## No 2 (2014)

**Year:**2014**Articles:**66**URL:**http://journals.rudn.ru/miph/issue/view/508

Articles

Nonexistence of Global Solutions of Quasi-linear Backward Parabolic Equations

###### Abstract

This paper deals with the nonexistence of global solutions of quasi-linear backward parabolic equations for p-Laplacian operators: ut = −div Dup−2Du + uq−1u, x,t ∈ Ω × (0,∞) with the Dirichlet boundary condition u = 0 on the boundary ∂Ω × (0,∞) and a bounded integrable initial function u(x,0) = u0(x), where Ω is a smoothly bounded domain in ℝN. We also consider this problem in the case of Ω = ℝN. The problem is analyzed using the test function method, developed by E. L. Mitidieri and S. I. Pohozaev [Mitidieri E., Pohozaev S. I. A Priory Estimates and the Absence of Solutions of Non- linear Partial Diﬀerential Equations and Inequalities // Proceedings of the Steklov Institute of Mathematics, 2001. - Vol. 234, No 3. - 362 p. - (in russian).] It is based on deriving a priory estimates for solutions by an algebraic analysis of the integral form of inequalities with an optimal choice of test functions. With the help of this method, we obtain the nonexistence conditions based on the weak formulation of the problem with test functions of the form: φ(x,t) = ±u±(x,t) + εδφR(x,t),for ε > 0,δ > 0, where u+ and u− are the positive and negative parts of the solution u of the problem respectively and φR is a standard cut-oﬀ function whose support depends on a parameter R > 0.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):11-26

Upstream Traffic Analysis in Passive Optical Networks

###### Abstract

Nowadays, access network evaluation is being conducted in both directions, such as high bit rate access development for providing high quality of service and decrease length of cooper wiring in local line networks. Rapidly developing passive optical network (PON) technology is a future optical technology that enables high-speed data transfer of multiservice traﬃc using optical ﬁbers. The distribution network of the technology uses passive optical splitter/combiner. There are several advantages, such as cost duration of the access system, volume reduction of the network control, long transmitting distance and no need for the following network upgrade. The use of a classical Erlang multiservice model for the modeling of a PON is restricted due to a speciﬁc functioning process of the optical network units (ONUs). In the present paper, we propose an upstream traﬃc multiservice model considering the functioning process of ONUs. The functioning of an ONU is modeled by the step Markov process with the transition rate of each ONU ON to OFF-state and vice versa. These results are used in the blocking probability analysis of the model.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):27-35

A Mathematical Model of Switch Working in OBS Network with FDL and Deflection Routing

###### Abstract

One of the main development trends of telecommunication networks is the process of transport networks fotonization that should lead to creation of the whole Optical Transport Network (All-Optical Network, AON). This conception let not to increase reserves that are required for meeting growing demand for data transfer. According to OBS technology packets in ingress node are gathered in bursts. Collisions appears in case of two or more bursts at a time transfer to the same output wavelength. Fiber delay lines (FDL), deﬂection routing and wavelength conversion are used for their correction. Through the use of FDL bursts are hold for some period of time and using deﬂection routing they can transfer en reroute but not the main route to the receiver. The whole version of wavelengths let to modify any incoming wavelengths to the outgoing one. In this article the switch in OBS network with FDL, the whole wavelength conversion and deﬂection routing are regarded. Also the system of equations in the global balance for the steady-state blocking probabilities and formulas for calculation of main productivity characteristics of separate optical ﬁber are derived.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):36-42

Analysis of a Finite-Capacity M|G|1|r Queue with Threshold Overload Control

###### Abstract

One of the main challenges faced by telecommunications industry today is an issue of searching for the most effective overload control mechanisms on SIP servers. Generally, overload occurs in SIP networks when SIP servers have insufficient resources to handle all SIP messages they receive to handle all incoming SIP traffic. Such problems can decrease performance of SIP server or even cause its crash. The IETF offers several solutions depending on types of overloads: to increase the number of SIP servers, through 503 (Service Unavailable) response code (IETF RFC 3261), rate-based overload control, loss-based overload control. However, SIP servers are still vulnerable to overload. In this paper we have built and analyzed the M|G|1|r queue with one level hysteretic input load control. Stationary distribution has been achieved based on the Embedded Markov chain method. Approach that allows computation of probability of loss and an average length of queue is developed. Another important parameter, the return time from overloading states to normal state is also considered. A numerical example illustrating the control mechanism that minimizes this characteristic is given to demonstrate some optimization issues.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):43-50

Analysis of Two-Channel Multi-Flow Queuing System with Resequence Customers and Distributions of Phase Type

###### Abstract

The two-channel finite-capacity queuing system with several Poisson flows of customers of different types is considered. The service time is distributed according to phase-type which depends on the type of customers and the device of which it is served. On leaving the system there is a buffer in which there is a resequence of customers according to order of their receipt. Functioning of the system is described by uniform Markov process. In the assumption that intensity of flows and service of customers are positive and finite the final probabilities of statuses of Markov process exist, are strictly positive, don’t depend on initial distribution and match the stationary probabilities. For search of these probabilities the equilibrium system of equations is removed. Then possibility of convergence of the received equations to the similar equations for queuing system with resequence of customers with one Poisson flow of summary intensity and the subsequent determination of the type of customers just before arrival on service is set. The last circumstance allowed using results of the previous operations of authors for calculation of stationary distribution of queue length. As a result the recurrent matrix algorithm was developed for calculation of probabilities of statuses of considered system in the conditions of a stationary operation mode.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):51-60

Time Characteristics of Queuing System with Renovation and Reservice

###### Abstract

This article is devoted to time characteristics of queuing system with recurrent input flow, one server, exponential service time distribution and infinite queue. The mechanism of renovation with reservice (repeated service) is introduced. It means that a packet at the moment of the end of its service with some probability may just leave the system or with complementary probability will drop all other packets in the system and return for service. Assuming that we know the steady-state probability distribution of number of packets (calculated with help of embedded by the moments of arrival Markov chain) the main emphasis of the article will be on system time characteristics such as steady-state distributions of time in system for serviced or dropped packets, average time characteristics - mean service time, mean waiting time for a dropped, serviced and an arbitrary packet.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):61-66

Numerical Algorithm for Simulation of Thermal Processes in Four Layer Cylindrical Object

###### Abstract

In the paper the algorithm is proposed for the numerical simulation of the thermal conductivity for the design and optimization of cryogenic cells pulsed (in the millisecond range) feeding the working gases into the electron-stringed source of multiply charged ions. Heating process comes when the electric current passed through one of the layer. A model of the cryogenic cell with four layers (materials) is investigated. The heat transfer into the object is described by the system of heat equations with temperature dependent discontinuous thermal coefficients. The discontinuous thermal coefficients are given by experimental data and approximated by the least-squares method using the polynomial analytical functions. Conjugation condition between materials is considered to be ideal. The results are reported for a common configuration of the cell. The parallel algorithm for modeling thermal processes into four layer model was developed and speedup of the algorithm in depending on number of CPUs is shown.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):67-71

The Influence of Territorial Heterogeneity and Falsifications on Integral Electoral Indices

###### Abstract

Available information of Russian public elections provides many opportunities for the analysis of distributions of various electoral indices. For example, the distribution of the number of polling station in turnout intervals (turnout polling station distribution) is often close to normal; it would be natural to expect if the rules by which voters make decisions on the participation of the elections are about the same for all voters. In practice it appears that turnout polling station distribution sometimes deviates significantly from the normal , and the differences between such distributions for different elections do not depend on the type of election and may occur for a brief period of time between two elections. These differences can be explained by the territorial heterogeneity in the electoral behavior of voters. However, the question arises why such territorial heterogeneity manifested in Moscow, but do not appear, for example, in Yekaterinburg. Also in Moscow, these “heterogeneity” appear very irregularly. The observed turnout polling station distribution has good explanation with model of ballot stuffing - cramming votes to one of the contenders (party or candidate). This model describes the observed behavior of not only the turnout polling station distribution, but also the behavior of other electoral indicators, for example - the distribution of votes. The article describes the results of a computer simulation of certain rules and vote counting. The results of simulations are compared with the actual electoral statistics.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):72-80

