Examples of mathematical models in technology presentation. Presentation for the lesson "compilation of mathematical models". Classification according to the type of parameter sets used

02.08.2020

Fundamentals of mathematical modeling

S.V. Zvonarev
Fundamentals of Mathematics
modeling
Lecture No. 2. Mathematical models and their classifications
Yekaterinburg
2012

The purpose of the lecture

Define the concept of a mathematical model.
To study the generalized mathematical model.
Consider classification mathematical models.
2 Mathematical model.
Generalized mathematical model.
.
The degree of correspondence of the mathematical model to the object.
Classification of mathematical models.
3

Mathematical model

MATHEMATICAL MODEL
4

Mathematical model

A mathematical model is a set of equations
or other mathematical relationships reflecting the main
properties of the object or phenomenon under study within the framework of the accepted
speculative
physical
models
and
peculiarities
his
interaction with the environment.
The main properties of mathematical models are:
adequacy;
simplicity.
The process of formulating a mathematical model is called
task setting.
The mathematical model is a mathematical analogue
the designed object. The degree of adequacy of its object
is determined by the formulation and correctness of solutions to the problem
design.
5

Math modeling

Mathematical model of a technical object -
set of mathematical equations and relations
between them, which adequately reflects the properties
of the object under study, of interest to the researcher
(engineer).
Mathematical modeling is ideal
scientific symbolic formal modeling, in which
the description of the object is carried out in the language of mathematics, and
the study of the model is carried out using those or
other mathematical methods.
Methods for finding the extremum of a function of many
variables with different constraints often
called
methods
mathematical
programming.
6

Generalized mathematical model

Elements of the generalized mathematical model:
set of input data (variables) X,Y;
mathematical operator L;
set of output data (variables) G(X,Y).
7

Input data

X is a set of variable variables, which
forms the space of variable parameters Rx
(search space), which is metric with
dimension
n,
equal
number
variable
parameters.
Y is a set of independent variables (constants),
which forms the metric space of the input
Ry data. When each component
space Ry is given by the range of possible
values,
lots of
independent
variables
displayed
some
limited
subspace of the space Ry.
8

Independent Variables Y

They define the environment for the functioning of the object, i.e.
external
terms,
in
which
will be
work
designed object. These may include:
technical parameters of the object that are not subject to
change in the design process;
physical
environmental disturbances,
the design object interacts;
With
which
tactical parameters that must be achieved
design object.
9

Math operator and output

Mathematical operator L is a complete system
mathematical operations that describe numerical or
logical relationships between the sets of input and
output data (variables). He is defining
operations on input data.
Set of output data (variables) G(X,Y)
is a set of criterion functions,
including (if necessary) the objective function.
Output data of the considered generalized model
form a metric space of criterial
RG indicators.
10

Nonlinearity of mathematical models

Nonlinearity of mathematical models
‒ violation of the principle
superpositions, i.e. when any linear combination of solutions is not
is the solution to the problem. Thus knowledge about the behavior of the part
object does not yet guarantee knowledge of the behavior of the entire object.
Majority
real
processes
and
relevant
them
mathematical models are not linear. Linear models are responsible
very special cases and, as a rule, serve only the first
approaching reality.
Example - population models immediately become non-linear,
if we take into account the limited available population
resources.
11

The degree of correspondence of mathematical models to the object

Difficulties:
The mathematical model is never identical
the object in question and does not convey all of its properties and
features.
The mathematical model is an approximate description
object and is always approximate.
The accuracy of matching is determined by the degree of matching,
adequacy of the model and object. Ways:
Using experiment (practice) to compare models and
choosing the most suitable one.
Unification of mathematical models due to the accumulation of sets
finished models.
Transfer of finished models from one process to another,
identical, similar.
Using the minimum number of approximations and accounting
disturbing influences.
12

Classification of mathematical models

CLASSIFICATION
MATHEMATICAL MODELS
13

Classes of mathematical models

Mathematical models are divided into classes in
depending on:
complexity of the modeling object;
model operator;
input and output parameters;
modeling goals;
method of studying the model;
objects of study;
belonging of the model to the hierarchical level
object descriptions;
the nature of the displayed properties;
calculation procedure;
use of process control.
14

Classification by object complexity

AT
simple
models
at
modeling
not
considered internal structure object, not
stand out
constituents
his
elements
or
subprocesses.
The object system is a correspondingly more complex system,
which is a collection of interrelated
elements, separated from environment and
interacting with it as a whole.
15

Classification by model operator

mathematical
model
called
linear if the operator provides
linear
addiction
weekend
parameters
from
values
input
parameters.
mathematical
model
called
non-linear if the operator provides
non-linear
addiction
weekend
parameters
from
values
input
parameters.
The mathematical model is simple if the model operator is
algebraic
expression,
reflective
functional
dependence of output parameters on input ones.
Model including systems of differential and integral
relations is called complex.
A model is called algorithmic when it is possible to construct
some imitator of the behavior and properties of an object using an algorithm.
16

Classification by input and output parameters

Classification by the nature of the process being modeled

deterministic,
which
correspond
deterministic processes that have strictly
unambiguous relationship between physical quantities,
characterizing the state of the system in any
moment
time.
deterministic
model
makes it possible to unambiguously calculate and predict
values of the output values according to the values of the input
parameters and control actions.
Indefinite, which come from the fact that
a change in the defining quantities occurs
randomly, and the values of the output quantities
are in probabilistic correspondence with the input
quantities and are not uniquely determined.
18

