All issues

The paper considers integro-differential equations of fractional order moisture transfer with the Bessel operator. The studied equations contain the Bessel operator, two Gerasimov – Caputo fractional differentiation operators with different orders $\alpha$ and $\beta$. Two types of integro-differential equations are considered: in the first case, the equation contains a non-local source, i.e. the integral of the unknown function over the integration variable $x$, and in the second case, the integral over the time variable τ, denoting the memory effect. Similar problems arise in the study of processes with prehistory. To solve differential problems for different ratios of $\alpha$ and $\beta$, a priori estimates in differential form are obtained, from which the uniqueness and stability of the solution with respect to the right-hand side and initial data follow. For the approximate solution of the problems posed, difference schemes are constructed with the order of approximation $O(h^2+\tau^2)$ for $\alpha=\beta$ and $O(h^2+\tau^{2-\max\{\alpha,\beta\}})$ for $\alpha\neq\beta$. The study of the uniqueness, stability and convergence of the solution is carried out using the method of energy inequalities. A priori estimates for solutions of difference problems are obtained for different ratios of $\alpha$ and $\beta$, from which the uniqueness and stability follow, as well as the convergence of the solution of the difference scheme to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme.

Charge transfer in DNA is simulated by a discrete Holstein model «quantum particle + classical site chain + interaction». Thermostat temperature is taken into account as stochastic force, which acts on classical sites (Langevin equation). Thus dynamics of charge migration along the chain is described by ODE system with stochastic right-hand side. To integrate the system numerically, algorithms of order 1 or 2 are usually applied. We developed «mixed» algorithm having 4th order of accuracy for fast «quantum» variables (note that in quantum subsystem the condition «sum of probabilities of charge being on site is time-constant» must be held), and 2nd order for slow classical variables, which are affecting by stochastic force. The algorithm allows us to calculate trajectories on longer time intervals as compared to standard algorithms. Model calculations of polaron disruption in homogeneous chain caused by temperature fluctuations are given as an example.

The possibility to simplify the integration of Langevin equation using the variation of correlation between augmentation was researched. The analytical expression for a set of numerical schemes is presented. It’s shown that asymptotic limits for squared velocity depend on step size. The region of convergence and the convergence orders were estimated. It turned out that the incorrect correlation between increments decrease the accuracy down to the level of first-order methods for schemes based on precise solution.

The paper presents the methodology of «line-by-line» calculations of the Earth and atmosphere thermal radiation. Intensity of radiation is computed by numerical integration of the radiative transfer kinetic equation and the system of the angular momentum equations using quasi-diffusion method. Data from HITRAN molecular spectroscopic database [Rothman et al., 2009] are used to calculate the atmosphere optical parameters.

In this paper we present Rayleigh formulas obtained from Kirchhoff integral formulas, which can later be used to obtain migration images. The relevance of the studies conducted in the work is due to the widespread use of migration in the interests of seismic oil and gas seismic exploration. A special feature of the work is the use of an elastic approximation to describe the dynamic behaviour of a geological environment, in contrast to the widespread acoustic approximation. The proposed approach will significantly improve the quality of seismic exploration in complex cases, such as permafrost and shelf zones of the southern and northern seas. The complexity of applying a system of equations describing the state of a linear-elastic medium to obtain Rayleigh formulas and algorithms based on them is a significant increase in the number of computations, the mathematical and analytical complexity of the resulting algorithms in comparison with the case of an acoustic medium. Therefore in industrial seismic surveys migration algorithms for the case of elastic waves are not currently used, which creates certain difficulties, since the acoustic approximation describes only longitudinal seismic waves in geological environments. This article presents the final analytical expressions that can be used to develop software systems using the description of elastic seismic waves: longitudinal and transverse, thereby covering the entire range of seismic waves: longitudinal reflected PP-waves, longitudinal reflected SP-waves, transverse reflected PS-waves and transverse reflected SS-waves. Also, the results of comparison of numerical solutions obtained on the basis of Rayleigh formulas with numerical solutions obtained by the grid-characteristic method are presented. The value of this comparison is due to the fact that the method based on Rayleigh integrals is based on analytical expressions, while the grid-characteristic method is a method of numerical integration of solutions based on a calculated grid. In the comparison, different types of sources were considered: a point source model widely used in marine and terrestrial seismic surveying and a flat wave model, which is also sometimes used in field studies.

