MATHEMATICAL MODELING: A COMPENDIUM OF NUMERICAL METHODS AND APPLICATIONS

Аннотация

This article is about mathematical modeling, and thus a set of numerical methods and programs. Mathematical models can take many forms, such as statistical models, differential equations, or game theory models. Keywords: Mathematical model, model, modeling, economic- mathematical methods. The science and practice of modern economics increasingly uses the achievements of applied mathematics, turning them from a tool of scientific research into an important tool for effectively solving complex economic problems. Modern economic theory includes mathematical models and methods as a natural and necessary element at both the micro and macro levels. The use of mathematics in economics makes it possible to separate and formally describe the most important and significant connections of economic variables and objects, to clearly and succinctly state the rules, concepts and conclusions of economic theory. Models and modeling play an important role in this. A model is such a material or imaginary object that replaces the real object in the research process in such a way that its direct study provides new knowledge about the real object. When building models, important factors that determine the phenomenon under study are identified, and parts that are not important for solving the problem are excluded. On the one hand, the models should be easy to learn, so they should not be too complex - therefore, they will necessarily be only simplified copies. However, on the other hand, the conclusions obtained from the study of models should be applied to real objects, so the model should reflect the important aspects of the real object being studied. Modeling is the process of building, learning and applying models. The modeling process includes the following three elements: 15 ACADEMIC RESEARCH IN MODERN SCIENCE International scientific-online conference a model that mediates the relationship subject (researcher); research object; between the learning subject and the object being studied. Modeling in scientific research began to be used in ancient times and gradually began to cover new areas of scientific knowledge, such as construction, architecture, astronomy, physics, chemistry, biology, and finally, social sciences. The first mathematical models were used by F. Keene, A. Smith, D. Ricardo. The 20th century brought great success and prestige to the modeling method of practically all areas of modern science. To study various economic phenomena, their simplified formal representations called economic models are used. Examples of economic models are consumer choice models, firm models, economic growth models, commodity and financial market equilibrium models, and many others. In economics, a mathematical model is a mathematical representation of economic objects or processes for the purpose of analysis or management, that is, a mathematical record of an economic problem. A mathematical model of an economic object is its representation in the form of a set of functions, equations, inequalities, logical relationships, graphs. Such a reflection combines a set of relations of the elements of the studied object with similar relations of the elements of the model. Methods of applying economic-mathematical models in practice are called economic-mathematical methods. The main stages of the modeling process have their own characteristics in various fields, including the economy. Let's analyze the sequence and content of one cycle of economic-mathematical modeling. Setting an economic problem and analyzing it qualitatively. This stage separates the most important features and properties of the modeled object and abstracts them from secondary ones; to 16 ACADEMIC RESEARCH IN MODERN SCIENCE International scientific-online conference study the structure of the object and the main connections connecting its elements; includes the formation of (at least preliminary) hypotheses explaining the state and development of the object. Building a mathematical model. This stage is the stage of formalizing the economic problem, expressing it in the form of certain mathematical connections and relations: functions, equations, inequalities, etc. Usually, first the main device (type) of the mathematical model is determined, and then the components of this device: exact list of variables and parameters, form of connections are determined. The purpose of this step is to determine the general properties of the model. Pure mathematical methods of research are used here. In the analytical study of the model, the existence and uniqueness of the solution, which variables (unknowns) can be included in the solution, the relationships between them, in which scope these variables change depending on the initial conditions, and the directions of their change and similar issues will be clarified. Analytical research of the model is better than empirical (numerical) research in that the conclusions obtained in it retain their validity at different specified values of