Undefined Models

Stochastic - values of all or individual parameters
models are determined by random variables given
probability densities.
Random - values of all or individual parameters of the model
are set by random variables given by estimates
probability densities obtained as a result of processing
limited experimental sample of these parameters.
Interval - values of all or individual parameters
models are described by interval values given by
the interval formed by the minimum and maximum
possible values of the parameter.
Fuzzy - values of all or individual parameters of the model
are described by the membership functions of the corresponding
fuzzy set.
19

Classification in relation to the dimension of space

One-dimensional.
Two-dimensional.
Three-dimensional.
This division is applicable for models, including
parameters
which
are included
coordinates
space.
20

Classification in relation to time

Static. If the system state is not

static. Static Simulation
serves to describe the state of an object in
fixed point in time.
Dynamic. If the system state
changes over time, then the models are called
dynamic. Dynamic Simulation
serves to study the object in time.
21

Classification according to the type of parameter sets used

Quality.
Quantitative.
Discrete.
Continuous.
Mixed.
22

Classification by modeling goals

Descriptive. The purpose of such models is to establish laws
changing model parameters. An example is a model of rocket movement after
launch from the surface of the earth.
Optimization. Such models are designed to determine
parameters optimal from the point of view of some criterion
of the simulated object or to find the optimal mode
control of some process. An example of such a model is
serve as a simulation of the process of launching a rocket from the surface of the Earth with
the purpose of lifting it to a given height in the minimum time.
Managerial. Such models are used to make effective
management decisions in various areas purposeful
23
human activities.

Classification by method of implementation

Analytical. Analytical methods are more convenient for
subsequent analysis of the results, but are applicable only for
relatively simple models. If the mathematical
problem admits an analytical solution, then it is considered
numerical is preferred.
Algorithmic. Algorithmic methods are reduced to
some
algorithm
implementing
computing
24
experiment using a computer.

Classification by objects of study

Objects with a high degree of information. if in process
modeling, complete systems of equations are known,
describing all aspects of the process being modeled and all
numerical values of the parameters of these equations.
Objects with zero information level. Mathematical
the model of such an object is built on the basis of statistical
experimental data.
Objects with known basic regularities.
Values of constants in mathematical equations of description
models are established from experience.
Objects whose behavior is known
empirical nature. They use methods
physical modeling using mathematical
experiment planning.
25

Classification according to the model belonging to the hierarchical level of the object description

Micro level
(typical
processes
are
mass transfer,
thermophysical,
hydrodynamic).
Modeling
carried out
in
purposes
synthesis
technological process for a single or several
aggregates.
Macro level. Modeling processes with more
high level of aggregation; models are used for synthesis
current process control for one
unit or technological complex as a whole.
Metalevel. Simulation of processes in the aggregate
aggregates and connecting them material and energy
streams. Such models serve for the synthesis of technological
complex as a whole, that is, for the synthesis of control
development.
26

Classification by the nature of the displayed properties of the model

Functional
models.
Are used,
for
descriptions
physical and information processes occurring during
the functioning of the object.
Structural
models.
Describe
compound
and
interconnections
elements of the system (process, object).
27

Classification by order of calculation

Direct. Used to determine kinetic,
static and dynamic patterns of processes.
Reverse
(inversion).
Are used
for
determining the value of input parameters or other
specified properties of the processed substances or
products, as well as to determine acceptable
deviations of processing modes (optimization problems
processes and device parameters).
Inductive.
Apply
for
clarifications
mathematical equations of kinetics, statics or
process dynamics using new hypotheses or
theories.
28

Classification by use of process control

Forecast models, or computational models without control.
The main purpose of these models is to predict the behavior
systems in time and space, knowing the initial state
and information about its behavior at the border. Examples -models
heat distribution, electric field, chemical
kinetics, hydrodynamics.
optimization models.
– Stationary models. Used at the design level
various
technological
systems.
Examples
‒
deterministic tasks, all input information in which
is completely definable.
– Non-stationary
models.
Are used
on the
level
design, and, mainly, for optimal
management of various processes - technological,
economic, etc. In these problems, some parameters are
random or contain an element of uncertainty.
29 Hypothesis.
Phenomenological model.
Approximation.
Simplification.
heuristic model.
Analogy.
Thought experiment.
Possibility demonstration.
30

Hypothesis

These models are trial
description of the phenomenon. If such a model is built, then
this means that she is temporarily recognized as the truth
and you can focus on other issues.
However, this cannot be the point in research, and
only a temporary pause: the status of the model can be
only temporary.
Examples:
Model solar system according to Ptolemy.
The Copernican model (improved by Kepler).
Rutherford's model of the atom.
Big Bang Model.
and etc.
31

Phenomenological model

This model contains a mechanism for describing the phenomenon.
However, this mechanism is not convincing enough and cannot be
confirmed by available data or poorly consistent with
available theories and accumulated knowledge about the object.
Therefore, phenomenological models have the status of temporary
solutions. The role of the model in the study may change with
over time, it may happen that new data and theories
confirm phenomenological models and they will be upgraded to
hypothesis status. Likewise, new knowledge may gradually
come into conflict with models-hypotheses of the first type and those
can be translated into the second.
Examples:
The calorific model.
Quark model of elementary particles.
and etc.
32

Approximation

A common practice when you can't
solve equations even with the help of a computer,
describing the system under study - use
approximations. The equations are replaced by linear ones.
The standard example is Ohm's law.
33

Simplification

This model discards parts that
can noticeably and not always controllably affect
result.
Examples:
Application of the model of an ideal gas to a non-ideal one.
Van der Waals equation of state.
Most models of solid state physics,
liquids and nuclear physics. The path from microdescription to
properties of bodies (or media) consisting of a large number
particles, very long. Many have to be discarded
details.
34

heuristic model

The heuristic model preserves only the qualitative
semblance of reality and gives predictions only "according to
order of magnitude."
It gives simple formulas for the coefficients
viscosity, diffusion, thermal conductivity, consistent
with reality in order of magnitude. But at
the construction of a new physics is far from immediately obtained
a model that gives at least a qualitative description of the object.
A typical example is the average length approximation
free path in kinetic theory.
35

