On the Prandtl Equation

The unique solvability of the airfoil (Prandtl) integro-differential equation on the semi-axis ⁺ = [0, ) is proved in the Sobolev space W _p ¹ and Bessel potential spaces H _p ^s under certain restrictions on p and s.     

On the uniqueness of a solution of the inverse problem for a simple-layer potential

We prove the uniqueness of a solution of the inverse problem for a simple-layer potential for star-shaped smooth surfaces in the case of the metaharmonic equation Δv − K ² v = 0. For the Laplace equation, an analogous statement is not true. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 892–899, July, 2008.     

On the uniqueness of a solution of the problem with oblique derivative for the equation Δnν = 0

We prove the uniqueness of a solution of the problem with oblique derivative for the equation Δⁿν = 0. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 835–841, June, 2006.     

The coronograph developed for the KORONAS-F orbital solar observatory

We present the design and optical scheme of the externally occulted coronograph developed for the KORONAS-F orbital solar observatory.     

Dirichlet-Neumann-impedance boundary value problems arising in rectangular wedge diffraction problems

Boundary value problems originated by the diffraction of an electromagnetic (or acoustic) wave by a rectangular wedge with faces of possible different kinds are analyzed in a Sobolev space framework. The boundary value problems satisfy the Helmholtz equation in the interior (Lipschitz) wedge domain, and are also subject to different combinations of boundary conditions on the faces of the wedge. Namely, the following types of boundary conditions will be under study: Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, Impedance-Dirichlet, and Impedance-Neumann. Potential theory (combined with an appropriate use of extension operators) leads to the reduction of the boundary value problems to integral equations of Fredholm type. Thus, the consideration of single and double layer potentials together with certain reflection operators originate pseudo-differential operators which allow the proof of existence and uniqueness results for the boundary value problems initially posed. Furthermore, explicit solutions are given for all the problems under consideration, and regularity results are obtained for these solutions.

     

Accelerating effect of silica on the indicator reaction o-dianisidine-H(2)O(2)

Reaction of oxidation of o-dianisidine (o-D) with H(2)O(2) which is widely used in catalytic methods of analysis in solution has been conducted on silica plates for thin-layer chromatography. The rate of the reaction catalyzed by model compounds (p-toluenesulphonyl chloride, methyl benzoate, benzoic acid, and acrylamide) is noticeably higher on silica than in solution in comparable conditions. The degree of acceleration varies depending on the catalyst and is more pronounced at its lower concentrations. By use of p-toluenesulphonyl chloride determination as an example it has been shown that the accelerating effect of silica enables to decrease the detection limit down to 0.07 nmol cm(-2) (as compared with 4 nmol.cm(-2) in solution); the accuracy is not diminished. It is concluded that catalytic indicator reactions on solid supports may represent high interest for analytical chemists.     

On wave diffraction by a half‐plane with different face impedances

The impedance wave diffraction problem by a half‐plane screen is revisited in view of its well‐posedness upon different impedance and wave parameters. The problem is analysed with the help of potential and pseudo‐differential operators. Seven conditions between the impedance and wave numbers are found under which the problem will be well‐posed in Bessel potential spaces. In addition, an improvement of the regularity of the solutions is shown for the previous seven conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Mixed boundary value problems of diffraction by a half‐plane with an obstacle perpendicular to the boundary

The paper is devoted to the analysis of wave diffraction problems modeled by classes of mixed boundary conditions and the Helmholtz equation, within a half‐plane with a crack. Potential theory together with Fredholm theory, and explicit operator relations, are conveniently implemented to perform the analysis of the problems. In particular, an interplay between Wiener–Hopf plus/minus Hankel operators and Wiener–Hopf operators assumes a relevant preponderance in the final results. As main conclusions, this study reveals conditions for the well‐posedness of the corresponding boundary value problems in certain Sobolev spaces and equivalent reduction to systems of Wiener–Hopf equations. Copyright © 2013 John Wiley & Sons, Ltd.