2
1 Modeling Systems
process can be very complex and requires in-depth knowledge in a specific subject
area.
Modeling can be divided into physical and abstract.
Physical modeling suggests that some other system with the same physical nature
is used as the source object model. This system is created on the basis of the similarity
theory, which allows us to state that the model retains the required properties of the
original object. In engineering practice, a prototype of an object is produced, and tests
are conducted, during which its output parameters and characteristics are determined.
The results of such tests can be extended to a real object also considering similarity
criteria.
A variety of physical modeling is analog modeling, based on replacement of the
original object with an object of a different physical nature, but with similar behavior.
Unlike physical modeling, abstract modeling describes objects based on its
abstract images—most often they are various signs, symbols, schemes, graphics,
etc.
The most important type of abstract modeling is mathematical modeling based
on the use of mathematical tools and mathematical logic. Mathematical modeling is
the process of establishing the correspondence between the original real system and
a certain mathematical model, as well as the study of this model, which allows one to
evaluate the characteristics of the real system. To analyze the results of mathematical
modeling, an interpreter is required, which may be a professional in a specific subject
area or a computer.
For the mathematical model, it is a characteristic that the processes of functioning
of the system under study are described in the form of some functional relationships
(algebraic, differential, integral, and other equations) and logical conditions. If the set
of parameters characterizing the model can be explicitly expressed from functional
dependencies by analytical methods, then it is said that an analytical solution can be
obtained. In the case when this is not possible, a solution can only be obtained by
using numerical methods.
The analytical solution can be obtained only for relatively simple systems. For
complex systems, there are often big math problems. Often, to use the analytical
method, the initial model is significantly simplified, which can affect the quality of
the solution.
When it is not possible to find a solution to equations in a general form, one can
apply a qualitative method, when in the absence of a solution in an explicit form, one
can find some of its properties (e.g., estimate the stability of a solution). If, however,
it is necessary to obtain a solution, the equations can be investigated by numerical
methods for specific initial data. In this case, computer modeling finds great use.
For computer simulation, it is typical to represent the mathematical model of the
system in the form of a computational algorithm.
Computer modeling can be divided into numerical, imitation, and statistical.
In numerical simulation, computational mathematics methods are used to build
a computer model, and a computational experiment consists of numerically solving
mathematical equations for given parameters and initial conditions.
1 Modeling Systems
process can be very complex and requires in-depth knowledge in a specific subject
area.
Modeling can be divided into physical and abstract.
Physical modeling suggests that some other system with the same physical nature
is used as the source object model. This system is created on the basis of the similarity
theory, which allows us to state that the model retains the required properties of the
original object. In engineering practice, a prototype of an object is produced, and tests
are conducted, during which its output parameters and characteristics are determined.
The results of such tests can be extended to a real object also considering similarity
criteria.
A variety of physical modeling is analog modeling, based on replacement of the
original object with an object of a different physical nature, but with similar behavior.
Unlike physical modeling, abstract modeling describes objects based on its
abstract images—most often they are various signs, symbols, schemes, graphics,
etc.
The most important type of abstract modeling is mathematical modeling based
on the use of mathematical tools and mathematical logic. Mathematical modeling is
the process of establishing the correspondence between the original real system and
a certain mathematical model, as well as the study of this model, which allows one to
evaluate the characteristics of the real system. To analyze the results of mathematical
modeling, an interpreter is required, which may be a professional in a specific subject
area or a computer.
For the mathematical model, it is a characteristic that the processes of functioning
of the system under study are described in the form of some functional relationships
(algebraic, differential, integral, and other equations) and logical conditions. If the set
of parameters characterizing the model can be explicitly expressed from functional
dependencies by analytical methods, then it is said that an analytical solution can be
obtained. In the case when this is not possible, a solution can only be obtained by
using numerical methods.
The analytical solution can be obtained only for relatively simple systems. For
complex systems, there are often big math problems. Often, to use the analytical
method, the initial model is significantly simplified, which can affect the quality of
the solution.
When it is not possible to find a solution to equations in a general form, one can
apply a qualitative method, when in the absence of a solution in an explicit form, one
can find some of its properties (e.g., estimate the stability of a solution). If, however,
it is necessary to obtain a solution, the equations can be investigated by numerical
methods for specific initial data. In this case, computer modeling finds great use.
For computer simulation, it is typical to represent the mathematical model of the
system in the form of a computational algorithm.
Computer modeling can be divided into numerical, imitation, and statistical.
In numerical simulation, computational mathematics methods are used to build
a computer model, and a computational experiment consists of numerically solving
mathematical equations for given parameters and initial conditions.