Like a system of equations, or arithmetic relations, or geometric shapes, or a combination of both, the study of which by means of mathematics should answer the questions posed about the properties of a certain set of properties of a real world object, as a set of mathematical relationships, equations, inequalities that describe the basic patterns inherent in the process, object or system being studied.

In automated control systems, a mathematical model is used to determine the operating algorithm of the controller. This algorithm determines how the control action should be changed depending on the change in the master in order for the control goal to be achieved.

Model classification

Formal classification of models

The formal classification of models is based on the classification of the mathematical tools used. Often constructed in the form of dichotomies. For example, one of the popular sets of dichotomies:

and so on. Each constructed model is linear or nonlinear, deterministic or stochastic, ... Naturally, mixed types are also possible: concentrated in one respect (in terms of parameters), distributed in another, etc.

Classification according to the way the object is represented

Along with the formal classification, models differ in the way they represent an object:

• Structural or functional models

Model hypotheses in science cannot be proven once and for all; we can only talk about their refutation or non-refutation as a result of experiment.

If a model of the first type is built, this means that it is temporarily accepted as truth and one can concentrate on other problems. However, this cannot be a point in research, but only a temporary pause: the status of a model of the first type can only be temporary.

Phenomenological model

The second type is the phenomenological model ( “we behave as if...”), contains a mechanism to describe the phenomenon, although this mechanism is not convincing enough, cannot be sufficiently confirmed by available data, or does not fit well with existing theories and accumulated knowledge about the object. Therefore, phenomenological models have the status of temporary solutions. It is believed that the answer is still unknown, and the search for the “true mechanisms” must continue. Peierls includes, for example, the caloric model and the quark model of elementary particles as the second type.

The role of the model in research may change over time, and it may happen that new data and theories confirm phenomenological models and they are promoted to the status of a hypothesis. Similarly, new knowledge can gradually come into conflict with hypothesis models of the first type, and they can be translated into the second. Thus, the quark model is gradually moving into the category of hypotheses; atomism in physics arose as a temporary solution, but with the course of history it became the first type. But the ether models have made their way from type 1 to type 2, and are now outside of science.

The idea of ​​simplification is very popular when building models. But simplification comes in different forms. Peierls identifies three types of simplifications in modeling.

Approximation

The third type of models is approximations ( “we consider something very large or very small”). If it is possible to construct equations that describe the system under study, this does not mean that they can be solved even with the help of a computer. A generally accepted technique in this case is the use of approximations (type 3 models). Among them linear response models. The equations are replaced by linear ones. A standard example is Ohm's law.

Thought experiment

m x ¨ = − k x (\displaystyle m(\ddot (x))=-kx),

Where x ¨ (\displaystyle (\ddot (x))) means the second derivative of x (\displaystyle x) by time: x ¨ = d 2 x d t 2 (\displaystyle (\ddot (x))=(\frac (d^(2)x)(dt^(2)))).

The resulting equation describes the mathematical model of the considered physical system. This model is called a "harmonic oscillator".

According to the formal classification, this model is linear, deterministic, dynamic, concentrated, continuous. In the process of its construction, we made many assumptions (about the absence of external forces, the absence of friction, the smallness of deviations, etc.), which in reality may not be met.

In relation to reality, this is most often a type 4 model simplification(“we will omit some details for clarity”), since some essential universal features (for example, dissipation) are omitted. To some approximation (say, while the deviation of the load from equilibrium is small, with low friction, for not too much time and subject to certain other conditions), such a model describes a real mechanical system quite well, since the discarded factors have a negligible effect on its behavior . However, the model can be refined by taking into account some of these factors. This will lead to a new model, with a wider (though again limited) scope of applicability.

However, when refining the model, the complexity of its mathematical research can increase significantly and make the model virtually useless. Often, a simpler model allows for better and deeper exploration of a real system than a more complex one (and, formally, “more correct”).

If we apply the harmonic oscillator model to objects far from physics, its substantive status may be different. For example, when applying this model to biological populations, it should most likely be classified as type 6 analogy(“let’s take into account only some features”).

Hard and soft models

The harmonic oscillator is an example of the so-called “hard” model. It is obtained as a result of a strong idealization of a real physical system. The properties of a harmonic oscillator are qualitatively changed by small perturbations. For example, if you add a small term to the right side − ε x ˙ (\displaystyle -\varepsilon (\dot (x)))(friction) ( ε > 0 (\displaystyle \varepsilon >0)- some small parameter), then we get exponentially damped oscillations if we change the sign of the additional term (ε x ˙) (\displaystyle (\varepsilon (\dot (x)))) then friction will turn into pumping and the amplitude of oscillations will increase exponentially.

