Introduction. This study investigates the possibility of increasing the accuracy of numerically solving boundary value problems using the modified Bubnov-Galerkin method with a linear ordinary differential equation, where the coefficients and the right-hand side are continuous functions. The order of the differential equation n must be less than the number of coordinate functions m.

Materials and Methods. A modified Petrov-Galerkin method was used to numerically solve the boundary value problem. It employs a system of linearly independent power-type basis functions on the interval [−1,1], each normalized by the unit Chebyshev norm. The system of linear algebraic equations includes only the linearly independent boundary conditions of the original problem.

Results. For the first time, an integral quadrature formula with a 22nd order error was developed for a uniform grid. This formula is used to calculate the matrix elements and coefficients in the right-hand side of the system of linear algebraic equations, taking into account the scalar product of two functions based on the new quadrature formula. The study proves a theorem on the existence and uniqueness of a solution for boundary value problems with general non-separated conditions, provided that n linearly independent particular solutions of a homogeneous differential equation of order n are known.

Discussion and Conclusion. The hydrodynamic problem in a viscous strong boundary layer with a third-order equation was precisely solved. The analytical solution was compared with its numerical counterpart, and the uniform norm of their difference did not exceed 5·10^‒15. The formulas derived using the generalized Bubnov-Galerkin method may be useful for solving boundary value problems with linear ordinary differential equations of the third and higher orders.

Introduction. Drilling mud losses are among the most common complications encountered during well drilling. Forecasting these losses is a priority as it helps minimize drilling fluid wastage and prevent wellbore incidents. Mud loss events are primarily influenced by the geological properties of the formations being drilled. Understanding the relationship between mud loss occurrences and the geological characteristics of the formations has both fundamental and practical significance. Given the complexity of predicting mud loss probabilities using traditional mathematical models, this study aims to develop a machine-learning-based system to predict the probability of mud losses based on well location and stratigraphic description.

Materials and Methods. Experimental data from 735 wells at the Shkapovskoye oil field, including well location coordinates, geological layer indices, and mud loss intensities, were prepared for computational analysis. The dataset was divided into training and testing subsets. The classification problem was addressed using four intensity classes with the following machine learning models: Decision Tree, Random Forest, and Linear Discriminant Analysis.

Results. Predictions generated by the three models were compared against the experimental data in the test set. The evaluation metrics included accuracy and recall. All three models achieved an average prediction accuracy of 91%. Linear Discriminant Analysis was identified as the most accurate model.

Discussion and Conclusion. High-accuracy predictions enable reliable forecasting of the probability and intensity of mud losses based on the location and stratigraphic description of new wells. The study presents three machine learning methods that demonstrated superior results in solving this problem.

Introduction. Mathematical tools integrated with satellite data are typically employed as the primary means for studying aquatic ecosystems and forecasting changes in phytoplankton concentration in shallow water bodies during summer. This approach facilitates accurate monitoring, analysis, and modeling of the spatiotemporal dynamics of biogeochemical processes, considering the combined effects of various physicochemical, biological, and anthropogenic factors impacting the aquatic ecosystem. The authors have developed a mathematical model aligned with satellite data to predict the behavior of summer phytoplankton species in shallow water under accelerated temporal conditions. The model describes oxidative[1]reduction processes, sulfate reduction, and nutrient transformations (phytoplankton mineral nutrition), investigates hypoxia events caused by anthropogenic eutrophication, and forecasts changes in the oxygen and nutrient regimes of the water body.

Materials and Methods. To simulate the population dynamics of summer phytoplankton species correlated with satellite data assimilation methods, an operational algorithm for restoring water quality parameters of the Azov Sea was developed based on the Levenberg-Marquardt multidimensional optimization method. The initial distribution of phytoplankton populations was obtained by applying the Local Binary Patterns (LBP) method to satellite images of the Taganrog Bay and was used as input data for the mathematical model.

Results. Using integrated hydrodynamic and biological kinetics models combined with satellite data assimilation methods, a software suite was developed. This suite enables short- and medium-term forecasts of the ecological state of shallow water bodies based on diverse input data correlated with satellite information.

Discussion and Conclusion. The conducted studies on aquatic systems revealed that improving the accuracy of initial data is one mechanism for enhancing the quality of biogeochemical process forecasting in marine ecosystems. It was established that using satellite data alongside mathematical modeling methods allows for studying the spatiotemporal distribution of pollutants of various origins, plankton populations in the studied water body, and assessing the nature and scale of natural or anthropogenic phenomena to prevent negative economic and social consequences.

Introduction. Many practically significant tasks reduce to nonlinear differential equations. In this study, one of the applications of neural networks for solving specific nonlinear boundary problems for complex-shaped domains is considered. Specifically, the focus is on solving a stationary heat conduction differential equation with a thermal conductivity coefficient dependent on temperature.

Materials and Methods. The original nonlinear boundary problem is linearized through Kirchhoff transformation. A neural network is constructed to solve the resulting linear boundary problem. In this context, derivatives of singular solutions to the Laplace equation are used as activation functions, and these singular points are distributed along closed curves encompassing the boundary of the domain. The weights of the network were tuned by minimizing the mean squared error of training.

 Results. Results for the heat conduction problem are obtained for various complex-shaped domains and different forms of dependence of the thermal conductivity coefficient on temperature. The results are presented in tables that contain the exact solution and the solution obtained using the neural network.

 Discussion and Conclusion. Based on the computational results, it can be concluded that the proposed method is sufficiently effective for solving the specified type of boundary problems. The use of derivatives of singular solutions to the Laplace equation as activation functions appears to be a promising approach.

Introduction. Detecting oil spills is a critical task in monitoring the marine ecosystem, protecting it, and minimizing the consequences of emergency situations. The development of fast and accurate methods for detecting and mapping oil spills at sea is essential for prompt assessment and response to emergencies. High-resolution aerial photography provides researchers with a tool for remote monitoring of water discoloration. Artificial intelligence technologies contribute to improving and automating the interpretation and analysis of such images. This study aims to develop approaches for identifying oil spilled on water surfaces using neural networks and machine learning techniques.

Materials and Methods. Algorithms capable of automatically identifying marine oil spills were developed using computer image analysis and machine learning methods. The U-Net convolutional neural network was employed for image segmentation tasks. The neural network architecture was designed using the PyTorch library implemented in Python. The AdamW optimizer was chosen for training the network. The neural network was trained on a dataset comprising 8,700 images.

Results. The performance of oil spill detection on water surfaces was evaluated using metrics such as IoU, Precision, Recall, Accuracy, and F1 score. Calculations based on these metrics demonstrated identification accuracy of approximately 83–88%, confirming the efficiency of the algorithms used.

Discussion and Conclusion. The U-Net convolutional network was successfully trained and demonstrated high accuracy in detecting marine oil spills on the given dataset. Future work will focus on developing algorithms using more advanced neural network models and image augmentation methods.