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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 the computation of Aden functions

Peter Maličký, Marianna Maličká (1991)

Applications of Mathematics

The paper deals with the computation of Aden functions. It gives estimates of errors for the computation of Aden functions by downward reccurence.

On the computation of Riccati-Bessel functions

Peter Maličký, Marianna Maličká (1990)

Aplikace matematiky

The paper deals with the computation of Riccati-Bessel functions. A modification of Miller method is presented together with estimates of relative errors.

On the Relation between the S-matrix and the Spectrum of the Interior Laplacian

A. G. Ramm (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

The main results of this paper are: 1) a proof that a necessary condition for 1 to be an eigenvalue of the S-matrix is real analyticity of the boundary of the obstacle, 2) a short proof that if 1 is an eigenvalue of the S-matrix, then k² is an eigenvalue of the Laplacian of the interior problem, and that in this case there exists a solution to the interior Dirichlet problem for the Laplacian, which admits an analytic continuation to the whole space ℝ³ as an entire function.

On uniqueness in electromagnetic scattering from biperiodic structures

Armin Lechleiter, Dinh-Liem Nguyen (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectric structure mounted on a perfectly conducting plate in three dimensions. Given that uniqueness of solution holds, existence of solution follows from a well-known Fredholm framework for the variational formulation of the problem in a suitable Sobolev space. In this paper, we derive a Rellich identity for a solution to this variational problem under suitable smoothness conditions on the material parameter. Under additional...

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