## Vol 20, No 4 (2016)

**Year:**2016**Articles:**13**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1223

Articles

Standard model alignment scenarios for Higgs bosons

#### Abstract

This paper describes the model NMSSM with effective explicit and spontaneous CP-violations, and additional mixing of CP-even and CP-odd Higgs bosons states. The neutral Higgs bosons masses and decay widths were calculated at fixed parameters of the model so one of the physical states is responsible the results of the LHC experiments. The calculation of decay widths produced in the one-loop approximation in the framework of the quantum field perturbation theory. We defined two scenarios for observables. The first scenario corresponds to the set of parameters leading to the lightest Higgs boson mass of 125 GeV. The second one with the restriction on the Higgs boson mass implemented in the electroweak baryogenesis, which leads to the experimental data agreement with the second physical mass condition.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):581-588

The nonlocal problem for a hyperbolic equation with bessel operator in a rectangular domain

#### Abstract

We consider a boundary value problem for a hyperbolic equation with Bessel differential operator in a rectangular domain with integral nonlocal boundary value condition of the first kind. The equivalence between boundary value problem with integral nonlocal condition of the first kind and a local boundary value problem with mixed boundary conditions of the first and third kinds is proved. The existence and uniqueness of solution of the equivalent problem are established by means of the spectral method. At the uniqueness proof the completeness of the eigenfunction system of the spectral problem is used . At the existence proof the assessment of coefficients of series, the asymptotic formula for Bessel function of the first kind and asymptotic formula for eigenvalues are used. Sufficient conditions on the functions defining initial data of the problem are received. The solution of the problem is obtained in explicit form. The solution is obtained in the form of the Fourier-Bessel series. Its convergence is proved in the class of regular solutions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):589-602

An approximate group classification of a perturbed subdiffusion equation

#### Abstract

A problem of the Lie point approximate symmetry group classification of a perturbed subdiffusion equation with a small parameter is solved. The classification is performed with respect to anomalous diffusion coefficient which is considered as a function of an independent variable. The perturbed subdiffusion equation is derived from a fractional subdiffusion equation with the Riemann-Liouville time-fractional derivative under an assumption that the order of fractional differentiation is close to unity. As it is follow from the classification results, the perturbed subdiffusion equation admits a more general Lie point symmetry group than the initial fractional subdiffusion equation. The obtained results permit to construct approximate invariant solutions for the perturbed subdiffusion equation corresponding to different functions of the anomalous diffusion coefficient. These solutions will also be the approximate solutions of the initial fractional subdiffusion equation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):603-619

Necessary optimality conditions of the second oder in a stochastic optimal control problem with delay argument

#### Abstract

The optimal control problem of nonlinear stochastic systems which mathematical model is given by Ito stochastic differential equation with delay argument is considered. Assuming that the concerned region is open for the control by the first and the second variation (classical sense) of the quality functional we obtain the necessary optimality condition of the first and the second order. In the particular case we receive the stochastic analog of the Legendre-Clebsch condition and some constructively verified conclusions from the second order necessary condition. We investigate the Legendre-Clebsch conditions for the degeneration case and obtain the necessary conditions of optimality for a special control, in the classical sense.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):620-635

On a problem for mixed type equation with partial riemann-liouville fractional derivative

#### Abstract

The uniqueness and existence of solutions of a nonlocal problem proved for an equation of mixed type. This equation contains diffusion equation of fractional order. The boundary condition contains a linear combination of generalized operators of fractional order with the Gauss hypergeometric function. The solution of the problem is given explicitly.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):636-643

An ordinary integro-differential equation with a degenerate kernel and an integral condition

#### Abstract

We consider the questions of one value solvability of the nonlocal boundary value problem for a nonlinear ordinary integro-differential equation with a degenerate kernel and a reflective argument. The method of the degenerate kernel is developed for the case of considering ordinary integro-differential equation of the first order. After denoting the integro-differential equation is reduced to a system of algebraic equations with complex right-hand side. After some transformation we obtaine the nonlinear functional-integral equation, which one valued solvability is proved by the method of successive approximations combined with the method of compressing mapping. This paper advances the theory of nonlinear integro-differential equations with a degenerate kernel.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):644-655

