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The study of complex processes in various spheres of human activity is traditionally based on the use of mathematical models. In modern conditions, the development and application of such models is greatly simplified by the presence of high-speed computer equipment and specialized tools that allow, in fact, designing models from pre-prepared modules. Despite this, the known problems associated with ensuring the adequacy of the model, the reliability of the original data, the implementation in practice of the simulation results, the excessively large dimension of the original data, the joint application of sufficiency heterogeneous mathematical models in terms of complexity and integration of the simulated processes are becoming increasingly important. The more critical may be the external constraints imposed on the value of the optimized functional, and often unattainable within the framework of the constructed model. It is logical to assume that in order to fulfill these restrictions, a purposeful transformation of the original model is necessary, that is, the transition to a mathematical model with a deliberately improved solution. The new model will obviously have a different internal structure (a set of parameters and their interrelations), as well as other formats (areas of definition) of the source data. The possibilities of purposeful change of the initial model investigated by the authors are based on the realization of the idea of strategic reflection. The most difficult in mathematical terms practical implementation of the author's idea is the use of simulation models, for which the algorithms for finding optimal solutions have known limitations, and the study of sensitivity in most cases is very difficult. On the example of consideration of rather standard discrete- event simulation model the article presents typical methodological techniques that allow ranking variable parameters by sensitivity and, in the future, to expand the scope of definition of variable parameter to which the simulation model is most sensitive. In the transition to the “improved” model, it is also possible to simultaneously exclude parameters from it, the influence of which on the optimized functional is insignificant, and vice versa — the introduction of new parameters corresponding to real processes into the model.

We study the class of first order differential equations in partial derivatives of the Clairaut-type, which are a multidimensional generalization of the ordinary differential Clairaut equation to the case when the unknown function depends on many variables. It is known that the general solution of the Clairaut-type partial differential equation is a family of integral (hyper-) planes. In addition to the general solution, there can be particular solutions, and in some cases a special (singular) solution can be found.

The aim of the paper is to find a singular solution of the Clairaut-type equation in partial derivatives of the first order with a special right-hand side. In the paper, we formulate a criterion for the existence of a special solution of a differential equation of Clairaut type in partial derivatives for the case, when the function of the derivatives is a function of a linear combination of partial derivatives of unknown function. We obtain the singular solution for this type of differential equations with trigonometric functions of a linear combination of $n$-independent variables with arbitrary coefficients. It is shown that the task of finding a special solution is reduced to solving a system of transcendental equations containing initial trigonometric functions. The article describes the procedure for evaluation of a singular solution of Clairaut-type equation; the main idea is to find not partial derivatives of the unknown function, as functions of independent variables, but linear combinations of partial derivatives with some coefficients. This method can be used to find special solutions of Clairaut-type equations, for which this structure is preserved.

The work is organized as follows. The Introduction contains a brief review of some modern results related to the topic of the study of Clairaut-type equations. The Second part is the main one and it includes a formulation of the main task of the work and describes a method of evaluation of singular solutions for the Clairaut-type equations in partial derivatives with a special right-hand side. The main result of the work is to find singular solutions of the Clairaut-type equations containing trigonometric functions. These solutions are given in the main part of the work as an illustrating example for the method described earlier. In Conclusion, we formulate the results of the work and describe future directions of the research.

We present the iterative algorithm that solves numerically both Urysohn type Fredholm and Volterra nonlinear one-dimensional nonsingular integral equations of the second kind to a specified, modest user-defined accuracy. The algorithm is based on descending recursive sequence of quadratures. Convergence of numerical scheme is guaranteed by fixed-point theorems. Picard’s method of integrating successive approximations is of great importance for the existence theory of integral equations but surprisingly very little appears on numerical algorithms for its direct implementation in the literature. We show that successive approximations method can be readily employed in numerical solution of integral equations. By that the quadrature algorithm is thoroughly designed. It is based on the explicit form of fifth-order embedded Runge–Kutta rule with adaptive step-size self-control. Since local error estimates may be cheaply obtained, continuous monitoring of the quadrature makes it possible to create very accurate automatic numerical schemes and to reduce considerably the main drawback of Picard iterations namely the extremely large amount of computations with increasing recursion depth. Our algorithm is organized so that as compared to most approaches the nonlinearity of integral equations does not induce any additional computational difficulties, it is very simple to apply and to make a program realization. Our algorithm exhibits some features of universality. First, it should be stressed that the method is as easy to apply to nonlinear as to linear equations of both Fredholm and Volterra kind. Second, the algorithm is equipped by stopping rules by which the calculations may to considerable extent be controlled automatically. A compact C++-code of described algorithm is presented. Our program realization is self-consistent: it demands no preliminary calculations, no external libraries and no additional memory is needed. Numerical examples are provided to show applicability, efficiency, robustness and accuracy of our approach.