Analogy

This
model
first
arose,
when
interaction in the neutron-proton system tried
explain through the interaction of the atom
hydrogen with a proton. This analogy led to
the conclusion that there must be exchange
forces of interaction between neutron and proton,
due to the transition of an electron between two
protons.
36

Thought experiment and demonstration of possibility

A thought experiment is a reasoning
which eventually lead to a contradiction.
Demonstration of possibility is also mental
experiments
With
imagined
entities
demonstrating,
what
supposed
phenomenon
consistent with the basic principles and internally
consistent. One of the most famous of these
experiments - Lobachevsky geometry.
37

Conclusion and Conclusions

The concept of a mathematical model is considered.
A generalized mathematical model has been studied.
The concepts are defined: nonlinearity of mathematical models and degree
correspondence of the mathematical model to the object.
The classification of mathematical models is presented.
38 Samarsky, A.A. Mathematical modeling / A.A. Samara,
A.P. Mikhailov. – M.: Science. Fizmatlit, 1997.
Tarasevich, N.N. Mathematical and computer modeling.
Introductory course / N.N. Tarasevich. – M.: Editorial URSS, 2001.
Introduction to mathematical modeling: Uch. allowance / under
edited by P.V. Trusova. – M.: University book, Logos, 2007. –
440 s.

slide 3

Math modeling

this is an approximate description of some class of phenomena, expressed in the language of some mathematical theory(using a system of algebraic equations and inequalities, differential or integral equations, functions, a system of geometric sentences, vectors, etc.).

slide 4

Model classification

Formal classification of models Formal classification of models is based on the classification of mathematical tools used. Often built in the form of dichotomies. For example, one of the popular sets of dichotomies: Linear or non-linear models[; Concentrated or distributed systems; Deterministic or stochastic; Static or dynamic; discrete or continuous. and so on. Each constructed model is linear or non-linear, deterministic or stochastic, ... Naturally, mixed types are also possible: concentrated in one respect (in terms of parameters), distributed models in another, etc.

slide 5

Classification according to the method of representing an object Structural or functional models Structural models represent an object as a system with its own device and mechanism of functioning. Functional models do not use such representations and reflect only the externally perceived behavior (functioning) of the object. In their extreme expression, they are also called "black box" models. Also possible combined types models, sometimes referred to as gray box models.

slide 6

Meaningful and formal models Almost all authors describing the process of mathematical modeling indicate that first a special ideal construction, a meaningful model, is built. And the final mathematical construction is called a formal model or simply a mathematical model obtained as a result of the formalization of this content model. The construction of a meaningful model can be carried out using a set of ready-made idealizations, that is, they provide ready-made structural elements for meaningful modeling.

Slide 7

Slide 8

Type 1: Hypothesis (this could be)

These models "represent a trial description of the phenomenon, and the author either believes in its possibility, or even considers it to be true." No hypothesis in science can be proven once and for all. Richard Feynman formulated this very clearly: If a model of the first type is built, then this means that it is temporarily recognized as true and one can concentrate on other problems. However, this cannot be a point in research, but only a temporary pause: the status of the model of the first type can only be temporary.

Slide 9

Type 2: Phenomenological pattern (behave as if...)

Phenomenological models have the status of temporary solutions. It is believed that the answer is still unknown and it is necessary to continue the search for "true mechanisms". The role of the model in research may change over time, it may happen that new data and theories confirm phenomenological models and they are promoted to the status of a hypothesis. Likewise, new knowledge may gradually come into conflict with models-hypotheses of the first type, and they can be transferred to the second.

Slide 10

Type 3: Approximation (consider something very large or very small)

If it is possible to construct equations describing the system under study, this does not mean that they can be solved even with the help of a computer. A common technique in this case is the use of approximations (models of type 3). Among them are linear response models. The equations are replaced by linear ones.

slide 11

Type 4: Simplify (omitting some details for clarity)

In a type 4 model, details are discarded that can noticeably and not always controllably affect the result. The same equations can serve as a model of type 3 (approximation) or 4 (we omit some details for clarity) - this depends on the phenomenon for which the model is used to study. So, if linear response models are used in the absence of more complex models, then these are already phenomenological linear models.

slide 12

Type 5: Heuristic model (no quantitative confirmation, but the model provides insight)

The heuristic model retains only a qualitative similarity to reality and makes predictions only "in order of magnitude". It gives simple formulas for the coefficients of viscosity, diffusion, thermal conductivity, consistent with reality in order of magnitude.

slide 13

Type 6: Analogy (we will take into account only some features)

Similarity, equality of relations; the similarity of objects, phenomena, processes, quantities ..., in any properties, as well as knowledge, taking into account only some of the features.

Slide 14

Type 7: Thought experiment (the main thing is to disprove the possibility)

view cognitive activity, in which the key situation for a particular scientific theory is played out not in a real experiment, but in the imagination. In some cases, a thought experiment reveals contradictions between the theory and "ordinary consciousness", which is not always evidence of the theory's inaccuracy.

slide 15

Type 8: Demonstration of the possibility (the main thing is to show the internal consistency of the possibility)

These are also thought experiments with imaginary entities, demonstrating that the alleged phenomenon is consistent with the basic principles and is internally consistent. This is the main difference from models of type 7, which reveal hidden contradictions. The substantive classification is based on the stages preceding mathematical analysis and calculations. Eight types of models according to R. Peierls are eight types of research positions in modeling.

slide 16

Main stages of mathematical modeling

1. Building a model. At this stage, some "non-mathematical" object is specified - a natural phenomenon, a structure, economic plan, manufacturing process etc. At the same time, as a rule, a clear description of the situation is difficult. First, the main features of the phenomenon and the relationship between them at a qualitative level are identified. Then the found qualitative dependencies are formulated in the language of mathematics, that is, a mathematical model is built. This is the most difficult part of the modeling.

