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On a 3D-Hypersingular Equation of a Problem for a Crack

Samko, Stefan (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 45DB05, 45E05, 78A45We show that a certain axisymmetric hypersingular integral equation arising in problems of cracks in the elasticity theory may be explicitly solved in the case where the crack occupies a plane circle. We give three different forms of the resolving formula. Two of them involve regular kernels, while the third one involves a singular kernel, but requires less regularity assumptions on the the right-hand side of the equation.

On a Hypersingular Equation of a Problem, for a Crack in Elastic Media

Gil, Alexey, Samko, Stefan (2010)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification 2010: 45DB05, 45E05, 78A45.We give a procedure to reduce a hypersingular integral equation, arising in 2d diffraction problems on cracks in elastic media, to a Fredholm integral equation of the second kind, to which it is easier and more effectively to apply numerical methods than to the initial hypersingular equation.

On convergence of quadrature-differences method for linear singular integro-differential equations on the interval

A. I. Fedotov (2001)

Archivum Mathematicum

Here we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with Cauchy kernel on the interval ( - 1 , 1 ) . We consider equations of zero, positive and negative indices. It is shown, that the method converges to exact solution and the error estimate depends on the sharpness of derivative approximation and the smoothness of the coefficients and the right-hand side of the equation.

On solutions of integral equations with analytic kernels and rotations

Nguyen Van Mau, Nguyen Minh Tuan (1996)

Annales Polonici Mathematici

We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*)     x(t) + a(t)(Tx)(t) = b(t), where T = M n ₁ , k ₁ . . . M n m , k m and M n j , k j are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial P T ( t ) = t ³ - t . By means of the Riemann boundary value...

On the A -integrability of singular integral transforms

Shobha Madan (1984)

Annales de l'institut Fourier

In this article we study the weak type Hardy space of harmonic functions in the upper half plane R + n + 1 and we prove the A -integrability of singular integral transforms defined by Calderón-Zygmund kernels. This generalizes the corresponding result for Riesz transforms proved by Alexandrov.

On the asymptotic convergence of the polynomial collocation method for singular integral equations and periodic pseudodifferential equations

A. I. Fedotov (2002)

Archivum Mathematicum

We prove the convergence of polynomial collocation method for periodic singular integral, pseudodifferential and the systems of pseudodifferential equations in Sobolev spaces H s via the equivalence between the collocation and modified Galerkin methods. The boundness of the Lagrange interpolation operator in this spaces when s > 1 / 2 allows to obtain the optimal error estimate for the approximate solution i.e. it has the same rate as the best approximation of the exact solution by the polynomials.

On the efficient use of the Galerkin-method to solve Fredholm integral equations

Wolfgang Hackbusch, Stefan A. Sauter (1993)

Applications of Mathematics

In the present paper we describe, how to use the Galerkin-method efficiently in solving boundary integral equations. In the first part we show how the elements of the system matrix can be computed in a reasonable time by using suitable coordinate transformations. These techniques can be applied to a wide class of integral equations (including hypersingular kernels) on piecewise smooth surfaces in 3-D, approximated by spline functions of arbitrary degree. In the second part we show, how to use the...

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