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A mass congestion of people always represents a potential danger and threat for their lives. In addition, every year in the world a very large number of people die because of the crush, the main cause of which is mass panic. Therefore, the study of the phenomenon of mass panic in view of her extreme social danger is an important scientific task. Available information, about the processes of her occurrence and spread refers to the category inaccurate. Therefore, the theory of fuzzy sets has been chosen as a tool for developing a mathematical model of the mechanism of transmitting panic state among people with various temperament species.

When developing an fuzzy model, it was assumed that panic, from the epicenter of the shocking stimulus, spreads among people according to the wave principle, passing at different frequencies through different environments (types of human temperament), and is determined by the speed and intensity of the circular reaction of the mechanism of transmitting panic state among people. Therefore, the developed fuzzy model, along with two inputs, has two outputs — the speed and intensity of the circular reaction. In the block «Fuzzyfication», the degrees of membership of the numerical values of the input parameters to fuzzy sets are calculated. The «Inference» block at the input receives degrees of belonging for each input parameter and at the output determines the resulting function of belonging the speed of the circular reaction and her derivative, which is a function of belonging for the intensity of the circular reaction. In the «Defuzzyfication» block, using the center of gravity method, a quantitative value is determined for each output parameter. The quality assessment of the developed fuzzy model, carried out by calculating of the determination coefficient, showed that the developed mathematical model belongs to the category of good quality models.

The result obtained in the form of quantitative assessments of the circular reaction makes it possible to improve the quality of understanding of the mental processes occurring during the transmission of the panic state among people. In addition, this makes it possible to improve existing and develop new models of chaotic humans behaviors. Which are designed to develop effective solutions in crisis situations, aimed at full or partial prevention of the spread of mass panic, leading to the emergence of panic flight and the appearance of human casualties.

The grid-characteristic method is successfully used for solving hyperbolic systems of partial differential equations (for example, transport / acoustic / elastic equations). It allows to construct correctly algorithms on contact boundaries and boundaries of the integration domain, to a certain extent to take into account the physics of the problem (propagation of discontinuities along characteristic curves), and has the property of monotonicity, which is important for considered problems. In the cases of two-dimensional and three-dimensional problems the method makes use of a coordinate splitting technique, which enables us to solve the original equations by solving several one-dimensional ones consecutively. It is common to use up to 3-rd order one-dimensional schemes with simple splitting techniques which do not allow for the convergence order to be higher than two (with respect to time). Significant achievements in the operator splitting theory were done, the existence of higher-order schemes was proved. Its peculiarity is the need to perform a step in the opposite direction in time, which gives rise to difficulties, for example, for parabolic problems.

In this work coordinate splitting of the 3-rd and 4-th order were used for the two-dimensional hyperbolic problem of the linear elasticity. This made it possible to increase the final convergence order of the computational algorithm. The paper empirically estimates the convergence in L1 and L∞ norms using analytical solutions of the system with the sufficient degree of smoothness. To obtain objective results, we considered the cases of longitudinal and transverse plane waves propagating both along the diagonal of the computational cell and not along it. Numerical experiments demonstrated the improved accuracy and convergence order of constructed schemes. These improvements are achieved with the cost of three- or fourfold increase of the computational time (for the 3-rd and 4-th order respectively) and no additional memory requirements. The proposed improvement of the computational algorithm preserves the simplicity of its parallel implementation based on the spatial decomposition of the computational grid.

Many properties of ordinary differential equations systems solutions are determined by the properties of the equations in variations. An ODE system, which includes both the original nonlinear system and the equations in variations, will be called an extended system further. When studying the properties of the Cauchy problem for the systems of ordinary differential equations, the transition to extended systems allows one to study many subtle properties of solutions. For example, the transition to the extended system allows one to increase the order of approximation for numerical methods, gives the approaches to constructing a sensitivity function without using numerical differentiation procedures, allows to use methods of increased convergence order for the inverse problem solution. Authors used the Broyden method belonging to the class of quasi-Newtonian methods. The Rosenbroke method with complex coefficients was used to solve the stiff systems of the ordinary differential equations. In our case, it is equivalent to the second order approximation method for the extended system.

As an example of the proposed approach, several related mathematical models of the blood coagulation process were considered. Based on the analysis of the numerical calculations results, the conclusion was drawn that it is necessary to include a description of the factor XI positive feedback loop in the model equations system. Estimates of some reaction constants based on the numerical inverse problem solution were given.

Effect of factor V release on platelet activation was considered. The modification of the mathematical model allowed to achieve quantitative correspondence in the dynamics of the thrombin production with experimental data for an artificial system. Based on the sensitivity analysis, the hypothesis tested that there is no influence of the lipid membrane composition (the number of sites for various factors of the clotting system, except for thrombin sites) on the dynamics of the process.

