Problems of Computational and Applied Mathematics
type: channel
description: 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 scientif…
publisher: Научно-исследовательский институт развития цифровых технологий и искусственного интеллекта
language: en
publication name: Problems of Computational and Applied Mathematics
issn: 2181-8460
items: Approximate Solution of Initial Value Problems for First-Order Differential Equations using a Combined Runge-Kutta And Piecewise Constant Argument Method
items: Numerical Study of the Dynamics of Derivatives of Various Orders of the Falkner–Skan Equation Depending on the Pressure Gradient
items: Numerical Modeling of Elasticity Theory Problems in terms of Stresses using the Finite Element Method
items: Membrane Fouling Characteristics During Filtration and Transport Processes in a Cylindrical Porous Filter
items: Modelling Heat and Mass Transfer in a Relief Pipeline with Constant and Varying Diameters
items: Investigation into the Dynamics of Transient Processes Associated with Flow Rate Changes at the end of a Pipeline Section, Both with and Without Resistance Force
items: Computational Modeling of a Passive-Pitch Low-Wind Vertical-Axis Wind Turbine
items: A Mathematical Model of Heat Transfer from a Liquid Coolant Flowing through a Cylindrical Pipeline Finned with Rectangular Fins
items: Investigation of the Propagation Velocity of Small Pressure Disturbances in a Gas–Liquid Medium with account for Gas Mass Concentration and Pipeline Wall Deformation
items: Mathematical Model and Analysis of Water Hammer Damping using an Air Vessel
items: Mathematical Modeling of Multiphase Mixture Inflow into a Stratified Reservoir and the Breakdown of the Layered Structure
items: Asymptotic Complexity Estimates of Hybrid Algorithms for the Numerical Solution of a Model Equation of Radon Volume Activity with a Variable-Order Fractional Derivative
items: An Application of Optimal Interpolation Formula with Derivative to Approximate Integration
items: On a Discrete System for Finding the Coefficients of Weighted Optimal Quadrature Formulas
items: An Explicit-Implicit Difference Scheme for a two-Dimensional Linear Hyperbolic System with Dynamic Boundary Conditions
items: Numerical Modeling of the Cauchy Problem for a Third-Order Singularly Perturbed Equation
items: Mathematical Modeling of the Hydrodynamic Process of In-Situ Leaching taking into account the Changes in Hydrodynamic Parameters of a Porous Medium
items: Fractional Model and Robust Numerical Algorithm for Coupled Heat and Moisture Transfer in Heterogeneous Porous Bodies
items: Forecasting the Environmental Health Index of Uzbekistan Regions using Machine Learning and Artificial Intelligence Methods
items: Mathematical Model of Groundwater Head Variation Processes in Heterogeneous Porous Media
items: Mixed-Composite-Type Equations as a Model of Anomalous Diffusion in Tumor Tissues
items: FracDynZe is a Computer Program for Studying the Dynamics of Cardiac Function using the Fractional Zeeman Oscillator
items: Graph Generation with a Prescribed Structure: From Deep Neural Networks to Quantum Models (A Case Study of Novel Drug Design)
items: Numerical Modeling of the Boundary Value Problem for a two-Parameter Singularly Perturbed Differential Equation using the Spectral-Grid Method
items: Solution of the Dirichlet Problem by the Moving Node Method
items: Finding the Extremum of the Error Functional in the Space of Periodic Functions
items: On the Solvability of the Pursuit Problem for Differential Games with Fractional Hilfer Derivatives
items: Numerical Modeling of Plane Elastoplastic Problems in Strains
items: Numerical Modeling of the Geofiltration Process on Irrigated Lands Taking into account Physical Factors
items: Properties of Solutions to Systems of Heat Conduction Equations with Nonlinear Boundary Conditions

Approximate Solution of Initial Value Problems for First-Order Differential Equations using a Combined Runge-Kutta And Piecewise Constant Argument Method
type: item
description: This study presents an efficient method for approximating the solutions of a certain class of first-order differential equations with variable coefficients. The approach constructs an auxiliary differential equation that combines the Runge-Kutta method with a piecewise constant argument, derived from the original initial value problem and parameterized by a positive integer ????. It is shown that, for sufficiently large ????, this auxiliary equation has a unique piecewise-smooth solution that approximates the considered initial value problem. Error estimates for the residual are derived to quantify the accuracy and to illustrate the influence of ????. The numerical results show that the method achieves higher accuracy with fewer computational steps than the classical Runge-Kutta and related schemes, and the framework extends to a broader class of nonlinear equations.
subject: Статьи
  topic: section
subject: initial value problem
  topic: keywords
subject: piecewise constant argument
  topic: keywords
subject: approximated solution
  topic: keywords
subject: Runge-Kutta method
  topic: keywords
subject: absolute error
  topic: keywords
creator: З.З. Жумаев
rights: Copyright (c) 2026 Z.Z.Djumaev https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 153
ending page: 163
doi: 10.71310/pcam.3_73.2026.11

