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[en] We present a brief analysis of the investigations carried out in the mechanics of coupled fields with regard for the thermomechanical behavior of thermally sensitive bodies carried out by the researchers of the Lviv scientific school in recent years.

[en] In this work, we consider the problem of interaction of elastic body with scalar field. The general solution of a uniform system of equations (of elasticity theory) for the static case is solved by using the Papkovich representation method. The contact problem is solved by using a special boundary-contact condition, in the case where the contact surface is a stretched spheroid. The uniqueness theorem for the solution is also proved. Solutions are obtained in the form of absolutely and uniformly convergent series.

[en] For a Kolmogorov-type ultraparabolic equation with two groups of spatial variables, we establish estimates for the increments of the classical fundamental solution of the Cauchy problem and its derivatives in the spatial variables.

[en] We prove the existence of the derivative of the acoustic single layer potential and study some properties of the operator generated by this derivative in generalized Hölder spaces.

[en] We study the motion of an incompressible polymeric fluid with volume charge and construct stationary solutions of the electrohydrodynamical equations.

[en] We find and analyze exact solutions to equations with logarithmic nonlinearity. To construct the soltuions, we use the classical group analysis methods, as well as the method of invariant manifolds and the Lagrangian coordinate method. Bibliography: 23 titles. Illustrations: 5 figures.

[en] We describe applications of asymptotic methods to problems of mathematical physics and mechanics, primarily, to the solution of nonlinear singularly perturbed problems in local domains. We also discuss applications of Padé approximations for transformation of asymptotic expansions to rational or quasi-fractional functions.

[en] We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudary-layer functions. The construction of the asymptotics is based on the regularization method for singularly perturbed problems developed by S. A. Lomov and adapted to singularly perturbed parabolic equations with two viscous boundaries by A. S. Omuraliev.

[en] We prove that every field of characteristic 0 whose cardinality does not exceed the bounding number 6 is a base field of some csp-ring.