# Vol 17, No 1 (2013)

**Year:**2013**Articles:**40**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1970

### Preface

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):10-14

10-14

### The Steklov nonlocal boundary value problem of the second kind for the simplest equations of mathematical physics

#### Abstract

The Steklov nonlocal boundary value problem of the second kind for the simplest equations of mathematical physics is studied. A priori estimates for the solutions of the considered problems are obtained by using the method of energy inequalities. Uniqueness and continuous dependence of the solutions on the input data follow from these estimates.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):15-23

15-23

### Boundary control for the processes, described by hyperbolic systems

#### Abstract

The boundary control problem for the system of hyperbolic equations with the mixed derivative is considered.The control is provided by the displacement (in the conditions of the first boundary-value problem).The coefficient matrices of different structure are explored for the system.The commutativity of these coefficients is the essential condition.If the matrices couldn't be brought to the diagonal form simultaneously,it's offered to use special differential operators for representation of the necessary problems solutions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):24-30

24-30

### The characteristic problem for the system of the general hyperbolic differential equations of the third order with nonmultiple characteristics

#### Abstract

We consider the well-posed characteristic problem for the system of the general hyperbolic differential equations of the third order with nonmultiple characteristics. The solution of this problem is constructed in an explicit form. The example of the analogue of Goursat problem for a particular system of the hyperbolic differential equations of the third order is given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):31-36

31-36

### A uniqueness criterion for solutions of the Dirichlet problem for a loaded equation with the Lavrent'ev-Bitsadze operator

#### Abstract

The first boundary value problem was considered for the second order loaded differential equation of mixed elliptic-hyperbolic type in a rectangular region. The local and nonlocal problems for the loaded partial differential equations of the individual and mixed types have been previously studied in areas where the hyperbolic part is the characteristic triangle. In this work, in contrast to the well-known ones, necessary and sufficient conditions of the uniqueness of this problem solution were found by the method of spectral analysis.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):37-45

37-45

46-52

### $L_p$-estimates of the nontangential maximal function for solutions a second-order elliptic equation solutions

#### Abstract

The work contains the survey of results related to the study of near the boundary behavior of the solution of the Dirichlet problem with the boundary function in $L_p,$ $p > 1$ for a second-order elliptic equation. There are new statements and some unsolved problems in this direction.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):53-69

53-69

### The Dunkl convolution operators and multipoint de la Vallee–Poussin problem

#### Abstract

The Dunkl operator as an object of mathematical physics is considered, we study the kernel and the surjectivity of Dunkl convolution operators in the space of entire functions and the space of entire functions of exponential type. The main result is the solution of the multipoint de la Vallee–Poussin problem for Dunkl convolution operators in the space of entire functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):70-81

70-81

### Solutions of anisotropic parabolic equations with double non-linearity in unbounded domains

#### Abstract

This work is devoted to some class of parabolic equations of high order with double nonlinearity which can be represented by a model equation \begin{gather*} \frac{\partial}{\partial t}(|u|^{k-2}u)= \sum_{\alpha=1}^n(-1)^{m_\alpha-1}\frac{\partial^{m_\alpha}}{\partial x_\alpha^{m_\alpha}} [|\frac{\partial^{m_\alpha} u}{\partial x_\alpha^{m_\alpha}}|^{p_\alpha-2} \frac{\partial^{m_\alpha} u}{\partial x_\alpha^{m_\alpha}}],m_1,\ldots, m_n\in \mathbb{N},\quad p_n\geq \ldots \geq p_1>k,\quad k>1. \end{gather*} For the solution of the first mixed problem in a cylindrical domain $ D=(0,\infty)$ $\times\Omega, \;\Omega\subset \mathbb{R}_n,$ $n\geq 2,$ with homogeneous Dirichlet boundary condition and finite initial function the highest rate of decay established as $t \to \infty$. Earlier upper estimates were obtained by the authors for anisotropic equation of the second order and prove their accuracy.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):82-89

82-89

### Solutions of anisotropic elliptic equations in unbounded domains

#### Abstract

In the paper the Dirichlet problem for an anisotropic quasilinear elliptic equations of the second order is considered. The upper estimates for the generalized solution of this Dirichlet problem are received, the closeness is proved for the isotropic case.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):90-96