The paper is devoted to the study of relationships between discrete and continuous models financial processes and their probabilistic characteristics. First, a connection is established between the price processes of stocks, hedging portfolio and options in the models conditioned by binomial perturbations and their limit perturbations of the Brownian motion type. Secondly, analogues in the coefficients of stochastic equations with various random processes, continuous and jumpwise, and in the coefficients corresponding deterministic equations for their probabilistic characteristics. Statement of the results on the connections and finding analogies, obtained in this paper, led to the need for an adequate presentation of preliminary information and results from financial mathematics, as well as descriptions of related objects of stochastic analysis. In this paper, partially new and known results are presented in an accessible form for those who are not specialists in financial mathematics and stochastic analysis, and for whom these results are important from the point of view of applications. Specifically, the following sections are presented.

• In one- and n-period binomial models, it is proposed a unified approach to determining on the probability space a risk-neutral measure with which the discounted option price becomes a martingale. The resulting martingale formula for the option price is suitable for numerical simulation. In the following sections, the risk-neutral measures approach is applied to study financial processes in continuous-time models.

• In continuous time, models of the price of shares, hedging portfolios and options are considered in the form of stochastic equations with the Ito integral over Brownian motion and over a compensated Poisson process. The study of the properties of these processes in this section is based on one of the central objects of stochastic analysis — the Ito formula. Special attention is given to the methods of its application.

• The famous Black – Scholes formula is presented, which gives a solution to the partial differential equation for the function $v(t, x)$, which, when $x = S (t)$ is substituted, where $S(t)$ is the stock price at the moment time $t$, gives the price of the option in the model with continuous perturbation by Brownian motion.

• The analogue of the Black – Scholes formula for the case of the model with a jump-like perturbation by the Poisson process is suggested. The derivation of this formula is based on the technique of risk-neutral measures and the independence lemma.

Dynamic scenarios leading to multistability in the form of continuous families of stable solutions are studied for a system of autonomous differential equations. The approach is based on determining the cosymmetries of the problem, calculating stationary solutions, and numerically-analytically studying their stability. The analysis is carried out for equations of the Lotka –Volterra type, describing the interaction of two predators feeding on two related prey species. For a system of ordinary differential equations of the 4th order with 11 real parameters, a numerical-analytical study of possible interaction scenarios was carried out. Relationships are found analytically between the control parameters under which the cosymmetry linear in the variables of the problem is realized and families of stationary solutions (equilibria) arise. The case of multicosymmetry is established and explicit formulas for a two-parameter family of equilibria are presented. The analysis of the stability of these solutions made it possible to reveal the division of the family into regions of stable and unstable equilibria. In a computational experiment, the limit cycles branching off from unstable stationary solutions are determined and their multipliers corresponding to multistability are calculated. Examples of the coexistence of families of stable stationary and non-stationary solutions are presented. The analysis is carried out for the growth functions of logistic and “hyperbolic” types. Depending on the parameters, scenarios can be obtained when only stationary solutions (coexistence of prey without predators and mixed combinations), as well as families of limit cycles, are realized in the phase space. The multistability scenarios considered in the work allow one to analyze the situations that arise in the presence of several related species in the range. These results are the basis for subsequent analysis when the parameters deviate from cosymmetric relationships.

The paper presents the results of applying a scheme of very high accuracy and resolution to obtain numerical solutions of the Navier – Stokes equations of a compressible gas describing the occurrence and development of instability of a two-dimensional laminar boundary layer on a flat plate. The peculiarity of the conducted studies is the absence of commonly used artificial exciters of instability in the implementation of direct numerical modeling. The multioperator scheme used made it possible to observe the subtle effects of the birth of unstable modes and the complex nature of their development caused presumably by its small approximation errors. A brief description of the scheme design and its main properties is given. The formulation of the problem and the method of obtaining initial data are described, which makes it possible to observe the established non-stationary regime fairly quickly. A technique is given that allows detecting flow fluctuations with amplitudes many orders of magnitude smaller than its average values. A time-dependent picture of the appearance of packets of Tollmien – Schlichting waves with varying intensity in the vicinity of the leading edge of the plate and their downstream propagation is presented. The presented amplitude spectra with expanding peak values in the downstream regions indicate the excitation of new unstable modes other than those occurring in the vicinity of the leading edge. The analysis of the evolution of instability waves in time and space showed agreement with the main conclusions of the linear theory. The numerical solutions obtained seem to describe for the first time the complete scenario of the possible development of Tollmien – Schlichting instability, which often plays an essential role at the initial stage of the laminar-turbulent transition. They open up the possibilities of full-scale numerical modeling of this process, which is extremely important for practice, with a similar study of the spatial boundary layer.

This work is devoted to molecular dynamic modeling of the thermal impact processes on the metal sample consisting of nickel atoms. For the solution of this problem, a continuous mathematical model on the basis of the classical Newton mechanics equations has been used; a numerical method based on the Verlet scheme has been chosen; a parallel algorithm has been offered, and its realization within the MPI and OpenMP technologies has been executed. By means of the developed parallel program, the investigation of thermodynamic equilibrium of nickel atoms’ system under the conditions of heating a sample to desired temperature has been executed. In numerical experiments both optimum parameters of calculation procedure and physical parameters of analyzed process have been defined. The obtained numerical results are well corresponding to known theoretical and experimental data.