# Vol 22, No 2 (2018)

**Year:**2018**Articles:**11**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1228

203-213

### Nonlinear dynamics of open quantum systems

#### Abstract

The evolution of a composite closed system using the integral wave equation with the kernel in the form of path integral is considered. It is supposed that a quantum particle is a subsystem of this system. The evolution of the reduced density matrix of the subsystem is described on the basis of the integral wave equation for a composite closed system. The equation for the density matrix for such a system is derived. This equation is nonlinear and depends on the history of the processes in the closed system. It is shown that, in general, the reduced density matrix trace does not conserve in the evolution processes progressing in open systems and the procedure of the trace normalization is necessary as the mathematical image of a real nonlocal physical process. The wave function collapse and EPR correlation are described using this approach.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):214-224

214-224

### A criterion for the unique solvability of the spectral Dirichlet problem for a class of multidimensional hyperbolic-parabolic equations

#### Abstract

In the cylindrical domain of Euclidean space for one class of multidimensional hyperbolic parabolic equations the spectral Dirichlet problem with homogeneous boundary conditions is considered. The solution is sought in the form of an expansion in multidimensional spherical functions. The existence and uniqueness theorems of the solution are proved. Conditions for the unique solvability of the problem are obtained, which essentially depend on the height of the cylinder.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):225-235

225-235

### Construction of Mikusinski operational calculus based on the convolution algebra of distributions. Methods for solving mathematical physics problems

#### Abstract

A new justification is given for the Mikusinsky operator calculus entirely based on the convolution algebra of generalized functions $D'_+$ and $D'_-$, as applied to the solution of linear partial differential equations with constant coefficients in the region $(x;t)\in \mathbb R (\mathbb R_{+})\times \mathbb R_{+}$. The mathematical apparatus used is based on the current state of the theory of generalized functions and its one of the main differences from the theory of Mikusinsky is that the resulting images are analytical functions of a complex variable. This allows us to legitimate the Laplace transform in the algebra $D'_{+} $ $( x\in \mathbb R_{+} )$, and apply the algebra to the region of negative values of the argument with the use of algebra $D'_{-}$. On classical examples of second-order equations of hyperbolic and parabolic type, in the case $x\in \mathbb R$, questions of the definition of fundamental solutions and the Cauchy problem are stated, and on the segment and the half-line $x\in \mathbb R_{+}$, non-stationary problems in the proper sense are considered. We derive general formulas for the Cauchy problem, as well as circuit of fundamental solutions definition by operator method. When considering non-stationary problems we introduce the compact proof of Duhamel theorem and derive the formulas which allow optimizing obtaining of solutions, including problems with discontinuous initial conditions. Examples of using series of convolution operators of generalized functions are given to find the originals. The proposed approach is compared with classical operational calculus based on the Laplace transform, and the theory of Mikusinsky, having the same ratios of the original image on the positive half-axis for normal functions allows us to consider the equations posed on the whole axis, to facilitate the obtaining and presentation of solutions. These examples illustrate the possibilities and give an assessment of the efficiency of the use of operator calculus.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):236-253

236-253

### Integral necessary condition of optimality of the second order for control problems described by system of integro-differential equations with delay

#### Abstract

We consider the optimal control problem that is described by the system of integro-differential equations of the Volterra type with delay and multipoint performance criterion. The first and the second variations of the performance criterion are calculated under the hypothesis that the control domain is open. The necessary condition of the first order optimality in the form analogous to the Euler equations is deduced from the equality of the first variation of performance criterion and zero along the optimal process. Next, the implicit necessary condition of the second order optimality is obtained, which helps to establish rather general but constructively verified necessary condition for the second order optimality. The obtained results are applicable for constructing easy-verifying necessary conditions of optimality for the singular (in the usual sense) controls.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):254-268

254-268

### On the question of the correctness of inverse problems for the inhomogeneous Helmholtz equation

#### Abstract

In the rectangular domain, the initial-boundary value problem for the Helmholtz equation and its non-local modifications are studied and the inverse problems for finding its right-hand side are studied. The solutions of direct problems with nonlocal boundary conditions and inverse problems are constructed in explicit form as the sums of orthogonal series in the system of eigenfunctions of the one-dimensional Sturm-Liouville spectral problem. The corresponding uniqueness theorems for the solution of all set problems are proved. Sufficient conditions for boundary functions are established, which are guaranteed by the existence and stability theorems for the solution of the proposed new problem statements.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):269-292

269-292

### Properties of stress-strain curves generated by the nonlinear Maxwell-type viscoelastoplastic model under loading and unloading at constant stress rates

#### Abstract

A physically nonlinear Maxwell-type constitutive relation for non-aging rheonomic materials is studied analytically to find out the set of basic rheological phenomena that it simulates, to indicate its application field and to develop identification techniques and ways of tuning and further modifications. Under minimal primary restrictions on two material functions of the relation, the general equation of theoretic stress-strain curves family produced by the model under loading and unloading at constant stress rates is derived and analyzed in uni-axial case. Intervals of monotonicity and convexity of loading and unloading curves, conditions for existence of extremum and inflection points, magnitudes of maximal strain, strain rate jumps and plastic strain arising as a result of loading- unloading cycle are considered and their dependences on material functions and on stress rate and maximal stress are examined. The main qualitative properties of stressstrain curves and unloading responses generated by the constitutive equation are compared to typical properties of test loading-unloading curves of viscoelastoplastic materials in order to elucidate capabilities of the model, to obtain necessary phenomenological restrictions which should be imposed on the material functions and to find convenient indicators of applicability (or non-applicability) that can (and should) be checked examining test data of a material.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):293-324

