# Contemporary Mathematics. Fundamental Directions

**Editor-in-Chief: Revaz Valerianovich Gamkrelidze,** Academician of the Russian Academy of Sciences, Professor, Doctor of Physical and Mathematical Sciences, Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russian Federation

**ISSN:** 2413-3639 (print), 2949-0618 (online). **Founded in** 2003 г. **Publication frequency**: quarterly. **Peer-Review: **double blind

**Open Access**: Open Access. **APC: **no article processing charge. **Indexation**: elibrary.ru, mathnet.ru, Google Scholar, Lens, Research4Life

**Publication language**: Russian

**Publisher**: Peoples’ Friendship University of Russia (RUDN University)

The journal "**Contemporary Mathematics. Fundamental Directions**" is published both in English and Russian.

The journal is devoted to the actual topics of contemporary mathematics.

The journal is focused on publication of surveys as well as articles containing novel results.

The English version is published by "Springer Science+Business Media, Inc." (USA) in the series "Journal of Mathematical Sciences." ISSN: 1072-3374 (print version) ISSN: 1573-8795 (electronic version)

## Current Issue

### Vol 68, No 4 (2022)

**Year:**2022**Articles:**11**URL:**https://journals.rudn.ru/CMFD/issue/view/1620**DOI:**https://doi.org/10.22363/2413-3639-2022-68-4

#### Full Issue

#### Articles

##### In uence of numerical diffusion on the growth rate of viscous ngers in the numerical implementation of the Peaceman model by the finite volume method

###### Abstract

A numerical model of oil displacement by a mixture of water and polymer based on the Peaceman model is considered. Numerical experiments were carried out using the DuMu^{x} package, which is a software library designed for modeling nonstationary hydrodynamic problems in porous media. The software package uses the vertex-centered variant of finite volume method. The effect of diffusion on the growth rate of ''viscous fingers'' has been studied. The dependencies of the leading front velocity on the value of model diffusion are obtained for three viscosity models. It is shown that the effect of numerical diffusion on the growth rate of ''viscous fingers'' imposes limitations on calculations for small values of model diffusion.

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):553-563

##### Boundary singular problems for quasilinear equations involving mixed reaction-diffusion

###### Abstract

We study the existence of solutions to the problem

\[\label{eng_A1} \begin{array}{rl} -\Delta u+u^p-M|\nabla u|^q=0 & \text{in }\;\Omega,\\ u=\mu & \text{on }\;\partial\Omega \end{array}\]

in a bounded domain \(\Omega\), where \(p>1\), \(1, \(M>0\), \(\mu\) is a nonnegative Radon measure in \(\partial\Omega\), and the associated problem with a boundary isolated singularity at \(a\in\partial\Omega,\)

\[\label{eng_A2} \begin{array}{rl} -\Delta u+u^p-M|\nabla u|^q=0 & \text{in }\;\Omega,\\ u=0 & \text{on }\;\partial\Omega\setminus\{a\}. \end{array}\]

The difficulty lies in the opposition between the two nonlinear terms which are not on the same nature. Existence of solutions to [eng_A1] is obtained under a capacitary condition \[\mu(K)\leq
c\min\left\{cap^{\partial\Omega}_{\frac{2}{p},p'},cap^{\partial\Omega}_{\frac{2-q}{q},q'}\right\}\quad\text{for
all compacts }K\subset\partial\Omega.\] Problem [eng_A2] depends on several critical exponents on \(p\) and \(q\) as well as the position of \(q\) with respect to \(\dfrac{2p}{p+1}\).

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):564-574

##### Construction of the planar vector fields with nonsimple critical point of prescribed topological structure

###### Abstract

The problem of constructing n-linear (n 2) plane vector elds with isolated critical point and given separatrices of prescribed types is considered. Such constructions are based on the use of vector algebra, the qualitative theory of second-order dynamic systems and classical methods for investigating their critical points. This problem is essentially an inverse problem of the qualitative theory of ordinary di erential equations, and its solution can be used to synthesize mathematical models of controlled dynamical systems of various physical nature.

