Математика /5. Математическое моделирование
Жамбаева А.К.
Костанайский государственный
университет им. А. Байтурсынова, Казахстан
Mathematical model
A mathematical model
is a powerful method of understanding the external world as well as of
prediction and control. Analysis of a mathematical model allows us to penetrate
the essence of the phenomena under study. The process of mathematical modeling,
that is, the study of a phenomenon using a mathematical model, can be divided
into four stages.
The first stage
consists in formulating the laws that relate the principal objects of the
model. This stage requires a broad knowledge of the facts pertaining to the
given phenomena and a deep understanding of the interrelations between the
phenomena. The final step of this stage is the notation in mathematical terms
of formulated qualitative conceptions of the relations among the objects of the
model.
The second stage
consists in investigating the mathematical problems that the mathematical model
leads to. Here the main question involves the solution of the direct problem,
that is, obtaining from an analysis of the model various output data
(theoretical conclusions) for further comparison with observations of the given
phenomena. At this stage, the mathematical apparatus required for analyzing the
mathematical model assumes an important role as does computer technology, a
powerful tool for obtaining quantitative output information through the
solution of complex mathematical problems. Often, the mathematical problems that
arise in connection with mathematical models for different phenomena are
identical; for example, the fundamental problem of linear programming covers
quite different kinds of situations. This justifies the examination of such a
typical mathematical problem as an independent subject of study abstracted from
the phenomena under consideration.
The third stage
consists in ascertaining whether or not the given hypothetical model satisfies
the criterion of practice, that is, determining whether the results of
observations agree with the theoretical consequences of the model within the
limits of observational accuracy. If the model is completely defined, that is,
all its parameters are given, then a determination of the deviations of the
theoretical conclusions from the observations provides a solution of the direct
problem as well as an estimate of the deviations. If the deviations exceed the
limits of the observational accuracy, the model cannot be adopted. The setting
up of a model often leaves some of the model’s characteristics undefined.
Inverse problems are
problems in which the characteristics (parametric or functional) of a model are
defined such that the output information can be compared within the limits of
observational accuracy with the results of observations of the phenomena under
study. If these conditions cannot be satisfied for a given mathematical model,
no matter how the characteristics are selected, the model is unsuitable for
investigating the phenomena in question. The use of the practice criterion in
evaluating a mathematical model allows us to determine whether the assumptions
on which the (hypothetical) model being studied is based are correct. This
method is the only way of studying phenomena of the macroworld or microworld
that are not directly accessible to us.
The fourth stage
consists in analyzing the model in conjunction with new data on the given
phenomena and updating the model. Data on the phenomena being studied are
continuously being refined in the course of the development of science and
technology, and a point is reached when conclusions that can be drawn from
existing mathematical models do not correspond to our knowledge of the
phenomenon. Thus, it becomes necessary to set up a new and more perfect
mathematical model.
The method of
mathematical modeling, which reduces the study of phenomena of the external
world to mathematical exercises, occupies a central place among other methods
of investigation, particularly since the advent of the electronic computer.
Computers make it possible to devise new technological facilities operating
under optimal conditions for the solution of complex scientific and
technological problems; they also make it possible to predict new phenomena.
Mathematical models themselves have proved to be an important means of control.
They are used in the most varied branches of knowledge and have become a
necessary apparatus in economic planning; they are also an important element in
automated control systems.
References
1. Popović,
Ž., "Basic mathematical models
in economic-ecological control", Economics and Organization Vol. 5,
No 3, University of Niš, Serbia. -2008, pp. 251 - 262
2. Gorelov, A.,
"Ecology - Science - Modeling", Nauka
Press, Moscow, 1985.