Slide 17

2. Solution mathematical problem, to which the model leads. At this stage great attention is given to the development of algorithms and numerical methods for solving a problem on a computer, with the help of which the result can be found with the required accuracy and within an acceptable time. 3. Interpretation of the obtained consequences from the mathematical model. The consequences derived from the model in the language of mathematics are interpreted in the language accepted in this field.

Slide 18

4. Checking the adequacy of the model. At this stage, it is found out whether the results of the experiment agree with the theoretical consequences from the model within a certain accuracy. 5. Model modification. At this stage, either the model becomes more complex so that it is more adequate to reality, or it is simplified in order to achieve a practically acceptable solution.

Slide 19

In this case, the following requirements must be met:

the model should adequately reflect the most essential (from the point of view of a certain problem statement) properties of the object, abstracting from its non-essential properties; the model must have a certain area of applicability, due to the assumptions adopted in its construction; the model should allow obtaining new knowledge about the object under study.

Slide 20

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Object (transport process)

Practical

Design scheme

Mathematical model

mathematical model

Algorithm

Program

At the first stage of mathematical modeling, the transition from the modeling object to the calculation scheme is carried out. A design scheme is a meaningful and/or conceptual model of an object. For example: a plan for the transportation of goods, a route map, a transport table, etc.

At the second stage, a search and a formalized description of the process (processes) of the design scheme by a mathematical model is carried out.

At the third stage, qualitative and quantitative analysis mathematical model including: 1) simplification, 2) resolution of contradictions, 3) correction.

At the fourth stage, an effective algorithm for mathematical modeling is developed, according to which, at the fifth stage, a program for the implementation of mathematical modeling is created.

At the sixth stage, practical recommendations are obtained by using the program. Practical recommendations is the result of using a mathematical model for a specific purpose in the study of an object (transport process).

The goals of mathematical modeling: 1) creation of models of transport processes for further designing optimal (in terms of time, cost) transport processes; 2) analysis of the properties of individual transport processes in order to estimate time and cost.

Types of mathematical modeling

	Parametric			simulation
				modeling
				modeling

static		dynamic



			Stationary		non-stationary

Parametric modeling is modeling without a strict connection with the object and process. Communication is carried out only by parameters, for example: mass, length, pressure, etc. There are abstractions: material point, ideal gas, etc.

Static parametric models do not contain the parameter "time" and allow obtaining the characteristics of the system in equilibrium. Dynamic parametric models contain the time parameter and allow you to get the nature of the transient processes of the system.

Simulation(Simulation) - mathematical modeling taking into account the geometric features of the simulation object (size, shape) as well as the density distribution with reference to the initial and boundary conditions (conditions at the boundaries of the object's geometry) to the objects.

processes

Program Algorithm

Stationary simulation allows you to get the characteristics of the object in the time interval tending to zero, that is, to "photograph" the characteristics of the object. Non-stationary modeling allows you to get the characteristics of the object over time.

Structure of the mathematical model

Input parameters	equations,	output parameters
	equations,
	dependencies, etc.

Properties of the mathematical model:

1) Completeness - the degree of reflection of the known properties of the object; 2) Accuracy - the order of coincidence of real (experimental) and characteristics found using the model;

3) Adequacy is the ability of the model to describe output parameters with a fixed accuracy for fixed input parameters (area of adequacy).

4) Profitability is an assessment of the cost of computing resources to obtain a result in comparison with a similar mathematical model;

5) Robustness - the stability of the mathematical model in relation to the errors of the initial data (for example, the data do not correspond to the physics of the process);

6) Productivity is the impact of the accuracy of the input data on the accuracy of the output data of the model;

7) Clarity and simplicity of the model.

Mathematical models (according to the method of obtaining)

Empirical Theoretical

Semi-empirical © FGBOU VPO USATU; cafe "Applied Hydromechanics" 16

Empirical mathematical models are obtained by processing and analyzing the results of experimental data. Identification is the correction of an existing mathematical model with empirical data.

Theoretical mathematical models are obtained by theoretical methods - analysis, synthesis, induction, deduction, etc.

Literature on the theory of mathematical modeling and mathematical models:

1) Zarubin V. S. Mathematical modeling in technology: textbook. for universities / V. S. Zarubin. - 3rd ed. - M .: Publishing house of MSTU im. N.E. Bauman. 2010. - 495 p.

2) Cherepashkov A. A., Nosov N. V. Computer technologies, modeling and automated systems in mechanical engineering: Textbook. for stud. higher educational establishments. - Volgograd: Publishing house "In-folio", 2009. - 640 p.

4. Mathcad as an application programming tool

Mathcad is a computer algebra system from the class of computer-aided design systems, focused on the preparation of interactive documents with calculations and visual support, it is easy to use and apply.

Mathcad was conceived and originally written by Allen Razdov of the Massachusetts Institute of Technology.

Developer: PTC. First issue: 1986.

Solving differential and algebraic equations numerically

methods;
Construction of two-dimensional and three-dimensional graphs of functions;

Use of the Greek alphabet;

Performing calculations in symbolic form;

Support for your own programming language

© FGBOU VPO UGATU; cafe "Applied Hydromechanics"

Numerical functions are designed to calculate the roots of equations by numerical methods of applied mathematics, solve optimization problems, solve differential equations by the Runge-Kutta method, etc.

