Problems of Computational and Applied Mathematics
type: channel
description: ISSN 2181-8460 eISSN 2181-046X Problems of Computational and Applied Mathematics is a peer-reviewed 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. 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 e…
publisher: Научно-исследовательский институт развития цифровых технологий и искусственного интеллекта
language: ru
publication name: Problems of Computational and Applied Mathematics
issn: 2181-8460
items: Relationship in Metric Classification Algorithms and Analysis of its Properties
items: Optimal Quadrature Formulas for Approximate Calculation of Fast Oscillating Integral
items: Study of the Dynamics of Derivatives of a Fourth-Order Differential Equation with a Small Parameter at the Highest Derivative
items: System for Finding Optimal Coefficients of Hermite-type Quadrature Formulas with third-Order Derivatives
items: Investigation of the Exponential Stability of the Numerical Solution of a Hyperbolic System with Negative Nonlocal Characteristic Velocities
items: An Optimal Quadrature Formula for Hadamard-Type Hypersingular Integrals with High Oscillation in the Sobolev Space
items: Construction of a Self-Similar Solution to Mutual Diffusion Problems
items: Modeling the Process of Filtering a low-Concentration Solution through a Porous Medium
items: Modeling the Dispersion of Pollutants in the Atmosphere, Accounting for Terrain and Meteorological Conditions
items: Numerical Modeling of Thin Plate Bending using a Discrete Version of the Pre-Integration Method
items: Comprehensive Mathematical Modeling of Thermo-Electro-Magneto-Elastic Processes in Anisotropic Thin Plates of Complex Shape based on the RFM Method
items: Numerical Solution of Plane Problems of the Theory of Elasticity Directly in Stresses
items: Fine-Tuned AlexNet for Roof Shape Classification in Uzbekistan: a Transfer Learning Approach
items: About Problems with Finding Outliers in a Single-Class Problem
items: Optimization of Approximate Integration Formulas for Periodic Function Classes
items: On an Optimal Interpolation Formula with Derivative in a Hilbert Space
items: On an Interpolation of a Function by Natural Splines
items: Numerical Study of Lyapunov Stability of an Upwind Difference Scheme for a Quasilinear Hyperbolic System
items: Construction of an Algebraic-Hyperbolic Natural Tension Spline of Eighth Order
items: Numerical Modeling of the Amplitude Dynamics of the Stream Function for Plane Poiseuille Flow
items: Numerical Solution of the Nonlinear Groundwater and Pressurized Water Filtration Problem
items: Anomalous Filtration of Liquid in a Plane-Radial Homogeneous Porous Medium
items: Degenerate Lotka-Volterra Mappings and their Corresponding Bigraphs as a Discrete Model of the Evolution of the Interaction of Two Viruses
items: Anomalous Solute Transport in an Element of a Fractured-Porous Medium with Memory Effects
items: Modified SHAP approach for interpretable prediction of cardiovascular complications
items: Quadratic programming in the resource allocation model in agriculture based on the quantum algorithm
items: Why are quantum computing technologies the future of artificial intelligence?
items: A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation
items: An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space
items: Spectral-grid approximation of an ordinary differential equation with a small parameter at the highest derivative

Relationship in Metric Classification Algorithms and Analysis of its Properties
type: item
description: The relationship between class objects is considered for analyzing the cluster structure of the training dataset. The uniqueness of the solution to the problem of minimal coverage of the training set by prototypes is discussed. Interest in the uniqueness issue is associated with the use of either a base metric as a distance measure for all objects or local metrics derived from the base one. The specific structure of object relationships is reflected in the non-spherical shape of the cluster configurations. For such clusters, no established quality assessments exist. The property of connectivity is explored and its application as a source of new knowledge during the construction of information models in specific subject areas is studied. The connectivity property is proposed to be evaluated through the search for the most distant cluster objects using Dijkstra’s algorithm. The input data is an adjacency matrix constructed based on the intersection information of hyperspheres centered at the cluster objects. The radii of these hyperspheres are defined as the distances to the nearest objects of the opposite classes. Examples of quantitative characteristics of clusters and their possible areas of application are provided. One such characteristic is the cluster curvature coefficient.
subject: Статьи
  topic: section
subject: connectivity relation
  topic: keywords
subject: compactness measures
  topic: keywords
subject: Dijkstra algorithm
  topic: keywords
subject: metric algorithms
  topic: keywords
creator: N.A. Ignatiev
creator: Sh.Sh. Ramazonov
rights: Copyright (c) 2025 N.A. Igntiev https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 122
ending page: 129
doi: 10.71310/pcam.4_68.2025.06

Optimal Quadrature Formulas for Approximate Calculation of Fast Oscillating Integral
type: item
description: The article investigates optimal quadrature formulas designed for the approximate evaluation of highly oscillatory integrals, which frequently arise in many applied problems of mathematical physics, signal theory, and computational mathematics. The work is based on the formulation and solution of Sard’s problem in a Sobolev space, where quadrature formulas are constructed taking into account not only the values of the integrand but also its derivatives at the nodal points. This approach makes it possible to significantly improve the accuracy of approximation. To determine the optimal coefficients of the quadrature formula, the Sobolev method is applied, which allows one to derive an analytical expression for the norm of the error functional. Based on the use of the extremal function and the Riesz representation theorem, a boundary value problem is constructed, the solution of which yields the explicit form of the optimal coefficients and an exact estimate of the error. The presented results have a solid theoretical foundation and confirm the effectiveness of the proposed method. The obtained optimal quadrature formulas provide high accuracy in computing integrals with highly oscillatory kernels, opening up prospects for their application in numerical methods for solving differential equations, modeling physical processes, and other areas of computational mathematics.
subject: Статьи
  topic: section
subject: Sobolev space
  topic: keywords
subject: optimal coefficients
  topic: keywords
subject: error functional
  topic: keywords
subject: extremal function
  topic: keywords
creator: X.M. Shadimetov
creator: F.A. Nuraliyev
creator: D.M. Mirkomilov
rights: Copyright (c) 2025 X.M. Shadimetov https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 110
ending page: 121
doi: 10.71310/pcam.4_68.2025.08

