# No 2.2 (2010)

**Year:**2010**Articles:**31**URL:**https://journals.rudn.ru/miph/issue/view/506

## Articles

### Numeric-Analytical Study of the Breather Type Solution on the Correctness Boundary

#### Abstract

Asymptotics when t →∞ for the Cauchy problem solution of the equation Utt = Uxx + i2Uttx + Uttxx with discontinuous initial data are proved. The found asymptotic formulae are in a good agreement with results of numerical experiments. Stability of the numerical methods used is studied as well. At the beginning we review other results in studying nonstandard linear equations arising in averaging equations describing wave propagation in periodic stratiﬁed media.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):5-9

5-9

### Symbolic Solving of Diﬀerential Equations with Partial Derivatives

#### Abstract

An algorithm for the symbolic solving of systems of linear partial diﬀerential equations by means of multivariate Laplace-Carson transform (LC) is produced. Considered is a system of K linear equations with M as the greatest order of partial derivatives and right hand parts of a special type, that permits a symbolic Laplace-Carson transform. Initial conditions are input. As a result of Laplace-Carson transform of the system according to the initial conditions, we obtain an algebraic system of equations. There exist eﬃcient methods to solve large size systems of such types. It gives a possibility to implement the method for solving the large PDE systems. A method to obtain compatibility conditions is discussed. The application of LC allows one to execute it in a symbolic way.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):10-14

10-14

15-18

### Multi-level LP-Structures in Rewriting Systems

#### Abstract

An algebraic system containing the semantics of a set of rules of the conditional equational theory (or the conditional term rewriting system) is introduced. The following basic questions are considered for the given model: existence of logical closure, equivalent transformations, construction of logical reduction. The obtained results can be applied to analysis and automatic optimization of the corresponding set of rules.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):19-23

19-23

### Symbolic and Numeric Schemes for Analysis of Deterministic Systems with Aftereﬀect

#### Abstract

There are considered problems of analysis for deterministic and stochastic processes in nonlinear dynamic systems with diﬀerent forms of delay. The main idea of all algorithms is a usage of a scheme of phase space extension. Such the scheme allows to pass on from equations with divergent arguments to equations without delay.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):24-29

24-29

### On Newton-Type Methods with Fourth and Fifth-Order Convergence

#### Abstract

In this paper, we suggest and analyze new three-step iterative methods for solving nonlinear equations. The analysis of convergence shows that the proposed methods are fourth and ﬁfth-order convergence. Several numerical examples are given to illustrate the eﬃciency and performance of the proposed methods. Comparison of diﬀerent methods is also given.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):30-35

30-35

### Computer Algebra Automation of the Qualitative Analysis of a Parametric System of Algebraic-Diﬀerential Equations

#### Abstract

Solutions of systems the ODE with fractional polynomial right parts in ﬁnite or inﬁnite range of time can reach inﬁnite values at ﬁnite or inﬁnite time. Correct deﬁnition of inﬁnite values of variables and derivatives made by enclosure into compact. Transformation is fulﬁlled automatically by Maple12 package. The oﬀered method used in numerical integration of the equations of Einstein with extremely wide range of values of variables and in unlimited area.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):36-39

36-39

### Common Lisp Version of the System of Computer Algebra REDUCE

#### Abstract

The work deals with implementation of the system of computer algebra REDUCE in the Common Lisp environment with the use of interpreters CMUCL and GCL. The issues of sharing the programs written in the programming languages Common LISP and Standard LISP as well as speciﬁc for CMUCL and GCL features of compilation and optimization of the system are considered.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):40-44

40-44

### About One Class of Finite Elements with Harmonic Basis Functions

#### Abstract

A new class of ﬁnite elements of high order approximation with vector basis functions is suggested. They satisfy both homogeneous equation with the div operator and homogeneous equation with the curl operator. To construct the ﬁnite elements two algorithms, developed by the authors before to obtain approximations of high order with the help of harmonic functions, are used. The main properties of the elements have been investigated.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):45-49

