# No 1 (2015)

**Year:**2015**Articles:**10**URL:**https://journals.rudn.ru/miph/issue/view/518

## Articles

### About Some Kind of Differential-Operator Equations with Variable Coefficients

#### Abstract

In this work a general method, allowing to find solutions of differential-operator equations of some type with variable coefficients by means of analitical vector-valued functions, is described. Examined equations include as particular case differential equations in partial derivatives, difference-differential and integral equations, and other functional-operator equations. Solutions are realized by uniformly converged functional vector-valued series, generated by set of solutions of ordinary differential equation of n-th order and some set of elements of locally convex space. Sufficient conditions of continuous dependence of solutions from generating set are found. Solution of Cauchy problem for examined equations is found as well and conditions of its uniqueness are specified. Besides that the so-called general solution of examined equations (the function of the most general view, from which any particular solution can be obtained) is found. The investigation is realized by means of characteristics (order and type) of operator and operator characteristics (operator type and operator order) of vector relative to operator. In this work in investigation a convergence of operator series relative to equicontinuous bornology is used.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):3-14

3-14

### Algorithm for Solving the Two-Dimensional Boundary Value Problem for Model of Quantum Tunneling of a Diatomic Molecule Through Repulsive Barriers

#### Abstract

Algorithm for solving the boundary value problems that describe the model of quantum tunneling of a diatomic molecule through repulsive barriers in s-wave approximation is presented. The boundary value problems are formulated and reduced to the one-dimensional ones for systems of coupled second-order differential equations by means of the Galerkin and Kantorovich methods. The description of elaborated algorithms and the calculated asymptotes of parametric basis functions, matrices of variable coefficients, and fundamental solutions of the systems of the coupled second-order differential equations needed for solving the boundary problems on a finite interval are given. The BVPs were solved by the elaborated set of programs implementing the finite element method. Analysis of benchmark calculations of quantum tunneling of a diatomic molecule model with the nuclei coupled by the Morse potential through Gaussian barriers and quantum transparency effect induced by metastable states embedded in continuous spectrum below dissociation threshold are presented.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):15-36

15-36

### On Solutions of the Maxwell’s Equations from the Viewpoint of Geometrical Optics

#### Abstract

Traditionally ideas of geometrical optics apply to research of the approximate solutions corresponding to high-frequency limit, but it is known that, e.g. jumps of solutions of the equations of Maxwell satisfy to Huygens’s law also. In the article we indicate the class of exact solutions of the Maxwell’s equations for which the approach of the geometrical optics can be still used. We consider solutions of the Maxwell’s equations with which it is possible to associate orthogonal system of coordinates of (x1,x2,x3) so that the directions of vectors ⃗E and ⃗e2 and also ⃗H and ⃗e3 are coincided. Conditions on Lamé coefficients of this system of coordinates are found: euh1 doesn’t depend on x2 and x3 and logarithmic derivatives eh1h3 h2 and μh1h2 h3 with respect to x1 don’t depend on x2 and x3 respectively. The ﬁrst condition means that x1-lines are rays of geometrical optics and it gives a reason to call such systems of coordinates as ray systems how it is accepted in geometrical optics. Thus the solution of the Maxwell’s equations can be described as a wave extending along a ray, that is as the solution of the two-dimensional hyperbolic equation. Necessary and suﬃcient conditions are found for association of such coordinates systems with the solution of the equations of Maxwell: the directions of vectors ⃗E and ⃗H don’t change over time, they are orthogonal each other and consist in involution, that is ⃗E × ⃗ H,rot ⃗E × ⃗ H = 0.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):37-44

37-44

### Numerical Method for Computation of Sliding Velocities for Vortices in Nonlocal Josephson Electrodynamics

#### Abstract

In this paper, a model of inﬁnite Josephson layered structure is considered. The structure consists of alternating superconducting and tunnel layers and it is assumed that (i) the electrodynamics of the structure is nonlocal and (ii) the current-phase relation is presented by sum of Fourier harmonics instead of one sinusoidal harmonic for the case of the sine-Gordon equation. The governing equation is a nonlocal generalization of the nonlinear Klein-Gordon equation with periodic nonlinearity that depends on external parameter of nonlocality λ. The velocity of vortices (2 π-kinks) in models of such kind are not arbitrary, but belong to some discrete set. The paper presents a method for computation of these velocities (called also “sliding velocities”) and the shapes of kinks. The estimation of error of the method is given. The results of computations are the families of 2 π-kinks parametrized by λ. It is observed that the 2 π-kinks corresponding to diﬀerent families for the same λ have nearly the same central part but diﬀer in asymptotics of the tails. The numerical algorithm has been incorporated into a program complex “Kink solutions” in MatLab environment. The complex enables to compute the shapes and velocities of 2 π-kinks for nonlinearities represented by sums of up to ten Fourier harmonics, as well as to model the propagation of these kinks.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):45-52