To resolve the question of the applicability of a rigid model, it is necessary to understand how significant the factors that we have neglected are. It is necessary to study soft models obtained by a small perturbation of the hard one. For a harmonic oscillator they can be given, for example, by the following equation:

m x ¨ = − k x + ε f (x , x ˙) (\displaystyle m(\ddot (x))=-kx+\varepsilon f(x,(\dot (x)))).

Here f (x , x ˙) (\displaystyle f(x,(\dot (x))))- some function that can take into account the friction force or the dependence of the spring stiffness coefficient on the degree of its stretch. Explicit function form f (\displaystyle f) We are not interested at the moment.

If we prove that the behavior of the soft model is not fundamentally different from the behavior of the hard one (regardless of the explicit type of perturbing factors, if they are small enough), the problem will be reduced to studying the hard model. Otherwise, the application of the results obtained from studying the rigid model will require additional research.

If a system maintains its qualitative behavior under small disturbances, it is said to be structurally stable. A harmonic oscillator is an example of a structurally unstable (non-rough) system. However, this model can be used to study processes over limited periods of time.

Versatility of models

The most important mathematical models usually have the important property versatility: Fundamentally different real phenomena can be described by the same mathematical model. For example, a harmonic oscillator describes not only the behavior of a load on a spring, but also other oscillatory processes, often of a completely different nature: small oscillations of a pendulum, fluctuations in the level of a liquid in U (\displaystyle U)-shaped vessel or a change in current strength in an oscillatory circuit. Thus, by studying one mathematical model, we immediately study a whole class of phenomena described by it. It is this isomorphism of laws expressed by mathematical models in various segments of scientific knowledge that inspired Ludwig von Bertalanffy to create a “general systems theory”.

Direct and inverse problems of mathematical modeling

There are many problems associated with mathematical modeling. First, you need to come up with a basic diagram of the modeled object, reproduce it within the framework of the idealizations of this science. Thus, a train car turns into a system of plates and more complex bodies from different materials, each material is specified as its standard mechanical idealization (density, elastic moduli, standard strength characteristics), after which equations are drawn up, along the way some details are discarded as unimportant, Calculations are made, compared with measurements, the model is refined, and so on. However, to develop mathematical modeling technologies, it is useful to disassemble this process into its main components.

Traditionally, there are two main classes of problems associated with mathematical models: direct and inverse.

Direct task: the structure of the model and all its parameters are considered known, the main task is to conduct a study of the model to extract useful knowledge about the object. What static load will the bridge withstand? How it will react to a dynamic load (for example, to the march of a company of soldiers, or to the passage of a train at different speeds), how the plane will overcome the sound barrier, whether it will fall apart from flutter - these are typical examples of a direct problem. Setting the right direct problem (asking the right question) requires special skill. If the right questions are not asked, a bridge may collapse, even if a good model for its behavior has been built. Thus, in 1879, the metal Railway Bridge across the Firth of Tay collapsed in Great Britain, the designers of which built a model of the bridge, calculated it for a 20-fold safety factor for the action of the payload, but forgot about the winds constantly blowing in those places. And after a year and a half it collapsed.

In the simplest case (one oscillator equation, for example), the direct problem is very simple and reduces to an explicit solution of this equation.

Inverse problem: many possible models are known, a specific model must be selected based on additional data about the object. Most often, the structure of the model is known, and some unknown parameters need to be determined. Additional information may consist of additional empirical data, or requirements for the object ( design problem). Additional data can arrive regardless of the process of solving the inverse problem ( passive observation) or be the result of an experiment specially planned during the solution ( active surveillance).

One of the first examples of a masterly solution to an inverse problem with the fullest use of available data was Newton’s method for reconstructing friction forces from observed damped oscillations.

Another example is mathematical statistics. The task of this science is to develop methods for recording, describing and analyzing observational and experimental data in order to build probabilistic models of mass random phenomena. That is, the set of possible models is limited to probabilistic models. In specific tasks, the set of models is more limited.

Computer simulation systems

To support mathematical modeling, computer mathematics systems have been developed, for example, Maple, Mathematica, Mathcad, MATLAB, VisSim, etc. They allow you to create formal and block models of both simple and complex processes and devices and easily change model parameters during modeling. Block models are represented by blocks (most often graphic), the set and connection of which are specified by the model diagram.