Model Queue Management on Routers

###### Abstract

Problems of modeling of active queue management (AQM) systems have been for a long time in the sphere of interests of authors. One of the areas of work was associated with a dynamic model of the control module Random Early Detection (RED) based on Poisson process driven stochastic differential equations. These equations are used in queuing theory quite recently, and not very well understood. As disadvantages of this approach the authors underlined its non-generic character. We describe the interaction between module RED and protocol TCP Reno. But its extension to other variants of the TCP protocol and the control module is not possible. Our group studied common approaches to modeling of such phenomena. As a result a method for randomization of one-step processes, allowing to obtain new models in a universal manner was developed. In this paper, the authors use this technique to model previously investigated RED module and protocol TCP Reno to demonstrate its applicability to this kind of problems. As a result an extended model of control module type RED for traffic type TCP Reno, was created. It contains previously studied model as a special case.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):81-92

Algorithm for Computing Wave Functions, Reflection and Transmission Matrices of the Multichannel Scattering Problem in the Adiabatic Representation using the Finite Element Method

###### Abstract

In adiabatic representation the multichannel scattering problem for a multidimensional Schr¨odinger equation is reduced to the boundary value problem (BVP) for a system of coupled self-adjoined second-order ordinary differential equations on a finite interval with homogeneous boundary conditions of the third type at the left and right boundary points in the framework of the Kantorovich method using adiabatic basis of surface functions depending on longitudinal variable as a parameter. The homogeneous third-type boundary conditions for the desirable wave functions of the BVP are formulated using the known set of linear independent regular and irregular asymptotic solutions in the open channels of the reduced multichannel scattering problem on an axis which involve the desirable reflection and transmission amplitude matrices, and the set of linear independent regular asymptotic solutions in the closed channels. The economical and stable algorithm for numerical calculation with given accuracy of reflection and transmission matrices, and the corresponding wave functions of the multichannel scattering problem for the system of equations containing potential matrix elements and first-derivative coupling terms is proposed using high-order accuracy approximations of the finite element method (FEM). The efficiency of the proposed algorithm is demonstrated by solving of the two-dimensional quantum transmittance problem for a pair of coupled particles with oscillator interaction potentials penetrating through repulsive Coulomb-type potentials and scattering problem of electron in a Coulomb field of proton and in the homogeneous magnetic field in the framework of the Kantorovich and Galerkin-type methods and studying their convergence.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):93-114

The Simplest Geometrization of Maxwell’s Equations

###### Abstract

For research in the field of transformation optics and for the calculation of optically inhomogeneous lenses the method of geometrization of the Maxwell equations seems to be perspective. The basic idea is to transform the coefficients of constitutive equations, namely the dielectric permittivity and magnetic permeability into the effective geometry of space-time (and the vacuum Maxwell equations). This allows us to solve the direct and inverse problems, that is, to find the permittivity and magnetic permeability for a given effective geometry (paths of rays), as well as finding the effective geometry on the base of dielectric permittivity and magnetic permeability. The most popular naive geometrization was proposed by J. Plebanski. Under certain limitations it is quite good for solving relevant problems. It should be noted that in his paper only the resulting formulas and exclusively for Cartesian coordinate systems are given. In our work we conducted a detailed derivation of formulas for the naive geometrization of Maxwell’s equations, and these formulas are written for an arbitrary curvilinear coordinate system. This work is a step toward building a complete covariant geometrization of the macroscopic Maxwell’s equations.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):115-125

On Straightening of Locally Deformed Waveguide

###### Abstract

Locally deformed planar wave guide, i.e. the strip limited to two curves, coinciding with couple of parallel straight lines out of some compact, is considered. By conformal map this strip can be straightened in a strip with rectilinear borders (straight waveguide). Thus the problem about initiation of electromagnetic oscillations in locally deformed waveguide can be reduced to a problem about excitement of a straight waveguide with non-homogeneous filling. This problem is simpler than an initial problem both for the theoretical analysis, and for practical calculations by, e.g., partial Galerkin method. For calculation of conformal map of the deformed strip on a straight strip is given the boundary problem for one of the map functions. Proved that this problem has the unique decision solution decreasing on infinity, and also that this solution is classical in case of smooth borders. For the solution of this problem the finite element method (FEM) is used, solutions for locally squeezed and locally stretched waveguides are given. Shown that entering corners in the boundary don’t change a character of map and a convergence of applied numerical method. It is shown that transformation coincides graphically with identical out of place of local stretching or compression; this is important for the formulation of partial radiation conditions.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):126-132

Calculating Phase Retardation Field in Smoothly Irregular Integrated-Optical Waveguide (Using Thin-Film Generalized Waveguide Luneburg Lens)

###### Abstract

Maxwell’s equations are extremely simple and elegant. However, the specific calculations require much more sophisticated approaches. Thus, in the problems of calculation of nonregular integrated optical waveguides a few basic techniques are used. The authors propose to use the method of adiabatic waveguide modes. This method follows in the footsteps of Luneburg works. Moreover, the method has a clear geometric interpretation. As well as Luneburg equations, the equations obtained by this correspond to the Hamilton equations on the cotangent bundle over the configuration space. Moreover, to calculate ray paths a simple geometrization is used, when the refractive index is represented as a metric of some efficient space. Thus, the phase function is evaluated as an action along the trajectory. Thin-film Luneburg lens is an interesting object in the general theoretical sense as well as in practical one. Its study allows to further describe a class of objects, but it is an essential element for the construction of a purely optical control devices. Thus, the authors consider the method of adiabatic modes most suitable for studies of such object as a thin-film generalized waveguide Luneburg lens.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):133-141

Object Recognition Based on Invariant Moments

###### Abstract

We investigate the properties of invariant moments of binary images, it is necessary for the formation of their set in order to recognize graphic images. Pattern recognition and distance measurement used Hu invariant moments. It is shown that the invariants have different sensitivity to changes in input data that defines the strategy of their choice. Experiments on pattern recognition of text characters, images of the aircrafts and the landing site in the form of a cross were carried out. The considered recognition algorithm works in real time, use only one camera, is invariant to rotation, shift and scale the object in the frame. The accuracy and completeness of recognition amounted to about 92% on a set of thousands of samples of each type. The results of the experimental determination of the quality of recognition of various objects based on their contour images, as well as the results of comparing recognition using a different set of invariant moments are presented. It is shown that the inclusion of the less sensitive invariant moments reduces the computation time, and lowers the computational error that occurs when fluctuations in the parameters of an object or scene in the frame take place. It is proposed to combine the method of invariant moments with probabilistic neural network, which will improve the quality of recognition, making it more fast, accurate and complete.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):142-149

Illness Severity Classification Based on Artificial Neural Networks

###### Abstract

The task of automatic classification of illness is very important since its solution greatly facilitates the work of a physician in a large amounts of data analysis and diagnosis. There are different approaches to solving this problem. One of them involves the use of production rules. Production rules empower knowledge representation in clinical medicine and effective in building diagnostic systems. Noting the prospects of applying production rules, however, it should be noted that the decision on the basis of their high complexity real problems requires large amounts of computation and restructure a system of rules at change of problem conditions. At the same time, as an effective alternative tool to analyze complex situations and classifications artificial neural networks (ANN) are widely used, which allow to perform the recognition and diagnosis of various phenomena and high complexity objects by training. This paper studies the possibility of using different neural networks to analyze an illness severity on the basis of precedential information. In particular, the problem of determining the severities of acute intervertebral disc protrusion and of acute acute asthma are solved. In order to improve recognition, the training set is expanded by creating additional precedents that that do not contradict the terms of the problem.The artificial neural networks of various configurations and with different activation functions are used: single-layer and multilayer perceptrons.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):150-156

