Vol 25, No 4 (2021)

Differential Equations and Mathematical Physics

Asymptotics of the eigenvalues of a boundary value problem for the operator Schrödinger equation with boundary conditions nonlinearly dependent on the spectral parameter

Hashimoglu I.F.


On the space H1 = L2(H, [0, 1]), where H is a separable Hilbert space, we study the asymptotic behavior of the eigenvalues of a boundary value problem for the operator Schrödinger equation for the case when one, and the same, spectral parameter participates linearly in the equation and quadratically in the boundary condition. Asymptotic formulae are obtained for the eigenvalues of the considered boundary value problem.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):607-615
pages 607-615 views

Hermitian metrics with (anti-)self-dual Riemann tensor

Krivonosov L., Luk'yanov V.


Equations of (anti-)self-duality for the components of the Levi–Civita connection of the Hermitian positive definite metric (not for the Riemann tensor) are compiled. With this well-known method, a simpler system of partial differential equations is obtained, which implies the (anti-)self-duality of the Riemann tensor. This system is of the 1st order, while the (anti-)self-duality conditions of the Riemann tensor are expressed by equations of the 2nd order. However, this method can obtain only particular solutions of the (anti-)self-duality equations of the Riemann tensor. The constructed equations turned out to be significantly different in the self-dual and anti-self-dual cases. In the case of self-duality, the equations are divided into three classes, for each of which a general solution is found. In the anti-self-dual case, we did not find the general solution, but gave two series of particular solutions. The connection between our solutions and Kähler metrics is shown. In the case of the (anti-)self-duality of the Levi–Civita connection for the Hermitian metric, a general form of parallel almost complex metric-preserving structures is obtained. These structures are all torsion free. For an arbitrary positive definite 4-metric, a general form of almost complex structures preserving this metric is found.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):616-633
pages 616-633 views

On the asymptotics of spectrum of an even-order differential operator with a delta-function potential

Mitrokhin S.I.


We study a sequence of differential operators of high even order whose potentials converge to the Dirac delta-function. One of the types of separated boundary conditions is considered. At the points of potential discontinuity, it is necessary to study the conditions of gluing for the correct determination of the corresponding differential equations solutions. For large values of the spectral parameter, asymptotic solutions of differential equations are furnished by the Naimark method. The conditions of gluing are studied, the boundary conditions are investigated, the equation for the eigenvalues of the considered differential operator is derived. The method of successive approximations is used to find the asymptotics of spectrum of studied differential operators, the limit of which determines a spectrum of operator with a delta-function potential.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):634-662
pages 634-662 views

Modeling of the “reaction–diffusion” transfer process in a nonlinear electromagnetic field

Uvarova L.A., Linn P.W.


The paper deals with a model of two-component mass transfer in an inhomogeneous spherical system. A model of two-component mass transfer in an inhomogeneous spherical system occurring in a nonlinear electromagnetic field is investigated. It is shown that the concentration in the inner region, at the boundary of the regions, as well as the concentration that crossed the border and got into the second region, depend on the nonlinearity parameter of the electrodynamic problem.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):663-675
pages 663-675 views

Mechanics of Solids

Creep and long-term fracture of a narrow rectangular membrane inside a high rigid matrix with proportional dependence on the transverse pressure on time

Lokoshchenko A.M., Fomin L.V., Basalov Y.G., Tretyakov P.M.


In this work, we studied the creep and long-term fracture of a narrow rectangular membrane in confined conditions (inside a high rigid matrix) with a proportional dependence on the magnitude of transverse pressure on time.

Deformation of the membrane is considered as a sequence of three stages. At first stage, the membrane is deformed under free conditions until it touches the longitudinal sides of the rigid matrix. At second stage, the membrane is deformed when it touches the longitudinal walls of matrix until it touches its transverse wall. At third stage, the membrane is already deformed while simultaneously touching the longitudinal and transverse walls of matrix. Membrane deformation occurs under creep conditions under two types of contact conditions: sliding of the membrane along the walls of matrix and adhesion of membrane to the walls of matrix. The dependence of the time until the fracture of membrane at different rates of increase in the magnitude of the transverse pressure is obtained.

The analysis of the gradual long-term fracture of the membrane is carried out using the kinetic theory of creep by Yu. N. Rabotnov, while the parameter of material damage in this problem has a scalar character. The solution of the system consisting of constitutive and kinetic equations showed that during the membrane deformation at the first stage regardless of the type of contact conditions, the level of damage accumulates in the membrane, which is close to its limiting value. In this regard, the creep processes of the membrane at second and third stages of deformation under both considered types of contact conditions practically coincide.

The obtained equations are used to analyze the creep and long-term fracture of a membrane made of 2.15Cr-1Mo steel, which is deformed under variable transverse pressure at a temperature of 600 \(^\circ\)C.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):676-695
pages 676-695 views

Plastic and creep deformations of thick-walled cylinder with a rigid casing under internal pressure

Firsov S.V.


The creep and plastic flow of cylindrical pressurized vessel with rigid casing was considered. To combine creep and plastic deformations the vessel was heated and subjected to the high inner pressure. The semi-analytical solution for plain strain problem of a thick-walled cylinder with rigid casing in the frame of small strain theory was obtained in this paper. This solution consists of analytical formula for displacement distribution with asking values of pressure and irreversible strains (plastic and creep) and a numerical solution for irreversible strain values. The Norton power law and advanced Mises condition for viscoplasticity, associated with flow rules have been used to describe creep and plastic behavior of medium.