45-52

### Computational Scheme for Solving Heat Conduction Problem in Multilayer Cylindrical Domain

#### Abstract

The computational scheme for solving heat conduction problem with periodic source function in multilayer cylindrical domain is suggested. The domain has a non-trivial geometry and the thermal coefficients are non-linear functions of temperature and have discontinuity of the first kind at the borders of the layers. The computational scheme is based on an algorithm for solving difference problem using the explicit-implicit method. The OpenCL realization of the suggested algorithm for calculations performed on a GPU is also compared to calculations performed using a CPU. It is shown that the scheme can be successfully applied to simulations of thermal processes in pulsed cryogenic cell, which is intended for pulse feeding the working gases into the working space of the ion source within the millisecond range. The results are given for a simulation of one of the particular cell structures, which is assumed to correspond to the practical realization. The computational scheme can be used for the optimization problem of the cell model parameters.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):53-59

53-59

### Stabilization of Redundantly Constrained Dynamic System

#### Abstract

This article addresses the issue of constraint stabilization in a dynamic system. The well known Lagrange’s equation of motion of second order is used for modelling the dynamics of a mechanical systems considered in this paper. It is known that Baumgarte’s method of constraint stabilization does not avoid the problem of singularity of mass matrices that may result from redundancy of constraints and as a result it fails to run simulations near and at singularity points. A generalized Baumgarte’s method of constraint stabilization is developed and the stability of the developed method is ascertained by Lyapunov’s direct method. The developed method avoids using the same correction parameters for all constraints under discussion. The usual Baumgarte’s method, which uses the same correction parameters, becomes a particular case of the one developed in this article. Moreover, a modified Lagrange’s equation is constructed in a way that explains all the details of its derivation. The modified Lagrange’s equation improves Lagrange’s equation of motion in such a way that, it addresses the issue of redundant constraints and singular mass matrices. As it is the case in Baumgarte’s method, the usual Lagrange’s equation is a particular case of the improved method developed in this paper. Besides, a numerical example is provided in order to demonstrate the effectiveness of the methods developed. Finally, the carried out simulations show asymptotic stability of the trajectories and run without problem at singularity points.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):60-72

60-72

### Dynamic Control of Constrained Systems and Inverse Problems of Dynamics

#### Abstract

The control problem of dynamic system, containing different physical elements, is solved. Using known dynamic analogies, processes in difficult system are described by the differential-algebraic equations of the classical mechanics. The corresponding differential-algebraic equations include the dynamic equations, the constraints equations and the formulation of purpose of control. Dynamics of system is described by Lagrange equations or by equations in the canonical variables, containing indeterminate multipliers in the right hand sides. The problem of definition of Lagrange multipliers or control functions corresponding to the constraints equations, is reduced to construction of the differential equations systems having partial integrals. Definition of solutions stability of the dynamics equations in relation to the constraints equations is given. The dynamic indicators considering deviations from the constraints equations are entered for ensuring asymptotic stability and constraints stabilization at the numerical solution of the differential equations. The expanded system of dynamics equations, consisting of the initial system dynamics equations and the constraints perturbations equations is under construction. The constraints perturbations equations, constructed on the modified dynamic indicators, allow to define stability conditions and constraints stabilization. Conditions of constraints stabilization, corresponding to the numerical solution of the dynamics equations are given by Euler method and Runge-Kutta method. The solution of a problem of stabilization of vertical position of the rod fixed by cylindrical joint on the cart, making rectilinear movement, is proposed. The control is performed by force acting on the cart and moment applied to the rod.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):73-82

73-82

### The Potential of Inflation and of Isotropization of Spinor Field

#### Abstract

The existence of inflation, when a scale factor was varying with exponential or with power low, is important requirement in the theory of early Universe. Also it was established through the use of astronomical observations, that in the present stage the Universe is expanding with acceleration. One of the possible explanation for accelerating expansion of the Universe is the assumption about the existence of “dark” energy, the nature of that is not clear now. In this case “dark” energy is dominating in the Universe. Its density exceeds the energy of all “usual” cosmic forms of the matter taken together. In many works “dark” energy is simulated by ideal fluid with negative pressure. Also there is some scientific interest in considering the models in which the process of the expansion is followed by the isotropization of the Universe on a large scale. In this work we consider the model of the evolution of the Universe that is filled by the spinor matter. The potential of inflation of spinor field is established according to its correspondence with ideal fluid. The potential of isotropization of spinor field is also established.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):83-90

83-90

### Svedeniya ob avtorakh

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):91

91

### Guidelines for Authors

**Discrete and Continuous Models and Applied Computational Science**. 2015;(1):92-93

92-93