Multidimensional Singular Integrals Research Articles

Peculiarities of matrix-element calculations with few Coulomb functions for particles' scattering processes

We present ultra high resolution data on single ionization of helium under impact of 1-MeV protons in comparison with theoretical calculations. Good agreement between theory and experiment [1] is obtained. Three initial trial helium wave functions are employed: a weakly correlated Roothaan-Hartree-Fock function, a simple Silverman-Platas-Matsen function ofthe configuration interaction family, and a strongly correlated function [2,3]. Multidimensional singular integrals which defining differential cross sections are calculated using special transform for each above trial function. Results of some calculations arepresented in Figure 1.

Properties of the symbol of multidimensional singular integrals in the weighted spaces and oscillating multipliers

Differential properties of symbols of multidimensional singular integrals in the weighted space of Bessel potentials on the sphere with the weighted functions, having singularities on a sphere are studied. The main results are applied to obtaining theorems on Fourier multipliers of spherical harmonic expansions.

Weighted Averages and Double Exponential Algorithms

This paper reviews two simple numerical algorithms particularly useful in Computational ElectroMagnetics (CEM): the Weighted Averages (WA) algorithm and the Double Exponential (DE) quadrature. After a short historical introduction and an elementary description of the mathematical procedures underlying both techniques, they are applied to the evaluation of Sommerfeld integrals, where WA and DE combine together to provide a numerical tool of unprecedented quality. It is also shown that both algorithms have a much wider range of applications. A generalization of the WA algorithm, able to cope with integrands including products of Bessel and similar oscillatory functions, is described. Similarly, the original DE algorithm is adapted with exceptional results to the evaluation of the multidimensional singular integrals arising in the discretization of Integral-Equation based CEM formulations. The new possibilities of WA and DE algorithms are demonstrated through several practical numerical examples.

The Superconvergence of Certain Two-Dimensional Cauchy Principal Value Integrals

Summary The composite rectangle (midpoint) rule for the computation of multi-dimensional singular integrals is discussed, and the superconvergence results is obtained. When the local coordinate is coincided with certain priori known coordinates, we get the convergence rate one order higher than the global one. At last, numerical examples are presented to illustrate our theoretical analysis which agree with it very well.

The Superconvergence of Certain Two-Dimensional Hilbert Singular Integrals

The composite rectangle (midpoint) rule for the computation of multi-dimensional singular integrals is discussed and the superconvergence results is obtained. When the local coordinate is coincided with certain priori known coordinates, we get the convergence rate one order higher than the global one. At last, numerical examples are presented to illustrate our theoretical analysis which agree with it very well.

Evaluation of Weakly Singular Integrals Via Generalized Cartesian Product Rules Based on the Double Exponential Formula

Various weakly singular integrals over triangular and quadrangular domains, arising in the mixed potential integral equation formulations, are computed with the help of novel generalized Cartesian product rules. The proposed integration schemes utilize the so-called double exponential quadrature rule, originally developed for the integration of functions with singularities at the endpoints of the associated integration interval. The final formulas can easily be incorporated in the context of singularity subtraction, singularity cancellation and fully-numerical methods, often used for the evaluation of multidimensional singular integrals. The performed numerical experiments clearly reveal the superior overall performance of the proposed method over the existing numerical integration methods.

On local properties of functions and singular integrals in terms of the mean oscillation

This paper is devoted to research on local properties of functions and multidimensional singular integrals in terms of their mean oscillation. The conditions guaranteeing existence of a derivative in the L p-sense at a given point are found. Spaces which remain invariant under singular integral operators are considered.

On the Direct Evaluation of Weakly Singular Integrals in Galerkin Mixed Potential Integral Equation Formulations

Weakly singular integrals over coincident triangles, arising in the Galerkin discretization of mixed potential integral equation formulations, are calculated using a direct evaluation method. The proposed method utilizes a series of coordinate transformations, together with a re-ordering of the integrations, in order to reduce the dimensionality of the original four-dimensional (4D) weakly singular integrals into 1D numerical integrations of smooth functions. The final formulas can be easily evaluated with a standard Gaussian quadrature rule, resulting in a scheme with great accuracy and efficiency properties. Numerical results for the comparison of the proposed method with both singularity subtraction and singularity cancellation methods, often used for the evaluation of multidimensional singular integrals, are presented, indicating the superior overall performance of the direct evaluation scheme.

