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MATHEMATICAL MODELING: A COMPENDIUM OF NUMERICAL
METHODS AND APPLICATIONS
Eshonkulova Feruza Abdirazok kizi
Teacher of the department of foreign economic activity
TASHKENT STATE UNIVERSITY OF ORIENTAL STUDIES
https://doi.org/10.5281/zenodo.13945436
Abstract.
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:
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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
subject (researcher);
research object;
a model that mediates
the relationship
between the learning
subject and the object
being studied.
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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 the
external and internal parameters of the model.
Nevertheless, models of complex economic objects are brought to
analytical studies with great difficulty. In cases where it is not possible to
determine the general properties of the model by analytical methods, and when
the simplification of the model leads to inappropriate results, numerical
methods of research are used. Preparation of preliminary data. Modeling
imposes strict requirements on the information system. At the same time, the
actual possibilities of obtaining information limit the choice of models intended
for practical use. Not only the practical possibility of information preparation
(within certain deadlines), but also the costs of preparing relevant information
arrays are taken into account. These costs should not exceed the benefits of
using additional information.
Numerical solution: This stage includes the development of algorithms for
the numerical solution of the problem, the creation of programs in EHMs and
direct calculations. Difficulties at this stage arise, first of all, from the large
volume of economic issues, the need to process very large information arrays. A
quantitative study can significantly complement the results of an analytical
study, and for many models it will be the only study performed. The class of
economic problems that can be solved by numerical methods is much wider
than the class of problems that can be analyzed analytically. Analysis of
numerical results and their application. At this final stage of the cycle, the
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question arises about the accuracy and completeness of the modeling results
and their level of practical application. Nevertheless, models of complex
economic objects are brought to analytical studies with great difficulty. In cases
where it is not possible to determine the general properties of the model by
analytical methods, and when the simplification of the model leads to
inappropriate results, numerical methods of research are used. Preparation of
preliminary data. Modeling imposes strict requirements on the information
system. At the same time, the actual possibilities of obtaining information limit
the choice of models intended for practical use. Not only the practical possibility
of information preparation (within certain deadlines), but also the costs of
preparing relevant information arrays are taken into account. These costs
should not exceed the benefits of using additional information.
Numerical solution: This stage includes the development of algorithms for
the numerical solution of the problem, the creation of programs in EHMs and
direct calculations. Difficulties at this stage arise, first of all, from the large
volume of economic issues, the need to process very large information arrays. A
quantitative study can significantly complement the results of an analytical
study, and for many models it will be the only study performed. The class of
economic problems that can be solved by numerical methods is much wider
than the class of problems that can be analyzed analytically. Analysis of
numerical results and their application. At this final stage of the cycle, the
question arises about the accuracy and completeness of the modeling results
and their level of practical application.
Mathematical methods of verification identify the incorrect structure of models
and thus narrow the class of models that may be correct. Informal analysis of
theoretical conclusions and numerical results obtained by means of the model,
comparing them with existing knowledge and real facts allows to notice the
shortcomings of the economic problem statement, the constructed mathematical
model, its information and mathematical support. Any econometric studies are
conducted based on data obtained as a result of statistical observation of
economic processes and events. Every economic event and process is
represented by macro or micro statistical indicators. Statistical indicators,
known from the science of statistical theory, consist of absolute, relative and
average, and they have their quantitative and qualitative aspects. So, economic
processes are represented by the above indicators.
References:
1.
Abdullaev O.M., Khodiev B.Yu., Ishnazarov A.I. Econometrics: Textbook.-T.:
ECONOMICS. 2018.
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2.
Buinachev S. K. Application of numerical methods in mathematical
modeling: a tutorial. – Ekaterinburg: Publishing House of the Ural University,
2014.
3.
Khasanov J. O. Numerical modeling of nondivergent cross-diffusion
problems in two component media// Contents of dissertation abstract of doctor
of philosophy (PhD) on physical-mathematical sciences. 2023.
