Problems of Computational and Applied Mathematics
ISSN 2181-8460 eISSN 2181-046X Problems of Computational and Applied Mathematics (PCAM - in uzbek Hisoblash va amaliy matematika muammolari) with ISSN: 2181-8460 (print medium) and 2181-046X (online medium), is a peer-reviewed open-access scholarly periodical scientific journal. Despite the fact that the journal is relatively new, it inherits a fairly long and rich history of the collection of scientific papers titled “Issues of Computational and Applied Mathematics” which was published by Institute of Cybernetics of the Academy of Sciences of Uzbekistan since 1970. Published by Laboratory Modelling of complex systems of the Digital Technologies and Artificial Intelligence Development Research Institute, Tashkent, Uzbekistan, PCAM focuses on the development and application of modern methods of mathematical modeling, computational mathematics, and numerical methods to solve current applied problems. Particular attention is paid to artificial intelligence, data mining, inverse and ill-posed problems, as well as parallel and distributed computing and modern programming technologies. Our commitment to interdisciplinary approaches aligns with the United Nations Sustainable Development Goals, particularly those in the areas of quality education, decent work and economic growth, industrial innovation, and responsible consumption and production. Many scientists from Uzbekistan and foreign countries published their researches in that collection. In 1983 each issue of the collection became thematic and contained articles related to one general topic. There were published 130 issues of the collection in total. Results of scientific researches in the fields of computational mathematics, mathematical physics, algorithmization, discrete mathematics, continuum mechanics, mathematical modeling and other areas of mathematics and its applications were widely presented within the collection. The total number of scientific papers published in the collection was about 2000. The scientific articles published in the collection were highly evaluated by scientists from foreign countries, and to this day they are indexed in well-known scientific journals, including “Mathematical Reviews” ”, “Zentralblatt MATH”, as well as in the Russian abstract journals “Mathematics”, “Informatics” and “Mechanics”. In 2015, the collection of scientific papers “Problems of Computational and Applied Mathematics” was turned into this periodical scientific journal “Problems of Computational and Applied Mathematics”. To date, more than 50 issues have been published. Since 2016, the journal has been included in the list of journals recommended by The Supreme Attestation Commission of the Republic of Uzbekistan, and since 2018 - in the list of peer-reviewed journals of The Supreme Attestation Commission under the President of the Republic of Tajikistan. The journal is also registered in the RSCI database. The journal team members are making efforts to get the journal indexed in international bibliometric databases such as Scopus, WoS etc. The journal team hopes that this journal will continue the glorious history of its predecessor in the coverage, dissemination and popularization of scientific achievements. We invite authors to actively participate in the formation of the next issues, and especially young scientists carrying out research in the scientific direction of the journal. We thank you for your attention and look forward to further fruitful cooperation. Editor-in-Chief: Normakhmad Ravshanov - Advisor to the Director for Science of the Research Institute for the Development of Digital Technologies and Artificial Intelligence, Doctor of Technical Sciences, Professor, Uzbekistan. Deputy Editor: Abdulla Azamov - Head of the Department of Applied Mathematics of the National University of Uzbekistan named after. M. Ulugbek, Doctor of Physical and Mathematical Sciences, Professor Uzbekistan. Mirsaid Aripov - professor at National University of Uzbekistan named after M. Ulugbek, Doctor of Physical and Mathematical Sciences, Professor Uzbekistan. Kholmatvay Shadimetov - professor at National University of Uzbekistan named after M. Ulugbek, Doctor of Physical and Mathematical Sciences, Professor Uzbekistan. Executive Secretary: Dilshot Akhmedov - senior researcher at the Research Institute for the Development of Digital Technologies and Artificial Intelligence, PhD in Technical Sciences, Associate Professor, Uzbekistan.