45-49

### Nonstandard Finite Diﬀerence Schemes for Reaction-Diﬀusion Equations

#### Abstract

A nonstandard, implicit ﬁnite diﬀerence scheme for reaction-diﬀusion equation was constructed. This scheme is an extension of the method of Mickens. Conditions of positivity and boundedness are established.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):50-54

50-54

### Numerical Modeling of Heat and Mass Transfer in a Porous Material

#### Abstract

The numerical research of the suggested phenomenological model of heat and moisture transfer in a porous material is performed. The model is described by a system of equations of four unknown functions - the water concentration wl, water vapor concentration wv, temperature T and source I as functions of the space variable x and time variable t. Diﬀerent cases of initial and boundary conditions are considered that correspond drying of a wet sample or wetting of a dry sample. Calculation results show that during the drying process of a wet sample the temperature of the sample decreases under an initial, room temperature and during the wetting process of a dry sample or the temperature is not changing or the heat in the sample is increasing depending on that whether a water or a hot water vapor is supplied to the boundary.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):55-58

55-58

### Fast Global Tracking for the CBM Experiment at FAIR

#### Abstract

Particle trajectory recognition is an important and challenging task in the Compressed Baryonic Matter (CBM) experiment at the future FAIR accelerator at Darmstadt. In this contribution, the status of the global track reconstruction software for the CBM experiment is presented. The global track reconstruction procedure is based on track following and Kalman Filter methods. The track reconstruction eﬃciency for central Au+Au collisions at 25 AGeV beam energy using events from the UrQMD model is at the level of 93-95%. Since CBM has to process terabytes of input data produced at high collision rates, it is extemly important to develop fast track reconstruction algorithms. Possibilities to speed up the algorithms have been studied. A signiﬁcant optimization of memory consumption and necessary combinatorics has been done. The usage of multithreading results in further acceleration. Overall, a factor 20 in speed could be achieved by these improvements.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):59-63

59-63

### Modeling of the Evolution of the Polaron States

#### Abstract

The evolution of polaron in a homogeneous environment is analyzed depending on parameters of the model and initial conditions which are selected in the form of various combinations of stationary polaron states. A computational scheme and results of the numerical modelling are presented. Work supported by RFBR grants 07-07-00313, 08-01-00800, 07-01-00738, 09-01-00770.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):64-69

64-69

### Identiﬁcation of Internal Points of Macromolecular System for the Deﬁnition of the Parameters of a Poisson-Boltzmann Equation

#### Abstract

Systems of overlapping spheres are widely used in macromolecular modeling, where atoms are represented by spheres. Study of geometric properties of such systems, like the surface area, the volume or the existence of internal cavities is important because of their physical applications. In this paper the cavity triangulation is applied to identify internal grid points for the numerical solution of the Poisson-Boltzmann equation describing the electrostatic potential of macromolecule.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):70-75

70-75

### Methods of e/π Identification with the Transition Radiation Detector in the CBM Experiment

#### Abstract

A problem of e/π identiﬁcation using n-layered transition radiation detector (TRD) in the CBM experiment is considered. With this aim, we elaborated algorithms and implemented various approaches. We discuss the characteristic properties of the energy losses by electrons and pions in the TRD layers and special features of applying artiﬁcial neural networks (ANN) and statistical methods to the problem under consideration. A comparative analysis is performed on the power of the statistical criteria and ANN.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):76-84

76-84

### The Thermal Spike Model in Materials at Irradiation by High Energy Heavy Ions with the Radiant Function Depending on the Ion Velocity