Numerical Study of the Dynamics of Derivatives of Various Orders of the Falkner–Skan Equation Depending on the Pressure Gradient
type: item
description: This paper studies the dynamics of derivatives of various orders of the Falkner–Skan (FS) equation in the presence of a longitudinal pressure gradient. The FS equation is a nonlinear singularly perturbed third-order equation, and the behaviour of its solution and derivatives for various values of the form parameter ????, which is related to the pressure gradient, remains largely unexplored. The first derivative determines the velocity profile of the main flow in the boundary layer and is important for hydrodynamic stability analysis. The FS equation is reduced to a Cauchy problem for three nonlinear first-order ordinary differential equations, solved by the fourth-order Runge–Kutta method in vector form. Results are obtained for positive, zero, and negative pressure gradients. At ???? = 0 the profile coincides with the Blasius profile; for ???? < 0 the boundary layer thickens and may separate; for ???? > 0 its thickness decreases.
subject: Статьи
  topic: section
subject: Falkner–Skan equations
  topic: keywords
subject: pressure gradient
  topic: keywords
subject: velocity profile
  topic: keywords
creator: М.А. Тиловов
rights: Copyright (c) 2026 M.A. Tilovov https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 139
ending page: 152
doi: 10.71310/pcam.3_73.2026.10

Numerical Modeling of Elasticity Theory Problems in terms of Stresses using the Finite Element Method
type: item
description: The paper addresses the formulation of plane elasticity problems in terms of stresses and their numerical solution by the Galerkin finite element method. Based on the equilibrium equations, geometric relations, and physical laws of elasticity, a system of differential equations describing the plane stress state of an elastic medium is derived. The Galerkin method yields the weak form of the boundary value problem and a discrete model suitable for computer implementation. The bilinear forms and local finite element matrices required to assemble the global algebraic system are constructed. The algorithms are implemented in C++ and FreeFEM++. The approach is verified on the classical Kirsch problem of stress distribution around a circular hole in an elastic plate: both implementations agree well with each other and with known analytical solutions, confirming the correctness of the formulation and the applicability of the method.
subject: Статьи
  topic: section
subject: stress-based formulation
  topic: keywords
subject: variational problem
  topic: keywords
subject: Galerkin method
  topic: keywords
subject: finite element method
  topic: keywords
subject: basis functions
  topic: keywords
subject: Kirsch problem
  topic: keywords
subject: FreeFEM++
  topic: keywords
creator: A.A. Khaldjigitov
creator: A.A. Bobonazarov
creator: R.A. Rakhmonova
rights: Copyright (c) 2026 A.A. Khaldjigitov https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 125
ending page: 138
doi: 10.71310/pcam.3_73.2026.09

Membrane Fouling Characteristics During Filtration and Transport Processes in a Cylindrical Porous Filter
type: item
description: This study presents a mathematical and numerical model for filtration flow, solute transport, adsorption, and membrane fouling in a cylindrical porous filter. The model combines axisymmetric Brinkman–Darcy flow equations with advection–diffusion–reaction equations, Langmuir adsorption, and the linear driving force (LDF) model. It also considers porosity changes, particle deposition, cake formation, osmotic pressure, and membrane resistance growth. A finite difference scheme with CFL-based stability control is used for simulations. Results show that adsorption and particle deposition significantly reduce porosity and membrane permeability, leading to increased hydraulic resistance and flux decline. The model provides a useful framework for predicting fouling dynamics and optimizing porous filtration systems.
subject: Статьи
  topic: section
subject: cylindrical porous filter
  topic: keywords
subject: membrane fouling
  topic: keywords
subject: Brinkman–Darcy model
  topic: keywords
subject: advection–diffusion–reaction equations
  topic: keywords
subject: adsorption kinetics
  topic: keywords
subject: Langmuir isotherm
  topic: keywords
subject: concentration polarization
  topic: keywords
subject: osmotic pressure
  topic: keywords
subject: cake layer formation
  topic: keywords
subject: porous media transport
  topic: keywords
subject: finite difference method
  topic: keywords
subject: filtration processes
  topic: keywords
creator: N. Ravshanov
creator: B.I. Boborakhimov
creator: Sh.Sh. Berdiyorov
rights: Copyright (c) 2026 B.I. Boborakhimov https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 104
ending page: 124
doi: 10.71310/pcam.3_73.2026.08

Modelling Heat and Mass Transfer in a Relief Pipeline with Constant and Varying Diameters
type: item
description: This work presents a comprehensive numerical and analytical study of hydrodynamic and thermal processes in closed pipelines of variable diameter, taking into account terrain relief and external thermal influences. The novelty lies in an improved quasi-onedimensional model that assesses the impact of geometric parameters (diameter, inclination angle, internal roughness) and thermophysical characteristics (material thermal conductivity, ambient temperature) on the pressure and temperature distribution along the pipeline. The model was validated against SST-turbulence-model simulations and Darcy–Weisbach calculations, with a maximum deviation below 2%. Hydraulic losses were found to depend mainly on pipe diameter and flow velocity, whereas thermal losses depend on the material’s thermal conductivity and the pipeline orientation. The proposed approach provides a reliable and computationally efficient basis for predicting the energy characteristics of fluid-transport systems and optimizing thermal networks.
subject: Статьи
  topic: section
subject: fluid flow
  topic: keywords
subject: heat transfer
  topic: keywords
subject: pipe flow
  topic: keywords
creator: Sh.A. Ravshanov
creator: M.I. Boborakhimova
creator: Sh.I. Chulliev
rights: Copyright (c) 2026 Sh.A. Ravshanov https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 90
ending page: 103
doi: 10.71310/pcam.3_73.2026.07

