CERTAIN ASPECTS OF MATHEMATICAL MODELLING

THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 125 CERTAIN ASPECTS OF MATHEMATICAL MODELLING Settiev Sh.R, Pardaeva N. A. Tashkent Perfect University, Tashkent, Uzbekistan https://doi.org/10.5281/zenodo.17725669 Abstract. Research using models is often the only feasible method for experimental study and solving critical practical problems. Mathematical modeling enables the formalization of complex processes in nature, technology, and society, replacing direct experiments with their analytical and numerical counterparts. This article examines the main stages and challenges in constructing mathematical models, issues of similarity and dimensionless parameters, as well as methods for solving and interpreting numerical calculations. Special attention is given to analyzing difficulties arising from equation nonlinearity, selection of numerical methods, and ensuring solution stability and convergence. Keywords: experiment, model, differential equations, numerical methods, stability, convergence. Annotatsiya. Modellar yordamidagi tadqiqotlar ko‘pincha eksperimental o‘rganish va muhim amaliy muammolarni hal qilishning yagona mumkin bo‘lgan usulidir. Matematik modellashtirish tabiat, texnika va jamiyatdagi murakkab jarayonlarni formallashtirish, bevosita tajribalarni ularning analitik va sonli analoglari bilan almashtirish imkonini beradi. Ushbu maqolada matematik modellarni qurishning asosiy bosqichlari va muammolari, o‘xshashlik va o‘lchovsiz parametrlar masalalari, shuningdek, sonli hisoblashlarni yechish va talqin qilish usullari ko‘rib chiqiladi. Tenglamaning chiziqlimasligidan kelib chiqadigan qiyinchiliklarni tahlil qilish, sonli usullarni tanlash, yechimning turg‘unligi va yaqinlashishini ta’minlashga alohida e’tibor qaratilgan. Kalit so‘zlar: tajriba, model, differensial tenglamalar, sonli usullar, turg‘unlik, yaqinlashish. Introduction. The primary purpose of modeling is to provide necessary insights into the nature of effects and various quantities related to real processes and phenomena based on the results of research conducted with the model. In most cases, modeling involves constructing models that adequately represent the real process, taking into account the research objectives and specific tasks. Typically, modeling tools are employed when direct study of the real object is challenging. Consequently, the investigation of a real process or phenomenon is substituted by examining its model in various forms, particularly through mathematical equations or functional relationships. The real object and its model are considered analogous if the results of analyzing one can be applied to the characteristics of the other, implying a transition from the outcomes of solving and analyzing the model to real processes. To accomplish this, it is essential to know the "transitional scales," which define the relationship between the model and the full-scale object. Methods and theoretical foundations of modeling. Parameters and indicators for two different but similar phenomena can be considered as numerical characteristics of the same phenomenon expressed in two different systems of measurement units. To maintain similarity during modeling, it is necessary to observe the corresponding THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 126 conditions. However, these conditions, which ensure the similarity of the phenomenon as a whole, are not always met. In such cases, the question arises about the magnitude of errors (scale effect) that occur when transferring the results obtained from the model to the real object. In such cases, the following theorem is used: For the similarity of two phenomena, it is necessary and sufficient that the numerical values of the dimensionless combinations forming the basis in these two phenomena be identical. If the similarity conditions are met, then to actually calculate all characteristics in the fullscale object based on dimensional parameter data from the model solution, it is necessary to know the conversion scales for all corresponding indicators. Applying this theorem to the system of equations describing the given model, we obtain so-called dimensionless equation systems, where the main parameters of the problem are dimensionless parameters - Mach, Strouhal, Reynolds, Froude numbers [1], and so on. It should be noted that the resulting solution of the system of equations, whether analytical or numerical, will be dimensionless. To obtain dimensional solutions, it is necessary to reverse the process using the formulas for dimensionless parameters. This determines at which scales, speeds, or angles of inclination the given solution is valid. Stages of mathematical modeling, problems and their solutions. Research using models is often the only possible way to experimentally study and solve the most critical practical problems. The first stage of mathematical modeling of natural phenomena involves considering the fundamental laws of physics - the laws of conservation of mass, energy, and momentum, the laws of thermodynamics, etc [2]. In the second stage, the resulting system of differential equations (regardless of type) is reduced to a dimensionless form. In the third stage, the dimensionless system of equations is solved by analytical methods if possible, or by appropriate numerical methods if an analytical solution is not available. In the fourth stage, the obtained solutions (analytical or numerical) are processed using information