and Computational Methods

#### 539.3 Asymptotic theory of thin two-layer elastic plates with layer slippage

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2019-1-326

The problem of deformation of thin two-layer plates, for which a slip condition is speci-fied at the interface between the layers, instead of the classical case of ideal contact, is considered. The method of asymptotic analysis of the general equations of the 3-dimensional theory of elasticity is used to solve this problem under the influence of transverse pressure, longitudinal and shear forces on the end surfaces. Asymptotic analysis is performed using a small geometric parameter representing the ratio of thickness to the characteristic length of the plate. Recurrent formulations of local quasi-one-dimensional problems of elasticity theory with slippage are obtained. For these problems, explicit analytical solutions are obtained. The averaged equations of elastic equilibrium of a two-layer plate with slippage of layers are derived. It is shown that, due to slippage, the order of the averaged equations of the theory of plates increases to 5 orders of magnitude, in contrast to the classical 4th order, which takes place in the theory of Kirchhoff – Love plates. Additional boundary conditions to this 5th order system are formulated and its analytical solution is obtained for the case of a rectangular plate under the influence of uniform pressure. A numerical analysis of the solution of the averaged problem is carried out. It is shown that the presence of layer slippage significantly increases the deflection of the plate in comparison with the conditions of ideal contact of the layers.

Димитриенко Ю.И., Губарева Е.А. Асимптотическая теория тонких двухслой-ных упругих пластин с проскальзыванием слоев. Математическое моделирование и численные методы. 2019. № 1. с. 3–26.

#### 539.3 Modeling of loads on a composite cylindrical shell with an elastic filler

**Dubrovin V. M. (Bauman Moscow State Technical University), Semyonov K. S. (Bauman Moscow State Technical University/RSC Energia)**

doi: 10.18698/2309-3684-2019-1-2742

A method for calculating the loads on a composite cylindrical shell, consisting of external and internal shells connected by a system of elastic transverse supports, is proposed. Between the shells is an elastic filler. The method takes into account the geometry and mechanical characteristics of the shells, the elastic characteristics of the transverse supports and the physico-mechanical properties of the material of the elastic aggregate. In solving the problem, it is assumed that the material of the elastic aggregate satisfies the basic relations of the theory of elasticity, and the elastic characteristics of the aggregate under dynamic loading correspond to the characteristics under static loading. This allows you to use the results to solve problems in both static and dynamic formulations. By choosing a different combination of characteristics of the shells and the elastic filler, it is possible to provide the most favorable loading conditions for both the inner and outer shells, depending on the statement of the problem. As an example, the loads on the inner shell were studied depending on the characteristics of the outer shell and the specific stiffness of the elastic filler. Similarly, estimates of the loads acting on the outer shell can be obtained.

Дубровин В.М., Семёнов К.С. Моделирование нагрузок на составную цилин-дрическую оболочку с упругим заполнителем. Математическое моделирование и численные методы, 2019, № 1, с. 27–42.

#### 517.956.4 On the numerical solution of the inverse problem of heat conduction with radiation

**Gribov A. F. (Bauman Moscow State Technical University), Zhidkov E. N. (Bauman Moscow State Technical University), Krasnov I. K. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2019-1-4353

The inverse problem of restoring the thermal conductivity coefficient of a nonlinear parabolic equation by the final temperature distribution, which serves as a mathematical model for the problem of determining structural defects, is investigated. An algorithm for numerical solution of the problem is proposed. A numerical example is considered.

Грибов А.Ф., Жидков Е.Н., Краснов И.К. О численном решении обратной зада-чи теплопроводности с излучением. Математическое моделирование и численные методы, 2019, № 1, с. 43–53.

#### 519.8 Stochastic model of repelling of attacks made by different types of means with a preemptive strike by one of the parties

**Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2019-1-5464

On the basis of the theory of continuous Markov processes, it was developed a stochastic model of reflection by the combat unit of the attack of two different types of enemy units with a preemptive strike of one of the opposing sides. The calculation formulas for calculating the current and final status are obtained. It is shown that the choice of the defending unit of tactics of firing does not depend on which of the opposing sides causes a preemptive strike, but it’s correct choice can significantly increase the probability of it’s victory. The model of two-way combat developed in this article can be used for estimation of the multi-purpose weapons systems combat effectiveness.

Чуев В.Ю., Дубограй И.В. Стохастическая модель отражения атаки разнотип-ных средств при упреждающем ударе одной из сторон. Математическое модели-рование и численные методы, 2019, № 1, с. 54–64.

#### 517.9 Methods of functional separation of variables and their application in mathematical physics

**Polyanin A. D. (Bauman Moscow State Technical University/Ishlinsky Institute for Problems in Mechanics/MEPhI)**

doi: 10.18698/2309-3684-2019-1-6597

A brief review of existing modifications of the method of functional separation of varia-bles is given. A new more general approach is proposed for construction of ex-act solu-tions of nonlinear equations of mathematical physics and mechanics, which is based on implicit transformations of integral type in combination using the split-ting principle. The effectiveness of this approach is illustrated on nonlinear diffusion equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on two or three arbitrary functions (such nonlinear equations are the most difficult to analyze). Many new exact solutions with functional separation of variables and generalized traveling wave type solutions are described. The obtained solutions can be used to test various numerical and approximate analytical methods of mathematical physics

Полянин А.Д. Методы функционального разделения переменных и их применение в математической физике. Математическое моделирование и численные методы, 2019, № 1, с. 67–97.

#### 517.925:519.2:519.6 Modeling of a differential equations system with dynamical invariants

**Karachanskaya E. V. (Far–Eastern State Transport University/Pacific National University)**

doi: 10.18698/2309-3684-2019-1-98117

This article is devoted to the construction method of differential equations system with smooth functions as invariants. This method allows us to construct both deterministic differential equations system and Ito’s stochastic differential equations system. These stochastic differential equations are diffusion equations or jump-diffusion ones. The algorithm under consideration is based on the author’s previous research and has no parallel. We study the realization of the algo¬rithm in R2 and in R3 in detail. We represent a MathCad-program for construction of every type of differential equations system de-scribed above. We also show some examples of differential equa¬tions system with the given invariant constructed automatically be the computer program. Rele¬vance of our research is guaranteed by numerical solution of the created system of differential equa-tions. These systems are solved using well-known numerical techniques: Euler method, Euler-Murayama method and Monte-Carlo procedure. The numerical solution of a sys-tem of differential equations and the corresponding values of the invariant function are represented in diagram form. Programmed control with probability 1 for a stochastic SIR-model is an application of the presented theory and the MathCad-program. Results of this research can be used to construct models of dynamical systems with invariant and to further explore these systems.

Карачанская Е.В. Моделирование систем дифференциальных уравнений с ди-намическими инвариантами. Математическое моделирование и численные мето-ды, 2019, № 1, с. 98–117.