Additional examples

Malthus' model

According to the model proposed by Malthus, the growth rate is proportional to the current population size, that is, described by the differential equation:

x ˙ = α x (\displaystyle (\dot (x))=\alpha x),

Where α (\displaystyle \alpha )- a certain parameter determined by the difference between fertility and mortality. The solution to this equation is the exponential function x (t) = x 0 e α t (\displaystyle x(t)=x_(0)e^(\alpha t)). If the birth rate exceeds the death rate ( α > 0 (\displaystyle \alpha >0)), the population size is unlimited and growing very quickly. In reality this cannot happen due to limited resources. When a certain critical population size is reached, the model ceases to be adequate, since it does not take into account the limited resources. A refinement of the Malthus model can be a logistic model, which is described by the Verhulst differential equation:

x ˙ = α (1 − x x s) x (\displaystyle (\dot (x))=\alpha \left(1-(\frac (x)(x_(s)))\right)x),

where is the “equilibrium” population size, at which the birth rate is exactly compensated by the death rate. The population size in such a model tends to an equilibrium value x s (\displaystyle x_(s)), and this behavior is structurally stable.

Predator-prey system

Let's say that two types of animals live in a certain area: rabbits (eating plants) and foxes (eating rabbits). Let the number of rabbits x (\displaystyle x), number of foxes y (\displaystyle y). Using the Malthus model with the necessary amendments to take into account the eating of rabbits by foxes, we arrive at the following system, named models Trays - Volterra:

( x ˙ = (α − c y) x y ˙ = (− β + d x) y (\displaystyle (\begin(cases)(\dot (x))=(\alpha -cy)x\\(\dot (y ))=(-\beta +dx)y\end(cases)))

The behavior of this system is not structurally stable: a small change in the parameters of the model (for example, taking into account the limited resources needed by rabbits) can lead to a qualitative change in behavior.

For certain parameter values, this system has an equilibrium state when the number of rabbits and foxes is constant. Deviation from this state leads to gradually fading fluctuations in the number of rabbits and foxes.

The opposite situation is also possible, when any small deviation from the equilibrium position will lead to catastrophic consequences, up to the complete extinction of one of the species. The Volterra - Trats model does not answer the question of which of these scenarios is being realized: additional research is required here.

see also

Notes

1. “A mathematical representation of reality” (Encyclopaedia Britanica)
2. Novik I. B., On philosophical issues of cybernetic modeling. M., Knowledge, 1964.
3. Sovetov B. Ya., Yakovlev S. A., Modeling of systems: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2
4. Samarsky A. A., Mikhailov A. P. Math modeling. Ideas. Methods. Examples. - 2nd ed., rev. - M.: Fizmatlit, 2001. - ISBN 5-9221-0120-X.
5. Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4
6. Sevostyanov, A. G. Modeling of technological processes: textbook / A. G. Sevostyanov, P. A. Sevostyanov. - M.: Light and food industry, 1984. - 344 p.
7. Rotach V.Ya. Theory of automatic control. - 1st. - M.: ZAO "Publishing House MPEI", 2008. - P. 333. - 9 p. - ISBN 978-5-383-00326-8.
8. Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena(English) . Springer, Complexity series, Berlin-Heidelberg-New York, 2006. XII+562 pp. ISBN 3-540-35885-4. Retrieved June 18, 2013. Archived June 18, 2013.
9. “A theory is considered linear or nonlinear depending on what kind of mathematical apparatus - linear or nonlinear - and what kind of linear or nonlinear mathematical models it uses. ...without denying the latter. A modern physicist, if he had to re-create the definition of such an important entity as nonlinearity, would most likely act differently, and, giving preference to nonlinearity as the more important and widespread of the two opposites, would define linearity as “not nonlinearity.” Danilov Yu. A., Lectures on nonlinear dynamics. Elementary introduction. Series “Synergetics: from past to future.” Edition 2. - M.: URSS, 2006. - 208 p. ISBN 5-484-00183-8
10. “Dynamical systems modeled by a finite number of ordinary differential equations are called concentrated or point systems. They are described using a finite-dimensional phase space and are characterized by a finite number of degrees of freedom. The same system under different conditions can be considered either concentrated or distributed. Mathematical models of distributed systems are differential equations in partial derivatives, integral equations or ordinary equations with a retarded argument. The number of degrees of freedom of a distributed system is infinite, and an infinite number of data are required to determine its state.”
    Anishchenko V. S., Dynamic systems, Soros educational journal, 1997, No. 11, p. 77-84.
11. “Depending on the nature of the processes being studied in the system S, all types of modeling can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous. Deterministic modeling reflects deterministic processes, that is, processes in which the absence of any random influences is assumed; stochastic modeling depicts probabilistic processes and events. ... Static modeling serves to describe the behavior of an object at any point in time, and dynamic modeling reflects the behavior of an object over time. Discrete modeling is used to describe processes that are assumed to be discrete, respectively, continuous modeling allows us to reflect continuous processes in systems, and discrete-continuous modeling is used for cases when they want to highlight the presence of both discrete and continuous processes.”
    Sovetov B. Ya., Yakovlev S. A., Modeling of systems: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2
12. Typically, a mathematical model reflects the structure (device) of the modeled object, the properties and relationships of the components of this object that are essential for the purposes of research; such a model is called structural. If the model reflects only how the object functions - for example, how it reacts to external influences - then it is called functional or, figuratively, a black box. Combined models are also possible. Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 p.

