Singular Integrals are Perron Integrals of a Certain Type

1970 ◽  
Vol 22 (2) ◽  
pp. 260-264
Author(s):  
W. F. Pfeffer
Keyword(s):  
Topological Space ◽  
Singular Integrals ◽  
Convergence Theorems ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Basic Properties ◽  
Arbitrary Topological Space ◽  
Principal Value

In [7] a Perron-like integral was denned in an arbitrary topological space and many of its basic properties were established. In this paper we shall show (the theorem in § 2) that in a suitable setting the integral from [7] includes a class of so-called singular integrals, i.e., generalized forms of the Cauchy principal value of an integral. Thus, the powerful machinery of Perron integration, e.g., the monotone and dominant convergence theorems, can be automatically applied to these singular integrals.

scholarly journals Extended Error Expansion of Classical Midpoint Rectangle Rule for Cauchy Principal Value Integrals on an Interval

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Chunxiao Yu ◽  
Lingling Wei
Keyword(s):  
Singular Integrals ◽  
Riemann Integral ◽  
Piecewise Constant ◽  
Cauchy Principal ◽  
Numerical Examples ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value ◽  
Error Expansion ◽  
Superconvergence Results

The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.

A General Algorithm for Multidimensional Cauchy Principal Value Integrals in the Boundary Element Method

1990 ◽  
Vol 57 (4) ◽  
pp. 906-915 ◽  
Author(s):  
M. Guiggiani ◽  
A. Gigante
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Singular Integrals ◽  
Applied Mechanics ◽  
Full Generality ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value ◽  
Element Method

This paper presents a new general method for the direct evaluation of Cauchy principal value integrals in several dimensions, which is an issue of major concern in any boundary element method analysis in applied mechanics. It is shown that the original Cauchy principal value integral can be transformed into an element-by-element sum of regular integrals, each one expressed in terms of intrinsic (local) coordinates. The actual computation can be performed by standard quadrature formulae and can be easily included in any existing computer code. The numerical results demonstrate the accuracy and efficiency of the method, along with its insensitivity to the mesh pattern. This new method has full generality and, therefore, can be applied in any field of applied mechanics. Moreover, there are no restrictions on the numerical implementation, as the singular integrals may be defined on surface elements or internal cells of any order and type.

scholarly journals An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals

2011 ◽  
Vol 2011 ◽  
pp. 1-21
Author(s):  
Samir A. Ashour ◽  
Hany M. Ahmed
Keyword(s):  
Numerical Evaluation ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Finite Part ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value

Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In this work we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals , and extend Clenshow's algorithm to evaluate these integrals in a stable way.

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