On the approximate solution of some two‐dimensional singular integral equations

An approximation method for a wide class of two‐dimensional integral equations is proposed. The method is based on using a special function system. Orthonormality and good interaction with fundamental integral operators arising in partial differential equations are remarkable properties of this system. In addition, all the basis elements can easily be calculated by recurrence relations. Taking into account these properties we construct a numerical algorithm which does not require additional effort (such as quadrature) to compute the values of the fundamental operators on the basis elements. Copyright © 2001 John Wiley & Sons, Ltd.     

The approximate solution of singular integral equations

A computational scheme of collocation type is proposed for a singular linear integral equation with a power singularity in the regular integral and the justification is given. The results obtained are used to justify the approximate solution of the singular integral equation $$Kx \equiv a(t)x(t) + \frac{{b(t)}}{{\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{x(\tau )d\tau }}{{\tau - t}} + \frac{1}{{2\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{h|t,\tau )x(\tau )}}{{\left| {\tau - t} \right|^\delta }}d\tau = f(t)$$ by a modification of the method of minimal residuals.     

On projective methods of approximate solution of singular integral equations

The estimate for the rate of convergence of approximate projective methods with one iteration is established for one class of singular integral equations. The Bubnov-Galerkin and collocation methods are investigated.This author is also known as A. V. Dzhishkariani.     

On solution sequences of the singular Liouville equations with an integral constraint

In this paper,we analyze the blow-up behavior of sequences{uk}satisfying the following conditionsΔuk=|x|2αk V k eukinΩ,(0.1)whereΩR2,V k→V in C1,|Vk|≤A,0a≤Vk≤b,0≤αk→α∈(0,∞),andΩ|x|2αkeuk dx≤Λ1.(0.2)Furthermore,we assume that there exists someq∈(1,2)such that rq 2Br(p)|uk|qdx≤Λ2(0.3)for anyB r(p)Ω.As a result,we give a new proof of the concentration-compactness theorem for the mean feld equation.     

On the approximate solution of one-dimensional singular integral equations of first kind

We prove an estimate for the error in approximate solution of one-dimensional singular integral equations. The estimate is obtained by an approximation of the kernel. For a specific problem we give a comparison of numerical results. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

On parameters influencing the stability of quadrature methods for singular integral equations with conjugation

In this paper, we study an approximation method for solving singular integral equations with conjugation on an open arc. The stability of the method depends on the invertibility of certain operators which belong to well-known algebras. We investigate properties of these operators and show how to choose the parameters of the approximation method so that the Fredholm indices of the operators mentioned become equal to zero.     

On the approximate solution of operator equations

Numerische Mathematik - Prolongations and restrictions are used to derive error estimates when linear operator equations of the second kind are solved by discretisation methods.     

Optimization of direct methods for the approximate solution of weakly singular integral equations

For a class of weakly singular integral equations with power and logarithmic singularities, we establish the optimal order of the error of direct methods and indicate the procedure which realizes this order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1246–1254, September, 1994.     

On approximate solution for operator equations of Hammerstein type

The aim of this paper is to investigate a method of approximating a solution of the operator equation of Hammerstein type x + KF(x) = f by solutions of similar finite-dimensional problems which contain operators better than K and F. Conditions of convergence and convergence rate are given and an iteration method to solve the approximative equation is proposed and applied to a concrete example.     

Application of Faber polynomials to the approximate solution of singular integral equations with the Cauchy kernel

We construct an approximate solution of the singular integral equation with the Cauchy kernel with the use of Faber polynomials, prove its convergence to the exact solution, and indicate the order of the convergence rate.     

The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

A method for finding the numerical solution of a weakly singular Fredholm integral equation of the second kind is presented. The Taylor series is used to remove singularity and Legendre polynomials are used as a basis. Furthermore, the Legendre function of the second kind is used to remove singularity in the Cauchy type integral equation. The integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the Cauchy principal value sense without using numerical quadratures. Four examples are given to show the accuracy of the method.     

On the approximate solution of weakly singular integro-differential equations of Volterra type

Recently, the convergence rate of the collocation method for integral and integro-differential equations with weakly singular kernels has been studied in a series of papers [1–7]. The present paper belongs to the same series. We analyze the possibility of constructing approximate solutions of high-order accuracy on a uniform or almost uniform grid for weakly singular integro-differential equations of Volterra type.Translated from Differentsialnye Uravneniya, Vol. 40, No. 9, 2004, pp. 1271–1279.Original Russian Text Copyright © 2004 by Pedas.     

On the solution of singular integral equations with automorphic kernels by mechanical quadrature

Translated fromIssledovaniya po Prikladnoi Matematike, No. 19, 1992, pp. 111–120.     

Local theory of the collocation method for the approximate solution of singular integral equations,I

A new approach for investigating the collocation method for singular integral equations is given. Using Banach algebra techniques and a local principle established by I. Gohberg and N. Krupnik under suitable conditions the convergence of the Lagrange- and Mul thopp-collocation method is proved for one-dimensional singular integral equations and systems of such equations with piecewise continuous coefficients having a countable infinite number of discontinuities.     

Rational interpolation and approximate solution of integral equations

In the space of continuous periodic functions, we construct interpolation rational operators, use them to obtain quadrature formulas with positive coefficients which are exact on rational trigonometric functions of order 2n, and suggest an algorithm for an approximate solution of integral equations of the second kind. We estimate the accuracy of the approximate solution via the best trigonometric rational approximations to the kernel and the right-hand side of the integral equation.     

On a multilinear singular integral operator

In this paper, we prove that the maximal operatorsatisfiesis homogeneous of degree 0, has vanishing moment up to order M and satisfies Lq-Dini condition for some     

On approximate solution of the Dixon integral equation and some its generalizations

The paper is devoted to the study and numerical analytical solution of Fredholm-type integral equations of the second kind with symmetric kernels represented by homogeneous functions of degree (-1). The well-known Dixon equation and some its direct generalizations are specially considered. The equations are solved by passing to a Wiener–Hopf equation and applying the kernel averaging method. Results of numerical calculations are presented.     

On the quadrature methods for the numerical solution of singular integral equations

A Cauchy type singular integral equation can be numerically solved by the use of an appropriate numerical integration rule and the reduction of this equation to a system of linear algebraic equations, either directly or after the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind. In this paper two fundamental theorems on the equivalence (under appropriate conditions) of the aforementioned methods of numerical solution of Cauchy type singular integral equations are proved in sufficiently general cases of Cauchy type singular integral equations of the second kind.