## Vol 21, No 1 (2017)

**Year:**2017**Articles:**11**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1222

Articles

On a speed of solutions stabilization of the Cauchy problem for the Carleman equation with periodic initial data

#### Abstract

This article explores a one-dimensional system of equations for the discrete model of a gas (Carleman system of equations). The Carleman system is the Boltzmann kinetic equation of a model one-dimensional gas consisting of two particles. For this model, momentum and energy are not retained. On the example of the Carleman model, the essence of the Boltzmann equation can be clearly seen. It describes a mixture of "competing” processes: relaxation and free movement. We prove the existence of a global solution of the Cauchy problem for the perturbation of the equilibrium state with periodic initial data. For the first time we calculate the stabilization speed to the equilibrium state (exponential stabilization).

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):7-41

On existence of solution in $\mathbb R^n$ of stochastic differential inclusions with current velocities in the presence of approximations with uniformly bounded first partial derivatives

#### Abstract

Notion of mean derivatives was introduced by Edward Nelson for the needs of stochastic mechanics (a version of quantum mechanics). Nelson introduced forward and backward mean derivatives while only their half-sum, symmetric mean derivative called current velocity, is a direct analog of ordinary velocity for deterministic processes. Another mean derivative called quadratic, was introduced by Yuri E. Gliklikh and Svetlana V. Azarina. It gives information on the diffusion coefficient of the process and using Nelson’s and quadratic mean derivatives together, one can in principle recover the process from its mean derivatives. Since the current velocities are natural analogs of ordinary velocities of deterministic processes, investigation of equations and especially inclusions with current velocities is very much important for applications since there are a lot of models of various physical, economical etc. processes based on such equations and inclusions. Existence of solution theorems are obtained for stochastic differential inclusions given in terms of the so-called current velocities (symmetric mean derivatives, a direct analogs of ordinary velocity of deterministic systems) and quadratic mean derivatives (giving information on the diffusion coefficient) on $\mathbb R^n$. Right-hand sides in both the current velocity part and the quadratic part are set-valued but satisfy some natural conditions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):42-54

The evaluation of the order of approximation of the matrix method for numerical integration of the boundary value problems for systems of linear non-homogeneous ordinary differential equations of the second order with variable coefficients. Message 2. Boundary value problems with boundary conditions of the second and third kind

#### Abstract

We present the second message of the cycle from two articles where the rearrangement of the order of approximation of the matrix method of numerical integration depending on the degree in the Taylor’s polynomial expansion of solutions of boundary value problems for systems of ordinary differential equations of the second order with variable coefficients with boundary conditions of the second kind were investigated. Using the Taylor polynomial of the second degree at the approximation of derivatives by finite differences leads to the second order of approximation of the traditional method of nets in inner points of the integration domain. In the study of boundary value problems for systems of ordinary differential equations of the second order we offer the previously proposed method of numerical integration with the use of matrix calculus where the approximation of derivatives by finite differences was not performed. According to this method a certain degree of Taylor polynomial can be selected at random for the construction of the difference equations system. The disparity is calculated and the order of the method of approximation is assessed depending on the chosen degree of Taylor polynomial. It is theoretically shown that a) for the boundary value problem with boundary conditions of the second and third kind the order of approximation is linearly proportional to the Taylor polynomial used and less than this level by 1 without regard to its parity; b) at even degree the order of approximation at boundary points of the integration domain is less by 1 than the order of approximation of the inner points; c) at uneven degree the orders of approximation at boundary points and in inner points of the integration domain are the same and less than this level by 1. For even degree the method of increasing of the order of approximation by 1 at boundary points of the integration domain to the order of approximation in inner points is performed. The theoretical conclusions are confirmed by a numerical experiment for boundary value problems with boundary conditions of the third kind.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):55-79

The Dirichlet problem for a mixed-type equation with strong characteristic degeneracy and a singular coefficient