Contactless Control of Robotic Arm Through Human Gestures

###### Abstract

In this paper the problem of contactless control of robotic arm through human hand gestures is considered. A new method is proposed, which allows to recognize and track the positions of palm and fingertips of human hand in real time and on the basis of these data to implement the control of the wrist and three fingers of the robotic hand. As an input device for capturing hand gestures a depth sensor is used, which works on the principals of triangulation and structured light. Hand gesture recognition is performed by the gradual procession of each video frame, which represents depth images. In the first phase the position of a random point of the palm is detected. In the second phase hand image is extracted using the detected point of the palm. The positions of the fingertips and the lengths of the fingertips are calculated through the analysis of distances between hand contour points and the random point of the wrist. The tracking of the fingertips is achieved through the usage of k-nearest neighbor’s algorithm. The recognized point of the palm is used to control the wrist of the robotic arm in three-dimensional space. The positions of human hand fingertips and lengths of fingers are used to control the fingers of the robotic arm. The proposed method is tested on the computer model of a robotic arm.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):157-163

A Static Generalization of the Schwarzschild Solution, that Gives not Asymptotically Dipole Term

###### Abstract

In this paper we study the static axisymmetric solutions of the vacuum Einstein equations. Among static axisymmetric vacuum solutions of the most interest are the asymptotically flat solutions reducing to the Schwarzschild solution. The purpose of this paper is to obtain a static solution which turned out to be appropriate for describing the gravitational field around an axisymmetric mass distribution. In this paper the method of singular sources os considered and some new applications are presented. By mean of the method of singular sources it is possible to construct gravitational multipoles which generalize the Schwarzschild solution. The linearity of the gravistatic equations makes it possible to solve the problem of superposition of two or several known solutions. The obtained static vacuum axisymmetric generalization of the Schwarzschild solution near two points of horizon has coordinate singularities. In the obtained solution the dipole term is absent, and we have found the corresponding condition. If one considers axially symmetric solutions of gravistatics, then construction of gravitational multipoles becomes ambiguous. It means that different solutions can give asymptotically the same Newtonian limit.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):164-168

Algebraic Dynamics on a Single Worldline: Vieta Formulas and Conservation Laws

###### Abstract

In development of the old conjecture of Stuckelberg, Wheeler and Feynman on the so-called “one electron Universe”, we elaborate a purely algebraic construction of an ensemble of identical pointlike particles occupying the same worldline and moving in concord with each other. In the proposed construction one does not make use of any differential equations of motion, Lagrangians, etc. Instead, we define a “unique” worldline implicitly, by a system of nonlinear polynomial equations containing a time-like parameter. Then at each instant there is a whole set of solutions setting the coordinates of particles-copies localized on the unique worldline and moving along it. There naturally arise two different kinds of such particles which correspond to real or complex conjugate roots of the initial system of polynomial equations, respectively. At some particular time instants, one encounters the transitions between these two kinds of particles-roots that model the processes of annihilation or creation of a pair “particle-antiparticle”. We restrict by consideration of nonrelativistic collective dynamics of the ensemble of such particles on a plane. Making use of the techniques of resultants of polynomials, the generating system reduces to a pair of polynomial equations for one unknown, with coefficients depending on time. Then the well-known Vieta formulas predetermine the existence of time-independent constraints on the positions of particles-roots and their time derivatives. We demonstrate that for a very wide class of the initial polynomials (with polynomial dependence of the coefficients on time) these constraints always take place and can be naturally interpreted as the conservation laws for total momentum, angular momentum and (the analogue of) total mechanical energy of the “closed” system of particles.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):169-180

The Cylindrical Symmetric Configurations of the Interacting Scalar and Spinor Fields with Regard of Ideal Fluid

###### Abstract

We have investigated the properties of the static cylindrical symmetric configurations of the interacting scalar and spinor fields taking account of ideal fluid with the state equation P = We, where P is pressure, e is energy density, W - arbitrary dimensionless parameter. Parallel with the usual matter with positive W , we have considered some types of ideal fluids with negative pressure (W < 0), that are actively used at present in cosmology (dark matter, cosmic strings, domain walls, quintessential, cosmic vacuum, fantom matter). We have obtained the exact solutions to the equations of the interacting scalar and spinor fields, Einstein equations and ideal fluid motion equation with arbitrary W . We have written the conditions of the regular metrics on the axis of the symmetry of the system and the conditions of the regular (flat or string) metrics. We have considered the influence of the different types of the ideal fluid upon the formation of the soliton-like or string-like configurations in the = 1 3 (ultrarelative system of the interacting fields. We have established that in case of W 1 (space string’s gas), W = − 2 3 matter), W = −(random distribution of the domain walls), 3 W = − 4 3 (phantom matter), regular configurations of the system of the interacting fields and ideal fluid can exist only under some relation among constants in the equations.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):181-190

A Kalman Filter Method for the Charged Particles Trajectories Reconstruction in the CBM Experiment and Its Parallel Implementation at the JINR LIT Manycore Server

###### Abstract

The high-intensity heavy-ion beams that will be produced by the accelerators at the Facility for Antiproton and Ion Research (FAIR), together with the Compressed Baryonic Matter (CBM) experiment, now in preparation, offer outstanding possibilities for studying baryonic matter at superhigh densities and moderate temperatures under laboratory conditions. The CBM physics program is aimed at studying the structure and behavior of baryonic matter at densities comparable to those in the center of neutron stars. The program includes 1) setting a phase boundary between the hadronic and partonic matter, 2) determining the critical end point, and 3) searching for indications of the origin of chiral symmetry reconstruction at high pure baryonic densities. The task of a charge particle trajectories reconstruction is one of the most important tasks of the CBM experiment. It assumes a full on-line event reconstruction, that requires development of fast algorithms, which utilize the potential of modern CPU and GPU architectures in the most efficient way. In the current work the results of analysis of the Kalman filter based track reconstruction algorithm, which is implemented using different parallelization approaches, are presented and discussed. For the analysis a manycore server with two Intel Xeon X5660 CPUs and a NVidia GTX 480 GPU at LIT, JINR was used.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):191-196

Principles of Software Construction for Simulation of Physical Processes on Hybrid Computing Systems (on the Example of GIMM FPEIP Complex)

###### Abstract

We discuss the problem of elaborating the software systems designed for modeling of physical processes on computing systems with hybrid architecture, formulate the basic principles of construction of such complexes, and present an example of their implementation for the GIMM FPEIP complex. Complex GIMM FPEIP is intended for simulation of thermal processes in materials irradiated by heavy ion beams. The construction of GIMM FPEIP complex is done according to the requirements, intergrability of GIMM FPEIP into a complex hierarchical structure GIMM NANO and specification of the problem under solution. The complex includes the computational modules that provide parallel algorithms realized on the basis of MPI and CUDA technologies and meant for performing computations on hybrid computing systems. The GIMM FPEIP complex provides a possibility to include new computational modules and to expand the current database of physical parameters. In the construction of the complex, a module approach to the structure of the complex has been applied. This allowed realizing a number of common modules in the form of separate libraries with the possibility of their use in other software complexes. In particular, with the use of this approach, a 3D modeling complex GIMM FPEIVE was constructed. GIMM FPEIP and GIMM FPEIVE complexes were tested on the multi-core computing complex CICC JINR, on hybrid computing complex K100 (Keldysh Institute of Applied Mathematics) and on ”Lomonosov” supercomputer (Lomonosov Moscow State University).

