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Numerical solutions for the two-dimensional reaction-diffusion equation with nonlocal nonlinearity are obtained. The solutions reveal formation of dissipative structures. Structures arising from initial distributions with one and several centers of localization are considered. Formation of extending circular structures is shown. Peculiarities of formation and interaction of extending circular structures depending on  nonlocal interaction are considered.

Optical systems with two-dimensional feedback demonstrate wide possibilities for studying the nucleation and development processes of dissipative structures. Feedback allows to influence the dynamics of the optical system by controlling the transformation of spatial variables performed by prisms, lenses, dynamic holograms and other devices. A nonlinear interferometer with a mirror image of a field in two-dimensional feedback is one of the simplest optical systems in which is realized the nonlocal nature of light fields.

A mathematical model of optical systems with two-dimensional feedback is a nonlinear parabolic equation with rotation transformation of a spatial variable and periodicity conditions on a circle. Such problems are investigated: bifurcation of the traveling wave type stationary structures, how the form of the solution changes as the diffusion coefficient decreases, dynamics of the solution’s stability when the bifurcation parameter leaves the critical value. For the first time as a parameter bifurcation was taken of diffusion coefficient.

The method of central manifolds and the Galerkin’s method are used in this paper. The method of central manifolds and the Galerkin’s method are used in this paper. The method of central manifolds allows to prove a theorem on the existence and form of the traveling wave type solution neighborhood of the bifurcation value. The first traveling wave born as a result of the Andronov –Hopf bifurcation in the transition of the bifurcation parameter through the сritical value. According to the central manifold theorem, the first traveling wave is born orbitally stable.

Since the above theorem gives the opportunity to explore solutions are born only in the vicinity of the critical values of the bifurcation parameter, the decision to study the dynamics of traveling waves of change during the withdrawal of the bifurcation parameter in the supercritical region, the formalism of the Galerkin method was used. In accordance with the method of the central manifold is made Galerkin’s approximation of the problem solution. As the bifurcation parameter decreases and its transition through the critical value, the zero solution of the problem loses stability in an oscillatory manner. As a result, a periodic solution of the traveling wave type branches off from the zero solution. This wave is born orbitally stable. With further reduction of the parameter and its passage through the next critical value from the zero solution, the second solution of the traveling wave type is produced as a result of the Andronov –Hopf bifurcation. This wave is born unstable with an instability index of two.

Numerical calculations have shown that the application of the Galerkin’s method leads to correct results. The results obtained are in good agreement with the results obtained by other authors and can be used to establish experiments on the study of phenomena in optical systems with feedback.

Nowadays the random search became a widespread and effective tool for solving different complex optimization and adaptation problems. In this work, the problem of an average duration of a random search for one object by another is regarded, depending on various factors on a square field. The problem solution was carried out by holding total experiment with 4 factors and orthogonal plan with 54 lines. Within each line, the initial conditions and the cellular automaton transition rules were simulated and the duration of the search for one object by another was measured. As a result, the regression model of average duration of a random search for an object depending on the four factors considered, specifying the initial positions of two objects, the conditions of their movement and detection is constructed. The most significant factors among the factors considered in the work that determine the average search time are determined. An interpretation is carried out in the problem of random search for an object from the constructed model. The important result of the work is that the qualitative and quantitative influence of initial positions of objects, the size of the lattice and the transition rules on the average duration of search is revealed by means of model obtained. It is shown that the initial neighborhood of objects on the lattice does not guarantee a quick search, if each of them moves. In addition, it is quantitatively estimated how many times the average time of searching for an object can increase or decrease with increasing the speed of the searching object by 1 unit, and also with increasing the field size by 1 unit, with different initial positions of the two objects. The exponential nature of the growth in the number of steps for searching for an object with an increase in the lattice size for other fixed factors is revealed. The conditions for the greatest increase in the average search duration are found: the maximum distance of objects in combination with the immobility of one of them when the field size is changed by 1 unit. (that is, for example, with $4 \times 4$ at $5 \times 5$) can increase the average search duration in $e^{1.69} \approx 5.42$. The task presented in the work may be relevant from the point of view of application both in the landmark for ensuring the security of the state, and, for example, in the theory of mass service.

The movement of bed load along the closed conduit can lead to a loss of stability of the bed surface, when bed waves arise at the bed of the channel. Investigation of the development of bed waves is associated with the possibility of determining of the bed load nature along the bed of the periodic form. Despite the great attention of many researchers to this problem, the question of the development of bed waves remains open at the present time. This is due to the fact that in the analysis of this process many researchers use phenomenological formulas for sediment transport in their work. The results obtained in such models allow only assess qualitatly the development of bed waves. For this reason, it is of interest to carry out an analysis of the development of bed waves using the analytical model for sediment transport.

