Null‐field approach for Laplace problems with circular boundaries using degenerate kernels

In this article, a semi‐analytical method for solving the Laplace problems with circular boundaries using the null‐field integral equation is proposed. The main gain of using the degenerate kernels is to avoid calculating the principal values. To fully utilize the geometry of circular boundary, degenerate kernels for the fundamental solution and Fourier series for boundary densities are incorporated into the null‐field integral equation. An adaptive observer system is considered to fully employ the property of degenerate kernels in the polar coordinates. A linear algebraic system is obtained without boundary discretization. By matching the boundary condition, the unknown coefficients can be determined. The present method can be seen as one kind of semianalytical approaches since error only attributes to the truncated Fourier series. For the eccentric case, vector decomposition technique for the normal and tangential directions is carefully considered in implementing the hypersingular equation in mathematical essence although we transform it to summability to divergent series. The five advantages, well‐posed linear algebraic system, principal value free, elimination of boundary‐layer effect, exponential convergence, and mesh free, are achieved. Several examples involving infinite, half‐plane, and bounded domains with circular boundaries are given to demonstrate the validity of the proposed method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The Poisson integral and Green’s function for one strongly elliptic system of equations in a circle and an ellipse

For a strongly elliptic system of second-order equations of a special form, formulas for the Poisson integral and Green’s function in a circle and an ellipse are obtained. The operator under consideration is represented by the sum of the Laplacian and a residual part with a small parameter, and the solution to the Dirichlet problem is found in the form of a series in powers of this parameter. The Poisson formula is obtained by the summation of this series.     

Center conditions and bifurcation of limit cycles at degenerate singular points in a quintic polynomial differential system

The center problem and bifurcation of limit cycles for degenerate singular points are far to be solved in general. In this paper, we study center conditions and bifurcation of limit cycles at the degenerate singular point in a class of quintic polynomial vector field with a small parameter and eight normal parameters. We deduce a recursion formula for singular point quantities at the degenerate singular points in this system and reach with relative ease an expression of the first five quantities at the degenerate singular point. The center conditions for the degenerate singular point of this system are derived. Consequently, we construct a quintic system, which can bifurcates 5 limit cycles in the neighborhood of the degenerate singular point. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of singular point quantities at degenerate singular point is linear and then avoids complex integrating operations.     

Averaging of the Cauchy kernels and integral realization of the local residue

We consider a family of kernels of integral representations associated with toric varieties. These kernels generalizes, in particular, the Bochner-Martinelli form. We show that the integral representation formulas can be derived by averaging of the Cauchy kernels on some positive measures. We apply then the obtained result to get an integral realization of the local residue corresponding to each kernel of integral representation.     

An MRA approach to surface completion and image inpainting

The objective of this paper is to introduce a multi-resolution approximation (MRA) approach to the study of continuous function extensions with emphasis on surface completion and image inpainting. Along the line of the notion of diffusion maps introduced by Coifman and Lafon with some “heat kernels” as integral kernels of these operators in formulating the diffusion maps, we apply the directional derivatives of the heat kernels with respect to the inner normal vectors (on the boundary of the hole to be filled in) as integral kernels of the “propagation” operators. The extension operators defined by propagations followed by the corresponding sequent diffusion processes provide the MRA continuous function extensions to be discussed in this paper. As a case study, Green's functions of some “anisotropic” differential operators are used as heat kernels, and the corresponding extension operators provide a vehicle to transport the surface or image data, along with some mixed derivatives, from the exterior of the hole to recover the missing data in the hole in an MRA fashion, with the propagated mixed derivative data to provide the surface or image “details” in the hole. An error formula in terms of the heat kernels is formulated, and this formula is applied to give the exact order of approximation for the isotropic setting.     

NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichletor Neumann boundary condition in a three-dimensional unbounded domain, which is theoutside domain of a prolate spheroid. We express the Poisson integral formula and naturalintegral operator in a series form explicitly. Thus the original problem is reduced to aboundary integral equation on a prolate spheroid. The variational formula for the reducedproblem and its well-posedness are discussed. Boundary element approximation for thevariational problem and its error estimates, which have relation to the mesh size andthe terms after the series is truncated, are also presented. Two numerical examples arepresented to demonstrate the effectiveness and error estimates of this method.     

