Showing papers on "Singular integral published in 1982"

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Abstract: A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator δ into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.

The use of piecewise quadratic polynomials for the solution of singular integral equations of Cauchy type

Abstract: A method for the numerical solution of singular integral equations of Cauchy type is developed. The unknown function is expressed as a product of a weight function and a continuous function φ( t ). The continuous function φ( t ) is approximated by piecewise quadratic polynomials, and the singular integral equation is reduced to a linear algebraic system. Numerical examples are given, and comparisons are made with the widely used Gauss-type methods.

Resolution in Diffraction-limited Imaging, a Singular Value Analysis

Abstract: In a previous paper, methods of singular function expansions have been applied to the analysis of coherent imaging when the object and image domains are allowed to differ. In this paper the method is extended to incoherent illumination, restricting the analysis to the aberration-free case. While singular functions and singular values for coherent imaging are related in a simple way to the prolate spheroidal functions and their eigenvalues, such relations do not exist for the incoherent imaging case. In spite of this difficulty many properties of singular functions and singular values are derived in this paper and asymptotic estimates are obtained in the limit of large space-bandwidth product. For small values of the space-bandwidth product, the singular values are computed numerically and by means of these results it is shown that super-resolution, in the sense of improving on previous criteria in the presence of noise, can be achieved.

The Classical Collocation Method for Singular Integral Equations

Abstract: First of all a method of collocation, which we call “classical” collocation, is described for the approximate solution of complete singular integral equations with Cauchy kernel taken over the arc $( - 1,1)$. Secondly we demonstrate that, under reasonable conditions, the approximate solutions converge to the solution of the original equation.

Orthogonal Polynomials Associated with Singular Integral Equations Having a Cauchy Kernel

Abstract: The Chebyshev polynomials of both the first and second kind are of fundamental importance when considering the particular case of the singular integral equation $a(t)\phi (t) + ({{b(t)} / \pi })\lambda \int_{ - 1}^1 {({{\phi (\tau )} / {(\tau - t)}})} d\tau = f(t), - 1 < t < 1$, in which $a \equiv 0$ and $b \equiv - 1$ on $[ - 1,1]$. We identify the two sets of orthogonal polynomials which play a corresponding role for the singular integral equation with general a, b and consider some of the relationships between these two sets of polynomials.

An Extension of the Stochastic Integral

Abstract: Two related extensions of the stochastic integral are discussed. These extensions allow the integrand to anticipate the Brownian motion, and arise in the study of linear stochastic integral equations. The development is based on the homogeneous chaos expansion of the integrand. Some properties of these extended integrals, and their commutativity with the classical integrals, are derived.

Explicit solutions of integral equations via block pulse functions

Abstract: A new method to obtain the explicit solutions of integral equations via block pulse functions is proposed. The elegant Laplace transform technique is used to integrate the block pulse convolution matrix in terms of the block pulse functions. Based on these results, the solutions of integral equations can be recursively calculated. The first and second order integral equations of the first kind, and the integral equations of the second kind arc all solved via this new approach.

Singular value analysis of deformable systems

Abstract: Singular value analysis, balancing, and approximation of a class of deformable systems are investigated. The deformable systems considered herein include several important cases of flexible aerospace vehicles and are characterized by countably infinitely many poles and zeros on the imaginary axis. The analysis relies completely on the so-called asymptotic singular value decompositon of the Hankel operator associated with the impulse response of the system. A parametric study of a six-dimensional single-input single-output case is performed.

Solution of plane anisotropic elastostatical boundary value problems by singular integral equations

Abstract: Basis for the presented method is the knowledge of certain fundamental solutions for theinfinite anisotropic medium. By superimposing these singular solutions in a suitable fashion, the given boundary value problem can be formulated as a tensorial integral equation (singularity method).

An integral transform solution to the fundamental problem in resistivity logging

Abstract: This paper develops the analysis necessary for the computation of the response of a resistivity tool within a well bore as it traverses a thin‐invaded bed. The solution for the potential induced by a steady current source (point current electrode) is formulated in terms of both Fourier cosine and Fourier sine transforms with arbitrary coefficients. A suitable matching of the necessary boundary conditions results in a system of singular integral equations. An iterative solution (Neumann series) is obtained for these transform coefficients which, in turn, are used to determine the potential at an arbitrary point of measurement. The theory is applied to some typical focused resistivity tools, and the results are found to be in close agreement with similar results obtained via a resistor network (analog) solution.

Singular integral equations — The convergence of the Nyström interpolant of the Gauss-Chebyshev method

Abstract: Nystrom's interpolation formula is applied to the numerical solution of singular integral equations. For the Gauss-Chebyshev method, it is shown that this approximation converges uniformly, provided that the kernel and the input functions possess a continuous derivative. Moreover, the error of the Nystrom interpolant is bounded from above by the Gaussian quadrature errors and thus convergence is fast, especially for smooth functions. ForC∞ input functions, a sharp upper bound for the error is obtained. Finally numerical examples are considered. It is found that the actual computational error agrees well with the theoretical derived bounds.

Expansion series of integral functions occurring in unsteady aerodynamics

Abstract: Attention is given to two real integral functions which occur in the kernel of singular integral equations for subsonic unsteady lifting surfaces. The arguments k, r, and X of the functions correspond to the reduced frequency, spanwise distance, and modified coordinate in the flow direction, respectively. The value of the parameter nu in the functions depends on geometrical conditions. The considered investigation has the objective to present a series for general values of the nonnegative integer nu in order to compute efficiently the integral functions. The approach makes it possible to avoid any approximation or numerical quadrature.

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

Abstract: 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.

On the integral means of derivatives of the atomic function

Abstract: In this note we give upper and lower estimates on integral means of the atomic function and its derivatives over a circle of radius r as r approaches 1. From this we derive some known and new results.

A method of cauchy integral equation for non-coherent transfer in half-space

Abstract: A technique is presented which allows easy construction of solutions for various half-space problems arising in non-coherent radiative transfer with complete redistribution. By use of an inverse Laplace transform method, Wiener-Hopf integral equations are reduced to Cauchy-type singular integral equations. The factorization technique used by Case and Zweifel for coherent scattering can then be carried over to non-coherent transfer. The method is applied to the inhomogeneous integral equation for the source function of a two-level atom, previously solved by Ivanov. It is also applied to the conservative, homogeneous case and to singular Wiener-Hopf equations arising from asymptotic expansions in the limit of vanishing probability of collisional destruction ϵ. Consequences for the scaling laws in a finite slab are examined in a companion paper.