Man is a rational being. As one of the representatives of a living being, a person needs all things for his life, for his existence. However, unlike other living beings, man is a rational being, i.e. he has intelligence. Therefore, a person constantly studies the world around him, learns its laws, and based on them he creates favorable conditions and objects that improve his standard of living. To study the world around us, a person uses various methods, ranging from the simplest observations to methods that use various hypotheses and assumptions. An entire branch of science deals with this - methodology, which studies methods of cognition. In philosophy, methods of cognition are divided into two: metaphysical and dialectical. One of the general scientific methods of cognition is modeling methods. Here, without dwelling on general modeling methods, we will consider some principles of mathematical modeling, which has recently become one of the main modeling methods. To assess the importance and significance of mathematical modeling, it is enough to cite the words of Academician A.N. Tikhonov, one of the founders of mathematical modeling together with his student Academician A.A. Samarsky: “Mathematical modeling is the third way of knowledge.” The difference between the mathematical modeling method and other methods is that an object or process is described in the language of mathematics, i.e. various formulas, equations, etc. The study of models is also carried out using mathematical methods, model analysis, solving the equations that form the basis of the model, etc. Sometimes the qualitative analysis of the model itself without solving the equations provides valuable information about the object or process. Today it is difficult to show an area of research where mathematical modeling methods are not used. Extremely rich results are obtained on the basis of mathematical modeling in such areas as physics (of course, first of all), biology, chemistry, economics, ecology, social sciences, etc. Mathematical modeling methods are invaluable in the analysis of various technological processes, so it is natural to expand the use these methods in technical sciences.

      Of course, it is worth noting some advantages of mathematical modeling methods over other methods. The first is the cost-effectiveness of the method. There is no need to create experimental, industrial or semi-industrial installations to study the process, which, as a rule, require significant material costs. Sometimes the very creation of such installations is impossible. The second is the ability to predict the process after establishing general patterns. Another useful property of mathematical modeling is the ability to repeat the study of a process many times with a choice of different options for changing parameters.

      Let us look at some of the work in this area that is being carried out at Samarkand State University. Sharaf Rashidov. The Faculty of Mathematics has a Department of Mathematical Modeling. The department is staffed with highly qualified, experienced specialists. All members of the department have academic degrees, incl. four of them are doctors of science. The department offers doctoral studies in two specialties: 1) “Mathematical modeling. Numerical methods and software packages”, 2) “Mechanics of liquid and gas”. There are currently 12 doctoral students conducting research in the doctoral program.

     The department is implementing a fundamental research grant FZ-2020092877 “Compilation and numerical analysis of mathematical models of processes of anomalous transfer of substances and filtration of liquids in inhomogeneous porous media.” Doctoral students of the department whose dissertation topics are related to the subject of the grant are involved in the execution of the grant. A number of new results have been obtained on mathematical modeling of the processes of anomalous transfer of substances and filtration of liquids in inhomogeneous porous media. These questions are important in the analysis and design of petroleum and oil and gas fields. Let's note some of them. A new mathematical model of fluid filtration in unstable formations under elastic-plastic conditions has been compiled. Using this model, it was possible to create the fundamental principles for the development of deep-seated oil fields in the Fergana intermountain basin, in particular such fields as Mingbulak, Namangan, Gumkhana, etc. For this model, unloading waves were discovered, i.e. waves by Kh.A. Rakhmatullin, the characteristics of their propagation were determined. A new model of relaxation filtration of homogeneous liquids in fractured porous media has also been compiled. This model has important practical significance. Almost all oil fields of the Bukhara-Karshi oil and gas province have rocks with a fractured-porous structure. For this model, a new, unimodal phenomenon of the propagation of pressure discontinuities in an oil reservoir was discovered, which had not been observed before. Employees of the department have created a new four-phase model of filtration of various fluids in the formation. The model was adapted to the conditions of the largest oil, gas and condensate field Kukdumalak (Uzbekistan), various methods of influence were analyzed, theoretically substantiated and the “Cycling process” was proposed to oil workers as the most effective method. The use of this method at the field had a colossal effect; oil recovery from the field amounted to more than 55%, which is extremely high for the conditions of this field. Another development of the department is a mathematical model of acid action on the bottom-hole zones of oil and gas formations, taking into account changes in the structure of the pore space by undissolved particles that are released due to the reaction of acid with rock. A computational environment has been created to determine the technological characteristics of acid exposure, which was provided to the oil company Uzbekneftegaz. This medium is now widely used by oil producing enterprises of the republic to draw up a plan for geological and technical measures for acid exposure. Recently, research has been carried out on the anomalous transport and filtration of liquids in porous media. To model anomalous phenomena, a new mathematical apparatus is used - fractional differentiation. On the basis of this, it was possible to explain a number of new transfer phenomena, in particular “slow diffusion”, “fast diffusion”, the formation of transfer waves of matter and the liquid itself, etc.

      To summarize what has been said, we note that mathematical modeling as a method of cognition is an effective tool that allows one to study various processes from a wide range of areas of human activity, nature, etc. Therefore, developing the mathematical thinking of future specialists, “equipping” them with methods of mathematical modeling is our most important task.

Head of the Department of Mathematical Modeling,
Doctor of Physical and Mathematical Sciences, Professor
Bakhtiyor Khuzhayorov