Introduction. A modified degenerate kernel method for solving Fredholm integral equations of the second kind is proposed for the first time. The main idea is to expand the integral kernel into a Taylor series with respect to a single variable x, rather than with respect to two variables x and s, as in the classical method.

Materials and Methods. The kernel expansion is performed at the midpoint of the integration interval, which reduces the absolute values of the elements of matrix C, as well as enlarges the nonsingularity region of the matrix I–λC. A system of power basis functions is employed on the integration interval. Three theorems establishing sufficient conditions for the correctness of the proposed algorithm based on the degenerate kernel method are obtained. A definition of the factorial Chebyshev norm of a vector-valued function is introduced. The factorial norm for the system of partial derivatives of the integral kernel with respect to the variable x, together with the parameter λ, is included in the inequality of the third theorem, which provides a sufficient condition for the correctness of the algorithm. The inverse matrix arising in the numerical solution was computed using the IMSL library within a finite number of elementary operations.

Results. The proposed numerical algorithm was tested on three Fredholm integral equations with kernels exhibiting exponential growth or periodic sign changes. The numerical solutions coincide with the exact solutions to 15 significant digits in the uniform metric.

Discussion. A modified algorithm for the numerical solution of Fredholm equations of the second kind with double precision is proposed. The solution is represented as the sum of n+1 power-type terms vanishing at the midpoint of the interval [a, b] and the right-hand side function of the Fredholm equation.

Conclusion. The degenerate kernel method is of interest for both functional analysis and numerical methods. The integral kernel must possess sufficient smoothness with respect to the variable x.

Introduction. In the context of increasing anthropogenic pressure and imbalance of nutrients, the development of phytoplankton dynamics models becomes particularly relevant. The proposed approach integrates the description of oxygen and carbon dioxide cycles, which is critically important for assessing the risks of cyanobacterial water “blooms” and reservoir deoxygenation.

Materials and Methods. An approach to constructing a comprehensive mathematical model is considered. Unlike existing analogues, this model simultaneously accounts for the dynamics of three functionally distinct phytoplankton groups with their specific preferences for nitrogen and phosphorus sources, as well as the complete silicon cycle, which is critically important for diatoms. The model integrates gas exchange processes, enabling the assessment of plankton dynamics’ impact on the oxygen regime of the aquatic environment and acidity, which are key indicators of ecosystem health.

Results. Stationary solutions were obtained for the problem of plankton dynamics, considering the transformation of nutrient compounds, suspended and dissolved substances, including oxygen and carbon dioxide. Depending on external conditions (light, temperature, input nutrient concentrations), several qualitatively different stable stationary states are possible.

Discussion. The obtained results can be used for forecasting the consequences of reservoir eutrophication and water “blooms”; assessing the seasonal succession of plankton communities; developing strategies to reduce anthropogenic load on water bodies; and evaluating the role of marine and freshwater ecosystems in CO₂ absorption and oxygen production.

Conclusions. Understanding the conditions under which the system reaches a particular stationary state (equilibrium) allows us to predict the long-term consequences of anthropogenic impact and develop effective management solutions.

Introduction. The hydrodynamic impact of waves on a ship hull in shallow water represents one of the key challenges in modern marine hydrodynamics. Limited water depth leads to amplification of wave action, a shift of resonance frequencies, and an increase in motion amplitudes, which significantly intensifies the loads acting on the hull.

Materials and Methods. Numerical simulations were performed using the ANSYS Aqwa software package within the framework of linear potential flow theory. A hybrid approach was implemented, combining frequency-domain analysis for determining Response Amplitude Operators (RAO) and subsequent time-domain simulations of irregular waves based on the JONSWAP spectrum. Various water depths, wave parameters (significant wave height and peak period), and wave incidence angles were considered.

Results. It was found that a decrease in water depth results in an increase in vertical motion amplitudes of up to 40% and a shift of resonance frequencies toward longer waves. Maximum hydrodynamic loads occur under beam wave conditions, whereas for following wave directions, pitch motion plays a significant role in redistributing loads along the hull. Quantitative estimates of pressure distribution and integral forces acting on the hull were obtained, and critical combinations of wave parameters and water depth were identified.

Discussion. It is shown that the formation of extreme loads is governed not only by wave height but also by spectral characteristics, primarily the coincidence of the peak wave period with the natural frequencies of ship motions. The significant role of shallow-water effects in amplifying the hydrodynamic response and altering the structure of the wave field near the hull is demonstrated.

Conclusion. The results expand current understanding of ship hydrodynamics in shallow water and can be applied in ship design, selection of operational regimes, and development of decision-support systems to ensure navigation safety under complex environmental conditions.

Introduction. This paper examines a methodology for the automated extraction and graphical representation of knowledge from unstructured texts using modern language models. Such methods are becoming increasingly relevant because of the growing need to structure information and identify semantic relations that are difficult to capture manually.

Materials and Methods. The proposed approach combines locally deployed language models with specialized relationextraction tools. Local deployment enables data to be processed in a secure environment without reliance on external services. The methodology includes text preprocessing, entity and relation extraction, structuring, and visualization of the resulting knowledge graphs.

Results. Experimental testing on a corpus of Russian-language scientific articles demonstrated that the approach is applicable both to technical descriptions and to texts containing more abstract concepts. The developed web interface supports interactive visualization and comparative analysis of graphs constructed by different models, thereby improving the interpretability of the results. The approach is robust to textual noise and is applicable to scientific, technical, and regulatory tasks.

Discussion. The results show that the proposed methodology is not limited to a single algorithm and permits the combination of direct extraction, specialized models, multi-stage pipelines, and OWL ontologies. The quality of the resulting graphs depends substantially on the structure of the source text, preprocessing accuracy, and the selected postprocessing procedures; interactive visualization facilitates comparison of outputs generated by different models and supports the interpretation of semantic relations.

Conclusions. The proposed approach can be applied to the analysis of scientific, technical, and regulatory texts in a secure local environment. It is a natural continuation of the authors’ previous research on semantic-associative data analysis and synthesis and the associative-ontological approach. Further development should focus on ensemble schemes, logical validation, semantic inference, and integration with formal ontologies, thereby extending its applicability to information retrieval, research support, and complex-system modelling.