# Vol 27, No 3 (2021)

**Year:**2021**Articles:**7**URL:**https://journals.ssau.ru/est/issue/view/549

## Full Issue

## Статьи

### Well-posedness of the main mixed problem for the multidimensional Lavrentiev — Bitsadze equation

#### Abstract

It is known that the oscillations of elastic membranes in space are modelled with partial differential equations. If the deflection of the membrane is considered as a function of u(x; t); x = (x1; :::; xm);m > 2; then, according to the Hamilton principle, we arrive to a multidimensional wave equation.

Assuming that the membrane is in equilibrium in the bending position, we also obtain the multidimensional Laplace equation from the Hamilton’s principle.

Consequently, the oscillations of elastic membranes in space can be modelled with a multidimensional Lavrentiev — Bitsadze equation.

The main mixed problem in the cylindrical domain for multidimensional hyperbolic equations in the space of generalized functions is well studied. In the works of the author, the well-posedness of this problem for multidimensional hyperbolic and elliptic equations is proved, and the explicit forms of classical solutions are obtained.

As far as we know, these questions for multidimensional hyperbolic-elliptic equations have not been studied.

The mixed problem with boundary-value conditions for the multidimensional Lavrentiev — Bitsazde equation is ill-posed.

In this paper, we prove the unique solvability and obtain an explicit form of classical solution of the

main mixed problem with boundary and initial conditions for the multidimensional Lavrentiev — Bitsadze equation.

**Vestnik of Samara University. Natural Science Series**. 2021;27(3):7-13

### CHARACTERISTIC PROBLEM FOR A FOURTH-ORDER EQUATION WITH A DOMINANT DERIVATIVE

#### Abstract

In this article we consider the Goursat problem for an equation with a dominating fourth-order mixed derivative and prove its unique solvability. The equation under consideration can be interpreted as a generalized Boussinesq — Love equation, which arises when describing longitudinal waves in a rod, taking into account transverse deformations. To justify the solvability, we proposed a method that is based on the possibility of reducing the problem posed to two Goursat problems for second-order equations. One of the problems is the classical Goursat problem for the simplest hyperbolic equation, while the other equation is loaded, and the study of the Goursat problem for it is the main result of the work.

**Vestnik of Samara University. Natural Science Series**. 2021;27(3):14-21

### DECOMPOSITION OF TRAVELING WAVES PROBLEMS

#### Abstract

In the article, the traveling waves problem for singularly perturbed systems of semilinear parabolic equations is considered. An effective method for the order reduction of singularly perturbed systems is proposed. The obtained mathematical results are used to study traveling waves both for abstract partial differential equations and for a specific model that can arise in physics problems, chemistry, and biology.

**Vestnik of Samara University. Natural Science Series**. 2021;27(3):22-30

### PROBLEMS OF DIFFERENTIAL AND TOPOLOGICAL DIAGNOSTICS. PART 7. DIFFERENTIAL DIAGNOSTICS IN SOME SYSTEMS OF DIRECT AND INDIRECT CONTROL

#### Abstract

Proposed work is the seventh work of the cycle on differential and topological diagnostics. Diagnostics of

malfunctions in the system of indirect control of an object whose motion is described by nonlinear ordinary differential equations of the third order (the problem of B.V. Bulgakov) is considered. In the diagnostic algorithm used, the scope of control is selected, when a certain constant value is assigned to each faulty system, and according to certain rules, the numbers obtained in the process of integrating equations and characterizing the functional state of the system are compared with it. Troubleshooting is also considered in one system of direct control of the movement of the aircraft, which can be described by nonlinear differential equations of the second order. At the same time, a certain diagnostic algorithm is built and used in accordance with the previously developed methodology.

**Vestnik of Samara University. Natural Science Series**. 2021;27(3):31-45

### APPLICATION OF THE UMAT SUBROUTINE FOR SOLVING CONTINUUM MECHANICS PROBLEMS (OVERVIEW)

#### Abstract

This paper presents an overview of the application of the UMAT subroutine of the SIMULIA Abaqus multifunctional software package in solid mechanics and related areas. This subroutine is used to describe new user materials that are not available in the class of standard materials of the SIMULIA Abaqus package. This overview article provides examples of problems and constitutive equations of materials that are modeled using UMAT / VUMAT procedures. Various types of materials are presented, successfully described by means of user-defined UMAT and VUMAT procedures. A general description and experience of using the UMAT subroutine is given. The results of finite element modeling of the deformation of a

plate weakened by a central circular hole under uniform and uniaxial tension with steady-state creep in a damaged medium evolving according to a power law are presented in the coupled formulation of the problem (creep - damage). The distributions of stresses, strains, and damage fields at the tip of the defect under creep conditions are found, and an analysis is made of the effect of the damage accumulation process on the stress fields at the crack tip under steady-state creep conditions. The distributions of stresses and creep strains are demonstrated taking into account the accumulation of damage over time.

**Vestnik of Samara University. Natural Science Series**. 2021;27(3):46-73

### ON DISCRETE SYSTEMS WITH POTENTIAL OPERATORS

#### Abstract

The main purpose of this work is to study the potentiality of a discrete system obtained from the system of the form C(t, u)u_ (t)+E(t, u) = 0 with continuous time. The definition of potentiality of the corresponding discrete system is introduced. Necessary and sufficient conditions for its potentiality with respect to a given bilinear form are obtained. The algorithm for the construction of the corresponding functional—the analogue of the Hamiltonian action—is presented. The illustrative example is given.

**Vestnik of Samara University. Natural Science Series**. 2021;27(3):74-82

### DECOMPOSITION OF ENZYME KINETICS SYSTEM WITH FAST AND SLOW VARIABLES IN SUICIDE SUBSTRATE PROBLEM

#### Abstract

In this paper an example of cooperative phenomenon, in which the substrate is known as suicide substrate, because it binds to the active enzyme as a substrate, inactivates because enzyme turns it into an inhibitor and provides an irreversible reaction is considered. In this case the substrate “commits suicide”. The aim of the work is to apply the method of integral manifolds to the reduction of the system of kinetics of suicide substrate. Work describes in detail the rationale for the decomposition algorithm of the enzyme kinetics problem for dynamical systems with fast and slow variables and the construction of integral manifolds for such systems, this article presents the results of applying the above methods to systems of the suicide substrate kinetics and compares solutions for four equations graphically. Comparisons of solutions for four equations are given graphically, the graphs are created using Microsoft Excel.

**Vestnik of Samara University. Natural Science Series**. 2021;27(3):83-88