METHOD TO A BOUNDARY INTEGRAL EQUATION OF THE FIRST KIND

For two-dimensional boundary integral equations of the first kind with logarithmic kernels, the use of the conventional boundary element methods gives linear systems with dense matrix. In a recent work [J. Comput. Math., 22 （2004）, pp. 287-298], it is demonstrated that the dense matrix can be replaced by a sparse one if appropriate graded meshes are used in the quadrature rules. The numerical experiments also indicate that the proposed numerical methods require less computational time than the conventional ones while the formal rate of convergence can be preserved. The purpose of this work is to establish a stability and convergence theory for this fast numerical method. The stability analysis depends on a decomposition of the coefficient matrix for the collocation equation. The formal orders of convergence observed in the numerical experiments are proved rigorously.     

Three Dimensional Interface Problems for Elliptic Equations

The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension.The set of singular points consists of some singular lines and some isolated singular points.It is proved that near a singular line or a singular point,each weak solution can be decomposed into two parts,a singular part and a regular part.The singular parts are some finite sum of particular solutions to some simpler equations,and the regular parts are bounded in some norms,which are slightly weaker than that in the Sobolev space H~2.     

Boundary properties for several singular integral operators in real Clifford analysis

On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Hlder continuous property of some singular integral operators with one integral variable. Then we divide a singular integral operator with two variables into three parts and prove its Hlder continuous property on the boundary.     

On the Unique Continuation Properties for Elliptic Operators with Singular Potentials

Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato-Fefferman-Phong＇s class in Lipschitz domains. An elementary proof of the doubling property for u^2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the Bp weight properties for the solution u near the boundary.     

A COLLOCATION METHOD FOR THE CONDUCTIVITY PROBLEM WITH DISCONTINUOUS COEFFICIENT

In this paper, a new collocation BEM for the Robin boundary value problem of the conductivity equation ▽(γ▽u) = 0 is discussed, where the 7 is a piecewise constant function. By the integral representation formula of the solution of the conductivity equation on the boundary and interface, the boundary integral equations are obtained. We discuss the properties of these integral equations and propose a collocation method for solving these boundary integral equations. Both the theoretical analysis and the error analysis are presented and a numerical example is given.     

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTION FOR NONLINEAR SINGULAR 2mth-ORDER CONTINUOUS AND DISCRETE LIDSTONE BOUNDARY VALUE PROBLEMS

By fixed point theorem of a mixed monotone operators, we study Lidstone boundary value problems to nonlinear singular 2mth-order differential and difference equations, and provide sufficient conditions for the existence and uniqueness of positive solution to Lidstone boundary value problem for 2mth-order ordinary differential equations and 2mth-order difference equations. The nonlinear term in the differential and difference equation may be singular.     

Singular integral on bounded strictly pseudoconvex domain

Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.