Symbolic functions designed for analytical calculations, which are similar in structure to classical mathematical transformations.

TOL system variable - Calculation tolerance (default 10-3 ).

Setting ranged variables with a fixed step: x:=0, 0+0.01..10.

If the variable is an array, then you can access the array element by entering the index with the [ key.

Literature 1. Samarsky A. A., Mikhailov A. P. Mathematical modeling: Ideas. Methods. Examples. - M.: Nauka, Volkov E. A. Numerical methods. - M.: Nauka, Turchak L. I. Fundamentals of numerical methods. - M.: Science, Kopchenova N. V., Maron I. A. Computational mathematics in examples and problems. – M.: Nauka, 1972.

A bit of history from manipulation of objects to manipulation of concepts about objects replacement of the object, process or phenomenon under study with a simpler and more accessible equivalent for research the inability to take into account the entire set of factors that determine the properties and behavior of the object

The role of models The building is ugly, fragile or does not fit into the surrounding landscape Demonstration of circulatory systems in nature is inhumane Voltages, for example, in wings, may be too high It is uneconomical to assemble electrical circuits for measurements

Communication of the model with the original Creating a model involves the preservation of some properties of the original, and in different models these properties may be different. The cardboard building is much smaller than the real one, but allows us to judge its appearance; the poster makes the circulatory system understandable, although it has nothing to do with organs and tissues; the model aircraft does not fly, but the voltages in its body correspond to the flight conditions.

Why are models used? 1. A model is more accessible for research than a real object, 2. It is easier and cheaper to study a model than real objects, 3. Some objects cannot be studied directly: it is not yet possible, for example, to build a device for thermonuclear fusion or conduct experiments in the interior of stars, 4. experiments with the past are impossible, experiments with the economy or social experiments are unacceptable

Appointment of models 1. With the help of the model, it is possible to identify the most significant factors that form the properties of an object. Since the model reflects only some of the characteristics of the object - the original, then by varying the set of these characteristics in the model, it is possible to determine the degree of influence of certain factors on the adequacy of the behavior of the model

The model is needed: 1. In order to understand how a particular object is arranged: what is its structure, properties, laws of development and interaction with the surrounding world. 2. In order to learn how to manage an object or process and determine best ways management under given goals and criteria. 3. In order to predict the behavior of the object and evaluate the consequences of various methods and forms of impact on the object (meteorological models, models of the development of the biosphere).

Property of a correct model good model has a remarkable property: its study allows you to gain new knowledge about the object - the original, despite the fact that when creating the model, only some of the main characteristics of the original were used

Material modeling The model reproduces the main geometric, physical, dynamic and functional characteristics of the object under study, when a real object is compared with its enlarged or reduced copy, which allows research in the laboratory with the subsequent transfer of the properties of the studied processes and phenomena from the model to the object based on the theory of similarity (planetarium, models of buildings and devices, etc.). The research process in this case is closely related to the material impact on the model, i.e., it consists in a full-scale experiment. Thus, material modeling is, by its nature, an experimental method.

Types of ideal modeling Intuitive - modeling of objects that are not amenable to formalization or do not need it. Life experience a person can be considered as his intuitive model of the surrounding world. Sign - modeling that uses sign transformations as models different kind: diagrams, graphs, drawings, formulas, etc. and containing a set of laws by which you can operate with model elements

Mathematical modeling The study of an object is carried out on the basis of a model formulated in the language of mathematics and studied using certain mathematical methods. Mathematical modeling is a field of science that deals with modeling natural phenomena, technology, economic and public life with the help of a mathematical apparatus and, at present, implementing these models using a computer

Classification mat. models By purpose: descriptive optimization simulation By the nature of the equations: linear nonlinear By taking into account changes in the system over time: dynamic static By the property of the domain of definition of the arguments: continuous discrete By the nature of the process: deterministic stochastic

Mathematical model- this is a set of mathematical objects and relationships between them, adequately reflecting the properties and behavior of the object under study.

Mathematics in the most general sense deals with the definition and use of symbolic models. A mathematical model covers a class of undefined (abstract, symbolic) mathematical objects such as numbers or vectors, and the relationships between these objects.

A mathematical relation is a hypothetical rule relating two or more symbolic objects. Many relationships can be described using mathematical operations that relate one or more objects to another object or to a set of objects (the result of an operation). The abstract model, with its objects of an arbitrary nature, relations and operations, is defined by a consistent set of rules that introduce operations that can be used and establish general relationships between their results. The constructive definition introduces a new mathematical model, using already known mathematical concepts (for example, the definition of addition and multiplication of matrices in terms of addition and multiplication of numbers).

A mathematical model will reproduce appropriately selected aspects of a physical situation if a correspondence rule can be established linking specific physical objects and relations to certain mathematical objects and relations. It can also be instructive and/or interesting to build mathematical models for which physical world analogues do not exist. The most commonly known mathematical models are systems of integers and real numbers and Euclidean geometry; the defining properties of these models are more or less direct abstractions of physical processes (counting, ordering, comparison, measurement).

Objects and operations of more general mathematical models are often associated with sets of real numbers, which can be correlated with the results of physical measurements.

Mathematical modeling is a method of qualitative and (or) quantitative description of a process using the so-called mathematical model, in the construction of which a real process or phenomenon is described using one or another adequate mathematical apparatus. Mathematical modeling is an integral part of modern research.