Application of the energy-based criterion to the simulation of the fracture of the steel structures

#### Abstract

In this work we have developed energy balance model for inelastic deformation process of metals. Changes in the material structure are taking into account with the help of tensorial variable having the physical meaning of additional strain induced by initiation of defects. Introduction of such a parameter allows one to calculate the stored energy value and develop an energy-based fracture criterion. There were considered two ways of derivation of constitutive equations for plastic and structural strain. The first method was based on the principles of linear nonequilibrium thermodynamics, the second one is the analogue of the flow plastisity theory. Developed thermomechanical model includes equilibrium equation, geometric relation for strain tensor, Hooke’s law, constitutive equations for structural and plastic strain and energy balance equation. It is assumed that fracture in the material takes place when stored energy reaches critical value in some volume of the material. The application of such an approach to fracture problems of the metals is illustrated by two numerical examples. The first example is crack path simulation in the steel shaft with initial crack oriented at the certain angle to the shaft axis. The second example is simulation of the crack initiation and propagation in the steel bearing bracket. The obtained results are in agreement with the previously published results and could be used for simulation of fracture of real structures.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):656-674

Effect of anisotropy of surface plastic hardening on formation of residual stresses in cylindrical samples with semicircular notch

#### Abstract

We study effect of anisotropy of surface plastic hardening on formation of residual stresses in solid cylindrical samples and samples with semicircular notch. Experimentally determined one and/or two components of residual stresses in a hardened layer are used as an initial information. We describe calculation method for the rest of diagonal components of residual stresses and plastic strains tensors, off-diagonal components are not considered. We propose numerical method for calculation residual stresses in semicircular notch of surface hardened cylindrical sample. This task was reduced to boundary value problem of fictitious thermoelasticity where initial (plastic) strains are modeled with temperature strains. Solution was build with the use of finite element method. We studied in detail the effect of radius of notch and anisotropy parameters of hardening on the nature and magnitude of distribution of residual stresses depending on the depth of layer in the smallest cross section of cylindrical samples of EI961 alloy steel and 45 steel. It was determined that with small radii of notch lower then thickness of hardening layer the value of axial component of residual stresses (absolute value) is higher then in the sample without a notch. Developed method was experimentally verified for samples without notches and the correspondence between calculated and experimental data was determined on distribution of axial and circumferential residual stresses depending on depth of hardening layer. For samples with notches we compare numerical solutions from this work with known solutions of other authors.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):675-690

The use of linear fractional analogues rheological models in the problem of approximating the experimental data on the stretch polyvinylchloride elastron

#### Abstract

We considere and analyze the uniaxial phenomenological models of viscoelastic deformation based on fractional analogues of Scott Blair, Voigt, Maxwell, Kelvin and Zener rheological models. Analytical solutions of the corresponding differential equations are obtained with fractional Riemann-Liouville operators under constant stress with further unloading, that are written by the generalized (two-parameter) fractional exponential function and contains from two to four parameters depending on the type of model. A method for identifying the model parameters based on the background information for the experimental creep curves with constant stresses was developed. Nonlinear problem of parametric identification is solved by two-step iterative method. The first stage uses the characteristic data points diagrams and features in the behavior of the models under unrestricted growth of time and the initial approximation of parameters are determined. At the second stage, the refinement of these parameters by coordinate descent (the Hooke-Jeeves’s method) and minimizing the functional standard deviation for calculated and experimental values is made. Method of identification is realized for all the considered models on the basis of the known experimental data uniaxial viscoelastic deformation of Polyvinylchloride Elastron at a temperature of 20 ℃ and five the tensile stress levels. Table-valued parameters for all models are given. The errors analysis of constructed phenomenological models is made to experimental data over the entire ensemble of curves viscoelastic deformation. It was found that the approximation errors for the Scott Blair fractional model is 14.17 %, for the Voigt fractional model is 11.13 %, for the Maxvell fractional model is 13.02 %, for the Kelvin fractional model 10.56 %, for the Zener fractional model is 11.06 %. The graphs of the calculated and experimental dependences of viscoelastic deformation of Polyvinylchloride Elastron are submitted.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):691-706

