# Vol 26, No 3 (2022)

**Year:**2022**Articles:**10**URL:**https://journals.eco-vector.com/1991-8615/issue/view/5154**DOI:**https://doi.org/10.14498/vsgtu/v226/i3

## Full Issue

## Differential Equations and Mathematical Physics

### On a q-analogue of the Sturm–Liouville operator with discontinuity conditions

#### Abstract

n this paper, a *q*-analogue of the Sturm–Liouville problem with discontinuity condition on a finite interval is studied. It is proved that the *q*-Sturm–Liouville problem with discontinuity conditions is self-adjoint in ${L}_{q}^{2}(0,\pi )$. The completeness theorem and the sampling theorem are proved.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(3):407-418

### Uniform optimization of controlled systems with distributed parameters

#### Abstract

A constructive method is proposed for solving a spatiotemporal control problem in systems with distributed parabolic parameters under the conditions of the given accuracy of uniform approximation of the final state of a plant to the required spatial distribution of the controlled variable. The developed approach is based on the previously designed alternance method for constructing the parameterizable programmed control algorithms, which extended the results of the theory of nonlinear Chebyshev approximations to a wide range of optimization problems and uses the fundamental laws of the subject area. It is shown that in linear quadratic problem optimization the equations of optimal controllers with autonomous modal controls in the open domain of their definition and taking into account restrictions on the nature of the spatial distribution of the control actions specified by the conditions of technical implementation are reduced to linear feedback algorithms for the measured state of the plant with nonstationary transmission coefficients and the given dependence on the spatial arguments of the controlled value. The results obtained are extended to the problem of searching for time-invariant spatially distributed controls, considered as the desired design solutions for a plant.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(3):419-445

## Mechanics of Solids

### Prediction of individual deformation characteristics of structural elements by a “leader” product

#### Abstract

We propose a numerical method for predicting the individual deformation characteristics of structural elements by a “leader” product. The basis of this method is generalized one-parameter models. These models relate the integral characteristics of the stress state to the integral characteristics of the deformation state in the “generalized load – generalized displacement” coordinates. The scope of the method is structural elements of the same type, which are under identical conditions of external loading and are characterized by a large spread of deformation characteristics (generalized displacement). It is assumed that the operation of one structural element (prototype) begins some time earlier than the others. A hypothesis on the similarity of all realizations reduced to a single origin by a time shift in the “generalized displacement – time” coordinates is introduced. Using statistical information on the initial sections of “lagging” structural elements and the sample prototype, the statistical characteristics of the similarity parameter of the operated structural element are determined in relation to the “leader” product, and then its deformation characteristics are predicted.

In the paper, we investigate friction units and structural elements (rods, threaded connections) under creep conditions. Based on statistical correlation analysis of the experimental information, a verification of the similarity hypothesis usabilty for all implementations of the structural elements studied is carried out. The method was illustrated by an example of predicting the wear of the friction units of the front landing gear of the aircraft depending on the number of take-off and landing cycles. The method was also illustrated with an example of how to calculate the elongation of rods made of a polyvinylchloride compound under uniaxial loading and axial displacement of the screwing area of a threaded joint under creep conditions.

The experimental data for the generalized displacement of specific implementations are shown to not exceed the calculated limits of the confidence interval for mathematical expectation for all structural elements considered in prediction time intervals of one to four “basic” intervals, within which estimates of random parameters for specific structural elements were determined.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(3):500-519

## Mathematical Modeling, Numerical Methods and Software Complexes

### Mathematical modeling of the effect of spacers on mass transfer in electromembrane systems

#### Abstract

The transfer of ions near ion-exchange membranes causes concentration polarization, which significantly complicates mass transfer in electromembrane systems. Spacers are used to neutralize the effect of concentration polarization and increase mass transfer. Spacers reduce the thickness of the boundary layer by increasing the mixing depth of the solution and creating a normal component of convective transport; ions can reach membranes faster, and the current increases, from a hydrodynamic point of view. However, spacers significantly increase the hydrodynamic resistance and consequently the cost of pumping the solution.

For the first time, the main regularities of the transfer of salt ions in the desalination channel of an electrodialysis apparatus with spacers of various shapes and arrangements are determined, taking into account electroconvection, in overlimiting current modes. Namely, it is shown, using the current-voltage characteristic, that spacers of different shapes and locations are optimal at different stages of the desalination process.

