The grid-characteristic method is a promising numerical method for solving hyperbolic systems of equations, e.g., equations describing elastic and acoustic waves. This method has high precision and allows you to physically correctly simulate wave processes in heterogeneous media. The grid-characteristic method makes it possible to correctly take into account boundary conditions and conditions on surfaces with different physical characteristics. Most fully the advantages of the method are for one-dimensional equations, especially in combination with a fixed difference grid, as in conventional grid-based methods. However, in the multidimensional case using the algorithms of splitting with respect to spatial variables, the author has managed to preserve its positive qualities. The use of the method of Runge – Kutta type, or integro-interpolation method for hyperbolic equations makes it possible to effectively carry out the generalization of methods developed for linear equations, in nonlinear case, in particular, to enforce the difference analogs of the conservation laws, which is important for shock-capturing, for example, discontinuous solutions. Based on the author's variant of the grid-characteristic method were numerically solved several important problems of seismic prospecting, seismic resistance, global seismic studies on Earth and Mars, medical applications, nondestructive testing of railway lines, the simulation of the creation and characteristics of composite materials for the aerospace industry and other areas of practical application. A significant advantage of the constructed method is the preservation of its stability and precision at the strains of the environment. This article presents the results of numerical solution based on the grid-characteristic method to the problem of modeling elastic-plastic deformation in traumatic brain injury.

Introduction. The sediment transportation is the one of major processes that defines the magnitude and bottom surface changing rate of water basins. The most commonly used predictable researches in this field are based on mathematical models. Modeling gives possibilities to reduce and in some cases - to eliminate expensive and often dangerous experiments. Spatially one-dimensional models have been usually used to predict changes of water bottom topography. For real water systems with complicated coastal line, the flow vector is generally not orthogonal to the tangent line for the coastline at each of its points. It also may not coincide with the wind stress vector. Therefore, it is necessary to use spatially two-dimensional models of sediment transportation and effective numerical methods to solve many practically important problems associated with the prediction of bottom surface dynamics. Materials and Methods. The spatially two-dimensional model of sediment transport that satisfies the basic conservation laws (of material balance and momentum), which is a quasilinear parabolic equation, was earlier proposed by the authors (A.I. Sukhinov, A.E. Chistyakov, E.A. Protsenko, and V.V. Sidoryakina). The linear difference schemes were constructed and researched; the model and some practically important problems were solved. However, the theoretical research of the proximity of solutions for the original nonlinear initial-boundary value problem and the linearized continuous problem, on which basis a discrete model (difference scheme) was developed, remained in the shadow. The researching correctness of the linearized problem and the determination of sufficient conditions for positivity of solutions are caused special interest because only positive solutions of this sediment transport problem have physical value within the framework of the considered models. Research Results. The investigated nonlinear two-dimensional model of sediment transport in the coastal zone of shallow water basins takes into account the following physically significant conditions and parameters: bottom material porosity; critical value of the tangent stress at which bottom material transport is started; turbulent mixing; the dynamically varying bottom geometry; wind currents; and bottom friction. Linearization is carried out on the time grid; nonlinear coefficients of the parabolic equation are taken at the previous step of time grid. Then, a set of problems, connected by the initial data, are solved; final solutions of the linearized initial boundary value problems chain on a uniform time grid were constructed, and thus, the linearization of the initial 2D nonlinear model is carried out in total time interval. Earlier, the authors proved the existence and uniqueness of the linear problem solution. A priori proximity estimates for the solutions of linearising sequence of boundary value problems and initial non-linear task have been also obtained. Conditions of its positive solution and convergence to the nonlinear sediment transport problem are defined in the norm of the Hilbert space L1 with the rate O(τ), where the τ is a time step. Discussion and Conclusions. The obtained research results of the spatially two-dimensional nonlinear sediment transport model can be used for predicting the nonlinear hydrodynamic processes, improving their accuracy and reliability due to the availability of new accounting functionality of physically important factors, including the specification of the boundary conditions.

This work describes application of the integral transform method to solution of a quasi-static contact problem of the coating wear-out. Frictional heating and wear of the coating occurs during the sliding of a rigid body over its surface. The problem is considered in the framework of the coupled thermoelasticity theory. The solution of the problem is constructed in the form of contour quadratures of the inverse Laplace transformation. After the calculation of the quadratures the solution of the problem is constructed in the form of series over the poles of the integrands. Investigation of the poles of integrands is performed in dependence on four dimensionless parameters of the problem. The solutions obtained are studied in detail with respect to the dimensionless and dimensional parameters of the problem. Numerical examples of the obtained solutions for contact stresses, displacements, temperature and wear of the coating are presented.