The work is dedicated to investigation of possibility to increase the efficiency of solving external aerodynamic problems. Methodical questions of using locally-adaptive grids and wall functions for numerical simulation of turbulent flows past flying vehicles are studied. Reynolds-averaged Navier–Stokes equations are integrated. The equations are closed by standard $k–\varepsilon$ turbulence model. Subsonic turbulent flow of perfect compressible viscous gas past airfoil RAE 2822 is considered. Calculations are performed in CFD software FlowVision. The efficiency of using the technology of smoothing diffusion fluxes and the Bradshaw formula for turbulent viscosity is analyzed. These techniques are regarded as means of increasing the accuracy of solving aerodynamic problems on locally-adaptive grids. The obtained results show that using the technology of smoothing diffusion fluxes essentially decreases the discrepancy between computed and experimental values of the drag coefficient. In addition, the distribution of the skin friction coefficient over the curvilinear surface of the airfoil becomes more regular. These results indicate that the given technology is an effective way to increase the accuracy of calculations on locally-adaptive grids. The Bradshaw formula for the dynamic coefficient of turbulent viscosity is traditionally used in the SST $k–\omega$ turbulence model. The possibility to implement it in the standard $k–\varepsilon$ turbulence model is investigated in the present article. The calculations show that this formula provides good agreement of integral aerodynamic characteristics and the distribution of the pressure coefficient over the airfoil surface with experimental data. Besides that, it essentially augments the accuracy of simulation of the flow in the boundary layer and in the wake. On the other hand, using the Bradshaw formula in the simulation of the air flow past airfoil RAE 2822 leads to under-prediction of the skin friction coefficient. For this reason, the conclusion is made that practical use of the Bradshaw formula requires its preliminary validation and calibration on reliable experimental data available for the considered flows. The results of the work as a whole show that using the technologies discussed in numerical solution of external aerodynamic problems on locally-adaptive grids together with wall functions provides the computational accuracy acceptable for quick assessment of the aerodynamic characteristics of a flying vehicle. So, one can deduce that the FlowVision software is an effective tool for preliminary design studies, for conceptual design, and for aerodynamic shape optimization.

A hyperbolic model of a viscous heat-conducting gas is presented, in which the Maxwell – Cattaneo approach is used to hyperbolize the equations, which provides finite wave propagation velocities. In the modified model, instead of the original Stokes and Fourier laws, their relaxation analogues were used and it is shown that when the relaxation times $\tau_\sigma^{}$ и $\tau_w^{}$ tend to The hyperbolized equations are reduced to zero to the classical Navier – Stokes system of non-hyperbolic type with infinite velocities of viscous and heat waves. It is noted that the hyperbolized system of equations of motion of a viscous heat-conducting gas considered in this paper is invariant not only with respect to the Galilean transformations, but also with respect to rotation, since the Yaumann derivative is used when differentiating the components of the viscous stress tensor in time. To integrate the equations of the model, the hybrid Godunov method (HGM) and the multidimensional nodal method of characteristics were used. The HGM is intended for the integration of hyperbolic systems in which there are equations written both in divergent form and not resulting in such (the original Godunov method is used only for systems of equations presented in divergent form). A linearized solver’s Riemann is used to calculate flow variables on the faces of adjacent cells. For divergent equations, a finitevolume approximation is applied, and for non-divergent equations, a finite-difference approximation is applied. To calculate a number of problems, we also used a non-conservative multidimensional nodal method of characteristics, which is based on splitting the original system of equations into a number of one-dimensional subsystems, for solving which a one-dimensional nodal method of characteristics was used. Using the described numerical methods, a number of one-dimensional problems on the decay of an arbitrary rupture are solved, and a two-dimensional flow of a viscous gas is calculated when a shock jump interacts with a rectangular step that is impermeable to gas.