Научно-исследовательский институт развития цифровых технологий и искусственного интеллекта
Graph Generation with a Prescribed Structure: From Deep Neural Networks to Quantum Models (A Case Study of Novel Drug Design)
The discovery of new chemical compounds with specified properties is a challenging problem in drug development. Many studies encode molecules as string representations derived from molecular graphs; however, this approach is computationally expensive and does not readily extend to general (non-molecular) graphs. Advances in graph deep learning make it possible to train generative models directly on graph representations, avoiding costly search in the discrete and extremely large space of chemical structures. MolGAN is a family of implicit models for generating small molecular graphs that combines generative adversarial networks (GANs) with reinforcement learning (RL) to produce molecules with target chemical properties. This review considers four variants: the baseline MolGAN (up to 9 atoms), Large MolGAN (up to 20 atoms) with a graph-expansion mechanism that reduces the generation of disconnected graphs, a MolGAN variant based on WGAN as a more stable alternative to standard GAN training, and a hybrid MolGAN incorporating quantum computing modules. The paper describes the model architectures, compares their performance on established benchmarks, and discusses limitations and directions for future research.
Copyright (c) 2026 F.T. Adilova https://creativecommons.org/licenses/by/4.0
Numerical Modeling of the Boundary Value Problem for a two-Parameter Singularly Perturbed Differential Equation using the Spectral-Grid Method
The solutions of boundary value problems associated with two-parameter singularly perturbed differential equations are well known to exhibit the formation of two distinct boundary layers, typically occurring near the endpoints of the domain. The presence of these narrow regions, characterized by steep gradients in the solution, poses significant challenges for classical numerical techniques. As a result, standard finite difference, finite element, or low-order discretization methods often fail to capture the layer behavior accurately unless excessively fine meshes are employed, leading to increased computational cost and reduced efficiency. To address these difficulties, the present work proposes an efficient spectral-grid method for the numerical solution of boundary value problems involving two-parameter singular perturbations. The core idea of the proposed approach is to combine the high accuracy of spectral methods with a carefully designed grid that adapts to the boundary layer structure of the solution. By employing this spectral-grid framework, the original boundary value problem is transformed into an equivalent system of algebraic equations, which can be solved efficiently using standard linear algebra techniques. Extensive numerical experiments are carried out to assess the performance of the proposed method. The computed results are systematically compared with those available in the existing literature. These comparisons clearly demonstrate that the spectral-grid method developed in this study achieves superior accuracy while maintaining a relatively low computational cost.
two-parameter singularly perturbed differential equation
Copyright (c) 2026 S.K. Murodov https://creativecommons.org/licenses/by/4.0
Solution of the Dirichlet Problem by the Moving Node Method
The method of displaced nodes is applied when solving the Dirichlet problem for the Poisson equation in a rectangular domain. By approximating the Laplace operator using moving nodes, we obtain an approximate analytical solution to the Dirichlet problem, i.e., a solution expressed as a combination of analytic functions. Further improvement of accuracy is achieved by applying the method of lines. Within this approach, spatial variables are discretized using the moving nodes method, resulting in the original boundary value problem for the Poisson equation in a two-dimensional domain being transformed into an ordinary differential equation (ODE) - a two-point boundary value problem where the independent variable becomes one of the coordinates (e.g., ????), and for another coordinate (????). Each of the obtained ODEs describes the behavior of the solution along a line parallel to one of the axes and contains boundary conditions given on the opposite sides of the rectangular domain. Thus, the combination of the method of displaced nodes with the method of lines allows us to move from a two-dimensional problem to a one-dimensional one. To verify the effectiveness of the proposed approach, test problems with analytically given solutions were considered. The results confirm that the proposed approach ensures increased solution accuracy.
Copyright (c) 2026 U. Dalabaev https://creativecommons.org/licenses/by/4.0
Finding the Extremum of the Error Functional in the Space of Periodic Functions
The problem of constructing optimal quadrature formulas for approximate calculation of definite integrals and approximation of functions is one of the important problems of computational mathematics. These problems have been studied by many mathematicians and there are several methods for constructing optimal quadrature formulas. One of these methods is the method proposed by S.L. Sobolev with the concept of an extremal function of the error functional. This article is devoted to finding the main elements necessary for the process of constructing an optimal quadrature formula for real-valued periodic functions in a Hilbert space. The quadrature formula under consideration is a form of the norm of the error functional for obtaining the upper bound of the error. In turn, to find the form of the norm of the error functional, we need an extremal function corresponding to the error functional. This work is aimed at finding the extremal function of the error functional.