Study of the Dynamics of Derivatives of a Fourth-Order Differential Equation with a Small Parameter at the Highest Derivative
type: item
description: The article studies the dynamics of derivatives up to the highest derivative of an ordinary differential equation of the fourth order with a small parameter at the highest derivative. The solution of the differential problem is sought in the form of a series in Chebyshev polynomials of the first kind with unknown expansion coefficients. Substituting these series into the differential problem, we obtain a system of algebraic equations for finding the unknown expansion coefficients. Using these coefficients, we calculate the solution to the problem and its derivatives. To illustrate the high accuracy of the applied method, we use the method of trial functions. The essence of the method of trial functions is as follows. A certain function is selected, it can be chosen arbitrarily. Substituting it into the main differential equation, we find the right-hand side and satisfy the corresponding boundary conditions. The resulting problem is solved by the spectral method and the approximate solution is compared with the known trial function and its derivatives for different values of the small parameter and the approximating Chebyshev polynomials.
subject: Статьи
  topic: section
subject: Chebyshev polynomials
  topic: keywords
subject: small parameter
  topic: keywords
subject: high accuracy
  topic: keywords
creator: Ch.B. Normurodov
creator: N.T. Juraeva
creator: M.M. Normatova
rights: Copyright (c) 2025 Ch.B. Normurodov https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 97
ending page: 109
doi: 10.71310/pcam.4_68.2025.03

System for Finding Optimal Coefficients of Hermite-type Quadrature Formulas with third-Order Derivatives
type: item
description: In the world, special attention is given to developing various optimal computational methods for the approximate calculation of definite integrals. This article focuses on constructing composite optimal quadrature formulas in the space of differentiable functions using the Sobolev method. The quadrature formula consists of a linear combination of function values and its derivatives up to and including the third order at all nodes of the interval [0,1]. The accuracy of quadrature formulas is evaluated by the norm of the error functional of the quadrature formulas: The error of composite quadrature formulas is estimated through the norm of the error functional of the quadrature formula under onsideration and the norm of the function. The norm of the error functional of the quadrature formula will be determined through the extremal function of this quadrature formula. It is known that the norm of the error functional of a composite quadrature formula is expressed in terms of the coefficients of this quadrature formula. By minimizing this norm with respect to the coefficients, systems of linear equations will be obtained for finding the optimal coefficients of composite quadrature formulas.
subject: Статьи
  topic: section
subject: Sobolev space
  topic: keywords
subject: derivative optimal quadrature formula
  topic: keywords
subject: error functional
  topic: keywords
subject: optimal coefficients
  topic: keywords
creator: Kh.M. Shadimetov
creator: F.A. Nuraliev
creator: R.M. Edilbekova
rights: Copyright (c) 2025 R. Edilbekova https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 88
ending page: 96
doi: 10.71310/pcam.4_68.2025.07

Investigation of the Exponential Stability of the Numerical Solution of a Hyperbolic System with Negative Nonlocal Characteristic Velocities
type: item
description: In this work, the problem of stabilizing the equilibrium state for a hyperbolic equation with negative nonlocal characteristic velocity and measurement error is investigated. The formulation of the mixed control problem is presented. The weak solution, exponential stability of the equilibrium state of the mixed problem, and the definition of the Lyapunov function are given. The exponential stability of the equilibrium state of a mixed problem is given. The stability in the ????2 – norm is determined relative to the discrete perturbations of the equilibrium state of the initial-boundary difference problem. In a computational experiment, one hyperbolic equation with negative characteristic velocity was considered and its numerical solution was found using the constructed difference scheme. The graph of the norm ????2 of the numerical solution, demonstrating its stability, is shown.
subject: Статьи
  topic: section
subject: Hyperbolic equation
  topic: keywords
subject: nonlocal characteristic velocity
  topic: keywords
subject: stability
  topic: keywords
subject: explicit difference scheme
  topic: keywords
creator: R.D. Aloev
creator: V. Alimova
rights: Copyright (c) 2025 R.D. Aloev https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 75
ending page: 87
doi: 10.71310/pcam.4_68.2025.05

An Optimal Quadrature Formula for Hadamard-Type Hypersingular Integrals with High Oscillation in the Sobolev Space
type: item
description: This paper addresses the problem of constructing optimal quadrature formulas for the numerical evaluation of Hadamard-type hypersingular integrals with hihg oscillation. Such integrals frequently arise in the analysis of boundary-value problems for partial differential equations, particularly in hyperbolic and wave-type models, as well as in various applied fields such as elasticity theory, fluid dynamics, and electromagnetic scattering. Our primary focus is on deriving the analytical expressions for the coefficients of an optimal quadrature formula tailored to the structure of the singular and oscillatory kernel. The goal is to achieve minimal error in a rigorous sense by constructing formulas that are optimal in the sense of Sard, meaning that they minimize the worst-case error over a specified function class. To this end, we develop an optimal quadrature formula of the form (2) in the Sobolev space, applying the Sobolev method for constructing optimal quadrature formulas. This method leverages variational principles and operator-theoretic tools to systematically minimize the norm of the error functional. The resulting formulas are well-adapted to handle the challenging combination of hypersingularity and oscillation, and they offer both theoretical optimality and practical computational advantages.
subject: Статьи
  topic: section
subject: Sobolev space
  topic: keywords
subject: Sobolev method
  topic: keywords
subject: singular integrals
  topic: keywords
subject: hypersingular integrals
  topic: keywords
subject: error functional
  topic: keywords
subject: optimal quadrature formulas
  topic: keywords
subject: error estimation
  topic: keywords
creator: D.M. Akhmedov
creator: D.S. Buvasherov
rights: Copyright (c) 2025 Д.С. Бувашеров https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 65
ending page: 74
doi: 10.71310/pcam.4_68.2025.09