A compact finite difference scheme has been developed for modeling convection in a porous medium saturated with a fluid. We consider the problem for a rectangular domain with anisotropic permeability and thermal conductivity properties in terms of stream function and temperature deviation, taking into account Darcy's law. Boundary conditions of impenetrability and a linear distribution of temperature are set. This model is cosymmetric when certain conditions are imposed on the permeability and thermal conductivities. One parametric family of stationary convection regimes arises when mechanical equilibrium loses stability. A numerical method with a fourth-order finite difference approximation for spatial variables and a Runge – Kutta integrator for time has been developed. It has been proved that this scheme preserves cosymmetry. Numerical results for evaluating the critical Rayleigh number have been presented. We compare them with results obtained using a second-order finite-difference method. We show that critical Rayleigh numbers are repeated twice with very high accuracy, which proves cosymmetry preservation. Numerical evaluation of convective regimes and spectral properties are presented. The efficiency of the developed compact finite difference scheme on a nine-point stencil is assessed.

This paper is devoted to the problem of edge criticality identification and ranking in complex networks, which is a part of a modern research direction in the novel network science. The diffusion importance belongs to the set of acknowledged methods that help to identify the significant connections in the graph that are critical to retaining structural integrity. In the present work, we develop the Iterative Diffusion Importance algorithm that is based on the re-estimation of critical topological features at each step of the graph deconstruction. The Iterative Diffusion Importance has been compared with methods such as diffusion importance and degree product, which are two very well-known benchmark algorithms. As for benchmark networks, we tested the Iterative Diffusion Importance on three standard networks, such as Zachary’s Karate Club, the American Football Network, and the Dolphins Network, which are often used for algorithm efficiency evaluation and are different in size and density. Also, we proposed a new benchmark network representing the airplane communication between Japan and the US. The numerical experiment on finding the ranking of critical edges and the following network decomposition demonstrated that the proposed Iterative Diffusion Importance exceeds the conventional diffusion importance by the efficiency for 2–35% depending on the network complexity, the number of nodes, and the number of edges. The only drawback of the Iterative Diffusion Importance is an increase in computation complexity and hencely in the runtime, but this drawback can be easily compensated for by the preliminary planning of the network deconstruction or protection and by reducing the re-evaluation frequency of the iterative process.

The dual task of integral geometry – to define for a given averaging operator the function class where inversion of that operator is possible – is solved. Those classes are defined ambiguously. Full description of those classes is given in the form of minimal complimentary information necessary to know about the function. The possible to give a constructive description of the class is researched and in the case of a finite averaging system the inversion formulas are given.

An approach to the decrease of norm of the correction in Newton’s methods of optimization, based on the Cholesky’s factorization is presented, which is based on the integration with the technique of the choice of leading element of algorithm of linear programming as a method of solving the system of equations. We investigate the issues of increasing of the numerical stability of the Cholesky’s decomposition and the Gauss’ method of exception.

The different types of the reduced equations are known in the dynamics a heavy rigid body with a fixed point. Since the Euler−Poisson’s equations admit the three first integrals, then for the first approach the obtaining new forms of equations are usually based on these integrals. The system of six scalar equations can be transformed to a third-order system with them. However, in indicated approach the reduced system will have a feature as in the form of radical expressions a relatively the components of the angular velocity vector. This fact prevents the effective the effective application of numerical and asymptotic methods of solutions research. In the second approach the different types of variables in a problem are used: Euler’s angles, Hamilton’s variables and other variables. In this approach the Euler−Poisson’s equations are reduced to either the system of second-order differential equations, or the system for which the special methods are effective. In the article the method of finding the reduced system based on the introduction of an auxiliary variable is applied. This variable characterizes the mixed product of the angular momentum vector, the vector of vertical and the unit vector barycentric axis of the body. The system of four differential equations, two of which are linear differential equations was obtained. This system has no analog and does not contain the features that allows to apply to it the analytical and numerical methods. Received form of equations is applied for the analysis of a special class of solutions in the case when the center of mass of the body belongs to the barycentric axis. The variant in which the sum of the squares of the two components of the angular momentum vector with respect to not barycentric axes is constant. It is proved that this variant exists only in the Steklov’s solution. The obtained form of Euler−Poisson’s equations can be used to the investigation of the conditions of existence of other classes of solutions. Certain perspectives obtained equations consists a record of all solutions for which the center of mass is on barycentric axis in the variables of this article. It allows to carry out a classification solutions of Euler−Poisson’s equations depending on the order of invariant relations. Since the equations system specified in the article has no singularities, it can be considered in computer modeling using numerical methods.