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In this paper, a comparative analysis of the exact and heuristic algorithms presented by the method of branches and boundaries, genetic and ant algorithms, respectively, is carried out to find the optimal solution to the traveling salesman problem using the example of a courier robot. The purpose of the work is to determine the running time, the length of the obtained route and the amount of memory required for the program to work, using the method of branches and boundaries and evolutionary heuristic algorithms. Also, the most appropriate of the listed methods for use in the specified conditions is determined. This article uses the materials of the conducted research, implemented in the format of a computer program, the program code for which is implemented in Python. In the course of the study, a number of criteria for the applicability of algorithms were selected (the time of the program, the length of the constructed route and the amount of memory necessary for the program to work), the results of the algorithms were obtained under specified conditions and conclusions were drawn about the degree of expediency of using one or another algorithm in various specified conditions of the courier robot. During the study, it turned out that for a small number of points  $\leqslant10$, the method of branches and boundaries is the most preferable, since it finds the optimal solution faster. However, when calculating the route by this method, provided that the points increase by more than 10, the operating time increases exponentially. In this case, more effective results are obtained by a heuristic approach using a genetic and ant algorithm. At the same time, the ant algorithm is distinguished by solutions that are closest to the reference ones and with an increase of more than 16 points. Its relative disadvantage is the greatest resource intensity among the considered algorithms. The genetic algorithm gives similar results, but after increasing the points more than 16, the length of the found route increases relative to the reference one. The advantage of the genetic algorithm is its lower resource intensity compared to other algorithms.

The practical significance of this article lies in the potential possibility of using the results obtained for the optimal solution of logistics problems by an automated system in various fields: warehouse logistics, transport logistics, «last mile» logistics, etc.

The paper studies a multidimensional convection-diffusion equation with variable coefficients and a nonclassical boundary condition. Two cases are considered: in the first case, the first boundary condition contains the integral of the unknown function with respect to the integration variable $x_\alpha^{}$, and in the second case, the integral of the unknown function with respect to the integration variable $\tau$, denoting the memory effect. Similar problems arise when studying the transport of impurities along the riverbed. For an approximate solution of the problem posed, a locally one-dimensional difference scheme by A.A. Samarskii with order of approximation $O(h^2+\tau)$. In view of the fact that the equation contains the first derivative of the unknown function with respect to the spatial variable $x_\alpha^{}$, the wellknown method proposed by A.A. Samarskii in constructing a monotonic scheme of the second order of accuracy in $h_\alpha^{}$ for a general parabolic type equation containing one-sided derivatives taking into account the sign of $r_\alpha^{}(x,t)$. To increase the boundary conditions of the third kind to the second order of accuracy in $h_\alpha^{}$, we used the equation, on the assumption that it is also valid at the boundaries. The study of the uniqueness and stability of the solution was carried out using the method of energy inequalities. A priori estimates are obtained for the solution of the difference problem in the $L_2^{}$-norm, which implies the uniqueness of the solution, the continuous and uniform dependence of the solution of the difference problem on the input data, and the convergence of the solution of the locally onedimensional difference scheme to the solution of the original differential problem in the $L_2^{}$-norm with speed equal to the order of approximation of the difference scheme. For a two-dimensional problem, a numerical solution algorithm is constructed.

The paper deals with a heat wave motion problem for a degenerate second-order nonlinear parabolic equation with power nonlinearity. The considered boundary condition specifies in a plane the motion equation of the circular zero front of the heat wave. A new numerical-analytical algorithm for solving the problem is proposed. A solution is constructed stepby- step in time using difference time discretization. At each time step, a boundary value problem for the Poisson equation corresponding to the original equation at a fixed time is considered. This problem is, in fact, an inverse Cauchy problem in the domain whose initial boundary is free of boundary conditions and two boundary conditions (Neumann and Dirichlet) are specified on a current boundary (heat wave). A solution of this problem is constructed as the sum of a particular solution to the nonhomogeneous Poisson equation and a solution to the corresponding Laplace equation satisfying the boundary conditions. Since the inhomogeneity depends on the desired function and its derivatives, an iterative solution procedure is used. The particular solution is sought by the collocation method using inhomogeneity expansion in radial basis functions. The inverse Cauchy problem for the Laplace equation is solved by the null field method as applied to a circular domain with a circular hole. This method is used for the first time to solve such problem. The calculation algorithm is optimized by parallelizing the computations. The parallelization of the computations allows us to realize effectively the algorithm on high performance computing servers. The algorithm is implemented as a program, which is parallelized by using the OpenMP standard for the C++ language, suitable for calculations with parallel cycles. The effectiveness of the algorithm and the robustness of the program are tested by the comparison of the calculation results with the known exact solution as well as with the numerical solution obtained earlier by the authors with the use of the boundary element method. The implemented computational experiment shows good convergence of the iteration processes and higher calculation accuracy of the proposed new algorithm than of the previously developed one. The solution analysis allows us to select the radial basis functions which are most suitable for the proposed algorithm.