Investigation into the Dynamics of Transient Processes Associated with Flow Rate Changes at the end of a Pipeline Section, Both with and Without Resistance Force
type: item
description: Within a non-conservative formulation of the quasi-one-dimensional linearized mass and momentum conservation equations, transient processes are studied in an elementary undulating pipeline section caused by step-like changes in the boundary velocities of an incompressible fluid. To determine the average flow velocity, an incomplete telegrapher’s equation is derived and solved by separation of variables. The pressure distribution is then obtained by integrating the original system using the resulting velocity expression. Numerical experiments are carried out for constant values of the functions in the initial and boundary conditions. Flow regimes with constant, increasing, and decreasing mean pressure along the section are examined. The features of shock-wave propagation are demonstrated both with and without the friction force.
subject: Статьи
  topic: section
subject: fluid
  topic: keywords
subject: pipeline
  topic: keywords
subject: velocity
  topic: keywords
subject: pressure
  topic: keywords
subject: Fourier method
  topic: keywords
subject: speed of sound
  topic: keywords
subject: resistance force
  topic: keywords
subject: computational experiment
  topic: keywords
creator: Sh.Sh. Khozhikulov
creator: O.M. Begimov
creator: A.J. Obidjonov
rights: Copyright (c) 2026 Ш.Ш. Хожикулов https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 75
ending page: 89
doi: 10.71310/pcam.3_73.2026.06

Computational Modeling of a Passive-Pitch Low-Wind Vertical-Axis Wind Turbine
type: item
description: This paper presents a three-dimensional numerical study of a novel vertical-axis wind turbine (VAWT) with a unique aerodynamic profile and a passive blade-pitch mechanism. Rectangular blades are mounted on horizontal axes through articulated bearings and rotate freely up to 90∘, constrained by a pin-and-belt system: on the power stroke the blades hit a stopper to capture maximum energy, while on the return stroke they open to reduce drag, enabling efficient operation at low wind speeds (3–5 m/s) with stable torque. After a grid-independence study, the aerodynamics were analyzed in COMSOL Multiphysics by solving the RANS equations. Four turbulence models (SST, ????−????, ????−????, RNG) were compared; the SST model best captured flow separation and wake structures. Power-coefficient (????????) versus tip-speed-ratio (TSR) curves were obtained and validated against laboratory data (relative error < 5%).
subject: Статьи
  topic: section
subject: Navier–Stokes equations
  topic: keywords
subject: wind turbine
  topic: keywords
subject: Comsol Multiphysics
  topic: keywords
subject: computational fluid dynamics
  topic: keywords
subject: renewable energy
  topic: keywords
subject: velocity
  topic: keywords
subject: pressure
  topic: keywords
creator: S.A. Muzaffarov
creator: Kh.U. Maratov
creator: M.M. Hamdamov
rights: Copyright (c) 2026 S.A. Muzaffarov https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 61
ending page: 74
doi: 10.71310/pcam.3_73.2026.05

A Mathematical Model of Heat Transfer from a Liquid Coolant Flowing through a Cylindrical Pipeline Finned with Rectangular Fins
type: item
description: This paper develops a quasi-one-dimensional mathematical model of steady-state heat transfer from a thin rectangular fin installed on a cylindrical pipeline parallel to its generatrix. Unlike classical formulations, third-kind boundary conditions are specified at both fin bases, accounting for heat exchange with the coolant and the environment, as well as heat transfer through the fin’s lateral surfaces. An analytical solution of the heat-conduction equation describing the temperature distribution along the fin height is obtained. The influence of fin height and thickness and of the material’s thermophysical properties on the temperature field is analyzed numerically. Increasing the fin thickness is shown to slow the temperature decrease along the height and to raise the total heat removed. Formulas for the finning efficiency and for the streamwise change of the coolant temperature are proposed. The results confirm the model’s validity and its applicability for evaluating the finning efficiency of heat-exchange surfaces.
subject: Статьи
  topic: section
subject: fin
  topic: keywords
subject: heat transfer
  topic: keywords
subject: quasi-one-dimensional mathematical model
  topic: keywords
subject: steadystate thermal conductivity
  topic: keywords
subject: third-kind boundary conditions
  topic: keywords
subject: finning efficiency
  topic: keywords
subject: analog of Shukhov’s formula
  topic: keywords
creator: D.Y. Ergashev
creator: J.I. Khujaev
creator: S.S. Akhmadjonov
rights: Copyright (c) 2026 D.Y. Ergashev https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 50
ending page: 60
doi: 10.71310/pcam.3_73.2026.04