The paper proposed two-dimensional profile mathematical riverbed model, which allows to investigate the movement of sediment over a periodic bed. A feature of the mathematical model is the possibility of calculating the bed load transport according to an analytical model with the Coulomb–Prandtl rheology, which takes into account the influence of bottom surface slopes, bed normal and tangential stresses on the movement of bed material. It is shown that when the bed material moves along the bed of periodic form, the diffusion and pressure transport of bed load are multidirectional and dominant with respect to the transit flow. Influence of the effects of changes in wave shape on the contribution of transit, diffusion and pressure transport to the total sediment transport has been studied. Comparison of the received results with numerical solutions of the other authors has shown their good qualitative initiation.

Mathematical simulation of unsteady natural convection and thermal surface radiation within a rotating square enclosure was performed. The considered domain of interest had two isothermal opposite walls subjected to constant low and high temperatures, while other walls are adiabatic. The walls were diffuse and gray. The considered cavity rotated with constant angular velocity relative to the axis that was perpendicular to the cavity and crossed the cavity in the center. Mathematical model, formulated in dimensionless transformed variables “stream function – vorticity” using the Boussinesq approximation and diathermic approach for the medium, was performed numerically using the finite difference method. The vorticity dispersion equation and energy equation were solved using locally one-dimensional Samarskii scheme. The diffusive terms were approximated by central differences, while the convective terms were approximated using monotonic Samarskii scheme. The difference equations were solved by the Thomas algorithm. The approximated Poisson equation for the stream function was solved by successive over-relaxation method. Optimal value of the relaxation parameter was found on the basis of computational experiments. Radiative heat transfer was analyzed using the net-radiation method in Poljak approach. The developed computational code was tested using the grid independence analysis and experimental and numerical results for the model problem.

Numerical analysis of unsteady natural convection and thermal surface radiation within the rotating enclosure was performed for the following parameters: Ra = 10³–10⁶, Ta = 0–10⁵, Pr = 0.7, ε = 0–0.9. All distributions were obtained for the twentieth complete revolution when one can find the periodic behavior of flow and heat transfer. As a result we revealed that at low angular velocity the convective flow can intensify but the following growth of angular velocity leads to suppression of the convective flow. The radiative Nusselt number changes weakly with the Taylor number.

The work is dedicated to the use of the software package Turbulence Problem Solver (TPS) for numerical simulation of a wide range of laser problems. The capabilities of the package are demonstrated by the example of numerical simulation of the interaction of femtosecond laser pulses with thin metal bonds. The software package TPS developed by the authors is intended for numerical solution of hyperbolic systems of differential equations on multiprocessor computing systems with distributed memory. The package is a modern and expandable software product. The architecture of the package gives the researcher the opportunity to model different physical processes in a uniform way, using different numerical methods and program blocks containing specific initial conditions, boundary conditions and source terms for each problem. The package provides the the opportunity to expand the functionality of the package by adding new classes of problems, computational methods, initial and boundary conditions, as well as equations of state of matter. The numerical methods implemented in the software package were tested on test problems in one-dimensional, two-dimensional and three-dimensional geometry, which included Riemann's problems on the decay of an arbitrary discontinuity with different configurations of the exact solution.

Thin films on substrates are an important class of targets for nanomodification of surfaces in plasmonics or sensor applications. Many articles are devoted to this subject. Most of them, however, focus on the dynamics of the film itself, paying little attention to the substrate, considering it simply as an object that absorbs the first compression wave and does not affect the surface structures that arise as a result of irradiation. The paper describes in detail a computational experiment on the numerical simulation of the interaction of a single ultrashort laser pulse with a gold film deposited on a thick glass substrate. The uniform rectangular grid and the first-order Godunov numerical method were used. The presented results of calculations allowed to confirm the theory of the shock-wave mechanism of holes formation in the metal under femtosecond laser action for the case of a thin gold film with a thickness of about 50 nm on a thick glass substrate.

A classical for nonlinear dynamics model, Brusselator, is considered, being augmented by addition of a third variable, which plays the role of a fast-diffusing inhibitor. The model is investigated in one-dimensional case in the parametric domain, where two types of diffusive instabilities of system’s homogeneous stationary state are manifested: wave instability, which leads to spontaneous formation of autowaves, and Turing instability, which leads to spontaneous formation of stationary dissipative structures, or Turing structures. It is shown that, due to the subcritical nature of Turing bifurcation, the interaction of two instabilities in this system results in spontaneous formation of stationary dissipative structures already before the passage of Turing bifurcation. In response to different perturbations of spatially uniform stationary state, different stable regimes are manifested in the vicinity of the double bifurcation point in the parametric region under study: both pure regimes, which consist of either stationary or autowave dissipative structures; and mixed regimes, in which different modes dominate in different areas of the computational space. In the considered region of the parametric space, the system is multistable and exhibits high sensitivity to initial noise conditions, which leads to blurring of the boundaries between qualitatively different regimes in the parametric region. At that, even in the area of dominance of mixed modes with prevalence of Turing structures, the establishment of a pure autowave regime has significant probability. In the case of stable mixed regimes, a sufficiently strong local perturbation in the area of the computational space, where autowave mode is manifested, can initiate local formation of new stationary dissipative structures. Local perturbation of the stationary homogeneous state in the parametric region under investidation leads to a qualitatively similar map of established modes, the zone of dominance of pure autowave regimes being expanded with the increase of local perturbation amplitude. In two-dimensional case, mixed regimes turn out to be only transient — upon the appearance of localized Turing structures under the influence of wave regime, they eventually occupy all available space.