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):197-205

Parallel Algorithm and MPI Implementation of Numerical Study of Phase Transition in the 3D Thermal Spike Model

###### Abstract

We present an algorithm and parallel computer code for numerical investigation of the thermal processes and phase transitions in materials irradiated by the high energy heavy ion beams. We employ the modified thermal spike model based on the coupled heat conductivity equations for the electron gas and the ion lattice subsystems in the target sample. This system of equations is numerically solved in the cylindrical coordinate system in axially non-symmetric (3D) case. We utilize an expansion of the source function in spherical harmonics, a finite difference approximation and semi-explicit numerical scheme. The dynamics of phase transitions is implemented on the basis of the enthalpy approach. The mathematical formulation of the problem is given; a numerical scheme is described; a parallel algorithm is presented on the basis of the MPI technique (Message Passing Interface). The test calculations on the K100 multi-processor cluster (KIAM RAS, Moscow) with various dimension of the finite-difference mesh and with different number of parallel processors demonstrate efficiency of the C++/MPI code.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):206-210

Improvement of Locality of Parallel Algorithms of the Numerical Solutions of the Two-Dimensional Quasilinear Parabolic Equations

###### Abstract

The equations of parabolic type describe processes of nonlinear heat conductivity, diffusions of the loaded particles in plasma, diffusion and drift of impurity atoms in semiconductor structures, in chemical kinetics. At the numerical solution of practical tasks such there are the difficulties caused by the insufficient capacity and volume of memory of the personal computer. There is a problem of creation of parallel methods and algorithms for the numerical decision the parabolic equations on supercomputers. One of methods of the numerical solution of the multidimensional parabolic equations the locally one dimensional method is. Parallel realization of a locally one method for numerical solutions of the linear and quasi-linear two-dimensional parabolic equations on supercomputers with the distributed memory is offered. The parallel algorithm is constructed taking into account locality of data - the operations and data are distributed between processes in such a way that the considerable part of data is privatized by processes and doesn’t need communication operations. Results of numerical experiments are given.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):211-215

The Algorithms of the Mulri-Threaded Relevant LP-inference

###### Abstract

The relevant LP-inference, which is based on the solution of logical equations, is an effective tool that can be used for research and optimization of production-logical systems. It significantly reduces the number of executed queries to an external source of information (either to a database or an interactive user). The preference is given to testing the facts that are really needed in the inference. However, experiments have shown that the process of using the relevant LP-inference may require an excessive amount of computational resources of the computer. So, the relevant LP-inference method was modified to use parallel computing algorithms. This paper describes the implementation of a multi-threaded algorithms for relevant LP-inference and provides the pseudocodes of these algorithms. Multi-threading is a fundamentally new element in the implementation, which allows speeding up the process of constructing sets of facts that are required in the inference, and their further processing.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):216-219

Parallel Second Order Finite Volume Scheme for Maxwell’s Equations with Discontinuous Dielectric Permittivity on Structured Meshes

###### Abstract

A second order finite volume scheme on structured meshes is presented for numerical solution of time dependent Maxwell’s equations with discontinuous dielectric permittivity. The scheme is based on approaches of Godunov, Van Leer and Lax Wendroff and employs a special technique for gradient calculation near dielectric permittivity discontinuities. The scheme was tested for problems with linear and curvilinear discontinuities. Test results demonstrate second order of convergence and support second order of approximation in space and time. A parallel implementation of the scheme based on geometric decomposition was developed. Computational region was partitioned into subregions. Computations in each subregion were carried out independently using halo cells. Test results indicate linear scalability. Parallel implementation was applied to modelling photonic crystal devices. Computational results for photonic crystal waveguide with a bend correctly confirm bend configurations and frequencies with zero reflection.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):220-224

Simulation of the Gravitational Mixing on GPU

###### Abstract

Gravitational mixing induced by the Rayleigh-Taylor instability, arises in the system of two fluids/gases with different densities when the acceleration acts from a more dense material to the less dense one. In this case, the amplitude of small perturbations of the contact boundary increases over time, involving new flow regions into the mixing (Rayleigh, Proc. Of the London Math. Soc., 14, 1883; Taylor GI, Proc. Of the R. Soc. of London, A201, 1950). The numerical calculation of such problems requires the use of methods that can fully describe the discontinuous nature of the flow variables. One of such methods is the Godunov’s one (Godunov SK, Mat. Sb. (NS), 47 (89), 3, 1959), which is widely used and based on solving the Riemann problem for further calculation of the fluxes through the edges of cells. At the same time, we know that the exact solution of the Riemann problem is quite expensive in terms of computing resources. However, when using massively parallel architectures such as GPU, significant acceleration can be achieved due to the large number of computational processes which allows to perform calculations much faster. As part of the performed work two versions of a parallel algorithm were implemented for the calculation of mixing. The estimation of efficiency and speed up was made.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):225-229

Method of Artificial Viscosity on Unstructured Grids

###### Abstract

The paper presents a general description of a new method of artificial viscosity (AAV) for solving equations of gas dynamics. In a basis of a method are put research of stability finite difference schemes and classification of discontinuous desicions of gas dynamics equations. The method was adapted for the solution of problems on cartesian and unstructured meshes. With the help of the method a lot of gas dynamics problems were analyzed numerically, for example, the supersonic flow in a channel with a step was calculated.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):230-233

Comparative Study of Cluster and Neural Network Methods in the Problem of Protein Structure Analysis

###### Abstract

This work continues the previous study where the important problem of automatization of differentiation methods of the genetic protein structures according to their electrophoretic spectrums (EPS) was considered. The multicriterion problem of the agriculture cultivar identification by their spectra caused the idea of its solution by an artificial neural network (ANN) trained on an expert data base. In the given paper peculiarities of the neural net use as well as the purposefulness of cluster analysis applications for the EPS classifying are studied. A special model of multidimensional vectors adequately imitating the most essential characteristics of real data obtained after EPS digitalization, denoising and normalization is developed. A numerical experiment is fulfilled on such simulated data stream to study the influence of contamination and distortion factors on the ANN efficiency in order to suppress those factors and improve ANN functioning. Various methods of cluster analysis are also applied to simulated multidimensional data as either an ANN alternative or more soundly as a prior stage of a coarse data classification in some set of detached cultivar groups to be classified next by ANN.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):234-238

Modeling of Redistribution of Particles Arising from the Nonuniform Evaporation of a Thin Droplet

###### Abstract

When a colloidal droplet with pinned contact line evaporates on a substrate, surface tension produce a fluid flow. Usually the flow directs to the contact line and the solute in the drop is dragged by this flow, where it accumulates. The deposit remains on the edge of the drop and forms a solid ring after complete evaporation of the liquid. The conditions of evaporation and solution properties affect on deposit structure. We propose simple model of sessile thin drop desiccation under nonuniform evaporation. The model is based on mass conservation and has numerical solution. If the droplet is covered by a ”mask” with hole, evaporation primarily occurs under the hole so that surface tension drives a flow of liquid to replenish this loss. In the proposed physical model a radial flow velocity was studed and the redistribution of component in the droplet arising from the nonuniform evaporation under a mask was predicted. The effect of diffusion and hole radius on the component redistribution was studed too.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):239-243

Numerical Simulation of the Hydrated Electron Formation

###### Abstract

We consider the dynamic polaron model on the basis of a system of three nonlinear partial differential equations with appropriate initial and boundary conditions. Agreement of our numerical results with theoretical estimations confirms the correctness of numerical algorithm and computer code. A numerical simulation of formation of photo-excited electrons in water has been carried out. We show that the model provides a reasonable agreement with experimental data.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):244-247

About one Model of Computer Control on the Basis of Gaze Tracking

###### Abstract

The methodology of human-computer interaction based on the determination of the eyeball position and its movement direction. For interference and noise elimination from extraneous lighting devices infra-red illumination which allows receiving only one highlight on a cornea from an infra-red light-emitting diode is used. Thus the entrance video-stream in an infra-red range is processed in real time in two parallel processes. One process is used for definition of position of a highlight from a light-emitting diode on corneas of eyes, another process - for definition of position of a pupil and a direction of its moving. Aggregate of co-ordinates of a highlight of a pupil, position of a cornea and a direction of its moving allow remote to control of the computer. For prevention of loss of “object” the original technique of accumulation of the previous positions of an eyeball and forecasting of a direction of its moving is used. The given methodology of contactless control of the computer allows simulating pressing keys of the keyboard and/or mouse movement. The developed hardware and software system for gaze direction tracking be used as an alternative method of input medium, which is closer to the natural way of interaction with the environment, as well as the only way to work with a computer for the users with reduced mobility.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):248-252

Unilameller DMPC Vesicles Structure Analysis using Parallel Asynchronous Eifferential Evolution

###### Abstract

The Separated Form Factors model (SFF) developed previously for analysis of the small angle neutron scattering data, has been extended for numerical study of structure of polydispersed population of unilamellar DMPC vesicles using the small angle synchrotron scattering spectra (SAXS). Parameters of vesicle structure (average radius, polidispersity, bilayer thickness etc.) are fitted by means of the parallel Asynchronous Differential Evolution (ADE) method - the effective global minimization algorithm. Parallel computer implementation of our approach has been done using the MPI technique (Message Passing Interface). The numerical investigation of structure of polydispersed population of unilamellar vesicles of DMPC in the 40% water solution of sucrose has been performed. We show that accounting for the fluctuation of the bilayer thickness provides an agreement of our calculations with experimental data in the right part of SAXS spectra. On the basic of calculations with different models of internal bilayer structure, an appropriate form of the X-ray scattering length density across bilayer has been chosen. We present results of methodical calculations on the multiprocessor cluster (LIT, JINR, Dubna) demonstrating efficiency of the MPI-based parallel computer code of ADE.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):253-258