Copyright (c) 2026 S.S. Azamov https://creativecommons.org/licenses/by/4.0
On the Solvability of the Pursuit Problem for Differential Games with Fractional Hilfer Derivatives
In the early 1960s, seminal results in differential game theory, where the game is described by an ordinary differential equation in a finite-dimensional space, were obtained by academicians L.S. Pontryagin and N.N. Krasovsky. In the 21st century, differential games described by fractional differential equations have been actively studied. In recent years, fractional calculus has gained a strong position in the mathematical modeling of physical, economic, and applied problems. Numerous examples demonstrate the successful application of ordinary differential equations and partial differential equations with fractional derivatives. In particular, the works of A.A. Chikrii, M.Sh. Mamatov, and E.M. Mukhsinov consider the pursuit problem when the game is described by fractional differential equations. In this paper, we investigate the solvability of the pursuit problem for a differential game with Hilfer fractional derivatives in a Banach space of order ????, 0 < ???? < 1, and type ????, 0 6 ???? 6 1. Using Pontryagin’s first method and the strict separation theorem, we prove two theorems providing sufficient conditions for the solvability of the pursuit problem and for the optimality of the pursuit time. A game problem describing the relaxation process during glass formation in supercooled liquids is solved.
differential games with Hilfer fractional derivatives
Copyright (c) 2026 E.M. Mukhsinov https://creativecommons.org/licenses/by/4.0
Numerical Modeling of Plane Elastoplastic Problems in Strains
In this article, within the framework of the Saint-Venant compatibility conditions, plane problems of plasticity theory formulated in terms of strains are presented, aimed at investigating the stress-strain state of a rectangular plate under elastoplastic deformation. The numerical implementation of the formulated problem is carried out using the finite difference method, on the basis of which grid equations for the displacement and strain components at the nodes of the rectangular computational domain are constructed. Spatial discretization enables the transition from the differential form of the boundary-value problem to a system of algebraic finite-difference equations suitable for stable and efficient numerical solution. In the development of the finite-difference scheme, boundary conditions of various types are taken into account, adequately reflecting real loading regimes and fixation conditions of the plate. The system of finite-difference equations with respect to strains is solved using the alternating direction method, which ensures an increase in computational efficiency and improved convergence when solving problems with pronounced nonlinearity. The application of this method contributes to reducing approximation errors while maintaining high accuracy of the obtained results. A detailed comparison of numerical solutions obtained in formulations based on displacements and strains under identical boundary and loading conditions is performed. The analysis demonstrates their qualitative and quantitative agreement, manifested in the close correspondence of strain and stress distributions throughout the entire domain of the rectangular plate.
Copyright (c) 2026 U.Z. Djumayozov https://creativecommons.org/licenses/by/4.0
Numerical Modeling of the Geofiltration Process on Irrigated Lands Taking into account Physical Factors
This article examines the theoretical foundations and practical application of a mathematical model for forecasting groundwater movement and levels based on the analysis of modern literature. The mathematical model for determining the dynamics and levels of groundwater in irrigated agricultural lands has been improved based on Darcy’s law and convection-diffusion equations. The proposed model determines the velocity, direction, and concentration of groundwater, taking into account convection and diffusion processes, and also allows for more accurate prediction of groundwater level changes, taking into account water salinity, temperature, soil saturation, and other hydrogeological parameters, making the model effective not only for modeling water flow dynamics but also for optimizing irrigation systems and reducing environmental risks. This work makes a significant contribution to the development of an integrated management mechanism for irrigation systems and water resources not only from a theoretical but also from a practical point of view.
achieving maximum concentration of the medium
Copyright (c) 2026 B. Murodullaev https://creativecommons.org/licenses/by/4.0
Properties of Solutions to Systems of Heat Conduction Equations with Nonlinear Boundary Conditions
This article is devoted to the study of the properties of solutions of systems of heat conduction equations associated with nonlinear boundary conditions, to the construction of self-similar solutions and finding their asymptotics, as well as the construction of numerical solutions. Based on self-similar analysis, the conditions for the global nature of solutions over time and the formation of unbounded solutions were found. In particular, the values of the critical exponent of the Fujita type and the critical exponent of the globality of the solution were found. For the solutions of the system of heat conduction equations, associated with nonlinear boundary conditions, lower and upper estimates were obtained. It is also proposed to choose the initial approximation for the iterative process when solving a system of heat conduction equations associated with nonlinear boundary conditions.