#### Abstract

At passage of heavy ions through condensed media, their energy losses are generally spent on elastic and inelastic interactions. The SRIM-2008 computer program allows calculating the energy losses of heavy ions at their passage through condensed media. Of great importance for the investigation is the time from the ions hitting the target to its full stopping. The performed calculations (using the results of the SRIM-2008 program) have shown that the time of passage by a uranium ion of with the energy 700 MeV in a nickel target Δt ≈ 4 × 10−12c. In the previous investigations, the motion of an ion in a material was not considered and a source with the action time Δt ≈ 10−14s was used. In this paper the thermal spike model with a new source considering the motion of an ion within a material is proposed.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):85-89

85-89

### SLIPM - MAPLE Programm for Numerical Solution of Sturm-Liouville Partial Problem with the Help of the Continuous Analogue of Newtons Method

#### Abstract

A computer program SLIPM (Sturm-Liouville Problem in MAPLE) has been designed in the language of computer algebras MAPLE. It is intended for the numerical solution of Sturm-Liouville partial problem with the help of the continuous analogue of Newtons method, i.e., for calculating the eigenvalue of a linear second-order diﬀerential operator and a corresponding eigenfunction satisfying homogeneous boundary conditions. SLIPM is the development of the computer codes SLIP1 and SLIPH4 written in the Fortran language; it is added by a new procedure of calculating an initial approach to the solution and by new ways of calculating the initial value of the iterative parameter τ0.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):90-98

90-98

### Mathematical Aspects of Modeling of Required Operation Modes of Multi Purpose Isochronous Cyclotrons

#### Abstract

A mathematical and computer modelings of required operation modes for multipurpose isochronous cyclotrons is presented. The considered procedure is based on a calculation of current values in trim coils correcting the basic magnetic ﬁeld (Ij, j = 1,2,…,n) for a certain current level in the main coil (Imc ). A series of numerical and physical experiments on modelings and testing the main operation mode for the multipurpose isochronous cyclotron AIC-144 (proton, Eout ∼ 60.3∕60.7 MeV, Frf = 26.155∕26.25 MHz), proved both the necessity of taking into account the estimate of the stability of the sought solution, and the possibility of accelerating proton beams in the all range of working radii from the ion source to the ion-beam extraction system for small phase losses of protons in the range of isochronization of required magnetic ﬁeld.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):99-103

99-103

### Simulation of Physical and Chemical Phenomena and Processes in the Course of Evolution of Unicellular Organisms

#### Abstract

Based on the developed simulation technique of the functioning of animate systems the probabilities of the existence of genetically steady biological kinds of live objects are calculated. The possible reasons for probability discontinuous behaviour of speciation of unicells with the various genome sizes, adapted for diﬀerent environmental conditions (temperature and aggression) are shown. Evidence for larger expediency of symbiotic presence of mitochondrions in eukaryote cells in comparison with other energy sources, for example, polyphosphates is produced. The genetic reason of evolutionary success of syngenesis over agamic reproduction is revealed. A.Weismann dogma about larger evolutionary stability of gametal cells genome, than somatic is conﬁrmed. The calculations are carried out under the assumption of evolutionary conservatism of genetic stability systems of modern cells: reparations and apoptosis.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):104-107

104-107

### Stability and Bifurcations of Magnetic Flux Distributions in Josephson Junctions Described by Double Sine-Gordon Equation

#### Abstract

Aim of the work is mathematical modeling of static magnetic ﬂux distributions in long Josephson junctions taking into account the higher harmonics in the Fourier-decomposition of the Josephson current. Basic magnetic ﬂux distributions have been found; their stability in dependence on parameters of model has been investigated. Numerical results have been compared with results of numerical study of traditional model.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):108-112

108-112

### Simulation of Current Windings with Various Cable Cross-Section

#### Abstract

The work discusses the issues of modeling current windings of complex conﬁguration with various cable cross-sections. A convenient toolkit for specifying such windings has been developed. The problems are discussed which arise at the magnetic ﬁeld calculus from the current elements by using Biot-Savart law. The numerical methods taking into account the singularity of the applied integral equations are suggested.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):113-119