Mathematical modeling is a typical discipline located, as it is now often said, at the "junction" of several sciences. An adequate mathematical model cannot be built without deep knowledge of the object that is “served” by the mathematical model. Sometimes an illusory hope is expressed that a mathematical model can be created jointly by a mathematician who does not know the object of modeling, and a specialist in the “object” who does not know mathematics. For successful activity in the field of mathematical modeling, it is necessary to know both mathematical methods and the object of modeling. This is connected, for example, with the presence of such a specialty as a theoretical physicist, whose main activity is mathematical modeling in physics. The division of specialists into theoreticians and experimenters, which has been established in physics, will undoubtedly occur in other sciences, both fundamental and applied.

Due to the variety of applied mathematical models, their general classification difficult. In the literature, classifications are usually given, which are based on different approaches. One of these approaches is related to the nature of the process being modeled, when deterministic and probabilistic models are distinguished. Along with such a widespread classification of mathematical models, there are others.

Classification of mathematical models based on the features of the applied mathematical apparatus . It includes the following varieties.

Usually, such models are used to describe the dynamics of systems consisting of discrete elements. From the mathematical side, these are systems of ordinary linear or non-linear differential equations.

Mathematical models with lumped parameters are widely used to describe systems consisting of discrete objects or sets of identical objects. For example, the dynamic model of a semiconductor laser is widely used. In this model, two dynamic variables appear - the concentrations of minor charge carriers and photons in the active zone of the laser.

In the case of complex systems, the number of dynamic variables and, consequently, differential equations can be large (up to 102 ... 103). In these cases, various methods of system reduction are useful, based on the temporal hierarchy of processes, assessing the influence of various factors and neglecting the insignificant among them, etc.

The method of successive extension of the model can lead to the creation of an adequate model complex system.

Models of this type describe the processes of diffusion, heat conduction, propagation of waves of various natures, etc. These processes can be not only of a physical nature. Mathematical models with distributed parameters are widely used in biology, physiology and other sciences. Most often, the equations of mathematical physics, including nonlinear ones, are used as the basis of a mathematical model.

The fundamental role of the principle of greatest action in physics is well known. For example, all known systems of equations describing physical processes can be derived from extremal principles. However, in other sciences extremal principles play an essential role.

The extremal principle is used when approximating empirical dependencies by an analytical expression. The graphic representation of such a dependence and the specific form of the analytical expression describing this dependence is determined using the extremal principle, called the least squares method (Gauss method), the essence of which is as follows.

Let an experiment be carried out, the purpose of which is to study the dependence of some physical quantity Y from physical quantity x. It is assumed that the values x and y linked by functional dependency

The form of this dependence needs to be determined from experience. Let us assume that as a result of the experiment, we obtained a number of experimental points and built a dependence graph at from X. Usually, the experimental points on such a graph are not located quite correctly, they give some spread, i.e., they reveal random deviations from the visible general pattern. These deviations are associated with measurement errors that are inevitable in any experiment. Then the problem of smoothing the experimental dependence, which is typical for practice, arises.

To solve this problem, a calculation method is usually used, known as the method of least squares (or the Gauss method).

Of course, the listed varieties of mathematical models do not exhaust the entire mathematical apparatus used in mathematical modeling. The mathematical apparatus of theoretical physics and, in particular, its most important section, elementary particle physics, is especially diverse.

The areas of their application are often used as the main principle of the classification of mathematical models. With this approach, the following areas of application are distinguished:

physical processes;

technical applications, including managed systems, artificial intelligence;

life processes(biology, physiology, medicine);

large systems associated with the interaction of people (social, economic, environmental);

humanities (linguistics, art).

(The areas of application are listed in descending order according to the level of adequacy of the models).

Types of mathematical models: deterministic and probabilistic, theoretical and experimental factorial. Linear and non-linear, dynamic and static. continuous and discrete, functional and structural.

Classification of mathematical models (TO - technical object)

The structure of a model is an ordered set of elements and their relationships. A parameter is a value that characterizes a property or mode of operation of an object. The output parameters characterize the properties of the technical object, and the internal parameters characterize the properties of its elements. External parameters are parameters External Environment, which affects the functioning of the technical object.

Mathematical models are subject to the requirements of adequacy, economy, universality. These claims are contradictory.

Depending on the degree of abstraction in the description physical properties The technical system distinguishes three main hierarchical levels: the upper or meta level, the middle or macro level, the lower or micro level.

The meta level corresponds to the initial stages of design, at which scientific and technical1 search and forecasting, development of a concept and technical solution, and development of a technical proposal are carried out. To build mathematical models of the metalevel, the methods of morphological synthesis, graph theory, mathematical logic, theory automatic control, the theory of queuing, the theory of finite automata.

At the macrolevel, an object is considered as a dynamic system with lumped parameters. Mathematical models of the macrolevel are systems of ordinary differential equations. These models are used in determining the parameters of a technical object and its functional elements.

At the micro level, an object is represented as a continuous medium with distributed parameters. To describe the processes of functioning of such objects, partial differential equations are used. At the micro level, elements of a technical system that are indivisible in terms of functional characteristics, called basic elements, are designed. At the same time, the base element is considered as a system consisting of a set of similar functional elements of the same physical nature, interacting with each other and being influenced by the external environment and other elements of the technical object, which are the external environment in relation to the base element.

According to the form of representation of mathematical models, invariant, algorithmic, analytical and graphical models of the design object are distinguished.

AT invariant form, the mathematical model is represented by a system of equations without regard to the method of solving these equations.

AT algorithmic in the form of model relations are associated with the chosen numerical solution method and are written in the form of an algorithm - a sequence of calculations. Algorithmic models include imitation, models designed to simulate the physical and information processes occurring in the object during its operation under the influence of various environmental factors.