An important role in the construction of mathematical models of dynamic systems is played by inverse problems, which in particular include the problem of parametric identification. Unlike classical models that operate with point values, interval models give upper and lower boundaries on the quantities under study. The paper considers an interpolation approach to solving interval problems of parametric identification of dynamic systems for the case when experimental data are represented by external interval estimates. The purpose of the proposed approach is to find such an interval estimate of the model parameters, in which the external interval estimate of the solution of the direct modeling problem would contain experimental data or minimize the deviation from them. The approach is based on the adaptive interpolation algorithm for modeling dynamic systems with interval uncertainties, which makes it possible to explicitly obtain the dependence of phase variables on system parameters. The task of minimizing the distance between the experimental data and the model solution in the space of interval boundaries of the model parameters is formulated. An expression for the gradient of the objectivet function is obtained. On a representative set of tasks, the effectiveness of the proposed approach is demonstrated.

A numerical method is proposed that approximates the equations of the dynamics of a weakly compressible viscous flow in the presence of a polymer component of the flow. The behavior of the flow under the influence of a static external periodic force in a periodic square cell is investigated. The methodology is based on a hybrid approach. The hydrodynamics of the flow is described by a system of Navier – Stokes equations and is numerically approximated by the linearized Godunov method. The polymer field is described by a system of equations for the vector of stretching of polymer molecules $\bf R$, which is numerically approximated by the Kurganov – Tedmor method. The choice of model relationships in the development of a numerical methodology and the selection of modeling parameters made it possible to qualitatively model and study the regime of elastic turbulence at low Reynolds $Re \sim 10^{-1}$. The polymer solution flow dynamics equations differ from the Newtonian fluid dynamics equations by the presence on the right side of the terms describing the forces acting on the polymer component part. The proportionality coefficient $A$ for these terms characterizes the backward influence degree of the polymers number on the flow. The article examines in detail how the flow and its characteristics change depending on the given coefficient. It is shown that with its growth, the flow becomes more chaotic. The flow energy spectra and the spectra of the polymers stretching field are constructed for different values of $A$. In the spectra, an inertial sub-range of the energy cascade is traced for the flow velocity with an indicator $k \sim −4$, for the cascade of polymer molecules stretches with an indicator $−1.6$.

The paper proposes and investigates a mathematical model of a distributed computing system of parallel interacting processes competing for the use of a limited number of copies of a structured software resource. In cases of unlimited and limited parallelism by the number of processors of a multiprocessor system, the problems of determining operational and exact values of the execution time of heterogeneous and identically distributed competing processes in a synchronous mode are solved, which ensures a linear order of execution of blocks of a structured software resource within each of the processes without delays. The obtained results can be used in a comparative analysis of mathematical relationships for calculating the implementation time of a set of parallel distributed interacting competing processes, a mathematical study of the efficiency and optimality of the organization of distributed computing, solving problems of constructing an optimal layout of blocks of an identically distributed system, finding the optimal number of processors that provide the directive execution time of given volumes of computations. The proposed models and methods open up new prospects for solving problems of optimal distribution of limited computing resources, synchronization of a set of interacting competing processes, minimization of system costs when executing parallel distributed processes.

The semiclassical approximation method is applied for solution construction of the Fokker–Planck equation with quadratic nonlocal nonlinearity and various coefficients in models of asset returns estimation. Analitical expressions determining nonlinear evolution operator are obtained in semiclasical approximation.

In our work we present generalization of well-known approach for construction of invariant feature vectors of images in computer vision applications. Basic feature of the suggested algorithm is replacement of commonly used Gaussian filter by convolution of image function with Green’s function of evolution operator, which inherits symmetries of this operator. The use of general filtration allows to obtain additional characteristics of invariant feature vectors.

Stochastically forced discrete dynamical systems are considered. Using first approximation systems, we study dynamics of deviations of stochastic solutions from deterministic equilibria. Necessary and sufficient conditions of the existence of stable stationary solutions of equations for mean-square deviations are derived. Stationary values of these mean-square deviations are used for the estimations of the dispersion of random states nearby stable equilibria and analysis of noise-induced transitions. Constructive application of the suggested technique to the analysis of various stochastic regimes in Ricker population model with Allee effect is demonstrated.

Stability of finite difference schemes of lattice Boltzmann method for modelling of 1D diffusion for cases of D1Q2 and D1Q3 lattices is investigated. Finite difference schemes are constructed for the system of linear Bhatnagar–Gross–Krook (BGK) kinetic equations on single particle distribution functions. Brief review of articles of other authors is realized. With application of multiscale expansion by Chapman–Enskog method it is demonstrated that system of BGK kinetic equations at small Knudsen number is transformated to scalar linear diffusion equation. The solution of linear diffusion equation is obtained as a sum of single particle distribution functions. The method of linear travelling wave propagation is used to show the unconditional asymptotic stability of the solution of Cauchy problem for the system of BGK equations at all values of relaxation time. Stability of the scheme for D1Q2 lattice is demonstrated by the method of differential approximation. Stability condition is written in form of the inequality on values of relaxation time. The possibility of the reduction of stability analysis of the schemes for BGK equations to the analysis of special schemes for diffusion equation for the case of D1Q3 lattice is investigated. Numerical stability investigation is realized by von Neumann method. Absolute values of the eigenvalues of the transition matrix are investigated in parameter space of the schemes. It is demonstrated that in wide range of the parameters changing the values of modulas of eigenvalues are lower than unity, so the scheme is stable with respect to initial conditions.