Investigation of the Propagation Velocity of Small Pressure Disturbances in a Gas–Liquid Medium with account for Gas Mass Concentration and Pipeline Wall Deformation
type: item
description: This paper investigates how the propagation velocity of small pressure disturbances in a gas–liquid medium varies with the mass concentration of air. Two cases are considered: a rigid pipeline and a pipeline accounting for wall deformation. A “water–air” mixture, in which gas bubbles are present in the liquid, is adopted as the model of the medium. Formulas for the propagation velocity with and without wall deformability are derived in accordance with Hooke’s law. The analysis covers bubbly and emulsion flow regimes at low mass concentrations of the phases. Analytical dependences of the velocity on the mass concentration of air are obtained and analyzed graphically. It is shown that even small gas concentrations markedly reduce the wave propagation velocity, while accounting for wall deformation further decreases the speed of sound.
subject: Статьи
  topic: section
subject: small pressure disturbances
  topic: keywords
subject: gas–liquid medium
  topic: keywords
subject: flow regimes
  topic: keywords
subject: gas mass concentration
  topic: keywords
subject: bubbly flow
  topic: keywords
subject: wall deformation
  topic: keywords
subject: pipeline
  topic: keywords
creator: O.M. Begimov
creator: I.K. Khujaev
creator: Kh.A. Mamadaliev
rights: Copyright (c) 2026 О.М. Бегимов https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 37
ending page: 49
doi: 10.71310/pcam.3_73.2026.03

Mathematical Model and Analysis of Water Hammer Damping using an Air Vessel
type: item
description: This paper presents an improved quasi-one-dimensional model of unsteady compressible fluid flow in main pipelines for studying water hammer damping. Its novelty lies in a rigorous nonlinear boundary condition based on the conservation of mass of the trapped gas in the air vessel (damper). The governing equations are integrated by the traveling-wave method with an implicit finite-difference scheme. Verification against a reference MATLAB Simulink model confirmed high accuracy (5–8% error). A multivariate analysis details how damper volume, initial gas pressure, and pipeline diameter affect the system’s inertia. Critical thresholds for destructive cavitation and “overdamping” are established, demonstrating the need for individual calibration of protective devices for reliable pipeline operation.
subject: Статьи
  topic: section
subject: water hammer
  topic: keywords
subject: gas-hydraulic damper
  topic: keywords
subject: air vessel
  topic: keywords
subject: mathematical modeling
  topic: keywords
subject: traveling wave method
  topic: keywords
subject: unsteady flow
  topic: keywords
subject: cavitation
  topic: keywords
subject: main pipeline
  topic: keywords
creator: B.B. Bakhtiyorov
creator: I.K. Khujaev
creator: N.V. Turopova
rights: Copyright (c) 2026 I.K. Khujaev https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 25
ending page: 36
doi: 10.71310/pcam.3_73.2026.02

Mathematical Modeling of Multiphase Mixture Inflow into a Stratified Reservoir and the Breakdown of the Layered Structure
type: item
description: A three-dimensional mathematical model is proposed for the inflow of a turbid (sediment-laden) river flow into a density-stratified reservoir, accounting for phase separation and the breakdown of the layered structure. The model is based on Kh.A. Rakhmatulin’s theory of interpenetrating continua: the carrier fluid and the dispersed particles each have their own velocity field, and the interphase drag represents particle settling. The vertical mixing coefficient depends on the local Richardson number, capturing stratification breakdown through Kelvin–Helmholtz instability; re-erosion near the bed is modeled by the Garcia–Parker formula. The problem is solved on a Harlow–Welch staggered grid by a five-stage operator-splitting scheme. A simulation of the subcritical plunging regime reproduces the inflow plume, the plunge point, the bottom underflow, and the intrusion layer, and is verified against classical criteria and mass balance.
subject: Статьи
  topic: section
subject: stratified reservoir
  topic: keywords
subject: multiphase flow
  topic: keywords
subject: Rakhmatulin model
  topic: keywords
subject: phase separation
  topic: keywords
subject: plunging
  topic: keywords
subject: Richardson number
  topic: keywords
subject: Kelvin–Helmholtz instability
  topic: keywords
subject: operator splitting
  topic: keywords
subject: re-erosion
  topic: keywords
subject: Garcia–Parker formula
  topic: keywords
creator: D.S. Yakhshibaev
creator: B.I. Boborakhimov
rights: Copyright (c) 2026 B.I. Boborakhimov https://creativecommons.org/licenses/by/4.0
date: 2026-07-02
publication date: 2026-07-02
starting page: 7
ending page: 24
doi: 10.71310/pcam.3_73.2026.01

Asymptotic Complexity Estimates of Hybrid Algorithms for the Numerical Solution of a Model Equation of Radon Volume Activity with a Variable-Order Fractional Derivative
type: item
description: The article considers hybrid CPU-GPU parallel implementations of numerical solution algorithms for the hereditary model equation of radon volume activity. The test example is a direct Cauchy problem for a nonlinear fractional differential equation with a Gerasimov-Caputo operator of variable order and variable coefficients. The importance of developing efficient algorithms for solving direct problems of the radon volume activity model is due to their use in solving corresponding inverse problems based on radon monitoring data in order to solve practical problems of identifying certain parameters of the geological environment. Based on data on the average execution time of the test problem, asymptotic estimates of the complexity of sequential and proposed parallel algorithms are given. It is shown that the use of hybrid parallel CPU-GPU algorithms provides a performance gain of up to 17 times and can give a significant advantage in solving problems with large amounts of experimental data, due to the use of a GPU node. It is also shown that asymptotically exact complexity estimates are: for memory, for all hybrid algorithms of order Θ(????2); for the hybrid implementation of a non-local explicit scheme of order Θ(????); for the hybrid implementation of a non-local implicit scheme of order Θ(????2).
subject: Статьи
  topic: section
subject: fractional derivatives
  topic: keywords
subject: time nonlocality
  topic: keywords
subject: variable nonlocality
  topic: keywords
subject: finite difference schemes
  topic: keywords
subject: parallel computing
  topic: keywords
subject: GPU
  topic: keywords
subject: asymptotic estimates
  topic: keywords
subject: algorithms efficiency
  topic: keywords
creator: D.A. Tverdyi
rights: Copyright (c) 2026 D.A. Tverdyi https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 155
ending page: 167
doi: 10.71310/pcam.2_72.2026.11