system of nonlinear heat conduction equations
Copyright (c) 2026 А.Р. Зарипова https://creativecommons.org/licenses/by/4.0
Numerical Modeling of Filtration and Transport Processes in a Cylindrical Porous Filter Using the Finite Volume Method
This paper explores the numerical modeling of flow during filtration and transport of dissolved components in a cylindrical porous medium. The mathematical formulation is based on the axisymmetric Brinkman-Darcy equations describing hydrodynamic processes, as well as a system of equations for convective-diffusion transport taking into account adsorption on a solid matrix. The permeability of the porous medium is specified using the Kozeny-Carman model. The numerical solution is implemented using the finite volume method, which guarantees the fulfillment of conservation laws in each control volume. A computational algorithm for determining the distributions of pressure, velocities, and component concentrations is developed. The influence of porous structure parameters and adsorption kinetics characteristics on the efficiency and characteristics of the filtration process is studied.
Copyright (c) 2026 B.I. Boborakhimov https://creativecommons.org/licenses/by/4.0
The Occurrence of the Phenomenon of Elastic Return During Unsteady Flow of a Rheologically Complex Fluid in a Flat Channel within the Oldroyd-B Model
This article examines unsteady flows of rheologically complex fluids in a flat channel using the Oldroyd-B model after the actuator is switched off. It is assumed that until the actuator is switched off, the fluid flow is steady under a constant pressure gradient. From a given point in time, the pressure gradient in the system is assumed to be zero. The problem is solved analytically using the Laplace transform. Based on the solutions obtained, the corresponding hydrodynamic regularities are determined, which are important for technical and technological processes.
Copyright (c) 2026 D.S. Yakhshibaev https://creativecommons.org/licenses/by/4.0
Mathematical Modeling of the Dispersion of Harmful Substances Released into the Atmosphere in Complex Urban Environments
This study addresses the mathematical modeling of the dispersion process of harmful substances released into the atmosphere in complex urban environments. A mathematical model has been developed to determine the spatial-temporal distribution of pollutant concentrations based on the advection-diffusion equation. The model accounts for the transport of substances by wind flow, turbulent diffusion, gravitational settling, and absorption by surfaces. Mathematical expressions for point, line, and area pollution sources have been presented. Parameters accounting for traffic congestion and diurnal variations have been introduced. A numerical solution algorithm based on the finite difference method has been developed. Computational experiments have revealed the patterns of harmful substance dispersion in urban environments.
Copyright (c) 2026 N. Ravshanov https://creativecommons.org/licenses/by/4.0
On Optimal Iterative and Direct Methods for Solving the Dirichlet Problem for the Poisson Equation
For the numerical solution of the Dirichlet problem for the Poisson equation, both direct and iterative methods have been developed. However, the required number of arithmetic operations for direct methods, as well as the number of iterations in iterative methods, often turns out to be very large. For this reason, the issues of high accuracy and efficiency of various methods remain relevant. In this work, a new high-accuracy and efficient method is proposed for the numerical solution of the Dirichlet problem for the Poisson equation – a discrete version of the preliminary integration method, which significantly surpasses existing direct and iterative methods in terms of the number of arithmetic operations. The efficiency of the proposed method is illustrated by tabular and graphical results.
a discrete version of the pre-integration method
Copyright (c) 2025 Sh.A. Ziyakulova https://creativecommons.org/licenses/by/4.0
Optimization of Approximate Computation of Integrals of Rapidly Oscillating Functions in the Sobolev SPACE OF Complex-Valued Functions
Integrals of rapidly oscillating functions arise mainly in the theory of special functions and Fourier analysis, but also in other applied and computational sciences and engineering, for example, in theoretical physics, acoustic scattering, quantum chemistry, transport theory, electromagnetism, telecommunications, mechanics, and so on.The computation of integrals of rapidly oscillating functions is often carried out using the Filon method. The Filon method resembles Simpson’s quadrature formula. However, whereas in Simpson’s method the entire integrand is approximated by a parabola, in the Filon method only the function f(x) is approximated by a parabola. In this way, Filon derived a quadrature formula with coefficients that depend onw. In this work, optimal quadrature formulas in the Sobolev space of complex-valued periodic functions will be constructed for the approximate computation of rapidly oscillating integrals.
discrete analogues of differential operators
Copyright (c) 2025 Kh.M. Shadimetov https://creativecommons.org/licenses/by/4.0
An Optimal Method for the Approximate Solution of the Hypersingular Integral Equations
Hypersingular integral equations are encountered in a number of fields, such as the dynamics of air and fluid flow, elasticity, and wave propagation theory. Analytical solutions for these integral equations exist, but the solutions themselves are also expressed through singular integrals. Therefore, it becomes necessary to develop formulas for the approximate calculation of these solutions, and such formulas have been developed. Among these formulas, very few are highly accurate. However, they are not optimal. Our work is devoted to constructing an optimal quadrature formula for the approximate calculation of the analytical solutions of a hypersingular integral equation.
