Introduction. This study investigates the possibility of numerically solving a boundary value problem with a nonlinear differential equation, continuous coefficients, and a right-hand side using the modified Bubnov-Galerkin method. In the problem formulation, the partial derivatives of the equationʼs coefficients are continuous functions of all arguments. The order of the nonlinear differential equation n is strictly less than the number of coordinate functions m.

Materials and Methods. To numerically solve the nonlinear boundary value problem, the modifi Petrov-Galerkin method and the uniqueness property of decomposing a smooth function into a system of linearly independent polynomial basis functions on the interval [−1,1] with a unit Chebyshev norm for each function in the system are used. The system of linear algebraic equations includes linearly independent boundary conditions. The matrix elements and the right-hand side of the system depend on the simple iteration index s. The coefficient  vector of the solution decomposition into basis functions also depends on the index s. The inverse matrix of the system was computed using the Msimsl linear algebra library in Fortran.

Results. Sufficient conditions for the existence and uniqueness of the solution to the boundary value problem with a nonlinear differential equation using the simple iteration method have been formulated. When the sufficient conditions are met, the decomposition coefficients decrease absolutely as the basis function index increases.

Discussion and Conclusion. Three boundary value problems with a second-order nonlinear equation and one problem with a third-order equation were solved exactly. The analytical solutions were compared with numerical solutions, with the uniform norm of the difference having an order of 10^‒13, 10^‒11, 10^‒10, 10^‒10, respectively. The modified Bubnov-Galerkin method allows for solving each branch of a multivalued function in boundary value problems with nonlinear differential equations.

Introduction. This paper examines a two-dimensional spatial model of multifractional sediment transport, specifically focusing on shallow water zones. This process can be described using an initial-boundary value problem for a parabolic equation with nonlinear coefficients. The study employs a temporal grid linearization method with a step size τ, where nonlinear coefficients are calculated with a “lag” at the previous time layer. Previously, the well-posedness conditions for the linearized sediment transport problem were established, and a conservative and stable finite-difference scheme was developed and analyzed, with numerical implementations for both model and real-world problems (the Sea of Azov, the Taganrog Bay, and the Tsimlyansk Reservoir). However, the convergence of solutions of the linearized problem to the solution of the original nonlinear initial-boundary value problem for multifractional sediment transport had not yet been explored. The research results presented in this paper fill this gap. Earlier, the author, together with A.I. Sukhinov, conducted similar studies in the case where sediment fraction composition was not considered. These studies formed the basis for obtaining new results.

Materials and Methods. The derivation of inequalities guaranteeing the convergence of the solutions of a sequence of linearized problems to the solution of the original nonlinear problem is carried out using the method of mathematical induction, with the application of differential equation theory.

Results. The conditions for the convergence of solutions of the linearized multifractional sediment transport problem to
the solution of the nonlinear problem in the Banach space L₁
 norm at a rate O(τ) of are determined.

Discussion and Conclusion. The obtained research results can be used for forecasting nonlinear hydrophysical processes, improving their accuracy and reliability due to the new functional capabilities that account for physically significant factors.

Introduction. The Azov Sea is a shallow semi-enclosed sea where satellite altimetry (SA) faces challenges in ensuring accurate sea level measurements. This study focuses on verifying Sentinel-3 altimetry data in the coastal areas of the Azov Sea using observational platform data and a three-dimensional hydrodynamic model.

Materials and Methods. The study is based on a comparison of sea surface heights (SSH) obtained from the Sentinel-3 radar altimeter with tide gauge data and modelling results. A three-dimensional hydrodynamic model, adapted to the conditions of the Azov Sea, was used, along with satellite data processed considering atmospheric and tidal corrections.

Results. The root mean square error (RMSE) between satellite-derived and reference data was found to be 85 mm. The analysis demonstrated that Sentinel-3 Doppler altimetry in SAR mode provides higher accuracy compared to traditional altimetry, particularly in coastal areas.

Discussion and Conclusion. The assessment of Sentinel-3 data confirms their reliability in modeling water levels in the Azov Sea. The comparative analysis methodology proposed in this study enables the identification of systematic errors in satellite data and facilitates their integration with modelling and in situ observations. The study confirms the effectiveness of Sentinel-3 data in determining sea levels in complex coastal conditions. The developed methodology can be applied to other coastal areas to assess satellite altimetry performance.

Introduction. In the modern development of intelligent transportation systems (ITS), an urgent task is the accurate estimation of the velocity limit of traffic flow on a highway. Despite existing solutions to this problem based on statistical mechanics methods and stochastic models, gaps remain in adapting these theories to real road segments of limited length. The traditional thermodynamic limit formula, used to calculate the average velocity of traffic flow, becomes inaccurate for small road segment lengths, limiting its applicability in practical traffic monitoring tasks. The aim of this study is a comparative analysis of various approaches to estimating the average velocity limit of traffic flow.

Materials and Methods. The study was conducted using the method of statistical mechanics and a stochastic model on a one-dimensional finite lattice. Numerical experiments with various parameter values (number of cells, traffic density, and movement probability) were used for analysis.

Results. The study revealed significant discrepancies between the results obtained using the statistical mechanics method and other approaches when the road segment length was small. The efficiency of the second and third approaches was confirmed for limited road segments, where they demonstrated greater accuracy and applicability.

Discussion and Conclusion. The research results have practical significance for the development of intelligent traffic management systems, especially for short road segments. The proposed approaches can be successfully integrated into modern monitoring systems to improve their accuracy. The theoretical significance of this work lies in advancing the methodology for traffic flow estimation while accounting for the specific conditions of real-world environments.

information from pilot charts. The relevance of this task is driven by the need to automate the processing of large volumes of cartographic data to create depth maps suitable for mathematical modelling of hydrodynamic and hydrobiological processes. The objective of this work is to develop the software tool LocMap, designed for the automatic detection and 52     recognition of depth values represented as numbers on pilot chart images.

Materials and Methods. The study employs deep learning methods, including convolutional neural networks (ResNet) for feature extraction, the Differentiable Binarization (DB) algorithm for text detection, and the Scene Text Recognition with a Single Visual Model (SVTR) architecture for text recognition.

Results. The developed software allows users to upload pilot chart images, perform preprocessing, detect and recognize depth values, highlight them in the image, and save the results in a text file. Testing results demonstrated that the system ensures high accuracy in recognizing depth values on pilot charts.

Discussion and Conclusion. The obtained results highlight the practical significance of the developed solution for automating the processing of pilot charts.