# No 4 (2013)

**Year:**2013**Articles:**20**URL:**https://journals.rudn.ru/miph/issue/view/534

## Articles

### Estimating the Norm of Solution of Singularly Perturbed Quasilinear Problems for ODE Systems with Nonlinear Normal Matrices on the Semiaxis

#### Abstract

Using the method of unitary transformation, the singularly perturbed quasi-linear systems of ordinary diﬀerential equations with nonlinear normal matrices on the semiaxis were studied, which in some cases can lead to the existence of countable number of additional boundary layers. For such system, most problems arise in the study of the stability of their solution especially in critical cases where the spectrum deﬁned by the matrix lies (or touches) the imaginary axis. The proposed method allows us to study the traditional Lyapunov functions. We have shown suﬃcient conditions for stability (and asymptotic stability) and given the evaluation of the norm of the solution for such problems, which clariﬁes or supplements previously known results. In addition in the paper we have included some non-trivial examples of nonlinear singularly perturbed problems for quasi-linear systems of ordinary diﬀerential equations with nonlinear normal matrices.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):5-10

5-10

### The Algorithm of Reducibility of Inhomogeneous Systems with Polynomially Periodic Matrix on the Basis of Spectral Method

#### Abstract

The paper is devoted to investigation of the class of linear and quasi-linear systems of ordinary diﬀerential equations, the matrix of which can be characterized as polynomially periodic. The main aim of this article is to generate a new algorithm of their splitting in order to create equivalent sets with almost diagonal matrix that are simpler to analyze. Another objective is formulating and proving of suﬃcient stability conditions or asymptotic stability of their trivial decision. The question is topical since the analysis of a considered class of non-autonomous systems with the use of known methods (for example, the method of functions of Lyapunov) is complicated. In addition, the usage of spectral and other methods while solving non-uniform sets might cause extra diﬃculties. The authors of the paper develop an analytical method which appears to be a summary of known classical theorems. At the heart of the oﬀered algorithm of reducibility lies one of options of splitting method, which is conducted by a spectrum of a deﬁning matrix in studied non-autonomous system (taking into account its splitting on diagonal and non-diagonal part) lies. The present article shows possibilities of reducibility of sets of the speciﬁed class depending on structure of a matrix spectrum. This simpliﬁes the analysis of questions of stability. Theorems of stability or asymptotic stability of the trivial decision of the transformed equivalent systems and the relevant initial systems that is development and generalization of a spectral method of research of stability for the class of non-autonomous systems considered in work are proved.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):11-17

11-17

### Analytical Methods for Studying the Stability of Linear and Quasi-Linear Systems with Polynomial Completeness of the Periodic Matrix

#### Abstract

We propose a method for the analysis of linear and quasi-linear model systems of ordinary diﬀerential equations (ODE) with polynomially periodic matrix in the presence of A0(t) deﬁning diﬀerent stable Jordan structure. With the help of a modern method of splitting algorithm (proposed in the nineties of the twentieth century), the new above mentioned classes of systems of ordinary diﬀerential equations are studied and a number of non-trivial theorems on reducibility to an equivalent system with an almost diagonal matrix are made, allowing suﬃcient conditions for the stability of solutions of such systems. The developed method is given the opportunity to explore a number of application-speciﬁc modeling problems that generalizes and reﬁnes the known results.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):18-23

18-23

### Nonexistence of Positive Solutions to Semilinear Elliptic Inequalities for Polyharmonic Operator

#### Abstract

In this paper, we study the nonexistence of positive solution for some higher-order semilinear elliptic inequality particularly involving polyharmonic operator: Δku(x) ≥ x1 α1 x2 α2… xn αnuq(x), where k ∈ ℕ,q > 1, x = (x1,x2,…,xn) and αi ∈ ℝ,i = 1,2,…,n. The purpose of this paper is to establish conditions on values of αi,i = 1,2,…,n for the nonexistence of positive solution to this problem in a bounded and unbounded domain. The main tools are a priori estimates and integral inequalities. Using the test function method, we derive ﬁrst a priori estimates for solutions of the inequality based on integral inequalities and on the weak formulation of the problem with an optimal choice of test functions and then we formulate the nonexistence condition of the solution of the problem. The choice of such functions is determined by the nonlinear characters of the problem and depends on the concept of solutions that we are dealing with.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):24-32