An Application of Optimal Interpolation Formula with Derivative to Approximate Integration
type: item
description: In this paper, an optimal interpolation formula with derivative constructed in the Sobolev space is considered. The formula interpolates an unknown function using its values and derivatives first three order at equally spaced nodes. Explicit expressions for the coefficients of the corresponding quadrature formula are derived by integrating the interpolation basis functions. A theorem giving the exact form of the coefficients is presented. Numerical experiments are carried out for several smooth functions, and the absolute errors of the approximate integration are analyzed for different values of ????. The results show that the proposed approach provides high accuracy and can be effectively used for numerical integration problems where derivative information is available. In addition, the stability of the proposed formula with respect to perturbations in the input data is examined, and its asymptotic behavior as the number of nodes increases is discussed. Comparisons with classical interpolation and quadrature formulas demonstrate the advantage of incorporating derivative information, especially for highly smooth functions. The method also provides a constructive framework for extending optimal interpolation formulas to higher-order derivatives and nonuniform meshes. These results contribute to the broader development of optimal computational schemes in Sobolev spaces and highlight potential applications in solving boundary value problems, differential equations, and numerical simulation tasks requiring high precision.
subject: Статьи
  topic: section
subject: interpolation
  topic: keywords
subject: spline
  topic: keywords
subject: quadrature formula
  topic: keywords
subject: integration
  topic: keywords
subject: approximation
  topic: keywords
creator: N.N. Olimov
rights: Copyright (c) 2026 Н.Н. Олимов https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 147
ending page: 154
doi: 10.71310/pcam.2_72.2026.10

On a Discrete System for Finding the Coefficients of Weighted Optimal Quadrature Formulas
type: item
description: Fourier integrals of functions appear primarily in the theory of special functions and Fourier analysis, but also in other applied and computational sciences and engineering, such as theoretical physics, acoustic scattering, quantum chemistry, transport theory, electromagnetism, telecommunications, mechanics, and others. This paper considers the problem of finding the coefficients of weighted optimal quadrature formulas. First, we solve the boundary value problem for the extremal function of the quadrature formula. Using the extremal function, we find the norm of the error functional. The norm of the error functional depends on the coefficients and nodes. Using the Lagrange method, we obtain a system of linear algebraic equations for finding the coefficients of weighted optimal quadrature formulas.
subject: Статьи
  topic: section
subject: Hilbert space
  topic: keywords
subject: extremal function
  topic: keywords
subject: squared norm of the error functional
  topic: keywords
subject: Fourier coefficients
  topic: keywords
subject: weighted quadrature formulas
  topic: keywords
creator: A.K. Boltaev
rights: Copyright (c) 2026 A.K. Boltaev https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 136
ending page: 146
doi: 10.71310/pcam.2_72.2026.09

An Explicit-Implicit Difference Scheme for a two-Dimensional Linear Hyperbolic System with Dynamic Boundary Conditions
type: item
description: This paper considers a two-dimensional linear hyperbolic system with dynamic boundary conditions and proposes a difference scheme for its numerical solution. An explicit–implicit directional splitting method is constructed, where discretization is performed explicitly in one direction and implicitly in the other, while preserving the dissipative structure of the boundary conditions. The stability of the scheme is established under the Courant–Friedrichs–Lewy condition and a linear matrix inequality. It is shown that the discrete energy decreases exponentially in time. Numerical experiments confirm the theoretical results, demonstrating monotonic decay of the discrete ????2-norm and good agreement with the exact solution. The proposed method is stable, dissipative, and computationally efficient, and can be effectively applied to two-dimensional hyperbolic systems with dynamic boundary conditions.
subject: Статьи
  topic: section
subject: hyperbolic system
  topic: keywords
subject: dynamic boundary condition
  topic: keywords
subject: difference scheme
  topic: keywords
subject: directional splitting
  topic: keywords
subject: explicit–implicit method
  topic: keywords
subject: exponential stability
  topic: keywords
subject: CFL condition
  topic: keywords
creator: R.D. Aloev
creator: M.Kh. Ovlaeva
creator: Ilyani Abdullah
creator: N.T. Issayeva
rights: Copyright (c) 2026 Р.Д. Алоев https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 122
ending page: 135
doi: 10.71310/pcam.2_72.2026.08