Hörmander multipliers on two-dimensional dyadic Hardy spaces

In this paper we are interested in conditions on the coefficients of a two-dimensional Walsh multiplier operator that imply the operator is bounded on certain of the Hardy type spaces H p , 0 < p < ∞ . We consider the classical coefficient conditions, the Marcinkiewicz–Hörmander–Mihlin conditions. They are known to be sufficient for the trigonometric system in the one and two-dimensional cases for the spaces L p , 1 < p < ∞ . This can be found in the original papers of Marcinkiewicz [J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939) 78–91], Hörmander [L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math. 104 (1960) 93–140], and Mihlin [S.G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 109 (1956) 701–703; S.G. Mihlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965]. In this paper we extend these results to the two-dimensional dyadic Hardy spaces.

Three-Dimensional Boundary Value Problems of Elastothermodiffusion with Mixed Boundary Conditions

Abstract We investigate the basic boundary value problems of the connected theory of elastothermodiffusion for three-dimensional domains bounded by several closed surfaces when the same boundary conditions are fulfilled on every separate boundary surface, but these conditions differ on different groups of surfaces. Using the results of papers [Kupradze, Gegelia, Basheleishvili, and Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Publishing Company, 1979, Russian original, 1976–Mikhlin, Multi-dimensional singular integrals and integral equations, 1962], we prove theorems on the existence and uniqueness of the classical solutions of these problems.

Quasiwidths and optimization of methods of mixed approximation of multidimensional singular integrals with kernels of hilbert type

We consider the problem of application of mixed methods to the construction of algorithms, optimal in accuracy, for the calculation of multidimensional singular integrals with Hilbert-type kernels. We propose a method for the optimization of cubature formulas for singular integrals with Hilbert-type kernels based on the theory of quasiwidths.

Multidimensional singular integral equations in Lp applied to three-dimensional thermoelastoplastic stress analysis

A multidimensional singular integral equations analysis in L p is presented, by reducing the equations to the equivalent Fredholm equations and giving some basic theorems. These equations are widely used in many fields of mathematical physics and their mathematical character in L p spaces has been extensively investigated. A numerical technique is further presented, known as the singular integral operators method, for the numerical evaluation of the multidimensional singular integrals used in three-dimensional thermoelastoplastic stress analysis. Finally, an application is given of the determination of the thermoelastic stress analysis for a hollow cylinder subjected to a special temperature distribution.

On the numerical solution of the multidimensional singular integrals and integral equations, used in the theory of linear viscoelasticity

In the present report, we investigate the formulation, for the numerical evaluation of the multidimensional singular integrals and integral equations, used in the theory of linear viscoelasticity. Some simple formulas are given for the numerical solution of the general case of the multidimensional singular integrals. Moreover a numerical technique is also established for the numerical solution of some special cases of the multidimensional singular integrals like the two - and three - dimensional singular integrals. An application is given to the determination of the fracture behaviour of a thick, hollow circular cylinder of viscoelastic material restrained by an enclosing thin elastic ring and subjected to a uniform pressure.

The solution of a thermoelasticity problem for inhomogeneous bodies in terms of stresses by the perturbation method

Using the general properties of positive-definiteness of the elasticity theory operator, the convergence in an energy metric is proved for the solution of a thermoelasticity problem for a three-dimensional continuously inhomogeneous isotropic body by the perturbation method. The convergence of a solution by the perturbation method was proved earlier /1/ for the plane problem of the theory of elasticity of inhomogeneous bodies on the basis of properties for representing harmonic functions in terms of an integral with a weak singularity and the properties of multidimensional singular integrals. A comparison is given in /2/ between the exact solution and a solution constructed by the perturbation method for a special case of the plane problem, and their agreement in the domain of convergence is noted. Another approach that relies on the theory of integral equations is described in /3/, where an analysis is given of three-dimensional boundary value problems of the linear theory of elasticity and thermoelasticity for homogeneous and piecewise-homogeneous media, including existence and uniqueness theorems for the solutions.

Finite-dimensional approximations of singular integrals and direct methods of solution of singular integral and integrodifferential equations

A brief survey is given of papers reviewed in RZh “Matematika” in the last 15 years (1965–1979) on applied methods of calculating various classes of one-dimensional and multidimensional singular integrals and quadrature methods of solving various classes of singular linear equations with integrals in the sense of principal values.

Evaluation of multi-dimensional singular integrals with Cauchy kernels

Evaluation of multi-dimensional singular integrals with Cauchy kernels

Corrections to the article ?Multidimensional Singular Integrals in Classes of General Leading Modules of Smoothness?

Corrections to the article ?Multidimensional Singular Integrals in Classes of General Leading Modules of Smoothness?

Multidimensional singular integrals in classes of common leading moduli of continuity

Multidimensional singular integrals in classes of common leading moduli of continuity