# Vol 22, No 4 (2018)

**Year:**2018**Articles:**10**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1230

### On differential operators an differential equations on torus

#### Abstract

In this paper, we consider periodic boundary value problems for a differential equation whose coefficients are trigonometric polynomials. The spaces of generalized functions are constructed, in which the problems considered have solutions, in particular, the solvability space of a periodic analogue of the Mizohata equation is constructed. A periodic analogue and a generalization of the construction of a nonstandard analysis are constructed, containing not only functions, but also functional spaces. As an illustration of the statement that not all constructions on a torus lead to simplification compared to a plane, a periodic analogue of the concept of a hypoelliptic differential operator is considered, where number-theoretic properties are significant. In particular, it turns out that if a polynomial with integer coefficients is irreducible in the rational field, then the corresponding differential operator is hypoelliptic on the torus.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(4):607-619

607-619

### Geometric solutions of the Riemann problem for the scalar conservation law

#### Abstract

For the Riemann problem $$ \left\{\begin{array}{l}u_t+(\Phi(u,x))_x=0,\\ u|_{t=0}=u_-+[u]\theta(x) \end{array}\right. $$ a new definition of the solution is proposed. We associate a Hamiltonian system with initial conservation law, and define the geometric solution as the result of the action of the phase flow on the initial curve. In the second part of this paper, we construct the equalization procedure, which allows us to juxtapose a geometric solution with a unique entropy solution under the condition that $\Phi$ does not depend on $x$. If $\Phi$ depends on $x$, then the equalization procedure allows us to construct a generalized solution of the original Riemann problem.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(4):620-646

620-646

### Mathematical modeling of creep and residual stresses relaxation in surface hardened elements of statically indefinable rod systems

#### Abstract

We propose a method for modeling stress-strain state in surface-hardened elements of statically indefinable rod systems under creep. A method we propose is considered for a three-element asymmetric rod system. The solution consists of two steps: reconstruction of the stress-strain state after the procedure of surface plastic hardening of the cylindrical elements of the system (pneumatic blasting with micro balls) and the method for calculating the relaxation of residual stresses in the hardened elements amidst the creep state of rod system (as a whole structure). Rheological relations are determined on the basis of a model describing the first and second phases of creep. The solution of both stages and special aspects of the problem is illustrated on a model example of creep of systems with hardened elements made of ZhS6U alloy at the temperature of 650 °C. For hardening the rods of this alloy, real experimental data were used for axial and circumferential residual stresses. The technique of reconstruction of the stress-strain state after pneumatic blasting treatment is illustrated in detail. To build a rheological model, experimental data were used for the uniaxial creep curves of the ZhS6U alloy under various constant stresses at the temperature of 650 °C. The numerical values of the model parameters are given in the article. The uniaxial model is generalized to a complex stress state. The main problem is solved numerically using discretization by spatial and temporal coordinates. The stationary asymptotic stress-strain state of the rod system is investigated, which corresponds to the steady-state creep stage, which was used to estimate the convergence of the numerical method. The dependencies of the kinetics of all components of the residual stress tensor in all three strengthened elements of the system due to creep under a given external load are obtained. A comparative analysis of the residual stress relaxation rate in different rods is performed. The algorithm and software for solving the problem is developed. The main results of the work are illustrated by the residual stresses graphs over the depth of the hardened layer. Issues of applying the results obtained in the work to practical problems of assessing the reliability of hardened rod systems are discussed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(4):647-668

647-668

### A numerical method of nonlinear estimation based on difference equations

#### Abstract

The article considers a new numerical method for estimating the parameters of nonlinear mathematical models based on difference equations describing the results of observations. The algorithm of the numerical method includes: - the construction of a linear-parametric discrete model of the process under study in the form of difference equations, the coefficients of which are known to be associated with the parameters of a nonlinear mathematical model; - the formation of a generalized regression model based on the difference equations; - the calculation of the initial approximation estimate and the iterative procedure for refining the mean-square estimates of the coefficients of the generalized regression model; - the calculation of the estimates of the parameters of the nonlinear mathematical model based on the mean-square estimates of the coefficients of the difference equations; - evaluation of the error of the results of calculations based on the methods of statistical processing of experimental data. Various approaches to the construction of systems of difference equations for mathematical models in the form of nonlinear functional dependencies are proposed. The relations underlying the iterative process of refining the coefficients of the generalized regression model constructed on the basis of difference equations are obtained. The procedure for estimating the error of the results of calculations of the parameters of nonlinear functional dependencies, which are known to be associated with the coefficients of the system of difference equations, is described. The application of the numerical method based on the difference equations is illustrated by the examples of estimation of the parameters of the mathematical model of the linear oscillator with attenuation, the model of free oscillations of the dissipative mechanical system with turbulent friction, as well as the parameters of the logistic trend described by the Verhulst (Pearl-Reed) function.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(4):669-701