Copyright (c) 2025 D.M. Akhmedov https://creativecommons.org/licenses/by/4.0
Numerical Solution of Plane-Radial Boundary Value Inverse Problem for the Equation of Non-Stationary Relaxation Filtration of Fluid in a Porous Medium
The paper addresses a numerical solution of a plane–radial boundary inverse problem for the relaxation filtration equation describing fluid flow in an elastically deformable porous medium. The study is motivated by the widespread occurrence of relaxation filtration in hydrogeology, oil and gas production, and subsurface hydromechanics, where reliable identification of medium parameters and reconstruction of unknown boundary actions are essential. Such boundary inverse problems are ill-posed: small perturbations in the input data may cause large deviations in the recovered solution, which makes stable and accurate numerical techniques crucial. The inverse problem is solved using De Souza’s marching method; however, computational experiments show that its accuracy strongly depends on the distance between the measurement point providing the “initial data” and the target boundary. As this distance increases, the error grows due to error accumulation and the intrinsic instability of boundary reconstruction. To improve stability and reduce errors, smoothing splines are employed. The spline approximation effectively suppresses high-frequency noise and stabilizes the recovery of boundary values. As a result, more stable numerical solutions with acceptable accuracy are obtained even under substantial data errors, demonstrating the promise of combining marching methods with smoothing procedures for ill-posed boundary inverse problems of relaxation filtration.
Copyright (c) 2025 D.Sh. Turaev https://creativecommons.org/licenses/by/4.0
Design and Operating Principle of a Cogenerative Fractal Solar Collector
For the first time, a fractal solar collector operating in the cogeneration mode is proposed. In this context, we are talking about a modern innovative system that uses fractal structures to optimize the processes of collecting, distributing and converting solar energy simultaneously into thermal and electrical energy. The detailed design and results of the conducted experiment of this system are given. The stages of the experiment are shown in order to gain a deeper understanding of the principles of operation of a collector operating in the cogeneration mode. The high efficiency and promising application of this system in the field of renewable energy have been experimentally proven. In addition, the results obtained demonstrate the possibility of using the proposed fractal collector as an alternative energy source, which opens up new ways to develop energy-efficient and environmentally friendly technologies of the future.
Copyright (c) 2025 N. Rustamov https://creativecommons.org/licenses/by/4.0
Model and Algorithms for Classifying Anomalous Phenomena based on the Convergence of Acoustic-Visual Signals
This paper proposes a Context-adaptive Audio-Visual Neural Network (CAVN) model for anomaly detection in public safety systems. Existing approaches primarily rely on visual data and employ simple fusion strategies for combining modalities, which leads to limitations in capturing complex semantic relationships. The proposed model consists of four main components: a visual feature extraction module based on SlowFast architecture, an audio feature extraction module based on Audio Spectrogram Transformer (AST), a fusion module based on bidirectional cross-attention mechanism, and a temporal context aggregation module based on Transformer encoder. The main scientific novelty of the model lies in the adaptive modality balancing mechanism, which dynamically adjusts the relative importance of modalities under different conditions (dark/bright, noisy/quiet). Experimental results demonstrate that the proposed CAVN model outperforms existing methods by in overall accuracy and by in dark conditions. Ablation studies confirmed the contribution of each module to the overall performance.