24-32

### Weighted Inequalities for Quasilinear Integral Operators on the Cone of Monotone Functions

#### Abstract

Criteria for the Hardy-type inequalities with quasi-linear operators on the cones of monotone functions on the semiaxis are obtained. We study the problem of ﬁnding necessary and suﬃcient conditions for the weighted Hardy-type inequalities for the quasi-linear operators on the cone of monotone functions. To this end we choose a composition of power type integral operations and investigate the characterization problem on its boundedness in the weighted Lebesgue (quasi) norms on the cones of non-negative monotone decrasing functions on the real semiaxis. The main method of the solution of the problem is the reduction method which allows to reduce the inequality on the cones of monotone functions to the corresponding inequalities on the cones of arbitrary non-negative functions, which adopt equivalent description in terms of the boundedness appropriated functionals depending on ingredients of the initial problem. As usual we obtain equivalence of the functionals and the best constants involving into initial inequalities, where the mulpiple constants of equivalence depend only of the parameters of summation. Unlike the initial problems of this area we study multiparametrical case increasing the number of weight functions and summation parameters. This case is new for the weighted inequalities on the cones of monotone functions.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):33-44

33-44

### On Asymptotic Behaviour of the Second Moment for the Spectral Estimate of a Homogeneous Field

#### Abstract

A homogeneous (stationary in wide sense) random ﬁeld with zero mean and real-valued components is given. The case of discrete parameter is considered. The matrix of second order periodograms, constructed by the sample is given. The asymptotic behaviour of the second moment of the second order spectral estimate of the homogeneous ﬁeld is considered in the paper.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):45-55

45-55

### On a Queuing System with an Active Queue Management

#### Abstract

We consider a data transfer system with an active queue management designed to prevent overloading, where fuzzy logic controller is used. We developed a mathematical model that takes into account the features of the data transfer system with an active queue management, which keeps the queue length in the range of values close to a given reference value of the queue length. The method of hysteretic control for incoming load with two thresholds was used as a basis of the model. The mathematical model is a queuing system with a threshold control, which is designed for the analysis of the possibility of hysteresis in modeling of systems with active queue management. The model was described by a Markov process, for which the numerical solution of the equilibrium equations was obtained, steady state probabilities were calculated. The main probabilistic measures are the following: the mean value and the standard deviation of a queue length, and the probability for the queue length of being within the speciﬁed limits from the reference value. The numerical analysis in the load range, which includes a system overload, indicated the adequacy of the constructed mathematical model with hysteretic control and system with an active queue management based on fuzzy logic controller.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):56-64

56-64

### Simulation of Impact Interaction of Uncharged Metallic Nanoclusters with Metallic Surface

#### Abstract

Investigation results of impact interaction of uncharged metallic nanocluster with metallic surface are presented. Simulation and investigations of impact processes are fulﬁlled by molecular dynamics methods and suitable software. Characteristic dimensions of surface layer produced by impact as a functions of cluster size, colliding energy and frequency of impulsive nanoclusters source have been analysed. It was found out that penetration depth and deposited layer thickness depend on number of particles in colliding nanoclusters and frequency of impulsive nanoclusters source. It was also discovered that deposited layer thickness in contrast to penetration depth ceases depending on number of particles in colliding nanoclusters N, and frequency of impulsive nanoclusters source ω and colliding energy E with increasing of N, ω and E. And at the same time deposited layer becomes heterogeneous in thickness and gets a funnel-shaped form. It is shown that realization of one of the choice of nanoclusters surface interaction (soft landing, droplet spreading and implantation) should be controlled by means of changing of both nanoclusters beam energy and number of atoms in clusters. Investigation results should be of interest in various ﬁelds of technologies developing nanomaterial with new physical and chemical properties.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):65-79

65-79

### MPI Implementation of the 2D and 3D Simulation of Phase Transitions in Materials Irradiated by Heavy Ion Beams within the Thermal Spike Model

#### Abstract

We present the MPI-based implementation of the method of 2D and 3D calculations of the evolution of temperature ﬁelds and the phase transitions dynamics in materials irradiated by high-energy heavy ion and by pulsed ion beams. We utilize a modiﬁed thermal spike model based on a system of coupled heat conductivity equation describing thermal processes in the electron and ion subsystems of the target sample. Such equations are numerically solved in the cylindrical coordinate system in axially symmetric (2D) and axially nonsymmetrical (3D) cases. The dynamics of phase transitions is realized on the basis of Stephan’s problem in the framework of the enthalpy approach. The mathematical formulation of the problem is given; a numerical scheme is described; a parallel algorithm is presented. The numerical results conﬁrm the eﬃciency of our approach and the corresponding MPI/C++ computer code. It is shown that the results of numerical simulations are in agreement with experimental estimations of the track sizes which appear in the target samples exposed to heavy ion.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):80-94