669-701

702-713

### On the some properties of a symmetric Grubbs’ copula

#### Abstract

We investigate one-sided Grubbs’ statistics for a normal sample. Those statistics are standardized maximum and standardized minimum, i.e. studentized extreme deviation statistics. The two-parameter distribution of these statistics is considered, which arises when the one abnormal observation (outlier) differs from the other observations of its variance. We derive the formula for calculating the probability density function of studentized outlier deviation from sample average. A new two-parameter copula is extracted from the joint distribution of Grubbs’ statistics. The Grubbs’ copula is proved to be symmetric. As a result, one-sided Grubbs’ statistics have the property of exchangeability. Computer simulation of scatterplots from Grubbs’ copula is being performed. The scatterplot analysis shows that the Grubbs’ copula describes the negative statistical dependence. To study the effect of the copula’s parameters on the strength of this dependence, the estimation of the Kendall’s tau rank correlation coefficient is performed. The estimation algorithm uses computer simulation and it is realized in the R-package. We find that the copula’s parameters

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(4):714-734

714-734

### Dynamic equilibria of a nonisothermal fluid

#### Abstract

In this paper, stationary dynamic equilibria of the rotating mass of a nonisothermal fluid are discussed within the accuracy limits of the Boussinesq approximation. It is demonstrated that, in this case, a fluid exhibits a finite number of counterflows, higher values of velocities than those specified on the boundary and the formation of zones of positive and negative pressures and temperatures.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(4):735-749

735-749

### Dynamic stability of heated geometrically irregular cylindrical shell in supersonic gas flow

#### Abstract

On the basis of the Love model, a geometrically irregular heated cylindrical shell blown by a supersonic gas flow from one of its main surfaces is considered. The continuum model of a thermoelastic system in the form of a thin-walled shell supported by ribs along the incoming gas flow is taken as a basis. The singular system of equations for the dynamic thermal stability of a geometrically irregular shell contains terms that take into account the tension-compression and the shift of the reinforcing elements in the tangential plane, the tangential forces caused by the heating of the shell and the transverse load, as standard recorded by the piston theory. The solution of a singular system of differential equations in displacements, in the second approximation for the deflection function, is sought in the form of a double trigonometric series with time coordinate variables. Tangential forces are predefined as the solution of singular differential equations of non-moment thermoelasticity of a geometrically irregular shell taking into account boundary forces. The solution of the system of dynamic equations of thermoelasticity of the shell is sought in the form of the sum of the double trigonometric series (for the deflection function) with time coordinate variable coefficients. On the basis of the Galerkin method, a homogeneous system for the coefficients of the approximating series is obtained, which is reduced to one fourth-order differential equation. The solution is given in the second approximation, which corresponds to two half-waves in the direction of flow and one half-wave in the perpendicular direction. On the basis of standard methods of analysis of dynamic stability of thin-walled structures are determined critical values of the gas flow rate. The quantitative results are presented in the form of tables illustrating the influence of the geometrical parameters of the thermoelastic shell-edge system, temperature and damping on the stability of a geometrically irregular cylindrical shell in a supersonic gas flow.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(4):750-761

750-761

### One approach to determination of the ultimate load-bearing capacity of mechanical systems with softening elements

#### Abstract

The fundamental provisions of the limiting load calculation theory are presented for a discrete mechanical system with softening elements. The method is based on the numerical determination of degenerate critical points for the potential function of the system. At these points there is a transition from the stability of the loading process to instability such as a catastrophe or a failure. This approach helps to avoid solving a large number of nonlinear equilibrium equations. The problem of determining the limiting internal pressure in a thin walled cylindrical tank is solved as an example. A unified potential specially defined for a flat square element of material in biaxial tension is used in developing a potential function of the system. It describes all stages of deformation including the softening stage.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(4):762-773

762-773

### On the uniqueness of the solution of the Cauchy problem for the equation of fractional diffusion with Bessel operator

#### Abstract

In this paper, we consider fractional diffusion equation involving the Bessel operator acting with respect to a spatial variable and the Riemann-Liouville fractional differentiation operator acting with respect to a time variable. When the order of the fractional derivative is unity, and the singularity of the Bessel operator is absent, this equation coincides with the classical heat equation. Earlier, a solution of the Cauchy problem has been considered for the considered equation and a uniqueness theorem has been proved for a class of functions satisfying the analog of the Tikhonov condition. In this paper, we have constructed an example to show that the exponent (power) at the condition of the uniqueness of the solution to the Cauchy problem cannot be raised under. Its increase leads to a non-uniqueness of the solution. Using the well-known properties of the Wright function, we have obtained estimates for constructed function, which satisfies the homogeneous equation and the zero Cauchy condition.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(4):774-784

774-784