Riemann Boundary Value Research Articles

Closed-Form Solvability of a Linear Conjugation Problem for Three-Dimensional Piecewise Analytic Vector

We consider a three-dimensional Riemann boundary value problem which has two particular solutions such that, for one of the components, the ratios of the boundary values coincide on the conjugation contour. It is shown that, under certain restrictions on the elements of the matrix coefficient of the problem, its general solution can be obtained in a closed form.

Riemann and Dirichlet Type Problems for Hyperanalytic Functions in Fractal Domains

In this paper we are interested in finding solvability conditions for Riemann and Dirichlet type boundary value problems with generalized Holder-continuous boundary data in the case of hyperanalytic functions defined in a domain of the complex plane bounded by an h-summable closed curve.

An electroded electrically and magnetically charged interface crack in a piezoelectric/piezomagnetic bimaterial

An interface crack between two semi-infinite piezoelectric/piezomagnetic spaces under the action of remote mechanical loading, magnetic and electric fluxes is considered. The bimaterial is polarized in the direction orthogonal to the crack faces, and all fields are assumed to be independent of the coordinate co-directed with the crack front. The crack faces are covered with mechanically soft electrodes of ferroelectric material having a prescribed electric charge and residual magnetic induction. Special presentations of electromagnetomechanical quantities via sectionally analytic functions are used. “Open” and contact zone interface crack models are considered. In the first case, the formulated problem is reduced to a Hilbert problem and in the second one to a sequence of combined Dirichlet–Riemann and Hilbert boundary value problems. In both cases, the solutions of the mentioned problems are presented in exact analytical form. The well-known phenomenon of the crack-face interpenetration is observed for the “open” model, and the transcendental equation for the contact zone length determination is derived for the contact zone model. For a numerical illustration of the obtained results, the influence of the mechanical loading and the intensity of electric and magnetic fluxes upon the crack opening, mechanical, electric and magnetic fields at the crack region are shown.

On contact zone models for an electrically limited permeable interface crack in a piezoelectric bimaterial

An electrically impermeable interface crack between two semi-infinite piezoelectric planes under remote mechanical tension-shear and electrical loading is studied. Assuming the stresses, strains and displacements are independent on the coordinate x2 the expressions for the elastic displacement and potential jumps as well as for the stresses and electrical displacement along the interface via a sectionally holomorphic vector function are found. Introducing an artificial contact zone at the right crack tip and assuming the materials possess the symmetry class 6 mm the problem is reduced for a wide range of bimaterial compounds to a combination of combined Dirichlet–Riemann and Hilbert boundary value problems which are solved analytically. From these solutions clear analytical expressions for characteristic mechanical and electrical parameters are derived. As particular cases of the above mentioned solution the classical (oscillating) and contact zone solutions are obtained. Further, a comparison with an associated solution for an electrically permeable crack has been performed. The fracture mechanical parameters for all models via the remote loads are found analytically and important relationships between these parameters are obtained. Due to these relationships an important algorithm of a numerical method applicable for the investigation of an interface crack in a finite sized piezoelectric bimaterial is suggested.

The first basic boundary value problem of Riemann's type for bianalytical functions in a plane with slots

Abstract The paper is devoted to the investigation of one of the basic boundary value problems of Riemann's type for bianalytical functions. In the course of work there was made out a constructive method for solution of the problem given in a plane with slots. There was also found out that the solution of the problem under consideration consists of consequent solutions of two Riemann's boundary value problems for analytical functions in a plane with slots. Besides, a picture of solvability of the problem is being searched and its Noether is identified.

THE FIRST BASIC BOUNDARY VALUE PROBLEM OF RIEMANN'S TYPE FOR BIANALYTICAL FUNCTIONS IN A PLANE WITH SLOTS

The paper is devoted to the investigation of one of the basic boundary value problems of Riemann's type for bianalytical functions. In the course of work there was made out a constructive method for solution of the problem given in a plane with slots. There was also found out that the solution of the problem under consideration consists of consequent solutions of two Riemann's boundary value problems for analytical functions in a plane with slots. Besides, a picture of solvability of the problem is being searched and its Noether is identified. Šiame darbe tyrinejamas uždavinys, kai ieškoma dalimis bianaliziniu funkciju, nykstančiu begalybeje, apribotu greta kontūro trūkio tašku ir šiame kontūre tenkinančiu dvi kraštines salygas. Parodoma, kad sprendžiamas uždavinys suvedamas i sprendima dvieju Rimano uždaviniu analizinems funkcijoms.

THE SECOND BOUNDARY VALUE PROBLEM OF RIEMANN'S TYPE FOR BIANALYTICAL FUNCTIONS WITH DISCONTINUOUS COEFFICIENTS

The paper is devoted to the investigation of one of the basic boundary value problems of Riemann's type for bianalytical functions with discontinuous coefficients. In the course of work there was made out a constructive method for solution of the problem in a unit circle. There was also found out that the solution of the problem under consideration consists in consequent solutions of two Riemann's boundary value problems for analytical functions in a unit circle. Besides, the example is constructed.

On obtaining a solution of the equation for an airfoil: PMM vol. 43, no. 3, 1979, pp. 475–479

A number of artificial procedures is used to reduce the basic, two-dimensional singular integral equation of the linear theory of thin airfoil of arbitrary aspect ratio, arbitrary form in the plane, and with the angle of attack varying along the wing span, to a form allowing the separation of variables. It is shown that the solution of the resulting equation can be reduced to solving certain homogeneous Riemann's boundary value problems admitting a solution by means of the Cauchy type integrals [1].