Construction of a Self-Similar Solution to Mutual Diffusion Problems
type: item
description: The article considers the problem of constructing self-similar solutions of a system of nonlinear differential equations modeling mutual diffusion processes in multicomponent media. An analysis of a mathematical model taking into account complex interactions of components and the nonlinear nature of transfer processes is carried out. Approximate solutions of the system are found, allowing one to describe the behavior of concentration profiles in various modes. Asymptotic representations of solutions for regular, unlimited and limited cases are obtained, and the behavior of a two-sided linear stationary equation arising at intermediate stages of analysis is studied. The results are of interest for diffusion theory and applied problems of mathematical modeling of complex physical processes.
subject: Статьи
  topic: section
subject: mutual diffusion
  topic: keywords
subject: self-similar solutions
  topic: keywords
subject: nonlinear differential equations
  topic: keywords
subject: approximate solution
  topic: keywords
subject: asymptotic analysis
  topic: keywords
subject: multicomponent media
  topic: keywords
subject: scale invariance
  topic: keywords
subject: stationary equation
  topic: keywords
subject: mathematical modeling
  topic: keywords
subject: diffusion processes
  topic: keywords
creator: S.Y. Muminov
rights: Copyright (c) 2025 S.Y. Muminov https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 56
ending page: 64
doi: 10.71310/pcam.4_68.2025.12

Modeling the Process of Filtering a low-Concentration Solution through a Porous Medium
type: item
description: This article presents an integrated mathematical model of radial filtration in a cylindrical filter, in which the flow movement is carried out from the outer layer to the inner cavity. Free flow areas are described by Navier-Stokes equations for incompressible fluid, where the filtration process through the porous layer is formalized based on Darcy’s law, and the pressure distribution is determined by the Poisson equation. The transfer of dissolved substances is modeled using the advection-diffusion equation. The model includes a mechanism for the evolution of porosity over time (sealing or clogging), which allows tracking the conjugate dynamics of the "pressure-velocity-porosityconcentration" parameters. The analysis showed that considering the vertical orientation of the cylindrical filter is a significant factor influencing the reduction of energy costs and increasing the selectivity of the process.
subject: Статьи
  topic: section
subject: mathematical model
  topic: keywords
subject: cylindrical filter
  topic: keywords
subject: Navier–Stokes equations
  topic: keywords
subject: Darcy’s law
  topic: keywords
subject: Poisson equation
  topic: keywords
subject: advection–diffusion
  topic: keywords
subject: hydrodynamics
  topic: keywords
creator: U. Saidov
creator: I. Juraev
creator: J. Turakulov
rights: Copyright (c) 2025 J. Turakulov https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 47
ending page: 55
doi: 10.71310/pcam.4_68.2025.10

Modeling the Dispersion of Pollutants in the Atmosphere, Accounting for Terrain and Meteorological Conditions
type: item
description: This work proposes an expanded 3D Euler model for calculating the spatiotemporal evolution of pollutant concentrations released into the atmosphere under changing meteorological conditions. The model, alongside the advection-diffusion equation, combines in a single system: Stokes’ law deposition (for solid particles), temperature-dependent chemical decomposition according to Arrhenius’ law, retention by plants (passive sink), precipitation washout (rainout), terrain-dependent resistance, and parameterization of turbulent diffusion based on Pasquill-Gifford stability classes. The numerical solution employs upwind/QUICK convective fluxes, central difference diffusion, and a time step stabilized by the CFL criterion. Scenarios covering various wind directions and speeds demonstrate the role of terrain in flow direction and atmospheric-related dispersion characteristics. The model is mass-balanced.
subject: Статьи
  topic: section
subject: advection-diffusion
  topic: keywords
subject: Pasquill-Gifford
  topic: keywords
subject: Monin-Obukhov length
  topic: keywords
subject: wet deposition
  topic: keywords
subject: sedimentation (Stokes)
  topic: keywords
subject: vegetation sinks
  topic: keywords
subject: terrain
  topic: keywords
subject: Upwind/QUICK
  topic: keywords
subject: CFL
  topic: keywords
creator: N. Ravshanov
creator: O. Juraboeva
creator: B. Boborakhimov
creator: Kh. Sharipov
rights: Copyright (c) 2025 N. Ravshanov https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 38
ending page: 46
doi: 10.71310/pcam.4_68.2025.01

Numerical Modeling of Thin Plate Bending using a Discrete Version of the Pre-Integration Method
type: item
description: Many applied problems are described by the Dirichlet problem for the Poisson equa tions. This problem is used in the following practical situations: membrane or plate deflection, electric potential, gravitational and magnetic potential, stationary tempera ture distribution (diffusion), and electron-optical systems. Although there are both direct and iterative methods for numerically solving the Poisson equation, the question of the effectiveness of using these methods remains relevant. In this work, the bending of a thin plate is studied using a discrete version of the method of preliminary integration over Chebyshev polynomials of the first kind. The proposed method is used to determine the bending of the plate. Numerical calculations are performed to demonstrate the depen dence of the bending of the plate on the distributed load, and the proposed method is shown to be highly accurate.
subject: Статьи
  topic: section
subject: plate bending
  topic: keywords
subject: load distribution
  topic: keywords
subject: Chebyshev polynomials
  topic: keywords
subject: discrete version of the preliminary integration method
  topic: keywords
subject: high accuracy
  topic: keywords
subject: efficiency
  topic: keywords
creator: Ch.B. Normurodov
creator: Sh.A. Ziyakulova
rights: Copyright (c) 2025 Ch.B. Normurodov https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 26
ending page: 37
doi: 10.71310/pcam.4_68.2025.04