The paper provides the developed approach to solve the eigenvalues and eigenfunctions problems
for the Helmholtz equation in domains with an arbitrary configuration. In developing the approach
for numerical solution of problems, the point-sources method (PSM) was used. The proposed
method is based on the analysis of the condition number of the PSM system or the error of the numerical
solution of problems. The concept of "eigenvalues criteria" is introduced. The research result is a
developed effective method - an algorithm for solving problems of eigenvalues and eigenfunctions
for the Helmholtz equation. It is shown that at the approach of the Helmholtz parameter to the problem
eigenvalue, the condition number of the PSM system and the error of the numerical solution rise
sharply. Therefore, we calculate the dependence of the condition number of the PSM system or error
of the problem numerical solution on the Helmholtz parameter. Then, according to position of the
maximum of the received dependences we find the eigenvalues of the Helmholtz equation in a given
domain.. After finding the eigenvalues, it is possible to proceed to the determination of the eigenfunctions.
At that, if the eigenvalue appears degenerate, that is some eigenfunctions correspond to it, then
it is possible to find all the eigenfunctions taking into account the symmetry of the solution domain.
The two-dimensional and three-dimensional test problems are solved. Upon the results obtained, the
conclusion about the efficiency of the proposed method is made.

The paper adds to the literature on propaganda wars. This area attracts practitioners as well as researchers from a variety of fields such as philosophy, social and political science, psychology. It also attracts IT researchers and mathematicians who develop and study models of propaganda wars. In this paper we apply the mathematical model of making choices by individuals to the problem of how the extent of social polarization affects the outcome of the propaganda battle. By the term “propaganda battle” we mean that each member of the society is subject to two competing flows of information. These two flows are generated by two competing parties and each flow consists of propaganda and rumor. That is each party runs propaganda via its own mass-media, and the rumor adds to propaganda as individuals get information from media and transmit it further through interpersonal communications with other individuals. The kind of society is considered which comprises two groups with diametrically opposite fundamental attitudes. The mathematical model has been investigated analytically and numerically. It is shown that moderate political polarization favors the side that runs more intensive propaganda. However, the advantage of stronger propaganda is impaired if the polarization is great enough, because neither media nor individuals can reassure their radical opponents.

Introduction. The paper is devoted to the research of effective numerical methods for solving eu-trophication problem of shallow waters taking into account water environment, spatially-nonuniform distribution of temperature and salinity, microturbulent diffusion, gravitational sedi-mentation, and spreading of biogenic pollution, oxygen, phyto- and zooplankton, etc. The simula-tion objects are shallow waters – the Azov Sea and Taganrog Bay. Materials and Methods. The mathematical model of eutrophication of shallow waters was devel-oped. Parallel implementation was performed for computationally laborious convection-diffusion problems, taking into account the architecture and parameters of supercomputers based on the de-composition methods of grid domains. We determined that the maximum acceleration was 43 times on 128 computational nodes. We developed two algorithms including the algorithm based on the k-means method for data distribution between processors in parallel implementation. Due to the using these algorithms, the efficiency of algorithm for solving the problem is increased on 15% compared to the algorithm of the standard partition of computational domain. Results. New mathematical models and software complex were developed for mathematical model-ing of eutrophication processes in shallow waters. The concentrations of pollutants and plankton calculated for different wind situations were taken into consideration, if the relative error did not exceed 30%. Due to expedition researches the primary verification of the model of ecosystem of the Azov Sea was performed. The problem of modeling and forecasting the state of water ecosystems of the Azov Sea in conditions of anthropogenic influence and comprehensive research of the unique water ob-ject was implemented. Because of the water object is shallow, it’s more affected by anthropogenic influence. The software complex, combining mathematical models and databases, was designed. Using this complex we researched conditions which are contributed by the eutrophication processes in shallow waters. Discussion and Conclusions. Due to the solving the water ecology problem we can forecast differ-ent scenario of changing the water quality in shallow waters, and to investigate the mechanisms of formation of zones with low oxygen content.

Introduction. The paper is devoted to research of the wave processes with free boundary based on the finite-difference method. Materials and methods. A mathematical model describing the dynamics of distribution of wave fluctuation was proposed on the basis of heterogeneous wave equation with the appropriate initial and boundary conditions. Discretization of the model was conducted using the integro-interpolation method taking into account the partial "filling" of computational cells. The adaptive modified alternating triangular iterative method of variational type with the highest rate of convergence in the class of two-layer iterative methods for solving the developed difference equations. Results. The developed discrete mathematical model for numerical simulation of wave propagation. The results of numerical experiments were obtained. The developed numerical algorithms and their computer implementation were used to research the dynamics of distribution of wave processes in the presence of the free boundary. Discussion and conclusions. The obtained results can be used for research of the dynamics of distribution of the wave processes with a free boundary and controlling in conducting experimental researches, evaluation and diagnosis, etc.

The article studies the possibility of usage of energy-efficient Epiphany microprocessor for solving actual applied problem of face detection at still image. The microprocessor is a multicore system with distributed memory, implemented in a single chip. Due to small die area, the micropro-cessor has significant hardware limitations (in particular it has only 32 kilobytes of memory per core) which limit the range of usable algorithms and complicate their software implementation. Common face-detection algorithm based on local binary patterns (LBP) and cascading classifier was adapted for parallel implementation. It is shown that Epiphany microprocessor having 16 cores can outperform single-core CPU of personal computer having the same clock rate by a factor of 2.5, while consuming only 0.5 watts of electric power.