Volume 64, Nº 6 (2024)

The problem of numerical differentiation of functions with large gradients is considered. It is assumed that for the original function of one variable the decomposition is valid as the sum of a regular component with bounded derivatives up to a certain order and a boundary layer component having large gradients and known with an accuracy of up to a factor. Such a decomposition, in particular, is valid for solution of a singularly perturbed boundary value problem. The topic of the study is relevant, since the application of classical polynomial formulas of numerical differentiation to functions with large gradients can lead to significant errors. The error of the formulas of numerical differentiation, according to the construction of exact ones on the boundary layer component of the original function, is estimated. The results of numerical experiments are presented, consistent with the obtained error estimates.

The design of fault-tolerant production and supply systems is one of the priority areas of development of modern operations research. The traditional approach to modeling such systems is based on the use of probabilistic models describing the choice of a possible scenario of actions in the event of failures in the production or transport network. Along with a number of advantages, this approach has a wellknown drawback. The occurrence of failures of an unknown nature that can jeopardize the operability of the entire modeled system significantly complicates its application. In this paper, we introduce the minimax problem of constructing fault-tolerant production plans (Reliable Production Process Design Problem, RPPDP), the purpose of which is to ensure the smooth functioning of a distributed production system with minimal guaranteed costs. It is shown that the RPPDP problem is NP-hard in the strong sense and remains intractable under fairly specific conditions. To find exact and approximate solutions with accuracy estimates for this problem, branch and bound methods have been developed based on the proposed compact model of mixed integer linear programming (MILP) and the author’s heuristics of adaptive large neighborhood search (ALNS) within the framework of extensions of the well-known Gurobi MIP-solver. High performance and complementarity of the proposed algorithms have been confirmed by the results of numerical experiments conducted on an open library of test examples developed by the authors, containing adapted problem statements from the PCGTSPLIB library.

A new efficient analytical-numerical method is developed for solving a problem for the potential vorticity equation in the quasi-geostrophic approximation with allowance for vertical diffusion of mass and momentum. The method is used to analyze small perturbations of ocean currents of finite transverse scale with a general parabolic vertical profile of velocity. For the arising spectral nonself-adjoint problem, asymptotic expansions of the eigenfunctions and eigenvalues are constructed for small wave numbers and the existence of a countable set of complex eigenvalues with an unboundedly decreasing imaginary part is shown. On the integration interval , a system of three neighborhoods is introduced and a solution in each of them is constructed in the form of power series expansions, which are matched smoothly, so that the eigenfunctions and eigenvalues are efficiently calculated with high accuracy. For a varying wave number, the trajectories of complex eigenvalues are computed for various parameters of the problem and the existence of double eigenvalues is shown. The complex picture of instability developing in the simulated flow depending on physical parameters of the problem is briefly described.

The article gives an analytical and topological description of the eigenvalue functional on the manifold of periodic potentials

The mathematical modeling of the ice-water phase transition during fluid flow inside a pipe with a small ice buildup on the wall at high Reynolds numbers is considered. As a mathematical model describing the dynamics of the phase transition, a double-deck boundary layer model and a phase field system are used. The results of numerical simulation are presented.

A mathematical model is proposed that describes the flow of a thin layer of liquid on an inclined, non-uniformly heated substrate. The Navier-Stokes system for a viscous incompressible liquid and relations representing generalized kinematic, dynamic and energy conditions at the interface for the case of evaporation are used as governing equations. The statement is given in a two-dimensional case for large Reynolds numbers. The problem is solved within the framework of the long-wave approximation. A parametric analysis of the problem is carried out, an evolutionary equation is obtained for finding the thickness of the liquid layer. An algorithm for a numerical solution is proposed for the problem of periodic flow of liquid down an inclined substrate. The influence of gravitational effects and the nature of heating of a solid substrate on the flow of a liquid layer is studied.