90-96

97-105

### On problem of nonexistence of dissipative estimate for discrete kinetic equations

#### Abstract

The existence of a global solution to the discrete kinetic equations in Sobolev spaces is proved, its decomposition by summability is obtained, the influence of its oscillations generated by the interaction operator is explored. The existence of a submanifold ${\mathcal M}_{diss}$ of initial data $(u^0, v^0, w^0)$ for which the dissipative solution exists is proved. It’s shown that the interaction operator generates the solitons (progressive waves) as the nondissipative part of the solution when the initial data $(u^0, v^0, w^0)$ deviate from the submanifold ${\mathcal M}_{diss}$. The amplitude of solitons is proportional to the distance from $(u^0, v^0, w^0)$ to the submanifold ${\mathcal M}_{diss}$. It follows that the solution can stabilize as $t\to\infty$ only on compact sets of spatial variables.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):106-143

106-143

### On a boundary value problem for a system of hyperbolic equations with the wave operator and a singular coefficient of lower derivatives

#### Abstract

The boundary value problem with data given on the parallel characteristics for the system of hyperbolic equations with the wave operator and the singular matrix coefficient at the lower derivative is considered in the characteristic square. This system of differential equations in the characteristic coordinates can be reduced to the system of Euler–Poisson–Darboux equations. Using the known solution of Cauchy problem with data given on the singularity line of matrix coefficient, we reduce the problem to the Carleman system of integral equations.The explicit solution of the considered boundary value problem is constructed using the results of previous research on the solvability of the systems of generalized Abel integral equations, made by the author.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):144-149

144-149

### On the problem with generalized operators of fractional differentiation for mixed type equation with two degeneracy lines

#### Abstract

The nonlocal problem for mixed-type equation with perpendicular lines of degeneracy is investigated for the case when the Dirichlet condition is given on the elliptic boundary, and the generalized derivatives of the solution values on the characteristics are pointwise related to the solution and its normal derivatives values on the lines of a parabolic degeneracy in its hyperbolic parts.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):150-158

150-158

### Blow-up of solutions of Cauchy problem for nonlinear Schrödinger equation

#### Abstract

In this work we study the effect of time finiteness of the existence of Cauchy problem for nonlinear Schrödinger equation solution. Together with the ill-posed Cauchy problem we consider its neighborhood in the space of operators, representing Cauchy problem. We explore the convergence of sequence of solutions of Cauchy problems with the operators, approximating the initial Hamiltonian.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):159-171

159-171

### Modified boundary element method for the solution of connected problems of mathematical physics

#### Abstract

The modified boundary element method for physico-mathematical modeling of the multifactorial processes is offered for discussion. Physical modeling based on the Onsager's theorem about the relationship between generalized forces and fluxes where we assume the coefficients of reciprocity are nonlinear. The approach is illustrated by the example of the strain diagram. Mathematical modeling is based on a modification of the BEM where all incorrect procedures of the numerical differentiation and integration were replaced by preliminary analytical calculations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):172-180

172-180

### On some classes of nonlinear integral equations with noncompact operators

#### Abstract

The work is devoted to the investigation of some classes of nonlinear integral equations of Hammerstein–Nemitski type with noncompact operators. Above mentioned class of equations, beside the theoretical interest has immediately an application in kinetic theory of gases. The existence theorems for positive solutions in different functional spaces are proved.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):181-188

181-188

### Thermodynamics of the viscous fluid from an observer's viewpoint

#### Abstract

The development of the non-equilibrium thermodynamics of the viscous fluid not using the local thermodynamic equilibrium hypothesis is considered. The theory is based on the causal mechanics of the heat conducting continuum, which includes the 1st law of thermodynamics as a theorem. The conditions of applicability of the 2nd law of thermodynamics and the dissipation of the kinetic energy problem are discussed. Main conclusions are illustrated using the example from the numerical analysis.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):189-198

189-198

### Plasticity condition related with level lines of strainstates surface, and features of its application in the idealplasticity theory

#### Abstract

The feature of solution formation of boundary valueproblems is considered in plane strain of ideal rigid-plastic bodywhen the plasticity condition is related with level lines ofstrain states surface of work-hardening incompressiblerigid-plastic body.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):199-206

