Introduction. The development, analysis, and modification of finite difference schemes tailored to the specific features of the considered problem can significantly enhance the accuracy of modeling complex systems. In simulations of various processes, including hydrodynamic phenomena in shallow water bodies, it has been observed that for problems with thirdtype (Robin) boundary conditions, the theoretical error order of spatial discretization drops from second-order to firstorder accuracy, which in turn decreases the overall accuracy of the numerical solution. The present study addresses the relevant issue of how the approximation of third-type boundary conditions affects the accuracy of the numerical solution to the heat conduction problem. It also proposes a finite difference scheme with improved boundary approximation for the heat conduction equation with third-type boundary conditions and compares the accuracy of the numerical solutions obtained by the authors with known benchmark solutions.
Materials and Methods. The paper considers the one-dimensional heat conduction equation with third-type boundary conditions, for which an analytical solution is available. The problem is discretized, and it is shown that under standard boundary approximation, the theoretical order of approximation error for the second-order differential operator in the diffusion equation is O(h). To improve the accuracy of the numerical solution under specific third-type boundary conditions, a finite difference scheme is proposed. This scheme achieves second-order accuracy O(h2), for the differential operator not only at interior nodes but also at the boundary nodes of the computational domain.
Results. Test problems were used to compare the accuracy of numerical solutions obtained using the proposed scheme and those based on the standard boundary approximation.
Discussion and Conclusion. Numerical experiments demonstrate that the proposed scheme with enhanced boundary approximation for the heat conduction equation under specific third-type boundary conditions exhibits an effective accuracy order close to 2, which corresponds to the theoretical prediction. It is noteworthy that the scheme with standard boundary approximation also demonstrates an effective accuracy order close to 2, despite the lower theoretical order of boundary approximation. Importantly, the numerical error of the proposed scheme decreases significantly faster compared to the scheme with standard boundary treatment.

Introduction. This study investigates the numerical solution of a two-dimensional hydrodynamic problem in a rectangular cavity using the method of initial velocity field damping and the method of accelerating the initial conditions in terms of stream function and vorticity variables. The damping method was applied at Reynolds numbers Re ≤ 3000, and the acceleration method was used for Re = 8000.
Materials and Methods. To speed up the numerical solution of the problem using an explicit finite-difference scheme for the vorticity dynamics equation, the method of initial condition damping and the method of n-fold splitting of the explicit difference scheme (with n = 100) were used. Compared to the traditional method of accelerating from stationary fluid, the initial velocity field damping method reduced the computation time by a factor of 57. The splitting method used a maximum time step proportional to the square of the spatial step, while maintaining spectral stability of the explicit scheme in the vorticity equation. The majority of computation time was spent solving the Poisson equation in the “stream function — vorticity” variables. By freezing the velocity field and solving only the vorticity dynamics equation, computation time was further reduced in the splitting method. The inverse matrix for solving the Poisson equation using a finite number of elementary operations were computed using the Msimsl library.
Results. Numerical solutions demonstrated the equivalence of the damping and acceleration methods for the initial velocity field at low Reynolds numbers (up to 3000). The equivalence of solutions obtained using the “stream function — vorticity” algorithm and the implicit iterated polyneutic recurrent method for accelerated initial conditions was numerically confirmed. For the first time, an initial horizontal velocity field was proposed, smooth at internal points and composed of two sine waves, with a stationary center of mass for the fluid in the rectangular cavity.
Discussion and Conclusion. An algorithm for numerically solving a two-dimensional hydrodynamic problem in a rectangular cavity using “stream function — vorticity” variables is proposed. The approximation of the equations in system (1) has sixth-order accuracy at internal grid points and fourth-order accuracy at boundary points. A novel damping method is introduced using an initial horizontal velocity field formed by smoothly connecting two sine waves. The proposed algorithms enhance the efficiency of solving hydrodynamic problems using an explicit finite-difference scheme for the vorticity equation