#### Abstract

In this paper we consider the first boundary value problem in a rectangular area for a mixed-type equation of the second kind with a singular coefficient. The criterion of the uniqueness of the problem solution is determined. The uniqueness of the problem solution is proved on the basis of completeness of the system of eigenfunctions of the corresponding onedimensional spectral problem. The solution of the problem is built explicitly as a sum of Fourier-Bessel. There is the problem of the small denominators that appears when justifying the uniform convergence of the constructed series. In this regard, an evaluation of separateness from zero with a corresponding small denominator asymptotic behavior is found. This estimate has allowed to prove the convergence of the series and its derivatives up to the second order, and the existence theorem for the class of regular solutions of this equation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):80-93

A conditions of solvability of the Goursat problem in quadratures for the two-dimensional system of high order

#### Abstract

In this paper we consider the Goursat problem for the two-dimensional system of high order. The purpose is to find sufficient conditions of solvability of the considered problem in quadratures. The method of finding solutions of these problems in explicit calculation based on factorization of equations is devised. As a result the initial problem is reduced to 5 simpler problems: to four Goursat problems for equation and the Goursat problem for hyperbolic system of the second order. The final result in terms of the coefficients of the original system is formulated in two theorems.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):94-111

On nonlocal problem with fractional Riemann-Liouville derivatives for a mixed-type equation

#### Abstract

The unique solvability is investigated for the problem of equation with partial fractional derivative of Riemann-Liouville and boundary condition that contains the generalized operator of fractional integro-differentiation. The uniqueness theorem for the solution of the problem is proved on the basis of the principle of optimality for a nonlocal parabolic equation and the principle of extremum for the operators of fractional differentiation in the sense of Riemann-Liouville. The proof of the existence of solutions is equivalent to the problem of solvability of differential equations of fractional order. The solution is obtained in explicit form.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):112-121

Mesoscopic models for definition of the large-scale elastic properties of the soft magnetic elastomers

#### Abstract

A pair of magnetizing particles embedded in a cylinder made of a highelasticity (hyperelastic) material is considered as a model of a mesoscopic structure element of a soft magnetic elastomer. In the presence of the magnetic field particles magnetize and the force interaction is arisen between them. Particles change position inside the elastomer matrix as the elastic resistance allows it. Equilibrium position of the particles inside the sample is determined by the balance of magnetic and elastic forces and corresponds to the minimum of total energy of the system. In its calculation both the non-linearity and heterogeneity of the magnetization of the particles and non-linearity of the elastic properties of the elastomer have been taken into account. This brings us to the real magnetorheological composite, that is a soft elastomer filled with a micron ferromagnetic particles. The considered system exhibits bistability: increase and decrease of the applied magnetic field, leads to change of the distance between the particles in hysteretic manner, from a few radii to the tight contact (collapse). This behavior significantly affects the ability of a mesoscopic element to resist external load. Collapse of the particles inside it by a magnetic field or compressive load causes sharp increase of stiffness. The dependence of mechanical characteristics of the system on the strength of an applied magnetic field is studied for the elements of different compliance. This dependence also has a hysteresis. Despite its simplicity, the model in a generally correct way describes the field-induced changes of the internal structure of soft magnetic elastomers. The obtained results are used for qualitative analysis of the macroscopic magnetomechanics of the composite, this is done with the aid of a homogenisation procedure based of Voigt’s hypothesis. The obtained dependence of the magnetic stiffness of soft magnetic elastomer on the external magnetic field agrees qualitatively with the published experimental results.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):122-136

Transient dynamics of 3D inelastic heterogeneous media analysis by the boundary integral equation and the discrete domains methods

#### Abstract

For the study of transients in 3D nonlinear deformable media we develope modeling methods which based on integral representations of 3D boundary value problem of elastic dynamics, numerical high-order approximation schemes of boundaries and collocation approximation of solutions. The generalized boundary integral equation method formulations using fundamental solutions of static elasticity, equation of state of elastoplastic media with anisotropic hardening and difference methods for time integration are represented. We take into account the complex history of combined slowly changing over time and impact loading of composite piecewise-homogeneous media in the presence of local perturbation solutions areas. With the use of this method and discrete domains method the solutions of applied problems of the propagation of non-linear stress waves in inhomogeneous media are received. Comparisons with the solutions obtained by the finite element method are represented also. They confirm the computational efficiency of the developed algorithms, as well as common and useful for practical purposes of the proposed approach.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):137-159