Numerical Modeling of the Cauchy Problem for a Third-Order Singularly Perturbed Equation
type: item
description: This article presents a highly accurate and efficient method—a discrete version of the preintegration method—for numerically solving the Cauchy problem for a singularly perturbed third-order equation. The method is based on expanding the highest derivative of the differential equation into a finite series in Chebyshev polynomials of the first kind with unknown expansion coefficients. All lower derivatives and an approximate solution to the differential equation are determined by preintegrating the series for the highest derivative using a discrete integration formula that reduces the order of the highest derivative. This yields the fundamental algebraic equations of the proposed method. By adding additional equations derived from three initial conditions to these fundamental equations, a system of linear algebraic equations is obtained for determining the unknown expansion coefficients for the proposed solution. The number of equations and the number of unknowns in the resulting algebraic system coincide. This system is solved by a standard method; in this paper, it is solved by the Gaussian method. The calculation results show that for arbitrary values of the small problem parameter, a slight increase in the number of Chebyshev polynomials leads to a geometrically progressive reduction in absolute errors. Thus, the proposed discrete version of the pre-integration method is not only computationally efficient but also highly accurate and sufficiently versatile for solving a wide range of problems involving singularly perturbed equations.
subject: Статьи
  topic: section
subject: Chebyshev polynomials
  topic: keywords
subject: preliminary integration method
  topic: keywords
subject: discrete version
  topic: keywords
subject: Cauchy problem
  topic: keywords
subject: singularly perturbed equations
  topic: keywords
subject: algebraic system
  topic: keywords
creator: E.E. Shakaeva
rights: Copyright (c) 2026 E.E. Shakaeva https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 109
ending page: 121
doi: 10.71310/pcam.2_72.2026.07

Mathematical Modeling of the Hydrodynamic Process of In-Situ Leaching taking into account the Changes in Hydrodynamic Parameters of a Porous Medium
type: item
description: This article presents a mathematical model and algorithm for numerically solving a problem for studying the hydrodynamic process of in-situ leaching in a heterogeneous porous medium, taking into account mass transfer kinetics, the law of layer deformation dependent on the elastic coefficient, changes in porosity depending on pressure, and the permeability of the ore reservoir for the purpose of developing ore deposits. The developed mathematical apparatus allows for a comprehensive study of the properties of the ore seam, optimization of the location of production and injection wells, determination of changes in porosity under pressure, and analysis and provision of a factor ensuring the protection of groundwater from pollution. The model under consideration is described using a system of second-order parabolic partial differential equations with initial, boundary, and internal conditions. Since the analytical solution to this problem was too complex or impossible for numerical integration, a second-order finite-difference scheme was used.
subject: Статьи
  topic: section
subject: in-situ leaching
  topic: keywords
subject: mathematical modeling
  topic: keywords
subject: filtration
  topic: keywords
subject: diffusion
  topic: keywords
subject: kinetics
  topic: keywords
subject: useful component
  topic: keywords
subject: numerical methods
  topic: keywords
creator: L.S. Usmonov
rights: Copyright (c) 2026 L.S. Usmonov https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 89
ending page: 108
doi: 10.71310/pcam.2_72.2026.06

Fractional Model and Robust Numerical Algorithm for Coupled Heat and Moisture Transfer in Heterogeneous Porous Bodies
type: item
description: This research presents a multidimensional mathematical model and a robust secondorder numerical algorithm for coupled heat and moisture transfer with pressure-driven gas flow. Utilizing fractional Caputo derivatives (0 < ???? ⩽ 1), the model accounts for system memory and anomalous diffusion in heterogeneous porous media, integrating convection, phase transitions, and pressure dynamics. It incorporates environmental interactions, internal heat/moisture sources, solar radiation, and spatially varying transfer coefficients. A high-order stable difference scheme ensures computational efficiency and numerical stability. The algorithm effectively predicts the spatiotemporal evolution of temperature, moisture, and pressure fields. Results highlight how material heterogeneity and diurnal solar radiation create localized zones of high temperature and moisture accumulation, significantly impacting the overall dynamics of storage and drying processes.
subject: Статьи
  topic: section
subject: mathematical model
  topic: keywords
subject: Caputo fractional derivative
  topic: keywords
subject: heat transfer
  topic: keywords
subject: moisture transfer
  topic: keywords
subject: internal heat and moisture release
  topic: keywords
subject: heterogeneous porous body
  topic: keywords
creator: I.U. Shadmanov
creator: A.E. Izzatulloev
creator: Suhendro Busono
rights: Copyright (c) 2026 I.U.Shadmanov https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 61
ending page: 88
doi: 10.71310/pcam.2_72.2026.05

Forecasting the Environmental Health Index of Uzbekistan Regions using Machine Learning and Artificial Intelligence Methods
type: item
description: Recent years have witnessed significant climatic changes and increasing environmental pressure globally, including in Uzbekistan, necessitating an objective regional assessment. This research develops an approach for forecasting the Environmental Health Index (EHI) by integrating statistical analysis and machine learning algorithms. Long-term meteorological data from 13 regions (temperature, humidity, wind speed) were normalized to construct the EHI, with the Random Forest regressor applied to model nonlinear dependencies. Model performance was validated using MAE, MSE, and ????2 metrics. For the first time, an EHI based on climatic factors was developed specifically for Uzbekistan using a data-driven approach with automated feature weighting. Results demonstrate a stable correlation between climate variables and environmental status, suggesting potential degradation in certain regions if current trends persist.
subject: Статьи
  topic: section
subject: machine learning
  topic: keywords
subject: Random Forest
  topic: keywords
subject: ecological forecasting
  topic: keywords
subject: climate factors
  topic: keywords
subject: regional analysis
  topic: keywords
subject: sustainable development
  topic: keywords
creator: N. Ravshanov
creator: Achmad Tirta Dharu Wahyu Pambudi
creator: Muhammad Safari
creator: F. Kamoliddinova
rights: Copyright (c) 2026 N. Ravshanov https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 42
ending page: 60
doi: 10.71310/pcam.2_72.2026.04