In real problems of exploration seismology we deal with a heterogeneity of the nature of elastic waves interaction with the surface of a fracture by the propagation through it. The fracture is a complex heterogeneous structure. In some locations the surfaces of fractures are placed some distance apart and are separated by filling fluid or emptiness, in some places we can observe the gluing of surfaces, when under the action of pressure forces the fracture surfaces are closely adjoined to each other. In addition, fractures can be classified by the nature of saturation: fluid or gas. Obviously, for such a large variety in the structure of fractures, one cannot use only one model that satisfies all cases.

This article is concerned with description of developed mathematical fracture models which can be used for numerical solution of exploration seismology problems using the grid-characteristic method on unstructured triangular (in 2D-case) and tetrahedral (in 3D-case) meshes. The basis of the developed models is the concept of an infinitely thin fracture, whose aperture does not influence the wave processes in the fracture area. These fractures are represented by bound areas and contact boundaries with different conditions on contact and boundary surfaces. Such an approach significantly reduces the consumption of computer resources since there is no need to define the mesh inside the fracture. On the other side, it allows the fractures to be given discretely in the integration domain, therefore, one can observe qualitatively new effects, such as formation of diffractive waves and multiphase wave front due to multiple reflections between the surfaces of neighbor fractures, which cannot be observed by using effective fracture models actively used in computational seismology.

The computational modeling of seismic waves propagation through layers of mesofractures was produced using developed fracture models. The results were compared with the results of physical modeling in problems in the same statements.

The paper considers the process of flow formation in an outflow of a binary gas mixture through a thin slit into vacuum. An approach to modeling the flows of rarefied gas mixtures in the transient regime is proposed based on the direct solution of the Boltzmann kinetic equation, in which the conservative projection method is used to calculate the collision integrals. Calculation formulas are provided; the calculation procedure is described in detail in relation to the flow of a binary gas mixture. The Lennard–Jones potential is used as an interaction potential of molecules. A software modeling environment has been developed that makes it possible to study the flows of gas mixtures in a transitional regime on systems of cluster architecture. Due to the use of code parallelization technologies, an acceleration of calculations by 50–100 times was obtained. Numerical simulation of a two-dimensional outflow of a binary argon-neon gas mixture from a vessel into vacuum through a thin slit is carried out for various values of the Knudsen number. The graphs of the dependence of gas mixture components output flow on time in the process of establishing the flow are obtained. Non-stationary regions of strong separation of gas mixture components, in which the molecular densities ratio reaches 10 or more, were discovered. The discovered effect can have applications in the problem of gas mixtures separation.

The work is devoted to the study of the influence of nonlinear processes in the boundary layer on the general nature of gas flows in microchannels of technical systems. Such a study is actually concerned with nanotechnology problems. One of the important problems in this area is the analysis of gas flows in microchannels in the case of transient and supersonic flows. The results of this analysis are important for the gas-dynamic spraying techique and for the synthesis of new nanomaterials. Due to the complexity of the implementation of full-scale experiments on micro- and nanoscale, they are most often replaced by computer simulations. The efficiency of computer simulations is achieved by both the use of new multiscale models and the combination of mesh and particle methods. In this work, we use the molecular dynamics method. It is applied to study the establishment of a gas microflow in a metal channel. Nitrogen was chosen as the gaseous medium. The metal walls of the microchannels consisted of nickel atoms. In numerical experiments, the accommodation coefficients were calculated at the boundary between the gas flow and the metal wall. The study of the microsystem in the boundary layer made it possible to form a multicomponent macroscopic model of the boundary conditions. This model was integrated into the macroscopic description of the flow based on a system of quasi-gas-dynamic equations. On the basis of such a transformed gas-dynamic model, calculations of microflow in real microsystem were carried out. The results were compared with the classical calculation of the flow, which does not take into account nonlinear processes in the boundary layer. The comparison showed the need to use the developed model of boundary conditions and its integration with the classical gas-dynamic approach.