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):575-595

##### Nonautonomous dynamics: classification, invariants, and implementation

###### Abstract

The work is a brief review of the results obtained in nonautonomous dynamics based on the concept of uniform equivalence of nonautonomous systems. This approach to the study of nonautonomous systems was proposed in [10] and further developed in the works of the second author, and recently - jointly by both authors. Such an approach seems to be fruitful and promising, since it allows one to develop a nonautonomous analogue of the theory of dynamical systems for the indicated classes of systems and give a classi cation of some natural classes of nonautonomous systems using combinatorial type invariants. We show this for classes of nonautonomous gradient-like vector elds on closed manifolds of dimensions one, two, and three. In the latter case, a new equivalence invariant appears, the wild embedding type for stable and unstable manifolds [14,17], as shown in a recent paper by the authors [5].

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):596-620

##### On a system of differential equations with random parameters

###### Abstract

An explicit formula for the mathematical expectation and second moment functions of a solution to a linear system of ordinary differential equations with a random parameter and a vector random righthand side is obtained. The problem is reduced to the deterministic Cauchy problem for systems of first-order linear partial differential equations. We obtain an explicit formula for a solution of linear systems of partial differential equations of the first order with constant coeffcients. An example is given showing that random factors can have a stabilizing effect on a linear system of differential equations.

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):621-634

##### A model of string system deformations on a star graph with nonlinear condition at the node

###### Abstract

In this paper, a model of deformations of Stieltjes strings system located along a geometric star graph with a nonlinear condition at the node is studied. This kind of condition arises due to the presence of a limiter for the movement of strings in the node under the in uence of an external load. In the present paper, the necessary and sufficient conditions for the extremum of the energy functional are established; existence and uniqueness theorems for the solution are proved; the critical loads at which the strings come into contact with the limiter are analyzed; the dependence of the solution on the limiter length is studied.

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):635-652

##### Explicit solution of a Dirichlet problem in nonconvex angle

###### Abstract

In the present work, we give an explicit solution of the Dirichlet boundary-value problem for the Helmholtz equation in a nonconvex angle with periodic boundary data. We present uniqueness and existence theorems in an appropriate functional class and we give an explicit formula for the solution in the form of the Sommerfeld integral. The method of complex characteristics [14] is used.

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):653-670

##### Homogenization of a parabolic equation in a perforated domain with a unilateral dynamic boundary condition: critical case

###### Abstract

In this paper, we study the homogenization of a parabolic equation given in a domain perforated by ''tiny'' balls. On the boundary of these perforations, a unilateral dynamic boundary constraints are specified. We address the so-called ''critical'' case that is characterized by a relation between the coefficient in the boundary condition, the period of the structure and the size of the holes. In this case, the homogenized equation contains a nonlocal ''strange'' term. This term is obtained as a solution of the variational problem involving ordinary differential operator.

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):671-685

##### Numerical analysis of stationary solutions of systems with delayed argument in mathematical immunology

###### Abstract

This work is devoted to the technology developed by the authors that allows one for fixed values of parameters and tracing by parameters to calculate stationary solutions of systems with delay and analyze their stability. We discuss the results of applying this technology to Marchuk-Petrov's antiviral immune response model with parameter values corresponding to hepatitis B infection. The presence of bistability and hysteresis properties in this model is shown for the first time.

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):686-703

##### Maslov complex germ and semiclassical contracted states in the Cauchy problem for the Schrödinger equation with delta potential

###### Abstract

We describe the semiclassical asymptotic behavior of the solution of the Cauchy problem for the Schrödinger equation with a delta potential localized on a surface of codimension 1. The Schrödinger operator with a delta potential is defined using the theory of extensions and is given by the boundary conditions on this surface. The initial data are selected as a narrow peak, which is a Gaussian packet localized in a small neighborhood of the point. To construct the asymptotics, we use the Maslov complex germ method. We describe the re ection of the complex germ from the carrier of the delta potential.

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):704-715

##### Existence of solution of a free boundary problem for reaction-diffusion systems

###### Abstract

In this paper, we prove the existence of solution of a novel free boundary problem for reaction-diffusion systems describing growth of biological tissues due to cell influx and proliferation. For this aim, we transform it into a problem with fixed boundary, through a change of variables. The new problem thus obtained has space and time dependent coeffcients with nonlinear terms. We then prove the existence of solution for the corresponding linear problem, and deduce the existence of solution for the nonlinear problem using the xed point theorem. Finally, we return to the problem with free boundary to conclude the existence of its solution.

**Contemporary Mathematics. Fundamental Directions**. 2022;68(4):716-731