Неустроева Н.В., Лазарев Н.П. «Дифференциальные уравнения, динамические системы и оптимальное управление. Задача сопряжения для упругих балок Бернулли–Эйлера и Тимошенко» Сибирские электронные математические известия, № 13, с. 26-48 (2016)

In this paper, we consider а junction problem for the system Euler–Bernoulli and Timoshenko elastic beams and а contact problem for the two connecting beams. Unique solvability of these problems is proved. Under the assumption that solutions are smooth we find the corresponding differential formulations of the initial variational problems. In particular junction conditions on the border of bonding interface obtained. The analytical solution for a beams with a cut is given.

Мейрманов А.М., Гриценко С.А., Герус А.А. «Дифференциальные уравнения, динамические системы и оптимальное управление. Усредненные модели изотермической акустики в конфигурации «жидкость–пороупругая среда»» Сибирские электронные математические известия, № 13, с. 49-174 (2016)

We consider a mathematical model of the isothermal acoustics in composite medium with two different components: liquid region and the elastic body perforated by a system of pores, filled the same liquid. The model is based on the classical axioms of continuum mechanics and contains rapidly oscillating coefficients that depend on a small parameter. Such a model, although precise enough, cannot, however, be used for numerical calculations. The problem is solved by using homogenization, i. e. the derivation of the equations not containing rapidly oscillating coefficients. Separately for the fluid and separately for poroelastic medium results already obtained previously. In this configuration of the two-component medium the main problem is the conditions of continuity at the common boundary between the liquid region and poroelastic region. In the present work are displayed six homogenized models of different complexity with the various coefficients characterizing the medium.

Холмуродов А.Е., Тошмуродова Г. «Дифференциальные уравнения, динамические системы и оптимальное управление. Об особых решениях одномерного уравнения SH волн в пористых средах» Сибирские электронные математические известия, № 13, с. 300-351 (2016)

Singular solutions of the IS equation for SH waves in an elasticporous medium are obtained. For expansion coefficients of wave fields a system of Volterra integral equations of the second kind are obtained. It is shown that at vanishing of proposity these coefficients are transformed into well known expressions for the coefficients of expansion of wave fields for an elastic model.