Introduction. The widespread use of technical systems with moving boundaries necessitates the development of mathematical modelling methods and algorithmic software for their analysis. This paper presents a systematic review of studies examining the oscillatory and resonance properties of mechanical systems with moving boundaries, such as hoisting cables, flexible transmission mechanisms, strings, rods, beams with variable length, and others.
Materials and Methods. A problem statement is formulated, and numerical methods are developed for solving nonlinear problems that describe wave processes and the resonance properties of systems with moving boundaries.
Results. An analysis is conducted on wave reflection from moving boundaries, including changes in their energy and frequency. It is shown that the energy of the system increases when the boundary moves toward the waves and decreases when moving in the same direction as the waves. Criteria are obtained to determine the conditions under which the boundary motion must be considered for accurate calculation of oscillation amplitudes. Numerical results demonstrate the influence of boundary speed and damping on the system dynamics.
Discussion and Conclusion. The findings have practical significance for the design and operation of mechanical systems with variable geometry. The results make it possible to prevent large-amplitude oscillations in mechanical objects with moving boundaries at the design stage. These problems have not been sufficiently studied and require further research

Introduction. Differential equations are often used in modelling across various fields of science and engineering. Recently, neural networks have been increasingly applied to solve differential equations. This paper proposes an original method for constructing a neural network to solve elliptic differential equations. The method is used for solving boundary value problems in domains with complex geometric shapes.
Materials and Methods. A method is proposed for constructing a neural network designed to solve partial differential equations of the elliptic type. By applying a transformation of the unknown function, the original problem is reduced to Laplace’s equation. Thus, nonlinear differential equations were considered. In building the neural network, the activation functions are chosen as derivatives of singular solutions to Laplace’s equation. The singular points of these solutions are distributed along closed curves encompassing the boundary of the domain. During the training process, the weights of the network are adjusted by minimizing the mean squared error.
Results. The paper presents the results of solving the first boundary value problem for various domains with complex geometries. The results are shown in tables containing both the exact solutions and the solutions obtained using the neural network. Graphical representations of the exact and the neural network-based solutions are also provided.
Discussion and Conclusion. The obtained results demonstrate the effectiveness of the proposed neural network construction method in solving various types of elliptic partial differential equations. The method can also be effectively applied to other types of partial differential equations

Introduction. This work addresses the scientific problem of studying natural-technological systems (NTS) of the Far North under conditions of climate change and anthropogenic impacts. The relevance of ensuring their stability is emphasized, which requires a comprehensive analysis of field data. Problems in automated processing methods of such specific data have been identified. The aim of the study is to develop automated methods for processing field data to reveal patterns. Python libraries for data analysis, processing, and visualization are used as tools.
Materials and Methods. The research object is described — the Main Building of the Yakutsk Thermal Power Plant (TPP) in permafrost conditions. The study materials include field data obtained from engineering-geological boreholes at the Yakutsk TPP, monitoring stations Chabyda and Tuymaada, as well as a section of the Amur-Yakutsk railway (AYR). The data include measurements of soil temperature and moisture, seasonal thaw layer dynamics, snow cover characteristics, and others. A detailed sequence of automated processing of primary data from XLS files using the pandas library is presented, including reading, cleaning, format conversion, filling or replacing missing values, removing duplicates, and saving processed data in CSV, JSON, and XLSX formats.
Results. Specific results of automated processing and systematization of primary field data are presented. Heterogeneous measurements were successfully unified into a single format, ensuring their proper use. A unique data array was formed based on empirical observations under the specific conditions of the Far North. The practical application of Python libraries for executing key stages of preprocessing and data preparation is demonstrated.
Discussion and Conclusion. It is shown that the application of a systematic approach and automated data processing significantly improves the quality and reliability of natural-technological system data analysis. Handling missing data and normalization enhance accuracy, and the final data formats are convenient for further modeling. The universality of Python is highlighted. Prospects for further research include applying machine learning, clustering, and modeling methods aimed at uncovering patterns and forecasting the behavior of natural-technological systems in the Far North under climate and anthropogenic influences