No 2.2 (2010)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.