Comprehensive Mathematical Modeling of Thermo-Electro-Magneto-Elastic Processes in Anisotropic Thin Plates of Complex Shape based on the RFM Method
type: item
description: This article presents a comprehensive mathematical modeling of thermo-electro magneto-elastic processes in anisotropic thin plates of complex shapes using the R function method (RFM). The model considers interrelated physical fields: thermal, elec trical, magnetic, and mechanical, acting on the structure. The R-function method is employed to describe geometrically complex domains and to set boundary conditions, allowing the formation of smooth solutions without explicitly defining the boundary con ditions. A system of differential equations of motion is constructed, taking into account the material anisotropy and the influence of external physical factors. The proposed model can be used to analyze the state of complex technical structures operating under multiphysical influences. The paper presents numerical results and a graphical interpre tation of the plates’ behavior under various loading conditions.
subject: Статьи
  topic: section
subject: R-function method
  topic: keywords
subject: anisotropic plate
  topic: keywords
subject: thermo-electro-magneto-elastic processes
  topic: keywords
subject: comprehensive modeling
  topic: keywords
subject: complex geometry
  topic: keywords
subject: multiphysical systems
  topic: keywords
subject: equations of motion
  topic: keywords
subject: numerical solution
  topic: keywords
creator: F.M. Nuraliev
creator: B.N. Tokhirov
rights: Copyright (c) 2025 F.M. Nuraliev https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 17
ending page: 25
doi: 10.71310/pcam.4_68.2025.11

Numerical Solution of Plane Problems of the Theory of Elasticity Directly in Stresses
type: item
description: Usually, the solution of a plane problem of the theory of elasticity in stresses is reduced to solving a biharmonic equation for the Airy stress function. In this paper, two (A and B) variants of plane boundary value problems of the theory of elasticity are formulated directly in terms of stresses. In the first case (A), the boundary value problem consists of two equilibrium equations and one Beltrami-Michell equation with the corresponding boundary and additional boundary conditions. In the formulation of the second plane boundary value problem (B), in contrast to the first, the equations of equilibrium differentiated with respect to x and y, respectively, are used. Symmetric finite-difference equations are constructed and the known Timoshenko-Goodier problem of stretching a rectangular plate with a parabolic load is solved for comparison. The discrete analogs of boundary value problems A and B are composed by the finite-difference method and the iterative method and the marching method are used to solve them. By comparing the numerical results of boundary value problems, A and B obtained by two methods, the validity of the formulated boundary value problems and the reliability of the obtained results are ensured.
subject: Статьи
  topic: section
subject: stress
  topic: keywords
subject: Beltrami-Michell equations
  topic: keywords
subject: equilibrium equations
  topic: keywords
subject: difference equations
  topic: keywords
subject: iterative method
  topic: keywords
subject: marching method
  topic: keywords
creator: A. Khaldjigitov
creator: U. Adambaev
creator: О. Tilovov
creator: R. Rakhmonova
creator: M. Makhmadiyorova
rights: Copyright (c) 2025 Khaldjigitov A. https://creativecommons.org/licenses/by/4.0
date: 2025-09-20
publication date: 2025-09-20
starting page: 8
ending page: 16
doi: 10.71310/pcam.4_68.2025.02

Fine-Tuned AlexNet for Roof Shape Classification in Uzbekistan: a Transfer Learning Approach
type: item
description: This study investigates the classification of rooftop shapes using convolutional neural networks (CNNs), with a particular focus on regional adaptation through transfer learning. Initial training of the AlexNet architecture utilizes a publicly available Zenodo dataset comprising satellite imagery of flat, gabled, and hipped roofs. To address generalizability constraints in specific geographic contexts, a custom dataset of Uzbekistan rooftops, sourced from OpenStreetMap and high-resolution Mapbox Static Images API tiles, enables fine-tuning. Experimental outcomes reveal enhanced classification accuracy for flat, gabled, and hipped roofs, underscoring the efficacy of transfer learning in mitigating domain shift due to architectural and environmental variations. Integration of open-source geospatial tools with transfer learning offers a replicable framework for addressing geographic bias in rooftop shape classification, adaptable to other underrepresented regions.
subject: Статьи
  topic: section
subject: roof shape classification
  topic: keywords
subject: convolutional neural networks
  topic: keywords
subject: transfer learning
  topic: keywords
subject: AlexNet
  topic: keywords
subject: satellite imagery
  topic: keywords
creator: S.U. Yuldashev
rights: Copyright (c) 2025 S.U. Yuldashev https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 133
ending page: 146
doi: 10.71310/pcam.3_67.2025.12

About Problems with Finding Outliers in a Single-Class Problem
type: item
description: Outlier detection is considered in a one-class classification problem. The challenges of detection are related to the choice of loss functions, optimization criteria, and transformations of various types of features. Examples are given from subject areas where such problems exist. The absence of clustering methods for outlier detection and the possibility of their unambiguous interpretation are justified. The diversity of analysis options is related to the presence of uncertainties in selecting clustering parameters. As additional domain knowledge, an analysis of the environment’s nature for outlier objects is provided. For this purpose, kernel density estimates of outliers are calculated over local areas of fixed size. A method is presented for determining the location of objects closest to the class center. The task of analyzing meteorological data for the city of Tashkent has been solved. To enable prompt response to environmental violations, a transition to solving a two-class classification problem is proposed. It is assumed that one class describes an acceptable state, while the other represents a deviation from environmental standards.
subject: Статьи
  topic: section
subject: loss functions
  topic: keywords
subject: unsupervised learning
  topic: keywords
subject: diverse data
  topic: keywords
creator: N.A. Ignatiev
creator: A.O. Toshpulatov
rights: Copyright (c) 2025 N.A. Ignatiev https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 125
ending page: 132
doi: 10.71310/pcam.3_67.2025.11