The Research of Loss Stability of Level of Psychical Reaction of a Human with the Power of Informational Influence on Him

###### Abstract

In recent years significantly increased the negative information and psychological impact on the individual and mass consciousness. Therefore, strongly expressed of emergence of aggression, anxiety, despair, hopelessness, depression, a criminal manifestations and mental illness. The research of stability of level of psychical reaction with personal characteristics of a human and with the power of informational influence on him is presented in the article. The adjoin method offered by Kudinov A.N. was applied to research of the loss of stability. The main advantage of the adjoin method is that for use to the problems of dynamic stability studies in various fields of science, technology, biology, medicine and psychology, if their equations can be reduced to the equations of second order, don’t demand the introduction of Lyapunov functions. The adjoin method permits to find the equilibrium positions and to check the stability of nonperturbed state. Also the research of stability by Lyapunov’s method on first approximation is conducted, as a result the stability conditions of psychical reaction with personal characteristics of a human and with the power of informational influence on him are presented.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):259-262

World Economic Crises, Waves of Fractal Dimension of Global Temperature of Earth and Kondratyev’s Cycles

###### Abstract

In this work data of world statistics shows that the crisis phenomena of world economy have cyclic character. The main contribution to studying of these processes was made by the outstanding Russian economist N.D. Kondratyev who put forward the theory of cycles of an economic environment lasting 40-60 years. Dynamics of world economy for the last 200 years confirms existence in it long-wave the kondratiev cycles. Now there was an opinion that so difficult phenomenon as a business cycle, it is impossible to explain only with influence of one or small number of factors. But, undoubtedly, to number of such most essential factors influence on world economy of a factor of fluctuation of global temperature of the atmosphere of Earth belongs. This work also is devoted to this question. In it existence of waves of fractal dimension of global temperature of the atmosphere of Earth is discussed with the periods of recurrence 61 year and nature of their possible communication with kondratyevsky cycles as these periods are rather close.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):263-266

Evolutionary Optimality in Structured Systems and its Applications to Biological and Medical Problems

###### Abstract

The well known to specialist in the field of mathematical biology principle of evolutionary optimality, rising to darvinian concept of natural selection, which is based upon mechanisms of survival of the most strong is formulated. Expressed in term of stability of established equilibria in model system, this principle allows to generalize it to the systems, which can be described by mathematical models, including both integro-and partial differential equations. In the presented article the ways to use the author’s evolutionary optimality theory, which initially was constructed for the dynamical systems in Banach spaces, for to find the selection functional, which are to be optimized, for several structured biological systems are indicated. Particularly it is shown that in the case of communities with age and with spatial structure the constructed functionals have a real biological interpretation. As a practical application of the constructed theory the central result of the theory of correlation adaptometry is formulated.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):267-271

Volatility in Classification

###### Abstract

The goal of the presented work consists in the construction of the new three-levels scheme of authomatical classification. This scheme is based on the newly introduced notion of volatility of separate clusters as well as of whole classification. The property is exactly defined and efficiently calculated. It describes the stability, exactness, validity of subsets of the given initial set - in essence, their possibility (or impossibility) to be selected as clusters. The suggested algorithm finds the clusters with arbitrary levels of volatility, including the conventional case of zero volatility. The clusters in USA, Russia and Sweden stock market (for crisis period of 2008-2010) and deputies clusters based on voting results in the 3rd State Duma between September 2001 and January 2002 (the period including the creation of the party ”United Russia” 01.12.2001) were constructed by the suggested algorithm. Analyzing clusters constructed basing on the voting results for every of the considered months, it has turned out that the clustering volatility was equal to zero in September and October, drastically increased in November and slightly decreased in December and January. But several indices (i.e. concordance of parties’ positions) did not show sensible jumps near this political ”bifurcation point”. The other considered various model examples demonstrated the results well-coordinated with geometrical intuition.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):272-280

Mathematical and Computational Oil Spills Modelling

###### Abstract

The model of oil spill evolution on the water surface based on quasi-linear convection-diffusion equation is considered. To solve the latter the method of finite differences is used. Two-dimensional problem is reduced to one-dimensional by alternating direction method. For one-dimensional case traditional difference approximations are briefly discussed. And also difference schemes obtained by computer algebra based method of automatic generation of difference schemes are considered. For generated schemes the order of approximation and linear numerical dissipation are estimated. For explicit schemes stability conditions are briefly discussed. In the absence of convection a numerical comparison of traditional implicit difference scheme and one implicit scheme generated automatically with similarity solution of quasi-linear diffusion equation is carried out. This comparison shows that generated implicit scheme allows to obtain the relative error less than for traditional scheme. Using obtained implicit scheme evaluation of oil slick thickness changes in the presence of oil evaporation is done. Two different models were used to estimate oil evaporation rate. The first one is based on hypothesis that oil evaporation is regulated by air boundary layer. The second assumes that oil evaporation is regulated by oil diffusion. Calculations show that the choice of model essentially influences on oil slick thickness and oil total volume changes in time.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):281-286

On 2D and 3D Localized Solutions with Nontrivial Topology

###### Abstract

Localized solutions of nonlinear field models with nontrivial topological properties are discussed. Existence of various systems of definitions of the topological objects, developed in this area of research historically, can potentially lead to the wrong conclusions about existence of such solutions. The classification allowing to define accurately and differentiate objects with different topological properties is proposed, which prevents from inferring wrong conclusions. Such classification is especially important for multidimensional solutions. Such solutions are divided into 2 classes: the topological solitons (TS) and topological defects (TD). Solutions of both types describe the localized distributions of field energy, but they differ in topological properties. We exemplify and compare stationary TSs and TDs in 2 and 3 spatial dimensions. Examples of TSs are: solitons in Heisenberg magnets, Belavin-Polyakov solitons/instantons, Skyrmions, “baby-skyrmions”. Examples of TDs are: sine-Gordon kinks, Nielsen-Olesen strings-vortices in the Abelian Higgs (AHM) model, ’t Hooft-Polyakov hedgehog-monopoles in the Georgi-Glashow model. We note some technical problems with TDs, which are not met in the case of TSs. Soliton analogs of Nielsen-Olesen TDs in the AHM have been found: they are TSs in the A3M model. We have started search for TSs in the SU(2)-Higgs model which is currently in progress.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):287-291

Critical Points and Points of a Bifurcation of the Rotating Magnetized Newtonian Polytropic with 0.9 ≤ n ≤ 1.6 Index

###### Abstract

In this paper, the presence of critical points and bifurcation points of rotating Newtonian polytropes with an index of 0.9 ≤ n ≤ 1.6 has been shown for the first time. The symbolic-numerical calculation error in metric L2 has reached the size of 10 −5 order. The approximate analytical solution of the problem to the above mentioned accuracy has been set forth. The critical value of polytropic curve index n = nk =1.54665 has been calculated which is the highest one among the critical points and bifurcation points.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):292-294

Solving the Hysteresis Loop Calculation Problem for Josephson Junction Stacks

###### Abstract

A detailed investigation of the IVC breakpoint and the breakpoint region width gives important information concerning the peculiarities of stacks with a finite number of intrinsic Josephson junctions. The current-voltage characteristics for a stack of n Josephson junctions is defined from solving the system of n nonlinear differential equations. The current voltage characteristic has the shape of a hysteresis loop. On the back branch of the Hysteresis loop, near the breakpoint Ib, voltage V(I) decreases to zero rapidly. The goal of this work is to accelerate the computation of IVC based on numerical solution of the system. A numerical-analytical method was proposed in. This method showed perfect results in IVC calculations for a stack of 9 and 19 intrinsic Josephson junctions and the computation time reduced by five times approximately. The question of choosing a change-over point from “analytical” to numerical calculation was open. In testing computations the change-over point was taken equal to 2Ib. In the case of periodic boundary conditions an equation, determining the approximate location of Ib, was obtained. This moment we succeeded to develop an algorithm determining the approximate value Ib in more complicated technically case of non-periodic boundary conditions with g = 1. All calculations were performed using the REDUCE 3.8 system.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):295-300