Analytical the model represents the explicit dependences of the desired variables on the given values (usually, the dependences of the output parameters of the object on internal and external parameters). Such models are obtained on the basis of physical laws, or as a result of direct integration of the original differential equations. Analytical mathematical models make it easy and simple to solve the problem of determining the optimal parameters. Therefore, if it is possible to obtain a model in this form, it is always advisable to implement it, even if it requires performing a number of auxiliary procedures. Such models are usually obtained by experimental design (computational or physical).

Graphic(circuit) model is represented in the form of graphs, equivalent circuits, dynamic models, diagrams, etc. To use graphical models, there must be a one-to-one correspondence rule conditional images elements of graphic and components of invariant mathematical models.

The division of mathematical models into functional and structural ones is determined by the nature of the displayed properties of the technical object.

Structural models display only the structure of objects and are used only in solving problems of structural synthesis. Parameters of structural models are signs of functional or structural elements that make up a technical object and in which one version of the object structure differs from another. These parameters are called morphological variables. Structural models take the form of tables, matrices and graphs. The most promising is the use of tree-like graphs of the AND-OR-tree type. Such models are widely used at the meta level when choosing a technical solution.

Functional models describe the processes of functioning technical objects and have the form of systems of equations. They take into account the structural and functional properties of the object and allow solving problems of both parametric and structural synthesis. They are widely used at all levels of design. At the meta level, functional tasks allow solving forecasting problems, at the macro level - choosing the structure and optimizing the internal parameters of a technical object, at the micro level - optimizing parameters basic elements.

According to the methods of obtaining functional mathematical models are divided into theoretical and experimental.

Theoretical models are obtained based on the description of the physical processes of the object's functioning, and experimental- based on the behavior of the object in the external environment, considering it as a "black box". Experiments in this case can be physical (on a technical object or its physical model) or computational (on a theoretical mathematical model).

When constructing theoretical models, physical and formal approaches are used.

The physical approach is reduced to the direct application of physical laws to describe objects, for example, the laws of Newton, Hooke, Kirchhoff, etc.

The formal approach uses general mathematical principles and is used in the construction of both theoretical and experimental models. Experimental models are formal. They do not take into account the entire complex of physical properties of the elements of the technical system under study, but only establish a connection found during the experiment between the individual parameters of the system, which can be varied and (or) measured. Such models provide an adequate description of the processes under study only in a limited region of the parameter space, in which the parameters were varied in the experiment. Therefore, experimental mathematical models are of a particular nature, while physical laws reflect the general patterns of phenomena and processes that occur throughout technical system, as well as in each of its elements separately. Consequently, experimental mathematical models cannot be accepted as physical laws. However, the methods used to build these models are widely used in testing scientific hypotheses.

Functional mathematical models can be linear and non-linear. Linear models contain only linear functions of quantities characterizing the state of the object during its operation, and their derivatives. The characteristics of many elements of real objects are non-linear. Mathematical models of such objects include non-linear functions of these quantities and their derivatives and refer to non-linear .

If the modeling takes into account the inertial properties of the object and (or) the change in time of the object or the external environment, then the model is called dynamic. Otherwise the model is static. Mathematical representation of the dynamic model in general case can be expressed by a system of differential equations, and static - by a system of algebraic equations.

If the impact of the environment on the object is of a random nature and is described by random functions. In this case, it is necessary to build probabilistic mathematical model. However, such a model is very complex and its use in the design of technical objects requires a lot of computer time. Therefore, it is used for final stage design.

Most design procedures are performed on deterministic models. A deterministic mathematical model is characterized by a one-to-one correspondence between an external influence on a dynamic system and its response to this influence. In a computational experiment, when designing, some standard typical actions on an object are usually set: stepped, impulse, harmonic, piecewise linear, exponential, etc. They are called test actions.