Optimization of Approximate Integration Formulas for Periodic Function Classes
type: item
description: This study explores the optimization of quadrature formulas for the approximate integration of periodic functions within a specific functional space. The research focuses on developing optimal quadrature formulas by deriving an analytical expression for the error associated with the integration process. By employing Fourier transform techniques and the concept of an extremal function, the study establishes a precise representation of the error. Additionally, optimal coefficients for the quadrature formula are determined to minimize this error, yielding an explicit solution. The results demonstrate enhanced accuracy compared to existing approaches, with the error characterized through a series expansion that reveals its asymptotic behavior. These findings advance the efficiency of numerical integration for periodic functions, offering potential applications in mathematical analysis, scientific computing, and related disciplines.
subject: Статьи
  topic: section
subject: optimal quadrature formula
  topic: keywords
subject: Hilbert space
  topic: keywords
subject: the error functional
  topic: keywords
subject: Fourier transform
  topic: keywords
creator: M.Kh. Shadimetov
creator: S.S. Azamov
creator: H.M. Kobilov
rights: Copyright (c) 2025 M.Kh. Shadimetov https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 116
ending page: 124
doi: 10.71310/pcam.3_67.2025.10

On an Optimal Interpolation Formula with Derivative in a Hilbert Space
type: item
description: This study is devoted to the development of a system of equations for determining the coefficients of optimal interpolation formulas that integrate derivative information within a Hilbert space framework. Conventional interpolation methods, which rely solely on pointwise function values, often prove inadequate for functions exhibiting complex or rapidly varying behavior. To overcome this limitation, the proposed approach incorporates derivative data into the interpolation process, enhancing both the stability and accuracy of the resulting formulas. By formulating the interpolation problem within a Hilbert space, we establish a robust framework for deriving these optimal coefficients. The core analytical contribution lies in the formulation of a system of equations, derived through variational principles and leveraging tools such as the Riesz representation theorem and convolution operations. Solving this system enables the explicit computation of the optimal coefficients, which are readily applicable to practical interpolation tasks. This methodology is particularly significant in numerical analysis, especially in scenarios where positional data, directional vectors, or object velocities are available, as the inclusion of derivative information is both intuitive and critical. Furthermore, the approach is applicable to data approximation, signal processing, and computational contexts requiring function reconstruction from sampled or noisy data.
subject: Статьи
  topic: section
subject: optimal coefficients
  topic: keywords
subject: derivative-based interpolation
  topic: keywords
subject: Hilbert space
  topic: keywords
subject: error minimization
  topic: keywords
subject: variational methods
  topic: keywords
creator: A.R. Hayotov
creator: A.Y. Nafasov
rights: Copyright (c) 2025 A.R. Hayotov https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 107
ending page: 115
doi: 10.71310/pcam.3_67.2025.09

On an Interpolation of a Function by Natural Splines
type: item
description: There are algebraic and variational approaches of construction in the spline theory. In algebraic approach splines are considered as some smooth piecewise polynomial functions. In the variational approach splines are elements of Hilbert or Banach spaces minimizing certain functionals. Then we study the problems of existence, uniqueness, and convergence of splines and algorithms for constructing them based on their own properties of splines. In this paper, we study the problem of natural spline functions in a Hilbert space. Here, using the Sobolev method, an algorithm is given for solving a system of linear algebraic equations for the coefficients of the natural spline functions. For ???? = 2, explicit expressions for the optimal coefficients of the natural spline function in the Hilbert space ????(2,0) are obtained.
subject: Статьи
  topic: section
subject: Hilbert space
  topic: keywords
subject: extremal function
  topic: keywords
subject: error functional
  topic: keywords
subject: spline function
  topic: keywords
creator: A.K. Boltaev
creator: O.F. Pardaeva
rights: Copyright (c) 2025 A.K. Boltaev https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 97
ending page: 106
doi: 10.71310/pcam.3_67.2025.08

Numerical Study of Lyapunov Stability of an Upwind Difference Scheme for a Quasilinear Hyperbolic System
type: item
description: This study addresses a mixed problem for a quasilinear system of hyperbolic equations expressed in Riemann invariants, incorporating dissipative nonlinear boundary conditions. A numerical approach is developed through an initial-boundary difference problem utilizing an upwind difference scheme. The stability of nonlinear difference schemes is investigated, with a focus on establishing a sufficient stability criterion based on Lyapunov vector functions. The proposed criterion extends prior theoretical work, where a discrete Lyapunov function was formulated to demonstrate the exponential stability of the steady state for the quasilinear system. Numerical computations for a model problem validate these theoretical findings. The research highlights the potential of adapting the direct Lyapunov method to analyze the stability of nonlinear hyperbolic systems by constructing a positive definite function that exhibits monotonic decay along system solutions.
subject: Статьи
  topic: section
subject: exponential stability
  topic: keywords
subject: hyperbolic system
  topic: keywords
subject: mixed problem
  topic: keywords
subject: difference scheme
  topic: keywords
subject: Lyapunov function
  topic: keywords
creator: R.D. Aloev
creator: A.S. Berdishev
creator: D.E. Nematova
rights: Copyright (c) 2025 R.D. Aloev https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 83
ending page: 96
doi: 10.71310/pcam.3_67.2025.07

Construction of an Algebraic-Hyperbolic Natural Tension Spline of Eighth Order
type: item
description: This paper first demonstrates that an algebraic-hyperbolic spline of eighth order minimizes the norm within a Hilbert space framework. Subsequently, employing Sobolev’s method, which involves constructing a discrete analogue of the differential operator, the spline function is developed. Unknown coefficients of the spline are computed according to predefined smoothness criteria and boundary conditions. As a result, the constructed spline exhibits exceptional smoothness, enhances interpolation accuracy, and precisely reproduces hyperbolic functions, linear polynomials, and constants. The findings indicate that this approach is highly effective for applications requiring smooth interpolation and accurate modeling of physical phenomena. Additionally, incorporating tension parameters enables precise adjustment of the spline’s stiffness or flexibility.
subject: Статьи
  topic: section
subject: Hilbert space
  topic: keywords
subject: generalized spline
  topic: keywords
subject: algebraic-hyperbolic spline
  topic: keywords
subject: convolution
  topic: keywords
subject: discrete analogue
  topic: keywords
creator: G.Sh. Abdullaeva
rights: Copyright (c) 2025 Г.Ш. Абдуллаева https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 67
ending page: 82
doi: 10.71310/pcam.3_67.2025.06

