Vol 215, No 1 (2016)

The present paper deals with the investigation of the conditions under which infinite branched continued fractions are stable under perturbations of their elements. We establish the formulas for the relative errors of the approximants of branched continued fractions with complex partial denominators and numerators that are equal to one. By using the technique of the sets of elements and the corresponding sets of values of the tails of approximants, we construct the sets of relative stability under perturbations, namely, the angular sets and the sets representing the exterior domains of circles on the even floors of the fraction and half planes on its odd floors. We also establish estimates for the relative errors of approximants of these branched continued fractions.

We continue our investigation of the problem of synthesis for a radiating system with flat aperture according to a prescribed power directivity pattern originated in [P. O. Savenko, J. Math. Sci.,208, No. 3, 366–381 (2015)]. As a characteristic feature of problems from this class, we can mention the non-uniqueness and bifurcations of their solutions. On the basis of the theory of branching, we present a technique of finding the nonzero solutions (in the first approximation). This technique enables us to determine the main properties of solutions, which significantly simplifies the problem of finding the optimal solutions of the problem of synthesis by numerical methods.

For nonlinear systems of reaction–diffusion type, we propose a technique for the construction and analysis of solutions based on the method of small parameter. The proposed technique enables us not only to analytically obtain approximate quasiharmonic low-amplitude solutions appearing as a result of bifurcations of spatially homogeneous states of the system but also to determine the type of bifurcation in the system. Examples of application of this approach to the analysis of bifurcations and the construction of solutions of a specific mathematical reaction–diffusion model are given.

A general variational theorem is formulated and proved for the dynamical linear problem of coupled mechanothermodiffusion with initial conditions in inhomogeneous anisotropic shells with distortions.

We propose a mathematical model of the frictional contact of bodies with periodic surface textures. The bodies interact in two stages. First, they are pressed to each other by a monotonically increasing nominal pressure. Then the nominal tangential stresses are applied to the bodies, which leads to the frictional slip of their surfaces in the vicinities of the gaps. The contact problem is reduced to a system of singular integral equations for the functions of heights of the intercontact gaps and the relative shift of the surfaces in the slip zones. The algorithm used for the solution of this problem is described.