Том 243, № 1 (2019)

We study the semilocal convergence of the combined Newton–Kurchatov method to a locally unique solution of the nonlinear equation under weak conditions imposed on the derivatives and first-order divided differences. The radius of the ball of convergence is established and the rate of convergence of the method is estimated. As a special case of these conditions, we consider the classical Lipschitz conditions.

We consider a boundary-value multipoint (in time) problem with Dirichlet condition for a second-order parabolic equation with power singularities and degenerations of any order in coefficients with respect to spatial variables in a certain set of points. The conditions of existence and uniqueness of the solution of the posed problem in Hölder spaces with power weight are established.

We consider a new approach to the geometrization of electromagnetism based on the specific interpretation of the Kaluza–Klein theory. It is proposed to represent the five-dimensional space of this theory as a formal tool for the analysis of a four-dimensional space with torsion. For this purpose, we study the consequences of parametrization of a specific space of this type along the lines of the corresponding vector field that characterizes its geometry. It is shown that, within the framework of the new approach, all results of the Kaluza–Klein theory can be also obtained for a four-dimensional space with torsion (with some distinctions and generalizations).

We consider a problem of forced resonance vibration and dissipative heating of a hinged flexible viscoelastic beam with piezoactuators in the presence of transverse shear strains. The effects of geometric nonlinearities, in-plane shear strains, and the conditions of heat exchange on the surfaces on the amplitude and temperature-frequency characteristics of the forced vibration of the beam and the thermal failure of the system are investigated. The possibility of active damping of the mode of flexural vibrations by piezoactuators is analyzed.

We reduce the elastoplastic problem of limit equilibrium of a transversely isotropic cylindrical shell weakened by an internal longitudinal plane crack of any configuration located in the field of residual stresses to the problem of elastic equilibrium of the same shell containing a through crack of unknown length. This problem, in turn, is reduced to a system of nonlinear singular integral equations. We propose an algorithm for the numerical solution of the obtained system together with the conditions of plasticity, boundedness of stresses, and uniqueness of displacements.

We propose a variational approach to the solution of the problem of optimization of a stationary axisymmetric thermal stressed state of a finite solid cylinder by controlling the distribution of volumetric heat sources. The proposed approach is based on the variational method of homogeneous solutions developed earlier for the solution of axisymmetric problems of the theory of elasticity for a cylinder. We study the influence of the height-to-radius ratio of the cylinder on the optimal values of the objective functional and the stressed state.

The nonaxisymmetric problem of heat conduction for a piecewise-homogeneous transversely isotropic space with two (thermally active and thermally insulated) internal inclusions located parallel to the plane of conjugation of two different transversely isotropic half spaces is reduced to a system of two two-dimensional singular integral equations. The solution of this system is constructed in the form of series in Jacobi polynomials. As a result, we obtain the dependences of the temperature distribution on the thermophysical properties of materials and on the distances between the inclusions and the interface of the half spaces. The quantitative and qualitative specific features of the temperature field in the neighborhood of inclusions are analyzed.