Fast approximation to Poisson integral based on trigonometric periodic multiresolution analysis

In this article, a novel fast numerical computational algorithm for Poisson integral is developed by means of periodic trigonometric multiresolution analysis (PTMRA). The approximation formula of Poisson integral is derived. Subsequently, we establish some error estimates of approximation Poisson integral. Finally, several numerical results are given. Comparing with the existing wavelet-based method, the proposed method gives superior results.     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

On Compactness of Regular Integral Operators in the Space <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>

In this paper we obtain a sufficient condition for quite continuity of Fredholm type integral operators in the space L₁(a, b). Uniform approximations by operators with degenerate kernels of horizontally striped structures are constructed. A quantitative error estimate is obtained. We point out the possibility of application of the obtained results to second kind integral equations, including convolution equations on a finite interval, equations with polar kernels, one-dimensional equations with potential type kernels, and some transport equations in non-homogeneous layers.     

Note on the integral mean value method for Fredholm integral equations of the second kind

We study the paper of Avazzadeh et al. [Z. Avazzadeh, M. Heydari, G.B., Loghmani, Numerical solution of Fedholm integral equations of the second kind by using integral mean value theorem, Appl. Math. Model. 35 (2011) 2374–2383] with the integral mean value method for Fredholm integral equations of the second kind. The objective of the note is threefold. First, we point out a basic error in the paper. Second, we find that the given numerical examples are only related to the special cases of Fredholm integral equations of the second kind with the degenerate kernels, which can be solved simply. Third, due to the basic error, our observations reveal that generally the suggested method should not be considered for a Fredholm integral equation of the second kind.     

Higher order Poisson kernels and Lp polyharmonic boundary value problems in Lipschitz domains

In this article, we introduce higher order conjugate Poisson and Poisson kernels, which are higher order analogues of the classical conjugate Poisson and Poisson kernels, as well as the polyharmonic fundamental solutions, and define multi-layer potentials in terms of the Poisson field and the polyharmonic fundamental solutions, in which the former is formed by the higher order conjugate Poisson and the Poisson kernels. Then by the multi-layer potentials, we solve three classes of boundary value problems(i.e., Dirichlet, Neumann and regularity problems) with L~p boundary data for polyharmonic equations in Lipschitz domains and give integral representation(or potential) solutions of these problems.     

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

On a partial integral which can be derived from poisson matrix

Consider a surface which is a common level of some functions. Suppose that this surface is invariant under a Hamiltonian system. The question is if a partial integral can be derived explicitly from the Poisson matrix of these functions. In some cases such an integral is equal to the determinant of the matrix. This paper establishes a necessary and sufficient condition for this to hold true. The partial integral that results is not trivial if the induced Poisson structure is non-degenerate at one point at least. Therefore, the invariant surface must be even-dimensional.     

3‐D Contact problems for elastic wedges with Coulomb friction

3‐D quasi‐static contact problems for elastic wedges with Coulomb friction are reduced to integral equations and integral inequalities with unknown contact normal pressures. To obtain these equations and inequalities, Green's functions for the wedges, where one face of the wedges is either stress‐free or fixed, are needed. Using Fourier and Kontorovich–Lebedev integral transformations, all the stresses and displacements in the wedges can be constructed in terms of solutions of Fredholm integral equations of the second kind on the semiaxis. The Green's functions can be calculated as uniformly convergent power series in (1‐2ν), where νis Poisson's ratio. An exponential decay of the kernels and right‐hand sides of the Fredholm integral equations provides the applicability of the collocation method for simple and fast calculation of the Green's functions. For a half‐space, which is a special case of an elastic wedge, the kernels degenerate and the functions reduce to the well‐known Boussinesq and Cerruti solutions. Analysing the contact problems reveals that the Green's functions govern the kernels of the above mentioned integral equations and inequalities. Under the assumption that the punch has a smooth shape, the contact pressure is zero on the boundary of the unknown contact zone. Solving the contact problems with the help of the Galanov–Newton method, the normal contact pressure, the contact zone and the normal displacement around the contact zone can be determined simultaneously. In view of the numerical results, the influence of the friction forces on the punch force and the punch settlement is discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Mixed Measures of Convex Cylinders and Quermass Densities of Boolean Models