199-206

### Coupled thermodynamic orhogonality in non-linear models of type-III thermoelasticity

#### Abstract

The present study is devoted to a derivation of non-linear constitutive equations for the non-linear Green-Naghdi type-IIIthermoelastic model on the basis of the principle of thermodynamic (or thermomechanical) orthogonality.The latter was proposed by Ziegler as an extention to the Onsager linearirreversible thermodynamics. It states that the irreversible constituent parts of thermodynamic currents (velocities)are orthogonal to the convex dissipation potential level surface in the space of thermodynamic forces for anyprocess of heat propagation in a solid. Non-linear constitutive laws of the heat propagationcomplying with the principle of thermomechanical orthogonality are obtained and discussed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):207-214

207-214

### Analytical solutions of problems of thermoelasticity for multilayered bodies with variable properties

#### Abstract

The technics for the construction of approximate analytical solutions for the quasistatic problems of thermoelasticity (plane-stressed state, plane deformation) for the multilayered bodies with variable within limits of each layer physical properties of medium. The recursive method is used for the construction of systems of coordinate functions, satisfying the boundary matching conditions, given as the equality of radial (normal) stresses and displacements in the layer-contact points.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):215-221

215-221

### Infinite motion in the classical functional mechanics

#### Abstract

In the paper the description of infinite movement in the functional formulation of classical mechanics is investigated. On the example of simple exactly solvable problems (passing through the barrier and falling in the center) the two classes of problems of scattering and singularity are considered. The functional mechanics corrections, arising from scattering, to the mean values and variance of canonical variables are calculated. In particular in the simplest case of transmission through the barrier the shift of the mean value coordinate by a constant arises , this constant depends on the parameters of the barrier, and logarithmic correction to the variance of the free motion coordinate. Also it is shown, that functional mechanics approach leads to the elimination of singularities in the kinetic energy of the falling in the center, which is equivalent to the solution of the Friedman equation in cosmology.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):222-232

222-232

### Study of curvilinear reinforcement rational structures in polar coordinate system

#### Abstract

The problem of curvilinear fibers rationalreinforcement for axially symmetric ring-shaped lamel in polarcoordinate system is solved by reference to the structural model.The effect of structural parameters for a construction limitstressing is studied.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):233-244

233-244

### On a class of fractional differential equations for mathematical models of dynamic system with memory

#### Abstract

Some differential equation with Riemann–Liouville fractional derivatives is considered. The class of these equations are proposed as a model fractional oscillating equation for the description, analysis and investigation of oscillatory processes in dynamic systems with memory. The obtainment such a kind of equations is based on the hypothesis supposed the existence of the non-ideal viscoelastic connection in the one-dimensional dynamic system, which is associated with the fractional analogy of Zener rheologic model of the viscoelastic body. It's shown, that the initial values problems with Cauchy type conditions is reduced equivalently to the Volterra type integral equations with the differentiable kernels. This circumstance allow to use the method of successive approximation to resolve that integral equations. It's indicated, that such a kind of differential equations may be interesting as mathematical models of nonlinear dynamic systems behavior.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):245-252

245-252

### Effect of the influence of rheological beam longitudinal strains on the disc motion state

#### Abstract

The paper analyzes the effect that the material of a simple rheological beam has on thedynamics of a moving disc. The hybrid system of the differential equations describing the motion of the systemdisc–rheological beam consisting of the integro-differential equation of beam longitudinal vibrationsand the Lagrange equations of the first kind, defining the motion of the disc, and the equationsof nonholonomic constraints following from the difference between the Lagrange coordinates of thedisc mass center and the beam point contacting with the disc is composed. The paper considersthe mode of the disc steady motion, allowing to integrate the equation of beam vibrations regardlessthe system of equations describing the motion of the disc. It is identified that when the disc movesat a low speed, and in the mode corresponding to the limit value of the relaxation time it causes physically inadequate strain in the beam. When relaxation time is null there is a steady mode of forced beamvibrations at moderate amplitudes.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):253-259