80-94

### Study Solutions of the Geodesic Equations for a Model of a Point Source of Gravity in the Empty Space

#### Abstract

In this paper the properties of solutions of the geodesic equations for a model of a point source of gravity, radiating heat are studied. Geodesic equations are constructed using a metric which is the solution of equations that represent the zero trace of the Ricci tensor. These equations are a generalization of Einstein’s equations in vacuum. They allow to obtain solutions in the form of non-stationary spherically symmetric metrics, whose components are a function of two variables. The ordinary system of diﬀerential equations of second order for surveying natural parameter consists of four equations. It can be partially integrated and reduced to a system of two second order diﬀerential equations. By substitution method the system is reduced to a pair of diﬀerential equations in partial derivatives of the two unknown variables. Finally, we obtain one quasi-linear equation. In the normal case, equations of this type form gaps with limited solutions. However, the numerical calculations show that the solutions can also become unrestricted due to the pecularities in the right parts.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):95-100

95-100

### The Algorithms of the Numerical Solution to the Parametric Two-Dimensional Boundary-Value Problem and Calculation Derivative of Solution with Respect to the Parameter and Matrix Elements by the Finite-Element Method

#### Abstract

The eﬀective and stable algorithms for numerical solution with the given accuracy of the parametrical two-dimensional (2D) boundary value problem (BVP)are presented. This BVP formulated for self-adjoined elliptic diﬀerential equations with the Dirichlet and/or Neumann type boundary conditions on a ﬁnite region of two variables. The original problem is reduced to the parametric homogeneous 1D BVP for a set of ordinary second order diﬀerential equations (ODEs). This reduction is implemented by using expansion of the required solution over an appropriate set of orthogonal eigenfunctions of an auxiliary Sturm-Liouville problem by one of the variables. Derivatives with respect to the parameter of eigenvalues and the corresponding vector-eigenfunctions of the reduced problem are determined as solutions of the parametric inhomogeneous 1D BVP. It is obtained by taking a derivative of the reduced problem with respect to the parameter. These problems are solved by the ﬁnite-element method with automatical shift of the spectrum. The presented algorithm implemented in Fortran 77 as the POTHEA program calculates with a given accuracy a set ∼ 50 of eigenvalues (potential curves), eigenfunctions and their ﬁrst derivatives with respect to the parameter, and matrix elements that are integrals of the products of eigenfunctions and/or the derivatives of the eigenfunctions with respect to the parameter. The calculated potential curves and matrix elements can be used for forming the variable coeﬃcients matrixes of a system of ODEs which arises in the reduction of the 3D BVP (d = 3) in the framework of a coupled-channel adiabatic approach or the Kantorovich method. The eﬃciency and stability of the algorithm are demonstrated by numerical analysis of eigensolutions 2D BVP and evaluated matrix elements which apply to solve the 3D BVP for the Schrödinger equation in hyperspherical coordinates describing a Helium atom with zero angular momentum with help of KANTBP program.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):101-121

101-121

### The Derivation of the Dispersion Equations of Adiabatic Waveguide Modes in the Thin-Film Waveguide Luneburg Lens in the Form of Non-Linear Partial Differential Equation of the First Order

#### Abstract

This paper presents a derivation of the dispersion equation for a three-layer integrated-optical Luneburg lens based on the method of adiabatic waveguide modes. From this equation there follows the relationship between the coeﬃcient of phase deceleration and function, which determines the thickness of the irregular waveguide layer. The dispersion equation is represented in the form of non-linear partial diﬀerential equation of the ﬁrst order with coeﬃcients, depending on parameters. Among these parameters are regular waveguide layer thickness and optical parameters of the pending Luneburg lens. To represent the dispersion equation in the form of diﬀerential equations in partial derivatives, it is necessary to calculate a symbolic form the determinant of a matrix of 12th order, which determines the solubility of the system of linear algebraic equations, resulting from the boundary conditions. To calculate this determinant in analytical form a procedure of reduction of the system of linear algebraic equations with the use of the computer algebra system Maple is proposed.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):122-131

122-131

### Modeling in the Adiabatic Waveguide Modes Model of Amplitude-Phase Transformation of the Electromagnetic Field by a Thin-Film Generalized Waveguide Luneburg Lens