293-324

### Mathematical modeling of dynamic deformation of elasto-viscoplastic shells of finite length by a ray method

#### Abstract

The paper presents a mathematical modeling of dynamic stress-strain state of the rotation of the shell elasto-viscoplastic material. We solve the modified system of S. P. Timoschenko’s partial differential equations by constructing a system of equations on moving surfaces of the gap with the initial conditions in the form of a shock at the end, written in the form of a power series in time, whose coefficients have initial conditions for the differential equations. The solution is presented in the form of the Taylor’s row series up to the fourth order in the shell coordinate. To simulate the waves reflected from the boundaries, the conditions at the boundary of two types (rigidly restrained and stress-free), independent of time, are introduced. A set of programs written in Fortran 90 on the Code::Blocks platform is developed. Two programs for the simulation of dynamic deformation of shell in elastic and elasto-viscoplastic state are implemented. We use the difference representation of the derivatives, the calculation of the integrals by the trapezoid method with a given step of partitioning the segment. The result of the programs is the grid functions of the coefficients of the Taylor rows, which are used to construct the displacement graphs as functions of time and the longitudinal coordinate of the shell.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):325-343

325-343

### Numerical study of mass transfer in drop and film systems using a regularized finite difference scheme in evaporative lithography

#### Abstract

Mass transfer in drying drops and films is interesting with practical point of view, since it is used in problems of evaporative lithography. Compensatory flows arise when conditions of nonuniform evaporation from the surface of the liquid layer are created and move colloidal particles in the region of fast evaporation. This makes it possible to obtain micro- and nanostructures of the required shape on a solid surface. Nonstationary model of mass transfer in drops and films is described in this paper. Feature of the model is to jointly take into account viscous, gravitational and capillary forces. To solve the unstable discrete problem on drying drop (film), a regularized finite difference scheme is proposed. A computer algorithm is developed on the basis of this scheme. We present a way of obtaining ring structures by using evaporative lithography method that based on the results of the computational experiments.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):344-363

344-363

### Chaotic regimes of a fractal nonlinear oscillator

#### Abstract

In the paper, a fractal nonlinear oscillator was investigated with the aim of identifying its chaotic oscillatory regimes. The measure of chaos for a dynamic system is the maximum Lyapunov exponents. They are considered as a measure of the dispersal of several phase trajectories constructed under different initial conditions. To determine the maximum Lyapunov exponents, algorithms are used which are related either to the study of time series (Benettin’s algorithm) or to the direct solution of an extended dynamical system (Wolff’s algorithm). In this paper, the Wolf algorithm with the Gram-Schmidt orthogonalization procedure was used as the method for constructing Lyapunov’s maximum exponents. This algorithm uses the solution of the extended initial dynamical system in conjunction with the variational equations, and the Gram-Schmidt orthogonalization procedure makes it possible to level out the component of the maximum Lyapunov exponent when computing vectors along phase trajectories. Further, the Wolf algorithm was used to construct the spectra of Lyapunov exponents as a function of the values of the control parameters of the initial dynamical system. It was shown in the paper that certain spectra of Lyapunov exponents contain sets of positive values, which confirms the presence of a chaotic regime, and this is also confirmed by phase trajectories.It was also found that the fractal nonlinear oscillator has not only oscillatory modes, but also rotations. These rotations can be chaotic and regular.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):364-379

364-379

### Modal identification of a boundary input in the two-dimensional inverse heat conduction problem

#### Abstract

A method for the approximate solution of a two-dimensional inverse boundary heat conduction problem on a compact set of continuous and continuously differentiable functions is proposed. The method allows us to reconstruct a boundary action that depends on time and a spatial coordinate. A modal description of the object is used in the form of an infinite system of linear differential equations with respect to the coefficients of the expansion of the state function in a series in eigenfunctions of the initial-boundary value problem under study. This approach leads to the restoration of the sought value of the heat flux density in the form of a weighted sum of a finite number of its modal components. Their values are determined from the temporal modes of the temperature field, which are found from the experimental data on the basis of the modal representation of the field. To obtain a modal description of the identified input and the temperature field in the form of their expansions into series in eigenfunctions of the same spatial dimension, the mathematical model of the object in the Laplace transform domain and the method of finite integral transformations are used. On this basis, a closed system of equations with respect to the unknown quantities is formed. The proposed approach allows us to construct a sequence of approximations that uniformly converge to the desired solution with increasing number of considered modal components. The problem of the temperature experimental design is solved. This solution ensures the minimization of the approximation error of the experimental temperature field by its model representation in the uniform metric of estimating temperature discrepancies on the control line at the final moment of the identification interval.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2018;22(2):380-394

380-394