253-259

### On a fine localization of the Mathieu azimuthal numbers by Cassini ovals

#### Abstract

The study is devoted to numerical and analytical problems concerning generating periodic and antiperiodic solutions of the angular (circumferential) Mathieu equation obtained for the circumferential harmonics of an elliptic cylinder. The Mathieu eigenvalues localization problem and computations of elliptic azimuthal numbers are discussed. First, the Sturm–Liouville eigenvalue problem for the angular Mathieu equation is reformulated as an algebraic eigenvalue problem for an infinite linear self-adjoint pentadiagonal matrix operator acting in the complex bi-infinite sequence space $l_2$. The matrix operator is then represented as a sum of a diagonal matrix and an infinite symmetric doubly stochastic matrix, which is interpreted as a finite perturbation imposed on the diagonal matrix. Effective algorithms for computations of the Mathieu eigenvalues and associated circumferential harmonics are discussed. Azimuthal numbers notion is extended to the case of elastic and thermoelastic waves propagating in a long elliptic waveguide. Estimations of upper and low bounds and thus localizations of the angular Mathieu eigenvalues and elliptic azimuthal numbers are given. Those are obtained by algebraic methods employing the Gerschgorin theorems and Cassini ovals technique. The latter provides more accurate solution of the Mathieu eigenvalues localization problem.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):260-269

260-269

### On a rigorous definition of microscopic solutions of the Boltzmann–Enskog equation

#### Abstract

N. N. Bogolyubov discovered microscopic solutions of the Boltzmann–Enskog equation in kinetic theory of hard spheres. These solutions have the form of sums of the delta-functions and correspond to the exact microscopic dynamics. However, this was done at the “physical level” of rigour. In particular, Bogolyubov did not discuss the products of generalized functions in the collision integral. Here we give a rigorous sense to microscopic solutions by use of regularization. Also, starting from the Vlasov equaton, we obtain new kinetic equations for a hard sphere gas.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):270-278

270-278

### The use of the generalized Pauli's theorem for odd elements of Clifford algebrato analyze relations between spin and orthogonal groups of arbitrary dimensions

#### Abstract

In the present paper we consider the use of generalized Pauli's theorem to prove the theorem about double cover of orthogonal groups by spin groups. We prove theorems about double cover of orthochronous, othochorous, special and special orthochronous groups by corresponding spin groups. We show the difference between the approaches using adjoint action and twisted adjoint action.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):279-287

279-287

### The influence of the characteristics of the external circuit on the form of the electric pulse in the tasks of direct piezoeffect

#### Abstract

The axisymmetric non-stationary problem of electroelasticity for a solid piezoceramic axially polarised cylinder is considered under the kinematic load in the form of well-known mechanical displacements of its face surfaces, as well as the electric potential. The new closed solution is constructed by vector eigenfunction decomposition method in the form of structural algorithm of finite transformations. The obtained calculation relationships allow analyzing the influence of the external circuit characteristics on the form and sizes of the induced electric pulse in nonstationary problems of direct piezoeffect.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):288-296

288-296

### On construction of quantum logical gate based on ESR

#### Abstract

A quantum computer is a computation device operated by means of quantum mechanical phenomena. There are many candidates that are being pursued for physically implementing the quantum computer.The quantum logical gate based on the electron spin resonance (ESR) was studied in ref. [3]. In this paper, we discuss a construction of Controlled-Controlled-NOT (CCNOT) gate by using the nonrelativistic formulation of ESR.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):297-304

297-304

### Note on complexity of quantum transmission processes

#### Abstract

In 1989, Ohya propose a new concept, so-called Information Dynamics (ID), to investigate complex systems according to two kinds of view points. One is the dynamics of state change and another is measure of complexity. In ID, two complexities $ C^{S} $ and $ T^{S} $ are introduced. $ C^{S} $ is a measure for complexity of system itself, and $ T^{S} $ is a measure for dynamical change of states, which is called a transmitted complexity. An example of these complexities of ID is entropy for information transmission processes. The study of complexity is strongly related to the study of entropy theory for classical and quantum systems. The quantum entropy was introduced by von Neumann around 1932, which describes the amount of information of the quantum state itself. It was extended by Ohya for C*-systems before CNT entropy. The quantum relative entropy was first defined by Umegaki for $ \sigma $-finite von Neumann algebras, which was extended by Araki and Uhlmann for general von Neumann algebras and *-algebras, respectively. By introducing a new notion, the so-called compound state, in 1983 Ohya succeeded to formulate the mutual entropy in a complete quantum mechanical system (i.e., input state, output state and channel are all quantum mechanical) describing the amount of information correctly transmitted through the quantum channel. In this paper, we briefly review the entropic complexities for classical and quantum systems. We introduce some complexities by means of entropy functionals in order to treat the transmission processes consistently. We apply the general frames of quantum communication to the Gaussian communication processes. Finally, we discuss about a construction of compound states including quantum correlations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):305-314

