Solution of one hypersingular integro-differential equation defined by determinants

The paper provides an exact analytical solution to a hypersingular inregro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with vatiable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.

Partial closure of the periodic system of variable width bridged slits in an isotropic medium under compression

The problem of compression of an isotropic medium by a periodic system of variable width slits comparable with elastic deformations is considered. The slit faces are under internal pressure. It is considered that the slits have end zones wherein material's tractions act. It is accepted that interaction of the slit's surface under applied loads may reduce the formation of contact zones of their surfaces. The formation of several contact areas of faces of each slit is studied. Therewith, it is considered that on some parts of a contact area there happens stick of the faces, on the remaining part there may happen slippage. The problem of equilibrium of a periodic system of slits with partially contacting faces under compressing load is reduced to the problem of linear conjugation of analytic functions. Determination of unknown parameters characterizing partial closing of a periodic system of variable width slits is reduced to the solution of the system of singular integro-differential equations. The contact stresses, the tractions in bonds between the slit faces in the end zones, the sizes of the contact areas, the adhesion and end zones are studied.

On A Problem of Linear Conjugation in the Case of Nonsmooth Lines and Some Measurable Coefficients

A boundary value problem of linear conjugation is considered for more general curves than those studied previously. A condition on the coefficient is found, under which the classical results are valid for these curves.

A Problem of Linear Conjugation for Analytic Functions with Boundary Values from the Zygmund Class

The solvability conditions are established for a problem of linear conjugation for analytic functions with boundary values from the Zygmund class 𝐿(ln+ 𝐿) α when the conjugation coefficient is piecewise-continuous in the Hölder sense. Solutions of the problem are constructed in explicit form.