Vol 25, No 4 (2021)

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.

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.

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.

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.

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.

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.