Translative integral formulas for curvature measures of convex bodies were obtained by Schneider and Weil by introducing mixed measures of convex bodies. These results can be extended to arbitrary closed convex sets since mixed measures are locally defined. Furthermore, iterated versions of these formulas due to Weil were used by Fallert to introduce quermass densities for (non-stationary and non-isotropic) Poisson processes of convex bodies and respective Boolean models. In the present paper, we first compute the special form of mixed measures of convex cylinders and prove a translative integral formula for them. After adapting some results for mixed measures of convex bodies to this setting we then use this integral formula to obtain quermass densities for (non-stationary and non-isotropic) Poisson processes of convex cylinders. Furthermore, quermass densities of Boolean models of convex cylinders are expressed in terms of mixed densities of the underlying Poisson process generalizing classical formulas by Davy and recent results by Spiess and Spodarev.      

圆内平面弹性问题的边界积分公式

根据双解析函数可以得到单位圆内平面弹性问题应力函数的边界积分公式,但式中包含强奇异积分,不能用于直接计算．将边界上的应力函数展开为Fourier级数,再利用广义函数论中的几个公式进行卷积计算,可以得到不含强奇异积分核的边界积分公式,通过边界的应力函数值和法向导数的积分,直接得到圆内应力函数值,并给出几个算例,表明该结果用于求解单位圆内平面弹性问题十分方便．     

On the reduction (mod 1) of completely monotone functions (0, ∞)

Summary The Euler-Maclaurin summation formula and its harmonic analysis (Poisson) are applied to the case of functions which are completely monotone on an open half-line. What thus results is a curious class of Fourier series, which can be determined explicitly and which represent completely monotone functions on the first half of the period. A by-product is the complete monotony (on the first half-period) of the Bernoulli functions, whether the index is integral or fractional.     

The Poisson Voronoi Tessellation III. Miles' Formula

This paper gives distributional properties of geometrical characteristics of a Voronoi tessellation generated by a stationary Poisson point process. The considerations are based on a well-known formula given by [10] describing size and shape of a cell of the Delaunay tessellation and on the close connection between Delaunay and Voronoi tessellation. Several results are given for the two-dimensional case, but the main part is the investigation of the three-dimensional case. They include the density functions of the angles perpendicular to the ‘typical’ edge, spanned by two neighbouring Poisson points and that spanned by two neighbouring faces, the angle between two edges emanating from the ‘typical’ vertex, the distance of two neighbouring Poisson points, the angle between two edges emanating from the ‘typical’ vertex of the Poisson Voronoi tessellation and some others. These density functions are given partly explicitely and partly in integral form.     

Green’s Function of Ordinary Differential Operators and an Integral Representation of Sums of Certain Power Series

The eigenvalues and eigenfunctions of certain operators generated by symmetric differential expressions with constant coefficients and self-adjoint boundary conditions in the space of Lebesgue squareintegrable functions on an interval are explicitly calculated, while the resolvents of these operators are integral operators with kernels for which the theorem on an eigenfunction expansion holds. In addition, each of these kernels is the Green’s function of a self-adjoint boundary value problem, and the procedure for its construction is well known. Thus, the Green’s functions of these problems can be expanded in series in terms of eigenfunctions. In this study, identities obtained by this method are used to calculate the sums of convergent number series and to represent the sums of certain power series in an intergral form.     

数值积分校正公式

通过两个例子说明一旦具有数值积分公式的余项表达式,只需利用代数精度概念即可确定余项里的中间点的具体数值,从而获得更高代数精度的数值积分校正公式.本文的方法可用于各类数值积分公式.