Mathematical Model of Groundwater Head Variation Processes in Heterogeneous Porous Media
type: item
description: This research develops a mathematical formulation and numerical solution for the spatial-temporal distribution of groundwater head in heterogeneous porous media. Based on Darcy’s and mass conservation laws, a 3D parabolic partial differential equation for anisotropic aquifers is derived using tensor descriptions. The model incorporates detailed boundary conditions (river interaction, infiltration, evapotranspiration) and satisfies Hadamard criteria for correctness. Heterogeneity is addressed through deterministic and stochastic depth-dependent hydraulic conductivity models. Numerical results are obtained via an explicit finite difference scheme using harmonic mean interface conductivities, verified by von Neumann stability analysis and the CFL condition. The algorithm supports parallel computing on multi-core processors and GPUs. These results are applicable to groundwater resource management, drainage system design, and contaminant transport modeling.
subject: Статьи
  topic: section
subject: hydrogeology
  topic: keywords
subject: seepage
  topic: keywords
subject: mathematical physics
  topic: keywords
subject: computational mathematics
  topic: keywords
subject: high-performance computing
  topic: keywords
creator: R. Qodirov
creator: B. Boborakhimov
rights: Copyright (c) 2026 B. Boborakhimov https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 27
ending page: 41
doi: 10.71310/pcam.2_72.2026.03

Mixed-Composite-Type Equations as a Model of Anomalous Diffusion in Tumor Tissues
type: item
description: This paper examines a boundary value problem for a degenerate elliptic equation in a planar domain bounded by a line segment and an analytic curve. The primary objective is to construct an explicit analytical solution using the Green’s function together with singleand double-layer potential methods. A rigorous proof of the existence and uniqueness of the solution under mixed boundary conditions is presented. The degeneracy on a portion of the boundary introduces significant analytical difficulties, necessitating the use of advanced techniques in the theory of elliptic equations with singular coefficients. Furthermore, the potential applications of the model in biomedical settings, particularly in oncology, are discussed. The equation captures anomalous diffusion processes in tumor tissues, incorporating the spatial heterogeneity of the medium. This makes the model a valuable tool for analyzing the distribution of drugs, oxygen, and other substances in biological structures characterized by pronounced anisotropy and heterogeneity.
subject: Статьи
  topic: section
subject: Green’s function
  topic: keywords
subject: potential
  topic: keywords
subject: Jordan arc
  topic: keywords
subject: mixed type equation
  topic: keywords
subject: Lipschitz condition
  topic: keywords
subject: biomedicine
  topic: keywords
subject: oncology
  topic: keywords
subject: tumor
  topic: keywords
subject: anisotropy
  topic: keywords
creator: N.K. Ochilova
rights: Copyright (c) 2026 Н.К. Очилова https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 16
ending page: 26
doi: 10.71310/pcam.2_72.2026.02

FracDynZe is a Computer Program for Studying the Dynamics of Cardiac Function using the Fractional Zeeman Oscillator
type: item
description: This paper examines the dynamic modes of the fractional Zeeman oscillator. The fractional Zeeman oscillator is a system of two ordinary differential equations with fractional derivatives, understood in the Gerasimov-Caputo sense, for which local initial conditions are valid. The fractional Zeeman oscillator describes the dynamics of heart contractions using an electrochemical potential. Due to the nonlinearity of the fractional Zeeman oscillator, a numerical algorithm based on a nonlocal explicit finite-difference scheme was used to obtain a solution. The numerical algorithm was implemented in the FracDynZe computer program, written in Python in the PyCharm environment. The software package allows for visualization and saving of simulation results. This article describes the FracDynZe computer program, which implements a numerical algorithm for a nonlocal explicit finite-difference scheme. Using FracDynZe, test examples demonstrate that the numerical scheme has first-order accuracy. Examples and their visualizations are provided for various values of the Zeeman fractional oscillator parameters. The FracDynZe computer program can be supplemented with a module for the qualitative analysis of the Zeeman fractional oscillator. For example, this module can implement the ability to construct bifurcation diagrams for studying regular and chaotic regimes.
subject: Статьи
  topic: section
subject: fractional derivatives
  topic: keywords
subject: modeling
  topic: keywords
subject: phase trajectory
  topic: keywords
subject: stability
  topic: keywords
subject: memory
  topic: keywords
subject: python
  topic: keywords
subject: numerical methods
  topic: keywords
creator: R.I. Parovik
creator: G.S. Israyiljanova
rights: Copyright (c) 2026 R.I. Parovik https://creativecommons.org/licenses/by/4.0
date: 2026-05-02
publication date: 2026-05-02
starting page: 5
ending page: 15
doi: 10.71310/pcam.2_72.2026.01