The paper presents the results of mathematical and simulation modeling of the salt ion transport process in electromembrane systems with spacers in overlimiting current modes. 2D direct numerical simulation was carried out for the coupled system of the Nernst–Planck–Poisson and Navier–Stokes equations without fitting parameters. The finite element method was used in combination with the method of successive approximations and segregation to solve boundary value problems for systems of nonlinear differential equations with partial derivatives. The novelty of the method lies in the fact that after discretization in time, the problem on each time layer is split into hydrodynamic and electrochemical problems, each of which is solved by the method of successive approximations until a complete mutual agreement.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(3):520-543

### General principle of maximum pressure in stationary flows of inviscid gas

#### Abstract

Within the framework of the Euler equations, the possibility of achieving extreme pressure values at the inner point of a stationary flow of a nonviscous gas is considered. The flow can be non-barotropic. The well-known (G.B. Sizykh, 2018) subsonic principle of maximum pressure (SPMP) cannot be applied in transonic and supersonic flow regions. Under the conditions of the classical principle of maximum pressure by C. Truesdell (1953), there is no restriction on the values of local Mach numbers, but it has a number of features that do not allow it to be used to verify numerical calculations in the same way as it can be done when using SPMP in subsonic regions. A previously unknown principle of maximum pressure is discovered: a function of derivative flow parameters is found, which must have a certain sign (different for minimum and for maximum pressure) at the point where the pressure reaches a strict or nonstrict local extremum. This principle of maximum pressure is called “general” (GPMP) because its conditions do not include barotropicity, restrictions on the values of local Mach numbers, and the assumption that the gas obeys the Mendeleev–Clapeyron equation. One of the consequences of GPMP is the conclusion that the requirement of barotropicity can be excluded from the conditions of Truesdell's principle of maximum pressure. It is proposed to use GPMP to verify numerical calculations of the ideal gas flow behind a detached shock wave formed in a supersonic flow around bodies and to verify numerical calculations of a viscous gas flow around bodies in regions remote from sources of vorticity, where the effect of viscosity can be neglected.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(3):544-555

### The hp-version of the least-squares collocation method with integral collocation for solving a biharmonic equation

#### Abstract

A new algorithm for the numerical solution of the biharmonic equation is developed. It is based on the first implemented hp-version of the least-squares collocation method (hp-LSCM) with integral collocations for a fourth-order elliptic equation in combination with modern methods of accelerating iterative processes for solving systems of linear algebraic equations (SLAE). The hp-LSCM provides the possibilities to refine the grid (h-version) and increase the degree of polynomials to the arbitrary order (p-approach). The convergence of approximate solutions obtained by the implemented version of the method is analyzed using an example of a numerical simulation of the bending of a hinged isotropic plate. The high accuracy and the increased order of convergence using polynomials up to the tenth order in the hp-LSCM are shown.

The effectiveness of the combined application of algorithms for accelerating iterative processes to solve SLAE that are combined with LSCM is investigated. In this paper, we consider the application of the following algorithms: preconditioning of SLAE matrices; the iteration acceleration algorithm based on Krylov subspaces; the prolongation operation on a multigrid complex; parallelization using OpenMP; a modified algorithm for solving local SLAEs. The latter is implemented with iterations over subdomains (which are cells) and makes it possible to more effectively solve overdetermined SLAEs in the LSCM in the case of solving a linear differential equation. The form of the matrices does not change at each iteration. Only the elements of the vectors of their right parts corresponding to the matching conditions are modified. The calculation time on a personal computer is reduced by more than 350 times with the combined use of all acceleration techniques compared to the case when only preconditioning was used.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(3):556-572

### Probabilistic models for the analysis of inverse extremal problems in combinatorics

#### Abstract

In an inverse extremal problem for a combinatorial scheme with a given value of the objective function of the form of a certain extreme value of its characteristic, a probabilistic model is developed that ensures that this value is obtained in its outcomes. Two types of such characteristics are considered, relating to each of the schemes or to a set of outcomes.

The pre-asymptotic analysis of such a model is carried out by the author's enumerative method. It is based on the construction of an iterative random process with iterations of successive stages of a numbered non-repetitive enumeration and the formation of outcomes of the scheme. The iterative development of the process is represented by a probabilistic graph.

The study of the outcomes of the scheme according to the model in the enumerative method is carried out in the following areas: visual numbering of the outcomes of the scheme, finding their number, establishing a one-to-one correspondence between the types and numbers of outcomes of the scheme, obtaining their probabilistic distribution (controlled by a random process of listing the outcomes of the scheme), and modeling them with this distribution.

Along with the direct study of circuits in these areas, algorithms are proposed to obtain results for them by partially recalculating them from the results of a similar analysis of more general, previously studied circuits without restrictions or with less restrictions on the values of the characteristics under consideration.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(3):571-591

## Short Communications

### On the theory of fourth-rank hemitropic tensors in three-dimensional Euclidean spaces

#### Abstract

The paper is devoted to problems concerning the tensors with constant components, hemitropic tensors and pseudotensors that are of interest from the point of view of micropolar continuum mechanics. The properties and coordinate representations of tensors and pseudotensors with constant components are discussed. Based on an unconventional definition of a hemitropic fourth-rank tensor, a coordinate representations in terms of Kronecker deltas and metric tensors are given. A comparison of an arbitrary hemitropic fourth-rank tensor and a tensor with constant components are discussed. The coordinate representations for constitutive tensors and pseudotensors used in mathematical modeling of linear hemitropic micropolar continuums are given in terms of the metric tensor.The covariant constancy of fourth-rank pseudotensors with constant components and hemitropic tensors is considered and discussed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2022;26(3):592-602