#### Abstract

R.K. Luneburg proposed a model of the three-dimensional propagation of electromagnetic radiation. V. Guillemin and S. Sternberg showed that the basic equations of Luneburg, which are the Lagrange equations, correspond to Hamilton’s equations on the cotangent bundle over a three-dimensional conﬁguration space. The model described is a “close relative” of the adiabatic guided modes model, proposed by the authors. In this model similary, two-dimensional ray equations for integrated optical waveguide correspond to Hamilton’s equations on four-dimensional phase space. In this model, the construction of quasi-classical solutions is the phase function of the Hamilton–Jacobi, for the initial phase function the initial Lagrangian manifold is constructed, which is transformed by means of the Hamiltonian ﬂow. As long as the Lagrangian manifold occurred in this process is uniquely projected on the conﬁguration space, we ﬁnd the phase function by calculating the action along the path.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):132-142

132-142

### Boundary Method of Weighted Residuals with Discontinuous Basis Functions for High-Accuracy Solving Linear Boundary Value Problems with Laplace and Poisson’s Equation

#### Abstract

In the present paper the method of least squares with T-elements for solving linear boundary value problems with Laplace and Poisson’s equations is developed. In this approach it is oﬀered to use discontinuous basis functions of a high-order approximation from special functional spaces, elaborated by the authors earlier. Advantage of the algorithm in comparison with Galerkin’s standard method is that, in the process of adaptive solving, it makes possible to condense economically a mesh and, moreover, to use diﬀerent order of approximation of the solution on each cell of partition of calculated region. In contrast to Galerkin’s method with discontinuous basis functions, a penalty parameter here is not required, and the matrix of a discretized problem also is symmetric and positively deﬁnite. Examples of calculations by means of the schemes providing computer accuracy of the solution of boundary value problems for polynomials up to seventh order inclusive are given. In a three-dimensional case h − p-convergence of approximate solution to the exact one is shown.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):143-153

143-153

### Self-Adjusting Control of Non-Impact Docking of Two Moving Objects

#### Abstract

The problem of non-impact docking of two moving objects is solved, one of which is driven, moving in the body mode, pursuing the principle of proportional navigation to docking with the second object, moving unpredictably. In this non-control force, including force environmental resistance, considered to be unknown. To automatically select the optimal values of the control features self-adapting method, implemented by the “principle of feedback on the quasi-acceleration” at discrete points in time, is proposed. Solution of the problem is obtained as in the case of a haunting body of permanent mass, so as of variable mass, when the movement of the body is managed by reactive force. In the second case, the amount of mass, which expended in the process of control, is estimated.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):154-160

154-160

### Illumination-Induced Degradation of a-Si:H Solar Cell Parameters

#### Abstract

Hydrogenated amorphous silicon is found to be a leading candidate for the fabrication of low cost solar cells. However presently there are two main factors that limit the large scale applications of a-Si:H solar cells as power sources. One of the central technological obstacle is the low conversion eﬃciency of the cells. The other obstacle for the large scale technological application of a-Si:H solar cells is degradation of critical material properties under the light exposure. In our experiments we have performed light soaking tests on pinpin structure samples to see if the stability of a-SI:H solar cells is improved in comparison with the stability of pin structures. The pattern of light induced degradation, i.e. the degree of degradation of a-Si:H pinpin solar cell parameters was studied on diﬀerent i-layer thickness using high intensity ( ∼10 AM 1.5) illumination. It was found that stacked cells do not show a uniform degradation pattern as in the case of single junction solar cells. In particular, the degradation in short-circuit current Isc of stacked cells shows a big diﬀerence for thick ( ∼500 nm) and thin ( ∼400 nm) pinpin cells. It was found the degradation of the stacked cells with thick bottom layers exhibit a degradation pattern similar to that of single junction cells, i.e. the degradation in eﬃciency comes from the ﬁll factor and the short circuit current, while open circuit voltage being degraded slightly. The degradation in short circuit current of cells with thin bottom layers is negligibly small.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):161-164