Presentation of Boolean Polynomials as ZDD-Diagrams

###### Abstract

Boolean Gr¨obner basis have shown their practical efficiency for different problems. Among them are algebraic cryptanalysis HFE (Hidden Field Equations), modeling of quantum computing and boolean satisfiability problem (SAT). Algorithms for computing Gr¨obner basis have exponential complexity for execution time as well as for memory usage. The more appropriate data structure was introduced, which is based on zero-suppressed binary decision diagrams (ZDD). Also we show the relation between ZDD and special recursive notation of polynomials. The recursive notation is the collection of equalities which have had one-to-one correspondence with graphical presentation of ZDDs. We prove lemma which gives the number of nodes estimation for ZDD which represent boolean polynomial with all monoms up to d degree of n variables. Furthermore, we present C++11 ZDD package providing possibility for addition and multiplication of boolean polynomials, multiplication by variable, presentation in compressed recursive form and graphical presentation. The package includes its own implementation of red-black trees, lists, and memory manager. ZDD package was developed for using as internal data structure of the boolean polynomials for computation of involutive Gr¨obner basis.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):301-305

Evaluation of Parallel Computations of Gr¨obner and Involutive Bases on the Massive SMP Computer

###### Abstract

In previous papers author presented realizations of two different approaches to parallelization of computation of Gr¨obner and involutive bases of polynomial systems with benchmarking on the 8-cores SMP computer: reduction-level parallelism with coefficients of polynomials in Z-ring and basis-level parallelism using modular basis computation and lifting. In this work further development of this algorithms described, benchmarking results and maximal speedup achieved on the massive 32-cores computer presented, scalability differences of algorithms investigated.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):306-309

The Volume Integral Equations Method in Magnetostatics Problems

###### Abstract

In this article, use of volume integral equations method for calculations of magnetic systems is considered. GFUN program package based on integral approach to magnetostatics applies the method of collocations and piecewise constant approximations of unknown variables within the elements to discretization of equations. The limitation of this approach is related to singularity of the integral equations kernel. An alternative to collocation method, replacing observation point by integration over discretization elements, is considered in this article. This approach enables one to use higher order approximations for unknown variables. In the context of the finite element method, the piecewise constant and piecewise linear approximations of unknown variables are considered. The problem of computing matrix elements for discretized systems of equations can be reduced to evaluation of sixth-order integrals, singular ones in the general case, over two different elements of the computational region. Possible methods are proposed for calculating this kind of integrals. Iterative processes for solving the arising nonlinear systems of discretized equation are discussed. The proposed approach enables one to build discretizations with higher precision of approximation for the initial volume integral equations of magnetostatics. The proposed method was used for 3D modeling of a dipole magnet. Comparison of results obtained for simulation of the dipole magnet using different versions of integral magnetostatics problem discretization are given.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):310-315

Modeling the Track Formation in Amorphous Iron Alloys Exposed to High-Energy Heavy Ions

###### Abstract

An important process in the fundamental radiation in solid state physics and in applications is the process of the track formation at irradiates by high-energy heavy ions of different in their physical and chemical properties materials. The development of modern methods of analysis and studies of the structure of extended defects stimulates initiation of new experimental and theoretical research in this area. In the experimental work the track diameters 11.1 MeV/amu ion 132Xe, 152Sm, 197Au and 8.2 MeV/amu 238U ions in a number of amorphous alloys of iron and boron were measured using a small angle scattering of synchrotron radiation. In this work, a three-dimensional model of the thermal spike modified with phase transitions of the fusion was introduced and used to estimate the diameter of tracks all of the above iontarget combinations whose values were compared with experimental data. Accounting to the phase transitions made in this work to evaluate the tracks diameters significantly improves the agreement of the simulation results with experimental data.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):316-319

Investigation of Solutions of Quasistationary States for the Quasipotential Equation

###### Abstract

The excited states of quantum systems are nonstationary, and they break up. These states are called unstable or quasi-stationary. Such states are already observed in the study of scattering problems, the accumulation of particles in the lens (the particle prefers to live inside the lens) is accompanied by a large delay τ (the lifetime of the quasi-level). Here the lifetime of the quasi-level τ = γ−1, width of the quasi-level Г = ~γ and complex energy level E = E1 − iE2,E2 = Γ/2. Investigation of the quasi-stationary states is carried out for the quasi-potential equation with piecewise-constant potentials at various values of the parameter of the equation ε and the potential parameters. A comparative analysis of solutions of the quasi-potential equation for the different values of ε with the solutions of the Schredinger equation is performed. Found that at ε → 0 the solutions of quasi-potential equation tend to the solutions of the Schredinger equation.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):320-323

A Method for Statistical Comparison of Histograms

###### Abstract

The problem of the testing the hypothesis that two histograms are drawn from the same distribution is a very important problem in many scientific researches. There are several approaches to formalize and resolve this problem. Usually, one-dimensional test statistics is used for this purpose. We propose an approach for testing the hypothesis that two realizations of the random variables in the form of histograms are taken from the same statistical population (i.e. two histograms are drawn from the same distribution). The approach is based on the notion “significance of deviation”, which has a distribution close to standard normal distribution if both histograms are drawn from the same distribution. This approach allows to estimate the statistical difference between two histograms using multi-dimensional test statistics. The distinguishability of histograms is estimated with the help of the construction a number of clones (rehistograms) of the observed histograms. The approach considered in the paper allows to perform the comparison of histograms with a test more powerful, in the cases considered, than those that use only one test statistic. Also, the probability of correct decision is used as an estimate of the quality of the decision about the distinguishability of histograms.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):324-330

About Calculation Singularity of High-Order Derivative for Identification of the Graphic Objects Shape

###### Abstract

Methods of calculation of high order derivatives are considered on a basis: interpolation formulas; “without difference methods of calculation of derivatives”; applications of convolution with replacement of differentiation by integration operation; differentiation with use of quadratures on C. Lanczos; the method of Numerova. The comparative analysis of methods of calculation of high order derivatives on accuracy of calculations with use as the sample of the derivatives calculated in package Maple with 20 digit decimal accuracy is carried out. It is shown that all methods are almost equivalent on accuracy and are reduced to convolution calculation between differentiated function and some window which coefficient depend on an applied method. For carrying out of experiments the special program complex is developed for calculation of high order derivative (up to 7th) the tabulated functions with various step. Grids with steps from 0.005 to 0.1 have been investigated. Irrespective of a method of calculation of derivatives it has been defined that optimum value of step mesh for 64 digit arithmetic’s the step is from 0.01 till 0.05. Value of smooth functions differs less than their accuracy of representation at smaller value of a step, and at greater step - the differentiation error increases. Results of experiments confirm N.N. Kalitkin’s theoretical conclusions.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):331-335

Description of a Program for Computing Eigenvalues and Eigenfunctions and Their First Derivatives with Respect to the Parameter of the Coupled Parametric Self-Adjoined Elliptic Differential Equations

###### Abstract

Brief description of a FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the coupled parametric self-adjoined elliptic differential equations with the Dirichlet and/or Neumann type boundary conditions on the finite interval. The original problem is projected to the parametric homogeneous and nonhomogeneous 1D boundary-value problems for a set of ordinary second order differential equations which is solved by the finite element method. The program calculates also potential matrix elements - integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Parametric eigenvalues (so-called potential curves) and matrix elements computed by the POTHEA program can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs. As a test desk, the program is applied to the calculation of the potential curves and matrix elements of Schr¨odinger equation for a system of three charged particles with zero total angular momentum.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):336-341

KANTBP 3.0: New Version of a Program for Computing Energy Levels, Reflection and Transmission Matrices, and Corresponding Wave Functions in the Coupled-Channel Adiabatic Approach

