Chapter 13. Quadrature formulas of the method of discrete vortices for one-dimensional singular integrals

A numerical technique, first reported in 1979 in refs.[1] and [2], for the numerical evaluation of two-dimensional Cauchy-type principal-value integrals, is extended in this paper to include several cubature formlas of the Radau and Lobatto types. For the construction of such a cubature formula the 2-D singular integral is considered as an iterated one, and the second-order pole involved in this integral analyzed into a pair of complex poles. Based on this procedure, the methods of numerical integration, valid for one-dimensional singular integrals, are extanded to the case of two-dimensional singular integrals. The cubature formulas of the Lobatto- and Radau-type are now formulated to include the cases where some of the desired abscissas may be chosen accordins to any appropriate criterion.Moreover, the theory developed is enlarged to include the case of a 2-D principal-value integral, containing a logarithmic singularity. The validity of the results is illustrated by considering certain numerical examples. Furthermore, a complete analysis of the convergence and the construction of error estimates is also presented.

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On composite formulas for mathematical expectation of functionals of solution of the Ito equation in Hilbert space

Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series ◽

10.29235/1561-2430-2019-55-2-158-168 ◽

2019 ◽

Vol 55 (2) ◽

pp. 158-168

Author(s):

A. D. Egorov

Keyword(s):

Hilbert Space ◽

Random Process ◽

Wiener Process ◽

Mathematical Expectation ◽

Quadrature Formulas ◽

Third Order ◽

One Dimensional ◽

Temporal Variables ◽

Nonlinear Functionals ◽

Ito Equation

This article is devoted to constructing composite approximate formulas for calculation of mathematical expectation of nonlinear functionals of solution of the linear Ito equation in Hilbert space with additive noise. As the leading process, the Wiener process taking values in Hilbert space is examined. The formulas are a sum of the approximations of the nonlinear functionals obtained by expanding the leading random process into a series of independent Gaussian random variables and correcting approximating functional quadrature formulas that ensure an approximate accuracy of compound formulas for third-order polynomials. As a test example, the application of the obtained formulas to the case of a one-dimensional wave equation with a leading Wiener process indexed by spatial and temporal variables is considered. This article continues the research begun in [1].The problem is motivated by the necessity to calculate the nonlinear functionals of solution of stochastic partial differential equations. Approximate evaluation of mathematical expectation of stochastic equations with a leading random process indexed only by the time variable is considered in [2–11]. Stochastic partial equations in various interpretations are considered [12–16]. The present article uses the approach given in [12].

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Quadrature Formulas of Quasi-Interpolation Type for Singular Integrals with Hilbert Kernel

Journal of Approximation Theory ◽

10.1006/jath.1998.3166 ◽

1998 ◽

Vol 93 (2) ◽

pp. 231-257 ◽

Cited By ~ 7

Author(s):

Du Jinyuan

Keyword(s):

Singular Integrals ◽

Quadrature Formulas ◽

Hilbert Kernel

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Uniform estimates for families of singular integrals and double Fourier series

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700028020 ◽

1986 ◽

Vol 41 (1) ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Elena Prestini

Keyword(s):

Fourier Series ◽

Hilbert Transform ◽

Singular Integrals ◽

Double Fourier Series ◽

Variable Coefficients ◽

Partial Sums ◽

One Dimensional ◽

Uniform Estimates ◽

Almost Everywhere ◽

The One

AbstractIt is an open problem to establish whether or not the partial sums operator SNN2f(x, y) of the Fourier series of f ∈ Lp, 1 < p < 2, converges to the function almost everywhere as N → ∞. The purpose of this paper is to identify the operator that, in this problem of a.e. convergence of Fourier series, plays the central role that the maximal Hilbert transform plays in the one-dimensional case. This operator appears to be a singular integral with variable coefficients which is a variant of the maximal double Hilbert transform.

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