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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, 78A45 We 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.

Best Constant in the Weighted Hardy Inequality: The Spatial and Spherical Version

Samko, Stefan — 2005

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26D10. The sharp constant is obtained for the Hardy-Stein-Weiss inequality for fractional Riesz potential operator in the space L^p(R^n, ρ) with the power weight ρ = |x|^β. As a corollary, the sharp constant is found for a similar weighted inequality for fractional powers of the Beltrami-Laplace operator on the unit sphere.

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

Gil, AlexeySamko, 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.

Mixed Fractional Integration Operators in Mixed Weighted Hölder Spaces

Mamatov, TulkinSamko, Stefan — 2010

Fractional Calculus and Applied Analysis

MSC 2010: 26A33 We study mixed Riemann-Liouville integrals of functions of two variables in Hölder spaces of different orders in each variables. We consider Hölder spaces defined both by first order differences in each variable and also by the mixed second order difference, the main interest being in the evaluation of the latter for the mixed fractional integral in both the cases where the density of the integral belongs to the Hölder class defined by usual or mixed differences. The...

On a Class of Fractional Type Integral Equations in Variable Exponent Spaces

Rafeiro, HumbertoSamko, Stefan — 2007

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 45A05, 45B05, 45E05,45P05, 46E30 We obtain a criterion of Fredholmness and formula for the Fredholm index of a certain class of one-dimensional integral operators M with a weak singularity in the kernel, from the variable exponent Lebesgue space L^p(·) ([a, b], ?) to the Sobolev type space L^α,p(·) ([a, b], ?) of fractional smoothness. We also give formulas of closed form solutions ϕ ∈ L^p(·) of the 1st kind integral equation M0ϕ = f, known as...

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