###### Abstract

Brief description of a FORTRAN 77 program for calculating energy values, refection and transmission matrices, and corresponding wave functions in a coupled-channel approximation of the adiabatic approach is presented. In this approach, a multidimensional Schr¨odinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with the homogeneous boundary conditions of the third type at the leftand right-boundary points for continuous spectrum problem, or a set of first, second and third type boundary conditions for discrete spectrum problem. The resulting system of these equations containing the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):342-349

Algorithms for Selection of J/ψ → e +e −, Decays Registered in the CBM Experiment

###### Abstract

A procedure for selection of J/ψ → e+e− decays registered by the CBM set-up is presented. The key problem is a fast and reliable electron/positron identification using the energy losses of charged particles in the Transition Radiation Detector. An analysis of the application features and a comparison of power of two methods to solve this task are given: an artificial neuron network (ANN) and a ωkn goodness-of-fit criterion. The choice of the approach based on the ωkn goodness-of-fit criterion is explained.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):350-353

Modeling Speach Features Via Simulated Annealing Algorithm

###### Abstract

Mel-Frequency Cepstral Coefficients are in so far the most popular speech features. However, depending on the length of a vocal tract (it is worth mentioning that length of a vocal tract is dependent on sex and other physiologic parameters of a speaker, such as height, and can vary from 13 cm to 18 cm) frequencies of central formants are shifted. The value of the shift can be as large as 25%. This huge difference can lead to a wrong recognition of a new utterance by a previously well-trained model when the utterance was said by a new speaker, thus the system becomes speaker-dependent. Alternative way is to use speaker independent features such as that obtained using Auditory Image Model (AIM) to describe input utterance. In our work we propose AIM based features which are calculated using simulated annealing algorithm. Using Monte-Carlo schemes we investigate statistical properties of maximum likelihood estimates of Gram-Charlier extension of normal density obtained via simulated annealing algorithm, also we compare different methods to solve aforementioned optimization problem.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):354-358

Discrete Modeling Using Stochastic Cellular Automata

###### Abstract

New approach to low-level discrete simulation of natural (especially biological) systems using stochastic block cellular automata is considered. The notion of a Markov system, which is a special case of the string rewriting systems, is introduced. A key feature of Markov systems compared with other string rewriting systems are the stochastic procedure of the splitting the string into substrings and stochastic simultaneous application of the substitutions system to all obtained substrings. In such automata cellular space forms a matrix, and block decomposition into horizontal and vertical components occurs in probabilistic way. Based on a Markov system model the notion of two-dimensional Markov automata, which is a special case of block stochastic cellular automata, is constructed. The characteristics and expressive capabilities of such systems are considered. As an application, the problem of constructing neural network low-level model is considered. With this purpose a model of excitable medium, supporting the inhibition mechanism of excitation, is proposed. Based on this model an artificial neuron, including a system of communication (axons, dendrites, synapses) is constructed. Simple feedforward neural network, that implements the logical operation of exclusive disjunction, is considered and numerically investigated.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):359-362

Complexes of Localized States in Ac-Driven Nonlinear Schr¨odinger Equation and in Double Sine-Gordon Equation

###### Abstract

Complexes of localized states are numerically analyzed in two dynamical systems: directly driven nonlinear Schr¨odinger equation (NLS) and double sine-Gordon equation (2SG). Both systems have a wide range of physical applications. Our numerical approach is based on the numerical continuation with respect to the control parameters of the quiescent (stationary) solutions and stability and bifurcation analysis of the linearized eigenvalue problem. Multisoliton complexes of the NLS equation are studied in the undamped and the weak damping regimes. We show that in the weak damping case the directly driven NLS equation holds stable and unstable multi-soliton complexes. The results are confirmed by means of direct numerical simulations of the time-dependent NLS equation. Properties of the multi-fluxon solutions of 2SG equation are studied depending on the parameter of the second harmonic. We show that the second harmonic changes properties and increases the complexity of coexisting static fluxons of 2SG equation. Results are discussed within the frame of the long Josephson junction model.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):363-368

Modeling of Bound States of Quantum Systems in a Two-Dimensional Geometry of Atomic Traps

###### Abstract

We present numerical modeling of bound states for two-particle quantum systems in twodimensional geometry of optical traps. We have investigated the dependence of bound states energies on the scattering length as well as the convergence of obtained numerical results to analytical solution from the work [Two Cold Atoms in a Harmonic Trap / T. Busch, B.-G. Englert, K. Rzazewski, M. Wilkens // Foundation of Physics. - 1998. - Vol. 28, No 4. - Pp. 549-559.] in zero-range approximation for the interatomic interaction.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):369-374

Mathematical Modeling of Fluid Dynamics in Evaporating Drop with Taking into Account Capillary and Gravitational Forces

###### Abstract

An one-dimensional mathematical model described evolution of drop shape and viscous fluid dynamics (vertically averaged radial flow of the liquid) as the result of evaporation from impermeable horizontal surface is presented in this work. The model considers the influence of volume and capillary forces. Non-steady-state approach has been used to describe the process mathematically. The continuity equation and the motion equation for the case of a system with a variable mass are solved numerically with using standard tools of mathematical package Maple. We carry out calculations for cases of different sizes of water drop volumes. Results of modeling have shown that profile of drop which is bigger in size of capillary length (Bond number is greater than unity) differs from shape of spherical segment. The surface of such drop is almost flat. We explain it by domination of the gravity over a surface tension force. The flow of compensation nature is present in evaporated drops of different volumes what is coordinated with experimental data of other authors. Thus radial fluid flow is result of work of both capillary and gravitational forces. The results which we have got will help to describe the coffee ring effect in macrodrops in future.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):375-380

The Optimal Control Problem for Linear Distributed Systems of Fractional Order

###### Abstract

Optimal control problem considered for the plant which described by one-dimensional transfer equation with Caputo fractional derivative. The equation defined on finite segment. Investigation evaluates for both of cases when controls enter into right part of equation and depend on spatial coordinates and time and when controls enter into boundary conditions and depends on time only. Two types of optimal control problem studied: 1) the problem of plant transfer from initial state to given one with minimal transfer time and control norm restriction; 2) the problem of plant transfer from initial state to given one with minimal control norm at given transfer time. It’s assumed that admissible controls belong to the function class which p-integrable in given domain. It’s shown that assigned optimal control problem can be reduced to the known problem of moments and to corresponding problem of conditional minimization for convex multivariable function. For the problem of moments conditions of statement possibility and solvability derived. This work can be useful for control systems development for plants which dynamics can reveal anomalous diffusion.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):381-385

Application of Method of Conjugate Equations to Research of Loss Stability of Shell Under the Action of Moving Loads

###### Abstract

Actual problem of the theory of stability is creation of strict and effective methods of research of the loss of stability of movement of systems with distributed parameters, in particular for continuous environments. This problem has a huge impact on both theoretical and applied uses. In the article is presented the research of the loss of stability of the equations which describe mathematical models of soil massifs and the bases. These models reflect the character of the soil under load, are based on the laws of structural mechanics and the theory of elasticity. The adjoin method offered by Kudinov A.N. was applied to research of the loss of stability. The main advantage of the adjoin method is that for use to the problems of dynamic stability studies in various fields of science and technology, if their equations can be reduced to the equations of second order, don’t demand the introduction of Lyapunov functions. The developed algorithm allows to find positions of balance of the decision and to check up stability of system by adjoin method. The research of stability of the solution of linear systems by using Lyapunov’s method of first approximation is presented in the article.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):386-389

Self-Organization of Adiabatic Shear Bands in Copper and Steel

###### Abstract

In this work we consider the self-organization process of adiabatic shear bands (ASB) formation in OFHC copper and HY-100 steel taking into account the strain hardening factor. We proposed the numerical approach which is based on the Courant-Isaacson-Rees scheme. Using this method we made a numerical investigation, in which it was shown that the strain hardening process leads to the increase in the localization time and to a decrease the number of ASB formed. Using the fact, that the processes of ASB formation are quasi-periodic, we obtained a numerical estimate of the distance between ASB and compared our numerical results with theoretical estimates by others.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):390-393

Modeling the Track Formation in Amorphous Iron Alloys Exposed to High-Energy Heavy Ions

