第二类端点奇异Fredholm积分方程的分数阶退化核方法

本文针对第二类端点奇异Fredholm积分方程构造基于分数阶Taylor展开的退化核方法，设计了两种计算格式，一是在全区间上使用分数阶Taylor展开式近似核函数，二是在包含奇点的小区间上采用分数阶插值，在剩余区间上采用分段二次多项式插值逼近核函数.讨论了两种退化核方法收敛的条件，并给出了混合插值法的收敛阶估计.数值算例表明对于非光滑核函数分数阶退化核方法有着良好的计算效果，且混合二次插值法比全区间上的分数阶退化核方法有着更广泛的适用范围.     

A Singular Expansion of Solution for a Regularized Compressible Stokes System

A compressible Stokes system is considered in a sector of the plane. The continuity equation is regularized by adding the diffusion term . We give a high-order expansion of corner singularities for the regularized system when the corner singularities for the Laplacian are implemented. A solution formula is constructed in an abstract way, and new associate singular functions are introduced for extracting high-order corner singularities. In the expansion, the smoother parts of the associate singular functions are used. Communicated by Rafael D. Benguriasubmitted 14/11/02, revised 12/08/03, accepted 04/10/03     

Intersection Local Times of Independent Brownian Motions as Generalized White Noise Functionals

A 'chaos expansion' of the intersection local time functional of two independent Brownian motions in R ^d is given. The expansion is in terms of normal products of white noise (corresponding to multiple Wiener integrals). As a consequence of the local structure of the normal products, the kernel functions in the expansion are explicitly given and exhibit clearly the dimension dependent singularities of the local time functional. Their L ^p -properties are discussed. An important tool for deriving the chaos expansion is a computation of the 'S-transform' of the corresponding regularized intersection local times and a control about their singular limit.     

Delta-waves for Semilinear Hyperbolic Cauchy Problems

This paper deals with the propagation of strong singularities for constant coefficient semilinear hyperbolic equations and systems. Limits of regularized solutions are computed as the initial data converge to derivatives of Dirac measures on lower dimensional submanifolds. A general method is given which applies whenever the fundamental solution to the principal part is an integrable measure. Particular cases are semilinear first order systems in one space variable and the semilinear Klein-Gordon equation in at most three space variables.     

Computation of the attractive force of an ellipsoid

A numerical method for computing the attractive force of an ellipsoid is proposed that does not involve separating subdomains with singularities. The sought function is represented as a triple integral such as the inner integral of the kernel can be evaluated analytically with the kernel treated as a weight function. The inner integral is approximated by a quadrature for the product of functions, of which one has an integrable singularity. As a result, the integrand obtained before the second integration has only a weak logarithmic singularity. The subsequent change of variables yields an integrand without singularities. Based on this approach, at each stage of integral evaluation with respect to a single variable, quadrature formulas are derived that do not have singularities at integration nodes and do not take large values at these nodes. For numerical experiments, a rather complicated test function is constructed that is the exact attractive force of an ellipsoid of revolution with an elliptic density distribution.     

Hilbert核奇异求积

该文用分离奇点的方法建立了含Hilbert核的奇异积分带重结点的求积公式,给出了求积公式余项的积分表示式。     

A wavelet method for solving singular integral equation of MHD

This paper presents Haar wavelet approximation to solve a singular integral equation which has singularities on a diagonal of the domain R=[-1,1]×[-1,1]. The singularities arise basically due to modified Bessel function K₀ which appears as a part of the kernel. Thus the integral equation is weakly (logarithmically) singular only. The problem is solved considering all the singularities of the kernel and results are examined for approximations of different orders. Our interest to solve the problem using Haar wavelet basis is due to its simplicity and efficiency in numerical approximation. The results show convergence trend as mesh is refined. Comparison is made with some available results obtained earlier by partial consideration of the singularities.     

Matrix-free interior point method

In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditioner is computed. A two-step approach is used to design a preconditioner. First, the Newton equation system is regularized to guarantee better numerical properties and then it is preconditioned. The preconditioner is implicit, that is, its computation requires only matrix-vector multiplications with the original problem data. The method is therefore well-suited to problems in which matrices are not explicitly available and/or are too large to be stored in computer memory. Numerical properties of the approach are studied including the analysis of the conditioning of the regularized system and that of the preconditioned regularized system. The method has been implemented and preliminary computational results for small problems limited to 1 million of variables and 10 million of nonzero elements demonstrate the feasibility of the approach.     

The interaction of punches on a transversely isotropic half-space

Three-dimensional contact problems on the interaction of two similar punches on an elastic transversely isotropic half-space (five elastic constants) are investigated, when the isotropy planes are perpendicular to the boundary of the half-space. In this connection the stiffness of the half-space boundary depends on the direction. The kernel of the integral equation of the contact problems is represented in a quadrature-free form using the theory of generalized functions. This form of the kernel enables it to be regularized at singular points and enables Galanov's method to be used to solve the contact problem with an unknown contact area.     