technology methods - graphs and histograms are constructed, relationships are calculated, and conclusions are formulated [3]. A thorough analysis of existing problems in mathematical modeling revealed the following. When constructing models of real phenomena at the formalization stage, a system of differential equations (in partial derivatives) is obtained, the solution of which presents various challenges: a) The number of variables in the system exceeds the number of equations. Solution: Introduce additional assumptions to equalize the number of variables and equations, although this may reduce the model's adequacy [4]. b) Nonlinearity of differential equations and absence of exact solutions. Solution: linearization or application of a combination of numerical methods. c) Choosing a numerical method and implementing it in a programming language. Problem: ensuring the stability and convergence of the methods used. Solution: testing the schemes on benchmark problems with known analytical solutions. d) Satisfaction of boundary conditions at each time step. Solution: This stage requires careful tuning of the algorithm and meticulous work by the researcher. e) Difficulties associated with machine precision and rounding errors. THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 127 Solution: proper selection of programming language and monitoring the behavior of all equation terms within the integration domain[5]. f) Interpretation of numerical calculations. Solution: It is essential to be able to distinguish between "good" and "anomalous" numerical results by evaluating their physical meaning and adequacy. Application of mathematical modeling in economics and physics. The general scheme of economic and mathematical modeling includes the following stages: problem formulation, formalization, solution, adequacy verification, and model modification [1]. During the task formulation stage, the object and purpose of the research are defined, along with the system characteristics that the model should represent. Formalization involves analyzing the object, identifying structural and functional elements, and determining essential characteristics that influence the achievement of the goal [6]. System parameters are divided into model parameters (known) and model variables (to be determined). After mathematically describing the relationships between the elements, the model itself is constructed. Before solving the model, the compatibility of the system of equations is verified. If the system is closed and correct, the solution is executed; otherwise, the model is refined or rejected [7]. At the solution stage, the calculation method and algorithm are selected. For an analytical solution, the result is expressed in formulas; for a numerical solution, it is expressed as a set of iterative steps. In linear programming problems, the simplex method is employed, which is integrated, for example, into MS Excel's "Solver" feature. However, problems involving a large number of variables require computer implementation. In physical modeling, differential equations are often solved numerically. Function approximation, iterative methods, finite-difference schemes, and finite-element methods are utilized [8], [9]. Approximate solutions inevitably contain errors. In many cases, model simplifications are applied - discarding minor terms or expanding functions into series with respect to a small parameter while retaining only the first terms, which makes the problem linear [10], [11]. As a result, the equation is reduced to a form that can be solved analytically or is suitable for numerical methods. Discussion, conclusion. Modeling is a universal tool for solving scientific and applied problems. It enables the identification of patterns, determination of optimal system parameters, and assessment of factor influences. Despite the challenges associated with nonlinearity, numerical errors, setting boundary conditions, and ensuring convergence, modeling remains the primary method for investigating complex processes in physics, economics, and engineering. The appropriate selection of a numerical method and accurate interpretation of results ensure the reliability of conclusions and broaden the capabilities for analyzing systems of any nature. Mathematical modeling holds significant fundamental and cognitive importance. It serves as a foundation for establishing interconnections in nature and economics, identifying common properties across various classes of phenomena, developing research methodologies, and solving applied problems. THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”, NOVEMBER 20, 2025 128 The modeling results enable the development of systematic approaches, techniques, and recommendations for scientific and practical purposes. REFERENCES 1. Сеттиев Ш., Ражабов Ж. - Этапы построения математических моделей. Журнал “Современная наука: актуальные проблемы и пути их решения”, 2015. 2. Ivanov S., Televnoy A. - Численно-аналитический метод преобразований для анализа нелинейных математических моделей полиномиальной структуры. Журнал “Vestnik ASTU”, 2022. 3. Fox W., West R. - Numerical Methods and Analysis with Mathematical Modelling. “CRC Press / Taylor & Francis Group”, 2024. 4. Koukoudakis N. et al. - Numerical methods in modeling with supersaturated designs. Журнал “Computer Methods in Applied Mathematics and Engineering”, 2025. 5. Suárez-Taboada M., Vázquez-Cendón M. 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