Copyright (c) 2025 B.I. Boborakhimov https://creativecommons.org/licenses/by/4.0
Regularization of the Initial-Boundary Value Problem for a Inhomogeneous Parabolic Equation with Changing Time Direction
In this paper, the Cauchy problem with boundary conditions for an inhomogeneous parabolic equation with changing time direction is investigated. To construct the solution, the method of separation of variables (Fourier method) is employed, leading to a spectral problem where the eigenvalues and eigenfunctions are determined. An a priori estimate for the solution is obtained, theorems of uniqueness and conditional stability in the correctness set are proved. A regularized solution corresponding to the approximate initial data is constructed. The regularization parameter is chosen based on the efficiency estimate derived from the norm of the difference between the exact and the regularized solutions. Numerical experiments were performed for several cases of given data, and the results are presented in tables and graphs. The results show that the regularized solution corresponding to the approximate data closely regularized solution for the exact data, confirming the robustness of the method to errors in the input data.
parabolic equation with changing time direction
Copyright (c) 2025 И.О. Хажиев https://creativecommons.org/licenses/by/4.0
Modeling the Spatio-Temporal Dynamics of a Reservoir Area (using the Kattakurgan Reservoir as an example) based on NDWI, NDVI, EVI Indices and Ensemble Learning Methods
The paper models the spatiotemporal dynamics of the Kattakurgan Reservoir area using satellite-based water and vegetation indices combined with ensemble machinelearning methods. Multi-year time series of NDWI, NDVI, and EVI are used together with meteorological variables (air temperature, humidity, precipitation) from the ERA5 and CHIRPS datasets on the Google Earth Engine platform for 2018–2023. The reservoir water-surface area, derived from satellite imagery, serves as the target variable for training and validating a Random Forest model. The study includes a train–test split, hyperparameter tuning, and performance evaluation using R2, RMSE, and MAE. The results indicate high accuracy: R2 = 0.962 for NDWI prediction, and values close to 1 for reservoir area, with RMSE about 5–7 km2 and MAE 3–5 km2. Forecasts of NDWI and reservoir area for 2024 are produced to assess expected changes in water extent and support water-management decisions and climate adaptation planning.
Copyright (c) 2025 N. Ravshanov https://creativecommons.org/licenses/by/4.0
Numerical Simulation of the Coupled Dynamic Problem of Thermoelasticity in Stresses
This article proposes a dynamic model of a coupled thermoelasticity problem under stress. A coupled boundary value problem is formulated, consisting of three differen tial equations for the stress and temperature tensor components, as well as a heat flux equation with corresponding initial and boundary conditions. Explicit and implicit finite difference equations are developed, solved by successive application of the sweep method along the coordinate axes and recurrence relations, respectively. The coupled dynamic thermoelasticity problem for an anisotropic rectangle under stress is solved numerically. A comparison of the results of explicit and implicit difference schemes demonstrates the reliability of the obtained numerical results and the validity of the proposed coupled dynamic boundary value problem of thermoelasticity under stress.
Copyright (c) 2025 A.A. Kalandarov https://creativecommons.org/licenses/by/4.0
Three-Dimensional Mathematical Model and Numerical Solution Algorithm for Monitoring and Predicting In-Situ Leaching Processes in Porous Medium
The article analyzes the hydrodynamic processes associated with underground mining, in particular, the acid extraction of precious metals from ore deposits. In order to comprehensively study, monitor and predict the behavior of the object, a mathematical model (MM) based on the filtration-convection and diffusion processes characteristic of underground fluid filtration was developed. This model includes the influence of various hydrodynamic parameters, in particular, the filtration coefficient and average porosity, which are functions of the pressure level and process kinetics. Analysis of the problem statement shows that the change in pressure in the ore deposit resulting from the pouring and extraction of the solution directly affects the permeability and porosity coefficients of the layer. The experimental results showed that the change in hydrodynamic parameters was proportional to the change in pressure, with an exponential behavior observed under high pressure and a linear behavior under low pressure. It should be noted that in the process of in-situ leaching (ISL), a chemical reaction occurred as a result of the reagent’s effect on the ore deposits, and the substance passed from one phase to another, as a result of which the hydrodynamic parameters of the pore medium (filtration and porosity coefficients) and pressure changes in the ore reservoir were observed.