Numerical Modeling of the Amplitude Dynamics of the Stream Function for Plane Poiseuille Flow
type: item
description: This paper investigates the dynamics of the complex amplitude of the stream function for plane Poiseuille flow using the spectral method. An approximate solution of the flow under consideration is sought as a finite series expansion in terms of Chebyshev polynomials of the first kind with unknown expansion coefficients. The eigenvalue spectrum of the Poiseuille flow is determined, and the most unstable eigenvalue (with the largest modulus) is selected. The components of the corresponding eigenvector, which represent the complex unknown coefficients of the desired series expansion, are also determined. Using these coefficients, the real and imaginary parts of the amplitude of the stream function for the perturbed flow are computed. The results of the computations are presented in both tabular and graphical form and demonstrate the high accuracy of the proposed approach.
subject: Статьи
  topic: section
subject: Poiseuille flow
  topic: keywords
subject: stream function amplitude
  topic: keywords
subject: spectral method
  topic: keywords
subject: Chebyshev polynomials
  topic: keywords
subject: high accuracy
  topic: keywords
creator: Ch.B. Normurodov
creator: M.A. Tilovov
creator: D.Ch. Normurodov
rights: Copyright (c) 2025 Ch.B. Normurodov https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 53
ending page: 66
doi: 10.71310/pcam.3_67.2025.05

Numerical Solution of the Nonlinear Groundwater and Pressurized Water Filtration Problem
type: item
description: This article discusses the numerical modeling of the problem of nonlinear groundwater filtration taking into account evaporation and infiltration. Particular attention is paid to the influence of nonlinear effects on the distribution and dynamics of filtration flows in porous media. A mathematical model based on a system of nonlinear differential equations describing the process of water filtration in soil has been developed. Numerical methods, including difference schemes and iterative algorithms, ensuring the stability and convergence of calculations, have been applied to solve the problem. Numerical modeling of various conditions describing the dynamics of groundwater has been performed taking into account changing humidity and the impact of external factors. An algorithm developed on the basis of a finite-difference solution of a system of equations for groundwater filtration in a two-layer model of aquifers is presented. The algorithm is implemented as software designed to solve various hydrogeological and melioration problems of forecasting. The program takes into account the influence of both natural and artificial factors on the process of filtration of ground and pressure waters. The calculation results are visualized in graphical form.
subject: Статьи
  topic: section
subject: mathematical model
  topic: keywords
subject: numerical modeling
  topic: keywords
subject: groundwater
  topic: keywords
subject: filtration
  topic: keywords
subject: infiltration
  topic: keywords
subject: evaporation
  topic: keywords
subject: finite difference method
  topic: keywords
creator: E.Sh. Nazirova
creator: Kh.A. Karabaeva
rights: Copyright (c) 2025 E.Sh. Nazirova https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 37
ending page: 52
doi: 10.71310/pcam.3_67.2025.04

Anomalous Filtration of Liquid in a Plane-Radial Homogeneous Porous Medium
type: item
description: The paper considers the radial problem of anomalous filtration of a homogeneous liquid in a plane-radial formulation. The filtration process was modeled by a differential equation with fractional derivatives with respect to time and pressure. This differential equation was derived on the basis of Darcy’s fractional relaxation law, which takes into account relaxation effects both in the filtration velocity and in the pressure gradient. Fractional derivatives are defined in the Caputo sense, which is preferable to other definitions of fractional derivatives, such as Riemann-Liouville, Grundwald-Letnikov, etc. The problem is solved numerically by the finite difference method. The integral representation of the fractional Caputo derivative is discretized using known quadrature formulas. The pressure profiles are determined for different values of the order of the fractional derivative with respect to time both with respect to pressure and filtration velocity, and the effect of process anomality on the filtration characteristics is estimated. The effect of changing the orders of fractional derivatives on the pressure distribution at different moments in time is established. The effect of relaxation times for the pressure gradient and filtration velocity on the pressure distribution in the medium at different moments in time is also estimated. A comparative analysis of the effect of the orders of fractional derivatives and relaxation times in Darcy’s law on the pressure distribution in the medium is carried out.
subject: Статьи
  topic: section
subject: anomalous filtration
  topic: keywords
subject: fractional derivative
  topic: keywords
subject: finite difference method
  topic: keywords
subject: porous media
  topic: keywords
subject: pressure
  topic: keywords
subject: relaxation time
  topic: keywords
creator: B.Kh. Khuzhayorov
creator: M.S. Zokirov
rights: Copyright (c) 2025 B.Kh. Khuzhayarov https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 28
ending page: 36
doi: 10.71310/pcam.3_67.2025.03

