Open Repository of Keldysh Institute of Applied Mathematics of RAS
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Integration of ODEs on Riemann surfaces with an arbitrary precision
No full textAbstract:
We consider analytical systems of ODEs with a real or complex time. Integration of such ODEs is equivalent to an analytical continuation of a solution
along some path, which usually belongs to the real axis. The problems that may appear along this path are often caused by singularities of the solution
that lie outside the real axis. It is possible to circumvent problematic parts of the path (including singularities) by going on the Riemann surface of the solution (i.e., in the complex domain). A natural way to realize this program is to use the method of Taylor expansions, which does not require a formal complexification of the system (i.e., a change of variables). We use two classical problems, i.e., the Restricted Three-Body problem, and Van der Pol equation, to demonstrate how Taylor expansions can be used for integration of ODEs with an arbitrary
precision. We obtained some new results in these problems.Note:
Research direction:Mathematical modelling in actual problems of science and technic
Autonomous spacecraft's on-board orbit determination
No full textAbstract:
Methods, algorithms and programs for on-board orbit determination using global satellite navigation system measurements are proposed and investigated. Algorithms for three-stage processing of measurements are developed to improve the reliability and accuracy.Note:
Research direction:Theoretical and applied problems of mechanic
The development of Butcher rooted trees theory for reduced (m,k)-method
No full textAbstract:
The extension of the Burcher rooted trees theory for reduced (m,k)-methods is proposed. The new concept of 'color' for the stages of the (m,k)-method is introduced. Based on this concept the general rules are formulated on how to obtain order conditions. Obtained theoretical results are in good compliance with known particular results from other researchers.Note:
Research direction:Mathematical problems and theory of numerical method
On the interaction of opposed relativistic flows of neutral dense plasma
No full textAbstract:
A three-dimensional numerical model that describes the interaction of plasma with electromagnetic field in the framework of Vlasov-Maxwell equations is used for simulating the interaction of opposing flows of relativistic neutralized dense plasma in vacuum. The influence of the initial velocity and the concentration of the plasma particles on the process of the interaction of flows are studied.Note:
Research direction:Mathematical modelling in actual problems of science and technic
The kinetic equations and some approaches to their analysis for the new model of clusterization-destruction processes
No full textAbstract:
The new model of clusterization-destruction processes is proposed. We have shown that nonlinear equation can be converted to linear. The analysis of finite- and infinite cases is fulfilled, also some particular examples is analysed.Note:
Research direction:Mathematical modelling in actual problems of science and technic
On modeling of radiation-induced thermo-mechanical effects in heterogeneous materials with complex dispersion structure
No full textAbstract:
A complex model for supercomputing the parameters of radiation-induced thermomechanical fields in heterogeneous media of complex dispersed structure is developed. A technique for calculating the parameters of the photon-electron cascade generated in the object by the interaction of radiation with matter is constructed. A geometric model of the medium with a direct resolution of its microstructure is worked out. A part of the geometric description of the medium is a model of the detecting system for the statistical evaluation of the energy deposit density of radiation. The ideal hydrodynamic Euler model of the compressible medium dynamics in a conservative form is chosen as the basis for the modeling of thermomechanical processes. The results of demonstration calculations of thermomechanical fields parameters are presented.Note:
Research direction:Mathematical modelling in actual problems of science and technic
Numerical steady state analysis of the Marchuk-Petrov model of antiviral immune response
No full textAbstract:
The problem of guaranteed computation of all steady states of the Marchuk-Petrov model with fixed values of parameters and analysis of their stability is considered. It is shown that the system of ten nonlinear equations, non-negative solutions of which are steady states, can be reduced to a system of two equations. Region of possible non-negative solutions is analytically localized. An effective technology for computing all non-negative solutions and analyzing their stability is proposed. The obtained results provide a computational basis for the study of chronic forms of viral infections using the Marchuk-Petrov model.Note:
Research direction:Mathematical modelling in actual problems of science and technic
On the current state of the theory of oscillations
No full textAbstract:
The main milestones of the development of the theory of oscillations are reflected, and the prospects for its development are also considered. A generalized definition for the concept 'oscillation', formulated on the basis of the theory of cyclic Markov random processes, is proposed.Note:
Research direction:Mathematical modelling in actual problems of science and technic
On the correlation of time series in ecology of aquatic systems
No full textAbstract:
A method for estimating the mutual influence of short time series, which is based on extension of the concept of spectral entropy to the spectra of cross-correlation functions is developed. Algorithms of calculations and programs are given. The time series of plankton abundance in the lake ecosystem, which is situated in the Republic of Belarus, are analyzed in order to identify mutual correlations. Correlations between the bacterioplankton abundance and variations in seasonal temperature are identified. The peculiarities of interrelations between the abundances of zooplankton/phytoplankton and temperature depending on the reservoir are analysed.Note:
Research direction:Mathematical modelling in actual problems of science and technic
Difference schemes of support operator method for equations of elasticity with azimuthal rotation
No full textAbstract:
In the work, for full displacement vectors and velocities, taking into account the azimuthal rotations, on irregular grids with minimal reasonable restrictions for their topological and geometric structure, the approximations of vector analysis operations in cylindrical geometry were constructed with respect to the difference schemes for elasticity theory problems. Taking into account the energy balance of the medium in the presence of azimuthal rotations, families of integrally consistent approximations of vector analysis operations were made, sufficient for discrete modeling of these processes, with respect to the curvature of space caused by the cylindrical geometry of the system. On (r,z)–regular grids with differential rotation along the azimuthal coordinate θ, the difference schemes of the support operator method for the equations of the elasticity theory in displacements were constructed and investigated. The considered approximations retain the properties of divergence, self-adjointness and sign-definiteness of differential operators, and are also applicable for solving nonstationary problems of hydrodynamics with allowance for elastic processes.Note:
Research direction:Mathematical modelling in actual problems of science and technic