Design and Analysis of Solar Water Supply System for Multi-Story Residential Buildings based On Boolean Programming
This paper presents the design and analysis of an innovative solar-powered water supply system for multi-storey residential buildings under conditions of high solar insolation and limited energy resources. Instead of the conventional approach where each apartment uses an individual low-power pump, a centralized architecture is proposed based on surface multistage pumps powered directly by DC photovoltaic modules. This enables operation without grid electricity and significantly reduces operating costs. Reliability is ensured through a duty-rotation strategy and ???? + 1 redundancy: two pumps operate simultaneously while one remains on hot standby. Water is stored in rooftop tanks with a total volume of 15–20 m3, providing stable supply during short-term fluctuations in solar irradiance. The mathematical model is formulated as a Boolean programming problem that captures pump on/off logic and discrete constraints on tank filling levels; its objective function combines energy consumption, annual operating costs, and a reliability metric, enabling multi-objective optimization. A numerical experiment for a four-storey building in Jizzakh with a daily demand of 34 m3 shows that the proposed system can reduce energy consumption by more than 95%, cut annual costs from 11.6 million to less than 0.3 million UZS, and improve reliability due to optimal pump rotation. The approach has strong practical value and can be adapted to diverse operating conditions, opening prospects for broad deployment in municipal infrastructure across regions with developing energy systems.
Copyright (c) 2025 R.Kh. Khamdamov https://creativecommons.org/licenses/by/4.0
FrOsFHN Software Package for Quantitative and Qualitative Analysis of the FitzHugh-Nagumo Fractional Oscillator with Variable Memory
This article describes and provides examples of the FrOsFHN software package for the quantitative and qualitative analysis of a nonlinear FitzHugh-Nagumo oscillator with variable memory (the fractional FitzHugh-Nagumo oscillator (FrOsFHN)). Variable memory was taken into account in the FitzHugh-Nagumo oscillator model equation using fractional-order derivatives of the Gerasimov-Caputo type with respect to time. The quantitative analysis in the software package was implemented based on a numerical algorithm for a nonlocal explicit finite-difference scheme of first-order accuracy, and the qualitative analysis was implemented using algorithms for visualizing the numerical solution in the form of oscillograms, phase trajectories, and bifurcation diagrams. To conduct the quantitative analysis, the user was able to select the functional dependence of the fractional derivative orders on time: a linear decreasing function, a trigonometric function, or a decreasing exponential function. To construct bifurcation diagrams, we implemented the ability to consider a parallel version of the algorithm in addition to the sequential version, leveraging the computing resources of the computer’s central processor. The software package also features the ability to save calculations to a text file, as well as result graphs. The software package was written in Python using the PyCharm environment.
nonlocal explicit finite-difference scheme
Copyright (c) 2025 R.I. Parovik https://creativecommons.org/licenses/by/4.0
A Corpus-Based Model of the Middle Turkic Language using Weighted Averaging
This paper formalizes and extends the classic concept of the "Middle Turkic"language (Karimov–Mutalov, 1992) by applying tools from corpus and computational linguistics. The proposed model frames the selection of an "optimal"form for a linguistic unit as a multi-criteria optimization problem over a set of factors, including frequency, prevalence, simplicity, cultural compatibility, and a matrix of interlingual mutual intelligibility. To ensure stability and interpretability, the model introduces soft (softmax) and "saturating" weights, domain and diachronic modifiers, and inter-language "fairness"constraints. The model achieves up to 89% accuracy in linguistic form selection, a 78% intelligibility level, high grammatical consistency (0.92), and a high assimilation rate (0.85). The paper concludes by discussing the model’s potential applications in machine translation, its limitations, and directions for future validation.
This paper presents a method of semantic modeling based on representing complex objects and phenomena as informational aggregates composed of core, designating, characterizing, and associative entities. Each entity is described by a set of attributes, and their interactions are governed by semantic operation algorithms that generate new informational structures. This approach enables a transition from formal syntactic data representation to meaningful semantic interpretation. The proposed method can be applied to the development of expert and cognitive systems that perform machine understanding and semantic information processing within the field of artificial intelligence.