113-119

### Optimal Control in Electrical Heating Processes

#### Abstract

It is suggested an eﬀective numerical and analytical method (NAM) for solving complicated nonlinear multidimensial boundary-value problem of optimal control at interaction of electromagnetic ﬁeld and heat one. NAM is applied for solution parabolic heat equation and also for electromagnetic ﬁeld (elliptical equation). The idea of method is based on approximation nonlinear solution by eigen-function of specially constructed simple linear operator with using smooth properties of inverce one for boundary value problem. The direct methods in space of control are used for solution optimal problem with help of homotopy chain, approximation, forecast and random search. Work was supported by RFBR Grant №09-01-90425,08-01-90101.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):120-123

120-123

124-131

### Fractal Model of Population Growninig

#### Abstract

In the study there was developed a fractal model of population growth. In the model the average population growth rate in separate segments of the growth curve dynamics is a function of fractal dimension of segments SDS which is deﬁned as a cubic equation. The data analysis has been performing since 1950 up to now. The data marks three periods in 12, 20, and 27 years in duration. The prognostic within our model shows that a population growth trend will cross the line of 7 bln of people in June, 2011; 8 bln of people there will be in 2024, and 8,5 bln - in 2030.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):132-138

132-138

### Elaboration of Methods and Algorithms of Calculation of Main Characteristics of Three-Dimensional Irregular Integrated-Optical Waveguides

#### Abstract

In the present paper the methods and algorithms permitting to calculate ﬁelds for various directed and radiant TE and TM modes of symmetrical and asymmetrical integrated-optical waveguides are presented. The description of the theoretical approaches and algorithm of calculation of the ﬁeld of a scattered radiation outside of an irregular integrated-optical waveguide in the system of visual programming Delphi is given. The dispersion dependences for TE and TM modes in the trigonometric form, and appropriate pictures of ﬁelds of the radiation TE modes of the substrates and the pictures of ﬁelds of electromagnetic radiation scattered in an integrated-optical waveguide with three-dimensional irregularities are also given.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):139-151

139-151

### On Two-Field Solitons in 2 and 3 Dimensions

#### Abstract

We study two- and three-dimensional stationary solitons with non-trivial topology in gauge-invariant nonlinear sigma models (NSMs) describing interaction of scalar unit SN ﬁelds with gauge vector SU(N − 1) ﬁelds, N = 2,3.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):152-156

152-156

157-160

### Mean Field Green Function Solution of the Two-Band Hubbard Model in Cuprates

#### Abstract

The paper discusses the generalized mean ﬁeld solution of the Green function matrix of the eﬀective two-band two-dimensional Hubbard model of the high-Tc superconductivity in cuprates, as recently modiﬁed to include appropriate boundary conditions at zero doping. Two main results are found. (i) Hybridization of normal state energy levels preserves the center of gravity of the unhybridized levels. (ii) Hybridization of superconducting state energy levels displaces the center of gravity of the unhybridized normal levels. The whole spectrum is displaced towards lower frequencies.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):161-166

161-166

### Vacuum Instability and Pair Production in Strong QED

#### Abstract

Strong external electromagnetic ﬁelds make the QED vacuum unstable which decays by emitting signiﬁcantly boson or fermion particle-antiparticle pairs. The recent progress in studying the particle-antiparticle pair production phenomenon is reported: 1. New exact formulas for production rates of boson and fermion pairs by a smooth potential step ϕ(x) ∝ tanhkz in three dimensions. 2. Exact expressions for reﬂection and transmission coeﬃcients, as well as for average numbers of produced pairs and for pair production intensities obtained via the studying scattering versus tunneling process by this potential. 3. On this basis, re-examining and justify the normal spin-statistics relation, a highly nontrivial task due to the vacuum instability.

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):167-170

167-170

### Our autors

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):171-175

171-175

### Pravila oformleniyastatey

**Discrete and Continuous Models and Applied Computational Science**. 2010;(2.2):176-176

176-176