Vol 26, No 3 (2020)

This article is devoted to the study of the solvability of some boundary value problems with involution.
In the space Rn, the map Sx=−x is introduced. Using this mapping, a nonlocal analogue of the Laplace operator is introduced, as well as a boundary operator with an inclined derivative. Boundary-value problems are studied that generalize the well-known problem with an inclined derivative. Theorems on the existence and uniqueness of the solution of the problems under study are proved. In the Helder class, the smoothness of the solution is also studied. Using well-known statements about solutions of a boundary value problem with an inclined derivative for the classical Poisson equation, exact orders of smoothness of a solution to the problem under study are found.

Proposed work is the fifth work of the cycle on differential and topological diagnostics. The article gives an estimate of the errors method of direction fields in the case of not accurate trajectory measurements, but trajectory measurements with an error are limited by the modulus of a given a smooth function of time, and in case this error is a random variable distributed according to the normal law with fixed parameters. We show that in these more complex cases, you can specify the “best” number of required trajectory measurements, in
which the proposed algorithms of diagnostics will work constructively, and the malfunction will be determined unambiguously.

The article investigates the residual stresses arising in a thermoelastic cylinder as a result of layer-by-layer deposition of material on its lateral surface. Residual stresses are defined as the limiting values of internal stresses developing during the technological process. Internal stresses are caused by incompatible deformations that accumulate in the body as a result of joining parts with different temperatures. For the analysis of internal stresses, an analytical solution of the axisymmetric quasi-static problem of thermoelasticity for a layer-by-layer growing cylinder is constructed. It is shown that the distribution of residual stresses depends
on the scenario of the surfacing process. In this case, the supply of additional heat to the growing body can significantly reduce the unevenness of the temperature fields and reduce the intensity of residual stresses. The most effective is uneven heating, which can be realized, for example, by the action of an alternating current with a tunable excitation frequency. This is illustrated by the calculations performed using the constructed
analytical solution.