The purpose of the paper is a research of a 2nd order finite difference scheme based on the Z-scheme. This research is the numerical solution of several two-dimensional differential equations simulated the incompressible medium convection.

One of real tasks for similar equations solution is the numerical simulating of strongly non-stationary midscale processes in the Earth ionosphere. Because convection processes in ionospheric plasma are controlled by magnetic field, the plasma incompressibility condition is supposed across the magnetic field. For the same reason, there can be rather high velocities of heat and mass convection along the magnetic field.

Ionospheric simulation relevant task is the research of plasma instability of various scales which started in polar and equatorial regions first of all. At the same time the mid-scale irregularities having characteristic sizes 1–50 km create conditions for development of the small-scale instabilities. The last lead to the F-spread phenomenon which significantly influences the accuracy of positioning satellite systems work and also other space and ground-based radio-electronic systems.

The difference schemes used for simultaneous simulating of such multi-scale processes must to have high resolution. Besides, these difference schemes must to be high resolution on the one hand and monotonic on the other hand. The fact that instabilities strengthen errors of difference schemes, especially they strengthen errors of dispersion type is the reason of such contradictory requirements. The similar swing of errors usually results to nonphysical results at the numerical solution.

At the numerical solution of three-dimensional mathematical models of ionospheric plasma are used the following scheme of splitting on physical processes: the first step of splitting carries out convection along, the second step of splitting carries out convection across. The 2nd order finite difference scheme investigated in the paper solves approximately convection across equations. This scheme is constructed by a monotonized nonlinear procedure on base of the Z-scheme which is one of 2nd order schemes. At this monotonized procedure a nonlinear correction with so-called “oblique differences” is used. “Oblique differences” contain the grid nodes relating to different layers of time.

The researches were conducted for two cases. In the simulating field components of the convection vector had: 1) the constant sign; 2) the variable sign. Dissipative and dispersive characteristics of the scheme for different types of the limiting functions are in number received.

The results of the numerical experiments allow to draw the following conclusions.

1. For the discontinuous initial profile the best properties were shown by the SuperBee limiter.

2. For the continuous initial profile with the big spatial steps the SuperBee limiter is better, and at the small steps the Koren limiter is better.

3. For the smooth initial profile the best results were shown by the Koren limiter.

4. The smooth F limiter showed the results similar to Koren limiter.

5. Limiters of different type leave dispersive errors, at the same time dependences of dispersive errors on the scheme parameters have big variability and depend on the scheme parameters difficulty.

6. The monotony of the considered differential scheme is in number confirmed in all calculations. The property of variation non-increase for all specified functions limiters is in number confirmed for the onedimensional equation.

7. The constructed differential scheme at the steps on time which are not exceeding the Courant's step is monotonous and shows good exactness characteristics for different types solutions. At excess of the Courant's step the scheme remains steady, but becomes unsuitable for instability problems as monotony conditions not satisfied in this case.

Pattern formation described by the scalar Fisher–Kolmogorov–Petrovsky–Piscounov equation with nonlocal competition loses and convection linear on coordinates is considered numerically. Initial function localized around a point is shown to transform in a function localized around a ring with symmetrically sited local maxima. The ring radius and number of maxima depend on convection.

The article discusses the model of the anthropomorphic type of mechanism of the exoskeleton with links of variable length. Four models of parts of variable length are considered comprehensively: the model link of the exoskeleton of variable length with a resilient member and a rigid strong core; the model of the telescopic link; the model link with the masses in the hinge-joint between them; the link model with an arbitrary number of masses. The differential equations of motion in the form of Lagrange equations of the second kind are made. On the basis of analysis of differential equations of motion for multi-link rod of a mechanical system type, exoskeleton revealed their structure, which allowed us to represent them in vector-matrix form. The General pattern of building matrices are established for the first time and the generalization of the expressions for elements of matrices in two-dimensional case are obtained. New recursive and matrix methods of composing of differential equations of motion are given. A unified approach to constructing differential equations of motion of the exoskeleton based on the developed recursive and matrix methods write differential equations of motion of the proposed exoskeleton. Comparison of the time of writing the differential equations of motion proposed methods, in comparison with the Lagrange equations of the second kind, in the system of computer mathematics Mathematica conducted. An analytical study of the model of the exoskeleton carried out. It was found that for mechanisms with n movable links of the Cauchy problem for systems of differential equations of motion for any initial conditions there is no single and unlimited continue. Control of the exoskeleton is accomplished using the torques which are located in the hinge-joints in the joints of the links and simulating control actions. Numerical investigation of a model of the exoskeleton is made, a comparison of results of calculations for exoskeletons with various models of units is held. A numerical study of the empirical evidence about the man and his movements is used. It is established that the choice structure of the exoskeleton model with lumped masses is more preferable to a model with perfectly rigid strong core. As an exoskeleton, providing comfortable movement of people, and you should repeat the properties of the musculoskeletal system.