Mathematical simulation for strain-stress state of optical telescope stable-size composite elements with finite-element method

#### Abstract

Designing issues of optical telescope stable-size composite elements have been considered. Staging of design calculation for composite structural elements has been described. Basic relations of composite material micromechanics have been represented. Specifics of mathematical simulation taking into account assumptions made have been described by example of developed framework of electro-optical system with one-side enforcement by ribs. Results of experimental determination of carbon-filled plastic characteristics used in design of stable-size optical telescope frame structure have been represented; advantages of finite-element method as one of the basic methods for solving the boundary value problem in applied mechanics have been reflected. Reasonableness of analytical approach using in the initial development stage in order to shorten the period of design has been demonstrated. The leading part of finite-element simulation has been determined in behavior prognostication of structures at different operating stages. Stable-size supporting composite frameworks developed taking into account defined sequence of structural design have been showed. Described staging of structure making has been allowed to process and systematize data during design and experimental execution, refine structural model parameters, increase the confidence level and verify it.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):707-729

On a computer implementation of the block Gauss-Seidel method for normal systems of equations

#### Abstract

This article focuses on the modification of the block option Gauss-Seidel method for normal systems of equations, which is a sufficiently effective method of solving generally overdetermined, systems of linear algebraic equations of high dimensionality. The main disadvantage of methods based on normal equations systems is the fact that the condition number of the normal system is equal to the square of the condition number of the original problem. This fact has a negative impact on the rate of convergence of iterative methods based on normal equations systems. To increase the speed of convergence of iterative methods based on normal equations systems, for solving ill-conditioned problems currently different preconditioners options are used that reduce the condition number of the original system of equations. However, universal preconditioner for all applications does not exist. One of the effective approaches that improve the speed of convergence of the iterative Gauss-Seidel method for normal systems of equations, is to use its version of the block. The disadvantage of the block Gauss-Seidel method for production systems is the fact that it is necessary to calculate the pseudoinverse matrix for each iteration. We know that finding the pseudoinverse is a difficult computational procedure. In this paper, we propose a procedure to replace the matrix pseudo-solutions to the problem of normal systems of equations by Cholesky. Normal equations arising at each iteration of Gauss-Seidel method, have a relatively low dimension compared to the original system. The results of numerical experimentation demonstrating the effectiveness of the proposed approach are given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):730-738

The quasi-one-dimensional hyperbolic model of hydraulic fracturing

#### Abstract

The paper describes a quasi-one-dimensional hyperbolic model of hydraulic fracture growth assuming for the hydraulic fracturing that stress intensity is much higher than fracture resistance. The mode under analysis, which accounts for convective and unsteady terms in the fluid flow equation, is a generalization of the Perkins-Kern-Nordgren local model. It has been proved that the obtained system of differential equations is a quasi-linear strictly hyperbolic system, for which the characteristics were found as well as their correlations. For the case of the Coriolis correction neglect, the Riemann invariants were found. Neglecting the injected fluid leak-off and viscosity, the Riemann waves, similar to simple plane waves in gas dynamics, were defined and their properties were studied. The evolutionism of fracture boundaries was investigated. The initial boundary value problem was set for fracture growth. It has been shown that the neglect of dissipative terms in the presented model allows constructing a simple wave theory analogous to the theory of one-dimensional gas dynamics for isentropic plane waves.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):739-754

Method of searching for global extremum of a continuous function on a simplex

#### Abstract

A non-convex problem of mathematical programming is considered, which permissible region is a simplex. A two-stage algorithm is proposed for approximate solution of the problem. The region of global optimum is determined using the Ψ-transform method at the first stage; local “fine-tuning” of the solution is performed at the second stage. The Ψ-transform was modified taking into account the special features of the problem under consideration. Ψ-function is determined according to the results of statistical tests implemented using the generator of random points uniformly distributed over the simplex. The proposed method of reflection of regular simplexes is used for fine-tuning of the solution. An example of application of the developed algorithm for solving the problem of optimization of component composition of the hydrocarbon mixture is presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2016;20(4):755-768