The Direct Method of Lines for Forward and Inverse Linear Elasticity Problems of Composite Materials in Star-Shaped Domains

In this paper, we generalize the direct method of lines for linear elasticity problems of composite materials in star-shaped domains and consider its application to inverse elasticity problems. We assume that the boundary of the star-shaped domain can be described by an explicit $C^1$ parametric curve in the polar coordinate. We introduce the curvilinear coordinate, in which the irregular star-shaped domain is converted to a regular semi-infinite strip. The equations of linear elasticity are discretized with respect to the angular variable and we solve the resulting semi-discrete approximation analytically using a direct method. The eigenvalues of the semi-discrete approximation converge quickly to the true eigenvalues of the elliptic operator, which helps capture the singularities naturally. Moreover, an optimal error estimate of our method is given. For the inverse elasticity problems, we determine the Lamé coefficients from measurement data by minimizing a regularized energy functional. We apply the direct method of lines as the forward solver in order to cope with the irregularity of the domain and possible singularities in the forward solutions. Several numerical examples are presented to show the effectiveness and accuracy of our method for both forward and inverse elasticity problems of composite materials.     

Parametrix,heat kernel asymptotics,and regularized trace of the diffusion semigroup

We establish a relationship between a path integral representation of the heat kernel and the construction of a fundamental solution to a diffusion-type equation by the parametrix method; this relationship is used to find the coefficients of a short-time asymptotic expansion of the heat kernel. We extend the approach proposed to the case of diffusion with drift and obtain two-sided estimates for the regularized trace of the corresponding evolution semigroup.     

A method for the localization of singularities of a solution to a convolution-type equation of the first kind with a step kernel

We consider the problem of the localization of singularities (delta-functions) of a solution to a convolution-type equation of the first kind with a step kernel. We propose a regularization method which allows one to calculate the number of singularities, to approximate their location, and to estimate the approximation error. We also adduce bounds for an important characteristic of the method, namely, the separability threshold. We prove the order optimality of the proposed method on classes of functions with singularities both with respect to the accuracy and the separability.     

Semi infinite Orthotropic Cantilevered Strips and the Foundations of Plate Theories

Elastostatic problems of semiinfinite orthotropic cantilevered strips with traction-free edges and loading at infinity are reduced to the solution of a single scalar Fredholm integral equation of the first kind with a generalized Cauchy kernel. The known complex variable method for equations with a Cauchy type kernel is extended to handle the singularities in the solution for the generalized Cauchy kernel. The reduced problem lends itself to a more efficient numerical solution scheme than all existing methods. Moments of stresses at the root of the cantilever are accurately evaluated and used for the correct formulation of displacement boundary conditions for a plate theory solution (or the actual interior solution) of the elastostatics of thin flat bodies.     

On the Singularities of Fractionalized Potential Functions

The singularities of harmonic functions generated under the action of an integral transform with a fractionalized kernel are studied. A simple relationship arises connecting pairs of singularities.     

Fractional Hermite degenerate kernel method for linear Fredholm integral equations involving endpoint weak singularities

In this article, the Fredholm integral equation of the second kind with endpoint weakly singular kernel is considered and suppose that the kernel possesses fractional Taylor''s expansions about the endpoints of the interval. For this type kernel, the fractional order interpolation is adopted in a small interval involving the singularity and piecewise cubic Hermite interpolation is used in the remaining part of the interval, which leads to a kind of fractional degenerate kernel method. We discuss the condition that the method can converge and give the convergence order. Furthermore, we design an adaptive mesh adjusting algorithm to improve the computational accuracy of the degenerate kernel method. Numerical examples confirm that the fractional order hybrid interpolation method has good computational results for the kernels involving endpoint weak singularities.     

On the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals

The behaviour of a solution to a Fredholm integral equation of the second kind on a union of open intervals is examined. The kernel of the corresponding integral operator may have diagonal singularities, information about them is given through certain estimates. The weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described.     

Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method

The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontinuous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, can then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data. The combination with finite element discretization in space is also studied.

     

Numerical method for an integral-algebraic system

We present a numerical method for solving the system of integral-algebraic equations arising in the study of the oblique derivative problem for the Laplace equation outside open curves on the plane. The problem describes the electric current in a semiconductor film with curvilinear electrodes in the presence of a magnetic field. The integral-algebraic system has singularities, and the kernel in the integral equation is represented in the form of a Cauchy integral. The numerical scheme is of the second approximation order despite the singularities.     

Regularized learning in Banach spaces as an optimization problem: representer theorems

We view regularized learning of a function in a Banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel Banach spaces, we prove the representer theorem for the minimizer of regularized learning schemes with a general loss function and a nondecreasing regularizer. When the loss function and the regularizer are differentiable, a characterization equation for the minimizer is also established.     

On the resolution of singularities of ordinary differential systems

We show how some differential geometric ideas help to resolve some singularities of ordinary differential systems. Hence a singular problem is replaced by a regular one, which facilitates further analysis of the system. The methods employed are constructive and the regularized systems can also be used for numerical computations. This revised version was published online in June 2006 with corrections to the Cover Date.