Degenerate Lotka-Volterra Mappings and their Corresponding Bigraphs as a Discrete Model of the Evolution of the Interaction of Two Viruses
type: item
description: This paper proposes a discrete model of the interaction between two airborne viruses, based on an operator acting in a four-dimensional simplex. The model describes the progression of an epidemic in a closed population, divided into five compartments: susceptible individuals, individuals in the latent stage of the first virus, those infected with the first virus, those infected with the second virus, and individuals who have recovered from the first virus. The mathematical structure of the model captures complex transitions between states and interactions between strains, including cases of co-infection. Special attention is given to the analysis of the sets of initial and final states of the disease, defined by systems of inequalities. Depending on the model parameters, these sets may lie on different faces of the simplex, representing various scenarios of epidemic onset and resolution. Two main epidemiological scenarios are considered: one involving complete recovery after infection with the first virus, and another involving progression to co-infection without full recovery. The model is applicable to the analysis of tuberculosis co-infection with viral hepatitis B and C and allows assessment of the influence of various parameters on patient survival during multi-drug therapy. Finally, a numerical experiment is conducted, presenting trajectories, phase portraits, and 30-day dynamics of disease spread, illustrating system behavior under different initial conditions and parameter settings.
subject: Статьи
  topic: section
subject: Lotka-Volterra mapping
  topic: keywords
subject: simplex
  topic: keywords
subject: skew-symmetric matrix
  topic: keywords
subject: convex hull
  topic: keywords
subject: trajectory
  topic: keywords
subject: partially oriented graph
  topic: keywords
subject: biograph
  topic: keywords
creator: U.R. Muminov
rights: Copyright (c) 2025 U.R. Muminov https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 15
ending page: 27
doi: 10.71310/pcam.3_67.2025.02

Anomalous Solute Transport in an Element of a Fractured-Porous Medium with Memory Effects
type: item
description: Fractured porous media (FPM) in theoretical investigations is considered as a system of fractures and adjacent porous blocks (matrix). In such media, solute transport occurs mainly through the fracture system with mass transfer in porous blocks. In this work, the problem of solute transport through an element of FPM is studied, taking into account the memory effects. The medium consists of a single fracture and a porous block (matrix) bordering it. The problem was solved numerically using the finite difference method with Caputo’s definition of fractional derivatives. Based on the numerical results, concentration profiles in the fracture and matrix were obtained. The influence of the fractional order of derivatives on the distribution of concentration is shown. The current, total and summing flow rates of the solute from the fracture into the matrix were also determined.
subject: Статьи
  topic: section
subject: diffusion
  topic: keywords
subject: fractional derivatives
  topic: keywords
subject: flow rate
  topic: keywords
subject: fractured-porous medium
  topic: keywords
subject: matrix
  topic: keywords
creator: B. Khuzhayorov
creator: T.O. Dzhiyanov
creator: Z.Z. Eshdavlatov
rights: Copyright (c) 2025 https://creativecommons.org/licenses/by/4.0
date: 2025-07-27
publication date: 2025-07-27
starting page: 5
ending page: 14
doi: 10.71310/pcam.3_67.2025.01

Modified SHAP approach for interpretable prediction of cardiovascular complications
type: item
description: This article explores the significance of modifying SHAP (SHapley Additive exPlana tions) values to enhance model interpretability in machine learning. SHAP values provide a fair attribution of feature contributions, making AI-driven decision-making more trans parent and reliable. However, raw SHAP values can sometimes be difficult to interpret due to feature interactions, noise, and inconsistencies in scale. The article discusses key techniques for modifying SHAP values, including feature aggregation, normalization, cus tom weighting, and noise reduction, to improve clarity and relevance in explanations. It also examines how these modifications align interpretations with real-world needs, ensur ing that SHAP-based insights remain practical and actionable. By strategically refining SHAP values, data scientists can derive more meaningful explanations, improving trust in AI models and enhancing decision-making processes. The article provides a structured approach to modifying SHAP values, offering practical applications and benefits across various domains.
subject: Статьи
  topic: section
subject: model interpretability
  topic: keywords
subject: feature importance
  topic: keywords
subject: normalization
  topic: keywords
subject: AI transparency
  topic: keywords
creator: D.K. Sharipov
creator: A.D. Saidov
rights: Copyright (c) 2025 https://creativecommons.org/licenses/by/4.0
date: 2025-05-15
publication date: 2025-05-15
starting page: 114
ending page: 122
doi: 10.71310/pcam.2_64.2025.10

Quadratic programming in the resource allocation model in agriculture based on the quantum algorithm
type: item
description: This paper discusses the application of quantum algorithms to the problem of op timizing resource allocation in agriculture. In particular, the possibility of using the Quantum Approximate Optimization Algorithm to solve this problem is investigated. This is a quantum optimization algorithm that can be effectively applied to combinato rial optimization problems. The work involves adapting the algorithm to the problem of optimizing resource allocation in agriculture, including modeling the optimization func tion and taking into account resource use constraints. It is assumed that the use of a quantum algorithm can offer new approaches to solving complex optimization problems in agriculture and contribute to increasing the efficiency of resource use and improving the sustainability of production.
subject: Статьи
  topic: section
subject: quantum algorithms
  topic: keywords
subject: QAOA
  topic: keywords
subject: optimization
  topic: keywords
subject: resource allocation
  topic: keywords
subject: mathematical model
  topic: keywords
subject: combinatorial optimization
  topic: keywords
creator: D.T. Mukhamedieva
creator: M.Kh. Raupova
rights: Copyright (c) 2025 https://creativecommons.org/licenses/by/4.0
date: 2025-05-15
publication date: 2025-05-15
starting page: 101
ending page: 113
doi: 10.71310/pcam.2_64.2025.09

Why are quantum computing technologies the future of artificial intelligence?
type: item
description: Quantum technologies are transforming industries by harnessing the principles of quantum mechanics. These innovations promise to revolutionize computing, communi cations, sensing, cryptography, and healthcare, offering solutions previously considered unattainable. Quantum computing is changing the way complex problems are solved: un like classical computers that use bits to represent information in binary code, quantum computers exploit the laws of quantum physics to perform calculations beyond the reach of even the most advanced classical systems. Quantum algorithms can efficiently solve logistical challenges, optimize global supply chains, accelerate risk analysis, and enhance decision-making in complex financial models. Quantum computers enable unprecedentedly fast simulation of molecular interactions, expediting drug discovery, enabling per sonalized medicine, and elevating disease diagnostics to a new level. All these advances lay the foundation for the future development of artificial intelligence, the prospects of which are discussed in this paper.
subject: Статьи
  topic: section
subject: quantum computing
  topic: keywords
subject: artificial intelligence
  topic: keywords
subject: quantum AI
  topic: keywords
subject: innovative strategies
  topic: keywords
subject: business innovation
  topic: keywords
subject: medicine
  topic: keywords
creator: F.T. Adilova
rights: Copyright (c) 2025 https://creativecommons.org/licenses/by/4.0
date: 2025-05-15
publication date: 2025-05-15
starting page: 87
ending page: 100
doi: 10.71310/pcam.2_64.2025.08

