1.2 Classification of Mathematical Models
5
they depend only on time. If in a mathematical model of a macro-level the number
of phase variables is of the order of 10
4 –10
5 , then such mathematical models are
referred to as a meta-level. The quantitative analysis of such a mathematical model
is very complicated, and it requires a significant amount of computational resources.
In this case, by combining and enlarging the elements of a complex system, they
strive to reduce the number of phase variables by excluding the internal parameters
of the elements from consideration, limiting themselves only to the description of
mutual relations between the enlarged elements.
The most common form of representation of a dynamic (evolutionary) mathematical model of a micro-level is the formulation of a boundary value problem
for differential equations of mathematical physics. This formulation includes partial
differential equations with initial and boundary conditions.
The most common form of representation of a dynamic macro-level model is a
model described by a system of algebraic–differential equations.
1.3 Basic Properties of Mathematical Models
The use of mathematical models to study the characteristics of the original object
will be effective if the properties of the mathematical model satisfy several requirements [1]. These properties include: completeness, adequacy, accuracy, robustness,
productivity, and efficiency.
The completeness of the mathematical model makes it possible to sufficiently
reflect those characteristics and features of the system that interest us from the point
of view of the goal of the computational experiment. That is, completeness shows
how the properties of the model correspond to the properties of the object of study.
The adequacy of a mathematical model is the ability of a mathematical model
to reflect the properties of a system with a relative error not worse than a given
one. In a general sense, the adequacy of a mathematical model is understood as
the correct qualitative and fairly accurate quantitative description of precisely those
characteristics of the system that are important in this particular case. A model that
is adequate when selecting some characteristics may be inadequate when choosing
other characteristics of the system.
The accuracy of the mathematical model makes it possible to provide an acceptable
agreement of the real values with the output parameters of the system found using a
mathematical model.
The robustness of a mathematical model characterizes its stability with respect to
the errors of the original data, the ability to level these errors and prevent them from
excessively affecting the result of a computational experiment.
The productivity of the mathematical model is associated with the reliability of
the source data. If they are the result of measurements, then the accuracy of their
measurements should be higher than for those parameters that are obtained using a
mathematical model. Otherwise, the mathematical model will be unproductive and
its use for the analysis of a particular system will lose its meaning.
5
they depend only on time. If in a mathematical model of a macro-level the number
of phase variables is of the order of 10
4 –10
5 , then such mathematical models are
referred to as a meta-level. The quantitative analysis of such a mathematical model
is very complicated, and it requires a significant amount of computational resources.
In this case, by combining and enlarging the elements of a complex system, they
strive to reduce the number of phase variables by excluding the internal parameters
of the elements from consideration, limiting themselves only to the description of
mutual relations between the enlarged elements.
The most common form of representation of a dynamic (evolutionary) mathematical model of a micro-level is the formulation of a boundary value problem
for differential equations of mathematical physics. This formulation includes partial
differential equations with initial and boundary conditions.
The most common form of representation of a dynamic macro-level model is a
model described by a system of algebraic–differential equations.
1.3 Basic Properties of Mathematical Models
The use of mathematical models to study the characteristics of the original object
will be effective if the properties of the mathematical model satisfy several requirements [1]. These properties include: completeness, adequacy, accuracy, robustness,
productivity, and efficiency.
The completeness of the mathematical model makes it possible to sufficiently
reflect those characteristics and features of the system that interest us from the point
of view of the goal of the computational experiment. That is, completeness shows
how the properties of the model correspond to the properties of the object of study.
The adequacy of a mathematical model is the ability of a mathematical model
to reflect the properties of a system with a relative error not worse than a given
one. In a general sense, the adequacy of a mathematical model is understood as
the correct qualitative and fairly accurate quantitative description of precisely those
characteristics of the system that are important in this particular case. A model that
is adequate when selecting some characteristics may be inadequate when choosing
other characteristics of the system.
The accuracy of the mathematical model makes it possible to provide an acceptable
agreement of the real values with the output parameters of the system found using a
mathematical model.
The robustness of a mathematical model characterizes its stability with respect to
the errors of the original data, the ability to level these errors and prevent them from
excessively affecting the result of a computational experiment.
The productivity of the mathematical model is associated with the reliability of
the source data. If they are the result of measurements, then the accuracy of their
measurements should be higher than for those parameters that are obtained using a
mathematical model. Otherwise, the mathematical model will be unproductive and
its use for the analysis of a particular system will lose its meaning.