Continuation of the Table “Classification of mathematical models

Types of mathematical models of technical objects
By taking into account the physical properties of TO	By the ability to predict results
dynamic	deterministic
Static	Probabilistic
continuous
Discrete
Linear
At this stage, the following steps are performed. A plan is drawn up for creating and using a software model. As a rule, the model program is created using computer simulation automation tools. Therefore, the plan indicates: type of computer; simulation automation tool; approximate costs of computer memory for creating a model program and its working arrays; the cost of machine time for one cycle of the model; estimates of costs for programming and debugging the model program. Then the researcher starts programming the model. As terms of reference description for programming simulation model. The specifics of model programming work depends on the modeling automation tools that are available to the researcher. There are no significant differences between creating a model program and ordinary offline debugging of program modules big program or a software package. In accordance with the text, the model is divided into blocks and subblocks. In contrast to the usual offline debugging of program modules, when debugging blocks and subblocks of a program model, the amount of work increases significantly, since for each module it is necessary to create and debug an external environment simulator. It is very important to verify the implementation of the module functions in the model time t and estimate the cost of computer time for one cycle of the model as a function of the values of the model parameters. The work is completed during autonomous debugging of the model components by preparing the forms for representing the input and output data of the simulation. Next, proceed to the second verification of the reliability of the system model program. During this check, the correspondence of operations in the program and the description of the model is established. For this, it is produced reverse translation programs into the model diagram (manual "scrolling" allows you to find gross errors in the model statics). After eliminating gross errors, a number of blocks are combined and complex debugging of the model begins using tests. Debugging for tests starts with a few blocks, then an increasing number of model blocks are involved in this process. Note that complex debugging of the model program is much more difficult than debugging application packages, since simulation dynamics errors in this case are much more difficult to find due to the quasi-parallel operation of various model components. Upon completion of the complex debugging of the model program, it is necessary to re-estimate the costs of computer time for one cycle of calculations on the model. In this case, it is useful to obtain an approximation of the simulation time for one simulation cycle. The next step is to compile technical documentation for a complex system model. By the end of the complex debugging of the model program, the result of the stage should be the following documents: description of the simulation model; description of the model program indicating the programming system and accepted notation; full scheme of the model program; complete recording of the model program in the modeling language; proof of the reliability of the model program (the results of complex debugging of the model program); description of input and output values with necessary explanations (dimensions, scales, ranges of values, symbols); evaluation of the cost of computer time for one simulation cycle; instructions for working with the model program. To check the adequacy of the model to the object of study, after compiling a formal description of the system, the researcher draws up a plan for conducting full-scale experiments with a system prototype. If there is no prototype of the system, then a system of nested IMs can be used, differing from each other in the degree of detail of imitation of the same phenomena. Then the more detailed model serves as a prototype for the generalized IM. If it is impossible to build such a sequence, either because of the lack of resources to perform this work, or because of insufficient information, then they do without checking the adequacy of the IM. According to this plan, in parallel with the debugging of the IM, a series of full-scale experiments on a real system is carried out, during which control results. Having control results and MI test results at his disposal, the researcher checks the adequacy of the model to the object. If errors are found during the debugging phase that can only be corrected in the previous phases, a return to the previous phase can take place. In addition to the technical documentation, the results of the stage are accompanied by a machine implementation of the model (a program translated in the machine code of the computer on which the simulation will take place). This is an important step in the creation of the model. In this case, you must do the following. First, make sure that the dynamics of the development of the algorithm for modeling the object of study is correct in the course of simulating its functioning (to verify the model). Secondly, to determine the degree of adequacy of the model and the object of study. The adequacy of a software simulation model to a real object is understood as the coincidence with a given accuracy of the vectors of the characteristics of the behavior of the object and the model. In the absence of adequacy, the simulation model is calibrated (“correct” the characteristics of the model component algorithms). The presence of errors in the interaction of model components returns the researcher to the stage of creating a simulation model. It is possible that in the course of formalization, the researcher oversimplified the physical phenomena, excluded from consideration a number of important aspects of the functioning of the system, which led to the inadequacy of the model to the object. In this case, the researcher must return to the stage of system formalization. In cases where the choice of the formalization method turned out to be unsuccessful, the researcher needs to repeat the stage of compiling a conceptual model, taking into account new information and emerging experience. Finally, when the researcher has insufficient information about the object, he must return to the stage of compiling a meaningful description of the system and refine it, taking into account the results of testing the previous system model. At the same time, the accuracy of the simulation of phenomena, the stability of the simulation results, the sensitivity of the quality criteria to changes in the model parameters are evaluated. It is very difficult to obtain these estimates in some cases. However, without the successful results of this work, neither the developer nor the IM customer will have confidence in the model. Different researchers, depending on the type of IM, have developed different interpretations of the concepts of accuracy, stability, stationarity, sensitivity of IM. So far, there is no generally accepted theory of imitation of phenomena on a computer. Each researcher has to rely on his experience in organizing the simulation and on his understanding of the features of the simulation object. The simulation accuracy of phenomena is an assessment of the influence of stochastic elements on the functioning of a complex system model. The stability of the simulation results is characterized by the convergence of the controlled simulation parameter to a certain value with an increase in the simulation time of a variant of a complex system. The stationarity of the simulation mode characterizes a certain equilibrium of processes in the system model, when further simulation is meaningless, since the researcher will not receive new information from the model, and continuing the simulation practically only leads to an increase in computer time. It is necessary to provide for such a possibility and develop a method for determining the moment when the stationary simulation mode is reached. The MI sensitivity is represented by the value of the minimum increment of the selected quality criterion, calculated from the simulation statistics, with sequential variation of the simulation parameters over the entire range of their changes. This stage begins with the design of an experiment that allows the researcher to obtain maximum information with minimum computational effort. Statistical substantiation of the experimental plan is required. Experiment planning is a procedure for choosing the number and conditions of experiments that are necessary and sufficient to solve the problem with the required accuracy. At the same time, the following is essential: striving to minimize the total number of experiments, ensuring the possibility of simultaneous variation of all variables; the use of a mathematical apparatus that formalizes many of the actions of experimenters; choosing a clear strategy that allows you to make informed decisions after each series of experiments on the model. Then the researcher proceeds to carry out working calculations on the model. This is a very time-consuming process that requires a large computer resource and an abundance of clerical work. It should be noted that already at the early stages of creating an IM, it is necessary to carefully consider the composition and volume of modeling information in order to significantly facilitate further analysis of the simulation results. The result of the work are the simulation results. This stage completes the technological chain of stages of creating and using simulation models. Having received the simulation results, the researcher proceeds to interpret the results. The following simulation cycles are possible here. In the first cycle of the simulation experiment, the IM provides in advance the choice of options for the system under study by setting the initial simulation conditions for machine program models. In the second cycle of the simulation experiment, the model is modified in the modeling language, and therefore re-translation and editing of the program is required. It is possible that in the course of interpreting the results, the researcher found the presence of errors either when creating the model or when formalizing the modeling object. In these cases, a return is made to the stages of constructing a description of the simulation model or to compiling a conceptual model of the system, respectively. The result of the stage of interpretation of the simulation results are recommendations for the design of the system or its modification. With the recommendations at their disposal, the researchers begin to make design decisions. The interpretation of the simulation results is significantly influenced by the imaging capabilities of the computer used and the simulation system implemented on it. 1. How is the classification of mathematical models based on the features of the applied mathematical apparatus. Mathematics abstract Development of an economic and mathematical model for optimizing the sectoral structure of production in the agricultural sector