This paper provides an overview of studies on nonlinear supratransmission and related phenomena. This effect consists in the transfer of energy at frequencies not supported by the systems under consideration. The supratransmission does not depend on the integrability of the system, it is resistant to damping and various classes of boundary conditions. In addition, a nonlinear discrete medium, under certain general conditions imposed on the structure, can create instability due to external periodic influence. This instability is the generative process underlying the nonlinear supratransmission. This is possible when the system supports nonlinear modes of various nature, in particular, discrete breathers. Then the energy penetrates into the system as soon as the amplitude of the external harmonic excitation exceeds the maximum amplitude of the static breather of the same frequency.

The effect of nonlinear supratransmission is an important property of many discrete structures. A necessary condition for its existence is the discreteness and nonlinearity of the medium. Its manifestation in systems of various nature speaks of its fundamentality and significance. This review considers the main works that touch upon the issue of nonlinear supratransmission in various systems, mainly model ones.

Many teams of authors are studying this effect. First of all, these are models described by discrete equations, including sin-Gordon and the discrete Schr¨odinger equation. At the same time, the effect is not exclusively model and manifests itself in full-scale experiments in electrical circuits, in nonlinear chains of oscillators, as well as in metastable modular metastructures. There is a gradual complication of models, which leads to a deeper understanding of the phenomenon of supratransmission, and the transition to disordered structures and those with elements of chaos structures allows us to talk about a more subtle manifestation of this effect. Numerical asymptotic approaches make it possible to study nonlinear supratransmission in complex nonintegrable systems. The complication of all kinds of oscillators, both physical and electrical, is relevant for various real devices based on such systems, in particular, in the field of nano-objects and energy transport in them through the considered effect. Such systems include molecular and crystalline clusters and nanodevices. In the conclusion of the paper, the main trends in the research of nonlinear supratransmission are given.

An approximate mathematical model of blood flow in an axisymmetric blood vessel is studied. Such a vessel is understood as an infinitely long circular cylinder, the walls of which consist of elastic rings. Blood is considered as an incompressible fluid flowing in this cylinder. Increased pressure causes radially symmetrical stretching of the elastic rings. Following J. Lamb, the rings are located close to each other so that liquid does not flow between them. To mentally realize this, it is enough to assume that the rings are covered with an impenetrable film that does not have elastic properties. Only rings have elasticity. The considered model of blood flow in a blood vessel consists of three equations: the continuity equation, the law of conservation of momentum and the equation of state. An approximate procedure for reducing the equations under consideration to the Korteweg – de Vries (KdV) equation is considered, which was not fully considered by J. Lamb, only to establish the dependence of the coefficients of the KdV equation on the physical parameters of the considered model of incompressible fluid flow in an axisymmetric vessel. From the KdV equation, by a standard transition to traveling waves, ODEs of the third, second and first orders are obtained, respectively. Depending on the different cases of arrangement of the three stationary solutions of the first-order ODE, a cnoidal wave and a soliton are standardly obtained. The main attention is paid to an unbounded periodic solution, which we call a degenerate cnoidal wave. Mathematically, cnoidal waves are described by elliptic integrals with parameters defining amplitudes and periods. Soliton and degenerate cnoidal wave are described by elementary functions. The hemodynamic meaning of these types of decisions is indicated. Due to the fact that the sets of solutions to first-, second- and third-order ODEs do not coincide, it has been established that the Cauchy problem for second- and third-order ODEs can be specified at all points, and for first-order ODEs only at points of growth or decrease. The Cauchy problem for a first-order ODE cannot be specified at extremum points due to the violation of the Lipschitz condition. The degeneration of the cnoidal wave into a degenerate cnoidal wave, which can lead to rupture of the vessel walls, is numerically illustrated. The table below describes two modes of approach of a cnoidal wave to a degenerate cnoidal wave.