###### Abstract

Nonlinear waves in liquid with gas bubbles are investigated taken into account liquid viscosity and compressibility and inter phase heat transfer. The nonlinear differential equation for long weakly nonlinear waves is obtained with the help of the reductive perturbation method. At the derivation of the equation higher order corrections in the asymptotic expansion are taken into account. This equation is the generalization of the Burgers equation and describes nonlinear waves in a liquid with gas bubbles in the case of dissipation main influence. The normal form is constructed for the equation with the help of the near-identity transformations. It is shown that the normal form equation is integrable under certain condition on parameters. In this case the equation for nonlinear waves is the second member of the Burgers hierarchy. Exact solution in the form of kink is obtained in the general case. Dependence of this solution on physical parameters is investigated. It is shown that the amplitude of this exact solution decreases when the bubbles radius in the unperturbed state and the liquid viscosity increase.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):394-398

Mathematical Modeling of Heat Frozen Earth

###### Abstract

In-situ leaching is a mining process used to recover minerals such as copper and uranium through boreholes drilled into a deposit, in situ. Water permeability of orebody is a necessary condition of in-situ leaching. Permafrost or cryotic soil is soil at or below the freezing point of water for two or more years. Most permafrost is located in high latitudes. Ground ice is not always present, but it frequently occurs and it may be in amounts exceeding the potential hydraulic saturation of the ground material. Under these conditions it is necessary for the successful leaching warm orebody and melt the ice. Mathematical model of heating permafrost is considered taking into account the Stefan condition at the boundary of melting. An equivalent formulation of the problem is shown. We proposed numerical algorithm for analyzing this process. Computation module is produced on an open architecture with the use of object-oriented programming language OpenFOAM. Verification of the computation module carried using the known exact solutions of simplified tasks. Evolution of permafrost melting in the case of one, three and four cylindrical heaters is presented. The main result of this study is the time required for melting the solid of permafrost by four heaters. The time to complete defrosting orebody in the space between the heaters is shown.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):399-403

The Generalization Known Methods to Approximate Various Sets of Discreet Data

###### Abstract

In the process of mathematical modeling a necessity often arises to smoothly approximate various dependencies which are defined discretely or graphically. Especially if the value of such dependencies are obtained as a result of complex experiments or cumbersome calculations. The inverse transform of continuous simulated objects in discrete digital format that is used for storage and computer processing also requires a certain ordering. It is assumed beforehand that the smooth approximation of discrete set of points on the plane is performed by linear analytical model. Interpolation conditions lead to a system of linear equations with a square matrix. When interpolating polynomials by breaking and rearranging the terms of a power series one can get such basic functions as Lagrange polynomials or Bernstein polynomials. Other methods of interpolation are Newton polynomials, Aitken iterative process, etc. However, these methods realize only some particular cases of all possible approximations of discrete data by arbitrary basis functions and are mainly focused on manual calculations. In computer calculations, it is desirable to find a general algorithm for solutions in order to avoid programming many particular cases. The problem of generalization of existing methods for approximation of discrete data sets (generalized algorithm) and bringing these discrete data to a common form (a discrete unified structure) is considered.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):404-409

The Boundary Value Problem for Elliptic Equation in the Corner Domain

###### Abstract

Modern accelerator systems and detectors contain magnetic systems of complex geometrical configuration. Design and optimization of the magnetic systems demand solving a nonlinear boundary-value problem of magnetostatic. The region in which the boundaryvalue problem is solved, consists of two sub-regions: a region of vacuum and a region of ferromagnetic. In view of the complex geometrical configuration of magnetic systems, the ferromagnetic/vacuum boundary can be nonsmooth, i.e. it contains a corner point near of which the boundary is formed by two smooth curves crossed in a corner point at some angle. For linear differential equations it is known that in such regions the solutions of the corresponding boundary-value problems can possess unlimitedly growing first derivatives near of the corner point. Some works consider a nonlinear differential equation of divergent type in the region with a corner and the opportunity of existence of solutions with unlimitedly growing module of gradient near the corner point is shown. The present work analyzes the region consisting of two sub-regions (ferromagnetic/vacuum) divided by a boundary with the corner point. In this region one considers a formulation of the magnetostatics problem with respect to two scalar potentials. Nonlinearity of the boundary-value problem is related to the function of magnetic permeability which depends upon the module of gradient of the solution to the boundary-value problem. In a case when the function of magnetic permeability at big fields satisfies certain conditions, in this work a theorem of limitation of the module of gradient of the solution near the corner point is proved.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):410-414

Self-Adaptation in Swarm Optimization Algorithms

###### Abstract

Evolutionary algorithms are in active development for last two decades, due to numerous studies in the field of mathematical biology, and the wide spread of massively parallel computing systems since numerical modeling of biological systems (with significant degree of parallelism) requires significant computations. Swarm optimization algorithms discussed in this article are based on modeling of collective behavior in large colonies of animals, such as ants, bacteria, bees. Such algorithms are universal and applicable to a wide range of computational problems. The present paper is devoted to the new approach to the construction of self adaptive swarm optimization algorithms, which automatically adjusts parameters of the algorithm in the process of its evolution. The idea of building self adaptive evolutionary algorithms is based on the using in the background to the main algorithm (e.g., bacterial foraging algorithm) auxiliary genetic algorithm, the purpose of which is to adjust the parameters of the basic algorithm, providing the highest possible rate of its convergence. The application of the proposed scheme of self-adaptation on the examples of bacterial foraging algorithm and bees algorithms is considered. The results of the numerical study of such algorithms on the standard test problems of continuous optimization, demonstrating the efficiency of the proposed scheme of self-adaptation, are described.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):415-418

Ionization Amplitude Extraction from the Solution of the Time-Dependent Schr¨odinger Equation by Means of the Probability Amplitude Surface Flux

###### Abstract

We have developed a new method for the obtaining of the amplitudes of ionization of atoms and molecules by the strong time-dependent field from the solution of the time-dependent Schr¨odinger equation (TDSE). The method is based upon the conjunction of the two approximate approaches for the amplitude computation that have been proposed earlier. One of these approaches makes it possible to confine oneself to evaluate the wavefunction on the small space region while the other allows to do it for the small time interval. The method that is being suggested here combines these advantages, so it enables to extract ionization amplitudes by means of solving the TDSE both on the small space region and for the time interval not exceeding the external ionizing field duration. It is shown that the method we are proposing yields results more exact compared to the precessors as well as does not suffers from their peculiar drawbacks. These statements validity has been demonstrated by the example of the one-dimensional problem with the model potential. The correct boundary conditions were provided by means of the exterior complex scaling. In the future the method might be utilized for the aid of the solution of much more complicated problems.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):419-430

Creating Classifiers using Artificial Neural Networks and the ADABOOST Principle

###### Abstract

The problem of constructing various types of object detectors in images is still an urgent task, despite the relatively strong set of methods described in the literature. One of the methods that have become standard for the construction of efficient and fast classification, is a Viola-Jones cascade, which is still fundamental to search for objects in the image in real time and which implementation has been included in the open-source computer vision library OpenCV. For the experiments in this study we used the database of images CMU Face Database. In practice, when we use of the algorithms in computer vision the computational complexity becomes a significant factor. Preferably, one should use threshold decision rules or Haarfeatures as classifiers, which gives small the computational complexity. In this paper, the approach to the construction of classifiers of comparable performance for the problem of detecting faces. For the construction of the detector were studied approach involves separating detection process into two stages: construction the descriptor of image, and classification stage. For the phase, which responsible for the classification, were considered two possibilities: a two-layer neural network, i.e. using multilayer perceptron as a “strong” classifier, and a cascade of several such networks of different size. For the phase, which responsible for forming the descriptor, we also have investigated two possibilities. First one - fixed Haar-basis, which gives us a feature-vector of the descriptor of input image. This basis was constructed using the ADABOOST principle. The second possibility, investigated in this paper, was the construction of the basis of fewer required Haarfeatures, every of which more accurately reflects the object characteristics, which was obtained by using Karhunen-Loeve transform. In order to get Haar-features from eigenvectors, they have been quantized. As a result, the classifier built with efficiency which comparable to the Haar cascade.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):431-436

Nashi avtory

###### Abstract

**Discrete and Continuous Models and Applied Computational Science**. 2014;(2):437-443