Regularization method for numerical modelling of a transport of pollutant in shallow water

Abstract: A new method for solving the passive scalar transport equation in the framework of hydrodynamic equations in the shallow water approximation is described. The method is similar in structure to the previously constructed quasi-gas-dynamic algorithm for the numerical simulation of compressible gas flows. Regularized equations and difference schemes based on them, including those for flows with an impurity source, are presented. Typical one-dimensional and two-dimensional test problems are considered. In conclusion, a generalization of the constructed approach for the numerical simulation of the passive scalar transport in the framework of the viscous incompressible fluid approximation is given.Note: Research direction:Mathematical modelling in actual problems of science and technic

To the construction of the thermodynamics of quantum nonextensive systems in the framework of the statistics of Tsallis

Abstract: Within the framework of quantum statistics of Tsallis, based on parametric non-additive entropy associated with the density matrix, thermodynamic equations for a large canonical quantum-mechanical ensemble are obtained. A generalization of the zero law of thermodynamics for independent quantum systems at their thermal contact is obtained, which introduces the so-called physical temperature different from the inversion of Lagrange multiplier β. Taking into account the generalized first law of thermodynamics and Legendre transformation, the modified thermodynamic relations in Tsallis statistics are considered. The second law of thermodynamics is discussed on the basis of the convexity property of Ratier−Kannappan discrimination information generalized to the quantum case. Spontaneous transitions between stationary states of a complex quantum-mechanical system are studied and Boltzmann's H-theorem is proved. The developed approach involves the use of nonextensive quantum thermodynamics in various contexts, in particular, concerning the simulation of quantum thermal effects in nanoelectronic devices, in materials science, biomedicine and other quantum technologies.Note: Research direction:Mathematical modelling in actual problems of science and technic

Normal form of a Hamiltonian system with a periodic perturbation

Abstract: Near a stationary solution we consider the Hamiltonian system with such perturbation, that the unperturbed Hamiltonian function is autonomous and the perturbation of the Hamiltonian function is periodic in time. First we remind the normal form of the autonomous Hamiltonian function. Second we describe the normal form of the periodic perturbation of the Hamiltonian function. It can always be reduced to the time independent Hamiltonian. It allows to compute the local families of periodic solutions to the initial system. The first approximations of some of these families are found by means of computation of the Newton polyhedron of the reduced normal form of Hamiltonian. We also discuss problems of the computer algebra arising in these computations.Note: Research direction:Mathematical modelling in actual problems of science and technic

On approximate construction of near equatorial orbits of the satellites of spheroidal celestial bodies

Abstract: The problem about a motion of material point (satellite) with negligible small mass nearby the equatorial plane of spheroidal body, in particular asteroid, is considered. For the small inclinations of satellite orbit in the first approximation dividing of motions is possible into equatorial and latitudinal components. Equatorial central motion, when a power function depends only on distance of satellite to origin of coordinates (centre of mass of asteroid), is constructed by offered early a semi-analytical method. The construction of latitudinal motion foresees the solution of linearized system of second order differential equations with periodic coefficients by numeral determination of monodromy matrix on the period of equatorial motion and its analytical continuation at time. Model problems are considered about perturbed motion of hypothetical almost equatorial satellites of Ceres and Vesta. The estimation of methodical accuracy is obtained by comparing to the numerical solution.Note: Research direction:Theoretical and applied problems of mechanic

The kinetic equations and some approaches to their analysis for the new model of clusterization-destruction processes

Abstract: 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

The basic property of the Jacobi integral for gravity assists maneuvers in the Solar system

Abstract: It is shown that the standard and bulky method of the general used revealing of the asymptotic velocity invariance during gravity assists maneuvers in the model of the circular restricted three body problem (RTBP), used in modern astrodynamics, can be significantly simplified. The refined forms of the Jacobi integral are presented, which allow, among others, to reveal the transparent relationship of the Jacobi integral and the patched conics method in a RTBP.Note: Research direction:Theoretical and applied problems of mechanic

Determination of the Sun position onboard the spacecraft by photoelectric solar sensor

Abstract: The preprint deals with the development of a mathematical model of the solar sensor on the basis of photoelectric elements to the spacecraft attitude motion to construct and maintain a constant solar orientation (CSO). A mathematical model that allows to form the output of the sensor in the form of analog currents (or voltages) depending on the Sun position relative to the sensor is developed, as well as algorithms that restore the Sun position by magnitude of output currents of photoelectric cells in the instrument coordinate system. A statistical analysis was carried out to estimate the performance of spacecraft control algorithms at construction and maintenance of the CSO mode together with a mathematical model of the sensor and algorithms for restoring the position of the Sun for different initial spacecraft orientation relative to the Sun.Note: Research direction:Mathematical modelling in actual problems of science and technic

Integration of ODEs on Riemann surfaces with an arbitrary precision

Abstract: 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

To the derivation of symmetry of matrix of kinetic coefficients Onsager in framework of the nonextensive statistical mechanics of Tsallis

Abstract: In the framework of the nonextensive statistical mechanics of Tsallis, Onsager symmetry relations for the kinetic coefficients in linear regression equations for even and odd (when the velocities of elementary particles change direction) small fluctuations of macroscopic state parameters are derived. At a macroscopic level, these relations reflect the invariance of microscopic equations of motion with respect to time reversal. As in the case of classical Gibbs statistics, this conclusion is based on the theory of equilibrium fluctuations of dynamic variables characterizing the system, and on the properties of their invariance with respect to time reversal. In addition, the Onsager postulate was used, according to which the attenuation of the equilibrium fluctuations of the thermodynamic parameters of the state is described by linear differential equations of the first order. Traditional reciprocity relations for extensive systems are obtained from the derived relations in the case when the deformation parameter q, included in the parametric entropy functional of Tsallis, is equal to one.Note: Research direction:Mathematical modelling in actual problems of science and technic

Schrödinger equation as a consequence of new Vlasov type equations

Abstract: It is well known that the Schrödinger equation can be reduced to the Hamilton–Jacobi equation in Bohmian mechanics. Corresponding new equations of the Vlasov and Lamb types are derived, and their stationary solutions are investigated.Note: Research direction:Mathematical problems and theory of numerical method