Some cubature formulas for singular integrals in physical geodesy

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 quadrature and cubature formulas for a class of multiple singular integrals

Ukrainian Mathematical Journal ◽

10.1007/bf02513067 ◽

1997 ◽

Vol 49 (12) ◽

pp. 1891-1898

Author(s):

V. V. Akimenko

Keyword(s):

Singular Integrals ◽

Cubature Formulas ◽

Multiple Singular Integrals

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ON BUILDING OF QUADRATURE AND CUBATURE FORMULAS FOR COMPUTING OF HYPER-SINGULAR INTEGRALS

University proceedings Volga region Physical and mathematical sciences ◽

10.21685/2072-3040-2018-1-8 ◽

2018 ◽

Author(s):

I. V. Boykov ◽

◽

V. A. Esaf'eva ◽

P. V. Aykashev ◽

◽

...

Keyword(s):

Singular Integrals ◽

Cubature Formulas

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Cubature formulas for the evaluation of surface singular integrals

BIT Numerical Mathematics ◽

10.1007/bf01930990 ◽

1979 ◽

Vol 19 (3) ◽

pp. 368-377 ◽

Cited By ~ 3

Author(s):

G. Tsamasphyros ◽

P. S. Theocaris

Keyword(s):

Singular Integrals ◽

Cubature Formulas

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Multi-parameter Singular Integrals II: General Theory

10.23943/princeton/9780691162515.003.0005 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

General Theory ◽

Singular Integral ◽

Singular Integrals ◽

Integral Operators ◽

Singular Integral Operators ◽

Main Issue

This chapter turns to a general theory which generalizes and unifies all of the examples in the preceding chapters. A main issue is that the first definition from the trichotomy does not generalize to the multi-parameter situation. To deal with this, strengthened cancellation conditions are introduced. This is done in two different ways, resulting in four total definitions for singular integral operators (the first two use the strengthened cancellation conditions, while the later two are generalizations of the later two parts of the trichotomy). Thus, we obtain four classes of singular integral operators, denoted by A1, A2, A3, and A4. The main theorem of the chapter is A1 = A2 = A3 = A4; i.e., all four of these definitions are equivalent. This leads to many nice properties of these singular integral operators.

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The Calder´on-Zygmund Theory I: Ellipticity

10.23943/princeton/9780691162515.003.0001 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

Classical Theory ◽

Singular Integral ◽

Singular Integrals ◽

Differential Operators ◽

Integral Operators ◽

Singular Integral Operators ◽

Single Parameter ◽

Partial Differential ◽

Translation Invariant ◽

Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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Multi-parameter Singular Integrals. (AM-189)

10.23943/princeton/9780691162515.001.0001 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

Partial Differential Equations ◽

Graduate Students ◽

General Theory ◽

Differential Equations ◽

Singular Integrals ◽

Main Tool ◽

Classical Theorem ◽

Quantitative Version ◽

Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

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