Algebraic-Trigonometric Optimal Formulas for Numerical Integration
In this article, we present a detailed study on the determination of the coefficients of a quadrature formula that involves derivatives, making use of the so-called the phi – function method. This method provides a systematic framework for constructing optimal quadrature formulas in the context of approximate integration. In particular, the phi – function approach not only simplifies the process of finding the required coefficients, but also ensures that the resulting formulas achieve a high degree of accuracy. Furthermore, the functional error associated with the constructed quadrature formula is carefully analyzed. The error analysis is supported by precise mathematical expressions, which confirm the reliability and effectiveness of the derived formula. By considering quadrature formulas with arbitrarily fixed nodes, the optimality conditions are rigorously examined, and the procedure for determining the corresponding components and coefficients is described in detail. As a key outcome of this investigation, explicit analytical expressions for the coefficients of the optimal quadrature formula are successfully obtained. Of particular interest is the case of equally spaced nodes, for which the derived quadrature formula naturally reduces to a classical Euler-Maclaurin type formula, demonstrating both the generality and the practical significance of the phi – function method.
Copyright (c) 2025 А.Р. Хаётов https://creativecommons.org/licenses/by/4.0
Interior Boundary Value Problem for a System of Second-Order Mixed-Type Equations
Mathematical models of many applied problems lead to the need to solve interiorboundary value problems for partial differential equations. The problem under study belongs to the class of ill-posed problems in mathematical physics. The issue of small denominators arises; the existence and uniqueness of the solution depend on the numerical properties of the problem data. Theorems on the uniqueness of the solution and its conditional stability on the well-posedness set are proven, a priori estimates for the solution are obtained, and approximate solutions are constructed using the regularization method. The development of approximate methods for their solution is based on the construction and analysis of numerical methods for solving problems with interior and boundary data for the basic (fundamental, model) equations of mathematical physics.
Copyright (c) 2025 K.S. Fayazov https://creativecommons.org/licenses/by/4.0
An Optimal Quadrature Formula Exact to the Exponential Function
Numerical integration of definite integrals plays a crucial role in both fundamental and applied sciences. The precision of approximate integral calculations depends on the initial data and specific conditions, which impose various requirements on the resulting computations. Traditional methods for numerical analysis of definite integrals, such as the quadrature formulas developed by Gregory, Newton-Cotes, Euler, Gauss, Markov, and others, are well-established. Since the mid-20th century, the theory of creating optimal formulas for numerical integration through variational methods has evolved. It is important to mention that optimal quadrature formulas exist in the sense defined by Nikolsky and Sard. This paper focuses on the challenge of constructing an optimal quadrature formula according to Sard’s approach. In this process, the method of phi-functions is applied. The error of the formula is estimated by integrating the square of the phi function from a specific Hilbert space. Afterward, the appropriate function is chosen such that the integral of its square within this interval is minimized. Finally, the coefficients of the optimal quadrature formula are determined using the resulting phi function.
Algorithm for Solving a Two-Level Game Problem of Dynamic System Trajectory Transfer
The game problem of transferring the trajectory of a dynamic system controlled by two players whose goals are opposite, from any point of a given set to a terminal set is considered. The given set is described by the control parameter of the first player. The first player chooses his control first, then, knowing the first player’s control, the second player chooses his control. The goal of the first player is to choose a control to prevent the trajectory of the dynamic system from entering the terminal set under any control of the second player, and the goal of the second player is to choose a control to transfer the trajectory of the dynamic system to the terminal set under any control of the first player. If the second player manages to transfer the system trajectory to the terminal set, then the additional goal of the first player is to maximize the minimum functional for the second player’s control, and the additional goal of the second player is to minimize the linear functional. An algorithm for solving the problem under consideration has been developed and substantiated. An illustrative example is given.
Copyright (c) 2025 A.R. Mamatov https://creativecommons.org/licenses/by/4.0
Numerical Modeling of Liquid Solution Filtration in a Cylindrical Porous Filter
A mathematical and numerical model is presented for the process of liquid solution filtration in a cylindrical porous filter. The model is based on the Brinkman-Darcy equations, describing fluid motion in a porous medium with porosity and permeability linked through the Kozeny-Carman relation. The transport of dissolved substances is governed by the advection-diffusion-reaction system, complemented by the LDF kinetics and the Langmuir isotherm for adsorption. The model also incorporates a clogging (colmatation) equation that accounts for particle deposition and temporal reduction of filter porosity. A finite-difference numerical algorithm with iterative pressure correction using the conjugate gradient method is developed, ensuring stability according to the Courant-Friedrichs-Lewy criterion. The proposed approach enables analysis of the coupled hydrodynamic, mass transfer, and adsorption phenomena, allowing for assessment of filtration efficiency and prediction of filter lifespan in liquid purification systems.