305-314

### Ultrametricity as a basis for organization of protein molecules: CO binding to myoglobin

#### Abstract

In this paper, the basic notions of ultrametric ($p$-adic) description of protein conformational dynamics and CO rebinding to myoglobin are presented. It is shown that one and the same model of the reaction — ultrametric diffusion type describes essentially different features of the rebinding kinetics at high-temperatures ($300{\div}200$ K) and low-temperatures ($180{\div}60$ K). We suggest this result indicates a special structural order in a protein molecule. Besides all the other structural features, it is organized by such a way that its conformational mobility changes self-similar from room temperature up to the cryogenic temperatures.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):315-325

315-325

### The energy levels and eigen wave functions of electrons in quantum rings in the magnetic field

#### Abstract

The solution of the eigenvalue problem for non interacting electrons of the quantum ring in the magnetic field is discussed. The potential shape of the quantum ring permitting analytical solution was proposed. The solution of the appropriate eigenvalue problem was found in the terms of the Heun functions and expression for the energy levels was obtained. It was pointed out that proposed potential might be considered as a single-well or double-well potential of concentric quantum rings.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):326-333

326-333

### Equation on the basis of one-dimensional chaotic dynamics

#### Abstract

Modified Klein–Gordon–Fock equations were obtained on the basis of one-dimensional chaotic dynamics and the original Lagrangians were found. The concepts of $m$-exponential map and groups with broken symmetry are introduced. A system of bitrial orthogonal functions is considered.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):334-342

334-342

### The ultrametrical dynamics for the closed fractal-cluster resource models

#### Abstract

The evolution scenario of the resource distribution in the fractal-cluster systems which are identified as organism on Burdakov's classification is suggested. In this model the resource distribution dynamics is determined by the ultrametric structure of the fractal-cluster space. Thus for each cluster there is the characteristic time of its transition to an equilibrium state defined by ultrametric size of the cluster. The general equation that describes that dynamics is presented. The numeric solution for that equation for the certain types of resource transformation between clusters is received. The problem of identification of parameters of model with reference to real systems is discussed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):343-351

343-351

### The random-disturbed dynamic models and the maximum entropy method

#### Abstract

In the work the behavior of random-disturbed equations is analysed on the basis of the Reynolds method and the maximum entropy principle. The stability of models is analysed. The general features of dynamics of Verhulst model, Volterra–Lotke model and Euler's equations of solid body rotation are revealed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):352-360

352-360

### Phase space curvature

#### Abstract

Electromagnetic field in classical and quantum mechanics is naturally representedby geometry of extended phase space, with extra coordinates of time and canonically conjugate momentum$p_0=-E$.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):361-368

361-368

### High temperature heat kernel expansion and its applications

#### Abstract

The algorithm constructed to build the high-temperature heat kernel expansion and the statistic sum on the noncompact Lie groups manifolds is discussed in the article. The method is based on the formalism of non-commutative integration which originated from the coadjoint orbits' approach to the problems of integration and quantization. Applications of presented method to the problems of quantum statistic mechanics and quantum field theory are also discussed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):369-378

369-378

### Star product on the Lie coalgebra and its application for calculation of quantum integrals of motion

#### Abstract

The article gives an algorithm for constructing quantum integrals of motion on the basis of well-known classic integrals.To construct quantum integrals, we apply star product of the operators' symbols, which is used in the quantization theory.A non-trivial example of the Klein–Fock equation is considered on the four-dimensional Lie group.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):379-386

379-386

### Space localization of the quantum particle

#### Abstract

It is shown, that in addition to the evolution quantum object motion wave equation in the integral form can describe the wave function reduction as a physical process. Such description is represented for the space localization process, taking place when the space coordinate is measured, and it is shown, that collapse arises, as the result of the quantum particle and corresponding measuring instrument interaction. This physical phenomenon mathematical image looks like the instantaneous transformation of the virtual paths set to the subset, determined by the measuring process conditions, when the macroscopic changes appears in the measuring instrument.In conventional quantum mechanics such Hilbert space collapse itself corresponds to such reduction phenomenon.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2013;17(1):387-397

387-397