Graph Generation with a Prescribed Structure: From Deep Neural Networks to Quantum Models (A Case Study of Novel Drug Design)
type: item
description: 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.
subject: Статьи
  topic: section
subject: reinforcement learning
  topic: keywords
subject: WGAN
  topic: keywords
subject: implicit generative models
  topic: keywords
subject: graph connectivity enforcement
  topic: keywords
subject: quantum–classical hybrid models
  topic: keywords
subject: molecular design
  topic: keywords
subject: molecule generation benchmarks
  topic: keywords
creator: F.T. Adilova
creator: R.R. Davronov
rights: Copyright (c) 2026 F.T. Adilova https://creativecommons.org/licenses/by/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 123
ending page: 142
doi: 10.71310/pcam.1_71.2026.11

Numerical Modeling of the Boundary Value Problem for a two-Parameter Singularly Perturbed Differential Equation using the Spectral-Grid Method
type: item
description: 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.
subject: Статьи
  topic: section
subject: spectral-grid method
  topic: keywords
subject: two-parameter singularly perturbed differential equation
  topic: keywords
subject: Chebyshev polynomials
  topic: keywords
subject: maximum absolute error
  topic: keywords
subject: boundary layers
  topic: keywords
subject: numerical approximation
  topic: keywords
subject: high-accuracy algorithm
  topic: keywords
creator: S.K. Murodov
rights: Copyright (c) 2026 S.K. Murodov https://creativecommons.org/licenses/by/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 113
ending page: 122
doi: 10.71310/pcam.1_71.2026.10

Solution of the Dirichlet Problem by the Moving Node Method
type: item
description: 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.
subject: Статьи
  topic: section
subject: movable node
  topic: keywords
subject: Poisson equation
  topic: keywords
subject: Dirichlet problem
  topic: keywords
subject: method of lines
  topic: keywords
creator: U. Dalabaev
creator: D. Khasanova
rights: Copyright (c) 2026 U. Dalabaev https://creativecommons.org/licenses/by/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 103
ending page: 112
doi: 10.71310/pcam.1_71.2026.09

Finding the Extremum of the Error Functional in the Space of Periodic Functions
type: item
description: 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.
subject: Статьи
  topic: section
subject: space
  topic: keywords
subject: extremal function
  topic: keywords
subject: generalized function
  topic: keywords
subject: operator
  topic: keywords
subject: optimal quadrature formula
  topic: keywords
subject: error functional
  topic: keywords
creator: S.S. Azamov
creator: D.B. Bekmurodova
rights: Copyright (c) 2026 S.S. Azamov https://creativecommons.org/licenses/by/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 94
ending page: 102
doi: 10.71310/pcam.1_71.2026.08

On the Solvability of the Pursuit Problem for Differential Games with Fractional Hilfer Derivatives
type: item
description: 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.
subject: Статьи
  topic: section
subject: differential games with Hilfer fractional derivatives
  topic: keywords
subject: Banach space
  topic: keywords
subject: pursuit problem
  topic: keywords
subject: pursuit time optimality
  topic: keywords
creator: E.M. Mukhsinov
creator: R.I. Hakimov
rights: Copyright (c) 2026 E.M. Mukhsinov https://creativecommons.org/licenses/by/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 82
ending page: 93
doi: 10.71310/pcam.1_71.2026.07

Numerical Modeling of Plane Elastoplastic Problems in Strains
type: item
description: 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.
subject: Статьи
  topic: section
subject: compatibility condition
  topic: keywords
subject: plasticity
  topic: keywords
subject: strains
  topic: keywords
subject: displacement
  topic: keywords
subject: finite-difference equations
  topic: keywords
subject: stress
  topic: keywords
subject: elimination and iteration methods
  topic: keywords
creator: U.Z. Djumayozov
creator: R.A. Rakhmonova
creator: M.N. Abdirakhmonova
rights: Copyright (c) 2026 U.Z. Djumayozov https://creativecommons.org/licenses/by/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 71
ending page: 81
doi: 10.71310/pcam.1_71.2026.06

Numerical Modeling of the Geofiltration Process on Irrigated Lands Taking into account Physical Factors
type: item
description: 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.
subject: Статьи
  topic: section
subject: groundwater level
  topic: keywords
subject: Darcy’s law
  topic: keywords
subject: achieving maximum concentration of the medium
  topic: keywords
subject: filtration rate
  topic: keywords
subject: temperature
  topic: keywords
subject: concentration
  topic: keywords
subject: saturation
  topic: keywords
subject: irrigation water
  topic: keywords
subject: level function
  topic: keywords
creator: N. Kurbonov
creator: B. Boborakhimov
creator: D. Khaknazarova
creator: B. Murodullaev
rights: Copyright (c) 2026 B. Murodullaev https://creativecommons.org/licenses/by/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 57
ending page: 70
doi: 10.71310/pcam.1_71.2026.05

Properties of Solutions to Systems of Heat Conduction Equations with Nonlinear Boundary Conditions
type: item
description: 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.
subject: Статьи
  topic: section
subject: system of nonlinear heat conduction equations
  topic: keywords
subject: global solution
  topic: keywords
subject: unbounded solution
  topic: keywords
subject: asymptotics
  topic: keywords
creator: A.R. Zaripova
rights: Copyright (c) 2026 А.Р. Зарипова https://creativecommons.org/licenses/by/4.0
date: 2026-03-07
publication date: 2026-03-07
starting page: 43
ending page: 56
doi: 10.71310/pcam.1_71.2026.04