161-164

### Charge Carriers Transport Mechanism in a-TNF Thin Layers

#### Abstract

The development of modern photocopying machines, the search for cheap, eﬃcient and reliable solar cells, the search for new conducting materials and molecular storage systems has motivated experimental and theoretical work on organic materials such as molecular crystals, polymers and low-dimensional organic compounds. The organic electron acceptor 2,4,7-trinitro-ﬂuorenone is used as sensitizer of photosensitive polymers, to extend the spectral range of their photosensitivity through the formation of charge-transfer complexes. Also, the thin ﬁlms of TNF, depending on conditions of samples preparation, can be obtained in amorphous, polycrystalline and crystalline forms and, therefore, this material can be useful to investigate the eﬀect of structure of organic materials on their electrical and photoelectrical properties. The eﬀect of trapping centres on the conductivity of amorphous 2,4,7-Trinitro-Nine- Fluorenone (a-TNF) is investigated by Space Charge Limited Current (SCLC), Thermally Stimulated Currents (TSC, TSD) and Transient photoconductivity methods. It is found that electron traps in a-TNF have a smoothly varying distribution centered at about Et = 0.29 ± 0.04 eV with a dispersion parameter σ = 0.11 ± 0.02 eV. The true activation energy at room temperature is Ea = 0.45 ± 0.03 eV. The zero-ﬁeld extrapolated activation energy is Eao = 0.65 ± 0.02 eV. It was suggested that the transport of charge carriers in a-TNF is controlled by traps. Concentration of traps and drift mobility of electrons were evaluated.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):165-169

165-169

### Neutrino Charge with its Gauge Field as a New Physical Base for New Models of Solar Activity and the All Totality of Phenomena Associated with Supernovae Explosions, Forming of Pulsars and their Following Evolution

#### Abstract

The neutrino charge with its gauge ﬁeld introduced in [Kopysov Yu. S., Stozhkov Yu. I., Korolkov D. N. (2001)] for the purpose of decreasing counting rates in solar neutrino detectors generates a lot of new phenomena in astrophysical objects. The physics of the new phenomena is determined by the value of the neutrino charge eν which carriers are neutrinos, quarks and neutrons, and also by almost degenerate neutrino condensate in substance of macroscopic objects. It is shown that the strongest restriction on the value of eν can be obtained by a method of thermal balance of the Sun developed in [Domogatsky G. V. (1968)]. The new interaction generated by a new gauge (“neutromagnetic”) ﬁeld, gives rise to the neutrino Dirac’s magnetic moment of new type. Restriction on its value at the obtained restrictions on eν is only 2 ÷ 3 orders of magnitude lower than the electronic Bohr magneton and on many orders of magnitude exceeds all possible estimates of the traditional anomalous neutrino magnetic moment! The new scenario of formation of solar activity at which new interaction can play a key role is oﬀered. The new model assumes two-story structure of a convective zone: external with the developed thermal convection and internal — the solar troposphere, — in which under the inﬂuence of tidal forces of planets whirls like a tornado of the terrestrial troposphere are formed. In these whirlwinds magnetic ﬁelds of the new (neutromagnetic) type are generated which interaction with substance generates also usual magnetic ﬁelds. The new class of the phenomena arises due to inclusion of the neutrino charge into the theory of collapsing and neutronizing stars. New opportunities for solving old problems are being opened on this pathway. In this regard it is desirable to have theoretical justiﬁcation of need of introduction of the neutrino charge. In this work the problem of extension of the standard uniﬁed model of electroweak interaction by means of inclusion of the second charge in the right sector of extended model is put forward. The possible solution of this problem is planned.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):170-180

170-180

### The Typical Pictures of the Rogue Waves Geometry in Computational Experiments

#### Abstract

We consider typical pictures of geometry of rogue waves in computational experiments. They are sudden and single waves with amplitude of more than 2 times larger signiﬁcant wave height. The suddenness of the occurrence of abnormally large waves in the ocean deﬁnes serious danger that they pose to ships and marine buildings. Now we have incontrovertible evidence of this phenomenon such as the instrumental recordings and photographs. The main method of studying the phenomenon of rogue waves in our work is computational experiment which is based on the full nonlinear equations of hydrodynamics of ideal liquid with free surface. We apply the method of conformal variables to the original system of equations. This method allows to do eﬃcient and accurate calculations using computers. It is shown that in the computational experiments with diﬀerent initial parameters we observed that the majority of rogue waves (about 95%) have a characteristic proﬁle of steep ridge throughout its life cycle. Other rogue waves represent another type of this phenomenon — “hole into the sea”. Also we present the results of comparison of rogue waves from computational experiments with well-known instrumental recordings “New Year wave”, etc.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):181-189

181-189

### Our Authors

**Discrete and Continuous Models and Applied Computational Science**. 2013;(4):190-192

190-192