The nonlinear Maxwell-type model for viscoelastoplastic materials: simulation of temperature influence on creep, relaxation and strain-stress curves

#### Abstract

The nonlinear Maxwell-type constitutive relation with two arbitrary material functions for viscoelastoplastic multi-modulus materials is studied analytically in uniaxial isothermic case to reveal the model abilities and applicability scope and to develop techniques of its identification, tuning and fitting. The constitutive equation is aimed at adequate modeling of the rheological phenomena set which is typical for reonomic materials exhibiting non-linear hereditary properties, strong strain rate sensitivity, secondary creep, yielding at constant stress, tension compression asymmetry and such temperature effects as increase of material compliance, strain rate sensitivity and rates of dissipation, relaxation, creep and plastic strain accumulation with temperature growth. The model is applicable for simulation of mechanical behaviour of various polymers, their solutions and melts, solid propellants, sand-asphalt concretes, composite materials, titanium and aluminum alloys, ceramics at high temperature and so on. To describe the influence of temperature on material mechanical behavior (under isothermic conditions), two scalar material parameters of the model (viscosity coefficient and “modulus of elasticity”) are considered as a functions of temperature level. The general restrictions on their properties which are necessary and sufficient for adequate qualitative description of the basic thermomechanical phenomena related to typical temperature influence on creep and relaxation curves, creep recovery curves, creep curves under stepwise loading and quasi-static stress-strain curves of viscoelastoplastic materials are obtained. The restrictions are derived using systematic analytical study of general qualitative features of the theoretic creep and relaxation curves, creep curves under step-wise loading, long-term strength curves and stress-strain curves at constant strain or stress rates generated by the constitutive equation (under minimal restrictions on material functions) and their comparison to typical test curves of stable viscoelastoplastic materials. It is proved that the viscosity coefficient and the “modulus of elasticity” of the model and their ratio (i.e. relaxation time of the associated linear Maxwell model) should be decreasing functions of temperature. This requirements are proved to provide an adequate qualitative simulation of a dozen basic phenomena expressing an increase of material compliance (a decrease of tangent modulus and yield stress, in particular), strengthening of strain rate sensitivity and acceleration of dissipation, relaxation, creep and plastic strain accumulation with temperature growth.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):160-179

A large-scale layered stationary convection of an incompressible viscous fluid under the action of shear stresses at the upper boundary. Velocity field investigation

#### Abstract

The exact solution of the definition of convective motions in a layered large-scale flows of a viscous incompressible fluid in a steady case is considered. It was shown that the received problem is, firstly, overdetermined and, secondly, a nonlinear (due to the presence of members of a convective derivative in a heat conduction equation). Also it was shown that the solution class choice can eliminate the override, and the specification of a boundary conditions can reduce the problem to the study of a thermal capillary convection (convection Benard-Marangoni). Then conditions of the counterflow appearance are defined, and their possible amount is investigated. In addition, the analysis of the nonvortex region in the test flow is made. And it was shown that under certain combinations of system parameters the vortex can change the direction.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):180-196

Obtaining exact analytical solutions for nonstationary heat conduction problems using orthogonal methods

#### Abstract

Through the interplay of orthogonal methods by L. V. Kantorovich, Bubnov-Galerkin and a heat balance integral method there have been obtained an exact analytical solution of a nonstationary heat conduction problem for an infinite plate under the symmetrical first-type boundary conditions. It was possible to obtain an exact solution through the employment of approximate methods due to the appliance of trigonometric coordinate functions, possessing the property of orthogonality. They enable us to determine eigenvalues not through the solution of the Sturm-Liouville boundary value problem, which supposes the second-order differential equation integration, but through the solution of a differential equation for an unknown function on time, which is the first-order equation. Due to the property of coordinate functions mentioned above, while determining constants of integration out of initial conditions it is possible to avoid solving large systems of algebraic linear equations with ill-conditioned matrix of coefficients. Thus, it simplifies both the process of obtaining a solution and its final formula and provides an opportunity to find not only an approximate, but also an exact analytical solution, represented by an infinite series.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2017;21(1):197-206