A discrete variant of the method pre-integration and its application to the numerical solution of a singularly perturbed equation
type: item
description: The article presents a numerical modeling approach for solving a non-homogeneous singularly perturbed fourth-order equation using by a discrete variant of the pre integration method. In the proposed method, the basis functions are Chebyshev poly nomials of the first kind. The highest derivative of the differential equation is expanded into a finite series in terms of these polynomials, with unknown expansion coefficients. All lower derivatives and the solution of the differential equation are represented in the form of the chosen expansion. The resulting series are substituted into the differential equation, leading to a system of discrete equations. For Chebyshev polynomials, there is a discrete integration formula that reduces the order of the derivative. By applying this formula, the discrete system is integrated four times, resulting in a system of algebraic equations where the number of equations is less than the number of unknown coefficients. The missing equations are obtained from the boundary conditions of the problem. The integration operation improves the smoothness of the approximating polynomials. For example, a zeroth-order polynomial, after four integrations, transforms into a fourth order polynomial. Numerical calculations demonstrate the high accuracy and efficiency of the proposed method, especially for very small values of the parameter e.
subject: Статьи
  topic: section
subject: Chebyshev polynomials
  topic: keywords
subject: discrete variant
  topic: keywords
subject: pre-integration
  topic: keywords
subject: high accuracy
  topic: keywords
subject: small parameter
  topic: keywords
creator: Ch.B. Normurodov
creator: E.E. Shakaeva
creator: Sh.A. Ziyakulova
rights: Copyright (c) 2025 https://creativecommons.org/licenses/by/4.0
date: 2025-05-15
publication date: 2025-05-15
starting page: 74
ending page: 86
doi: 10.71310/pcam.2_64.2025.07

An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space
type: item
description: The numerical computation of definite integrals plays a crucial role in various applied and theoretical disciplines. In many cases, exact analytical evaluation of integrals is in feasible due to the complexity of the integrand or the nature of the integration limits. Quadrature formulas provide an efficient approach for approximating definite integrals, relying on weighted sums of function values at selected nodes. Traditional quadrature formulas, such as those of Newton-Cotes, Gauss, aim to optimize accuracy by care fully choosing nodes and weights. However, optimal quadrature formulas can also be constructed in the sense of Sard, minimizing the error functional within a given func tion space. In this paper, we focus on constructing an optimal quadrature formula in the Sobolev space with arbitrarily fixed nodes. Unlike conventional approaches where coefficients are determined sequentially, we simultaneously optimize both function and derivative coefficients, improving overall accuracy and stability. The derivation employs the method of- functions, which allows us to express the quadrature formula’s co efficients explicitly and analyze its error properties. The obtained quadrature formula minimizes the error norm in the chosen function space, ensuring an improved approxi mation of definite integrals. Furthermore, when the nodes are equally spaced, our results generalize the well-known Euler-Maclaurin formula, demonstrating the effectiveness of our approach.
subject: Статьи
  topic: section
subject: quadrature formula
  topic: keywords
subject: Sobolev space
  topic: keywords
subject: optimal approximation
  topic: keywords
subject: Euler-Maclaurin formula
  topic: keywords
creator: A.R. Hayotov
creator: U.A. Berdimuradova
rights: Copyright (c) 2025 https://creativecommons.org/licenses/by/4.0
date: 2025-05-15
publication date: 2025-05-15
starting page: 64
ending page: 73
doi: 10.71310/pcam.2_64.2025.06

Spectral-grid approximation of an ordinary differential equation with a small parameter at the highest derivative
type: item
description: This paper presents a highly accurate and efficient spectral-grid method for the nu merical solution of a boundary value problem for a fourth-order ordinary differential equation with a small parameter at the highest derivative. The method is based on Chebyshev polynomials of the first kind and involves partitioning the integration interval into multiple subintervals, ensuring the continuity of the solution and its derivatives up to the third order across the subintervals. This is particularly important in the pres ence of boundary layers that arise due to the small parameter at the highest derivative, rendering the problem singularly perturbed. An analysis of the method’s accuracy and stability is carried out at various levels of grid subdivision and numbers of approximating polynomials. Numerical experiments are presented, demonstrating high precision in the computation of both the solution and its derivatives up to the fourth order, even for very small values of the parameter (down to 10−9). The results show that increasing the number of grid elements and Chebyshev polynomials leads to a geometric reduc tion in absolute errors, confirming the convergence of the proposed approach. Thus, the proposed spectral-grid method is not only computationally efficient but also sufficiently robust and flexible for solving a wide range of problems involving singularly perturbed high-order differential equations.
subject: Статьи
  topic: section
subject: spectral-grid method
  topic: keywords
subject: number of elements and polynomials
  topic: keywords
subject: small parameter
  topic: keywords
subject: high accuracy
  topic: keywords
subject: efficiency
  topic: keywords
subject: maximum absolute errors
  topic: keywords
creator: Ch.B. Normurodov
creator: S.K. Murodov
creator: E.E. Shakaeva
rights: Copyright (c) 2025 https://creativecommons.org/licenses/by/4.0
date: 2025-05-15
publication date: 2025-05-15
starting page: 54
ending page: 63
doi: 10.71310/pcam.2_64.2025.05