Four stages of the deformation process were considered: pressure increasing, pressure fixed on maximum value for a long time, pressure decreasing and relaxing stage with zero pressure. Two cases of maximum pressure values of 200 MPa and 320 MPa were studied. An additional case of elastoplastic deformation was considered to investigate the influence of creep on the deformational process. It has been observed that creep has a significant influence on stress and strain evolution in medium, especially on stages with maximum and zero pressure. Also, because of the creep plastic flow evolves slower and stoppes earlier on the loading stage. In the unloading stage, the plastic flow starts earlier and affects greater area due to greater irreversible strains. Creep leads to sufficient stress relaxation and stresses for two pressure cases get similar values at the end of the stage with maximum pressure value. At the end of the relaxing stage besides stresses displacement and deformation also became similar for the two cases.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):696-715
pages 696-715 views

Mathematical Modeling, Numerical Methods and Software Complexes

Mathematical modeling of parameter identification process of convection-diffusion transport models using the SVD-based Kalman filter

Kuvshinova A.N., Tsyganov A.V., Tsyganova Y.V.


The paper addresses a problem of mathematical modeling of the process of identifying the coefficients of a partial differential equation in convection-diffusion transport models based on the results of noisy measurements of the function values. Identification process is performed using a new method belonging to the class of recurrent parameter identification methods based on optimal discrete Kalman-type filtering algorithms. One-dimensional models with constant coefficients, boundary conditions of first kind, or mixed boundary conditions of first and third kind are considered.

The proposed method is based on the transition from the initial continuous model with a partial differential equation to the model described by the state-space linear discrete-time dynamic system and the application of the maximum likelihood method to it with construction of an identification criterion (likelihood function) based on the values calculated by the SVD algorithm of the Kalman filtering. This filter is based on the singular value decomposition of error covariance matrix and works stably even in cases when it is close to singular. The SVD filter has proven itself well in solving various problems of discrete filtering and parameter identification. It has several advantages over the traditionally used conventional Kalman filter. The main of which is robustness against machine roundoff errors.

Computer modeling of parameter identification has been processed with the MATLAB system using a specialized software package. The results of numerical experiments confirm the efficiency of the proposed method and its advantages compared to the similar one based on the conventional Kalman filter.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):716-737
pages 716-737 views

Models of stochastic dynamics of development of industrial enterprises with lagging internal and external investments

Saraev A.L., Saraev L.A.


The article proposes new stochastic models of the dynamic development of enterprises that restore their production at the expense of internal and external lagging investments. Systems of stochastic differential balance equations for such enterprises are established, describing random changes in factors of production and output. Proportional, progressive and digressive depreciation deductions are considered and their interaction with lagging internal and external investments is investigated. The conditions for achieving an equilibrium state of the enterprises work are formulated and the corresponding limiting values of the factors of production are calculated. Algorithms of the Euler–Maruyama method are obtained for numerical solutions of systems of stochastic differential equations of enterprise development. For each numerical implementation of these algorithms, the corresponding stochastic trajectories are constructed for the random functions of factors of production and output. A variant of the method for calculating the mathematical expectations of random functions of production factors is proposed, for which the corresponding system of differential equations is obtained. Numerical analysis of solutions of stochastic differential equations for the developed models showed good agreement with the known statistical data on the development of industrial enterprises.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):738-762
pages 738-762 views

Short Communications

Steady thermo-diffusive shear Couette flow of incompressible fluid. Velocity field analysis

Bashurov V.V., Prosviryakov E.Y.


An exact solution that describes steady flow of viscous incompressible fluid with coupled convective and diffusion effects (coupled dissipative Soret and Dufour effects) has been found. To analyze shear fluid flow an over-determined boundary value problem has been solved. The over-determination of the boundary value problem is caused by the advantage of number of equations in non-linear Oberbeck–Boussinesq system against number of unknown functions (two components of velocity vector, pressure, temperature and concentration of dissolved substance). Non-trivial exact solution of system consisting of Oberbeck–Boussinesq equations, incompressibility equation, heat conductivity equation and concentration equation has been built as Birich–Ostroumov class exact solution. Since the exact solution a priori satisfies the incompressibility equation the over-determined system is solvable. Existence of stagnation points is shown both in general flow and in secondary fluid motion without vorticity. Conditions of countercurrent appearance are found.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):763-775
pages 763-775 views

On a ordering of area tensor elements orientations in a micropolar continuum immersed in an external plane space

Murashkin E.V., Radayev Y.N.


The paper deals with the problems of ordering the reper orientations for a micropolar continuum immersed in an external plane space. Based on the concept of an elementary tensor volume (area) \(M\)-cells, an algorithm for comparing and matching external spatial orientations of \(M\)-cells is proposed. The process of continuous transfer of reper directions associated with a \(M\)-cell is considered. As a result, we can talk about the orientation of micropolar continuum itself and its boundary. The oriented continuum plays an important role in micropolar elasticity. This is especially true for the theory of hemitropic elastic media. The pseudotensor formulation of Stokes' theorem is discussed.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):776-786
pages 776-786 views

Model of a stochastic process in the space of random joint events

Biryukov A.A.


A model of the space of random joint events is being constructed. In space, along with the existence of a symmetric difference of joint events, a new postulate is introduced about the existence of a symmetric sum of random joint events. In the generated space, the stochastic equation of motion of the system and the expression for the probabilities of the system's transition between two events are modeled. The transition probability depends on the probabilities of compatibility of two, three, etc. events. The equation is equivalent to the Markov chain equation for incompatible events. The equation is equivalent to the equation of quantum theory if the events are compatible only in pairs.

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2021;25(4):787-796
pages 787-796 views

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