For the theory of mathematical modeling, it is necessary to know the purpose of modeling and represent the modeling object in mathematical form. The word "model" comes from the Latin modus (copy, image, outline). The simplest and most obvious example of modeling is geographic and topographic maps. Models are structural formulas in chemistry. The model as a means of cognition stands between logical thinking and the process or phenomenon being studied.

Modeling is the replacement of some object A with another object B. The replaced object is called the original, the replacing one is called the model. Thus, the model is a substitute for the original. Depending on the purpose of the replacement, the model of the same original may be different. In science and technology, the main purpose of modeling is to study the original using a simpler model of it. Replacing one object with another makes sense only if there is a certain similarity or analogy between them.

A mathematical model is an approximate representation, expressed in mathematical terms, of objects, concepts, systems or processes. Objects, concepts, systems or processes to be modeled are called objects of modeling (OM).

All objects and phenomena are interconnected to a greater or lesser extent, but during modeling most of the interrelations are neglected and the modeling object is considered as a separate system. If the modeling object is defined as a separate system, then it is necessary to introduce the principle of selectivity, ensuring the selection of the required connections with the external environment. For example, when modeling electronic circuits, thermal, acoustic, optical and mechanical interactions with the external environment are neglected and only electrical variables are considered. The principle of selectivity introduces an error into the system, i.e., a difference in the behavior of the model and the modeled object. The next important modeling factor is the principle of causality, which links input and output variables in the system.

To quantify the system, the concept of “state” is introduced. For example, the state of an electronic circuit refers to the values ​​of voltages and currents in an electronic circuit at a given time.

When deriving a mathematical model analytically, well-known categories are most often used: laws, structures and parameters.

If any variable y depends on another variable x, then the first quantity is a function of the second. This dependence is written in the form y = f(x) or y = y(x). In this notation, the variable x is called an argument. An important characteristic of a function is its derivative, the process of finding which is called differentiation. Equations that, according to mathematical rules, connect an unknown function, its derivatives and arguments are called differential. The process inverse to differentiation, which allows one to find the function itself from a given derivative, is called integration.

Let's consider a special case when the function is a path that depends on the argument - time. Then the derivative of the path with respect to time is the velocity, and the derivative of the velocity (or the second derivative of the path) is the acceleration. If, for example, the speed is known, then integration is used to find the path traveled by the body when moving in a certain time. If only the acceleration is known, then the integration operation is performed twice to find the path. In this case, after calculating the first integral, the speed becomes known.

The ultimate goal of creating mathematical models is to establish functional dependencies between variables. The functional dependence for each specific model can take a strictly defined form. When a device is simulated, the input of which receives a signal x y and the output signal y appears, the connection can be written in the form of a table. To do this, the entire range of changes in the input and output signals is divided into a certain number of sections. Each section of the range of variation of the input signal will correspond to a certain section of the range of variation of the output signal. In complex systems, where there are several inputs and several outputs, analytical dependencies are expressed by systems of differential equations.

* Laws are usually formulated for particular areas, such as Kirchhoff's and Newton's laws. Applying these laws to a system usually focuses our attention on a single area of ​​science and technology. By using Kirchhoff's laws and Maxwell's equations to analyze an electrical system, the researcher ignores other (for example, thermal) processes in the system.

Creating a mathematical model requires knowledge of the elements present in the system and their relationships. The parameters of the mathematical model (MM) are those included in the system of equations different odds. These coefficients, together with the equations and boundary conditions, form a complete MM.

Any mathematical model can be obtained as a result of: 1) direct observation of a phenomenon, its direct study and comprehension (models are phenomenological); 2) some process of deduction, when a new model is obtained as a special case from some more general model (such models are called asymptotic); 3) some process of induction, when the new model is a natural generalization of elementary models (such models are called composite, or ensemble models).

All systems exist in time and space. Mathematically, this means that time and the three spatial variables can be considered independent variables.

There are many signs of classification of mathematical models based on the use of certain variables as independent ones, presented in continuous or discrete form; MM is classified as follows:

1) models with distributed parameters (all independent variables are taken in continuous form);

2) models with lumped parameters (all independent spatial variables are discrete, and the time variable is continuous);

3) models with discrete parameters (all independent variables are taken in discrete form).

In Fig. 3.10 a... shows an approximate classification of models. All models can be divided into real and ideal (Fig. 3.10, a). This chapter discusses only ideal models, which are objective in their content (reflecting real reality), but subjective in form and cannot exist outside of it. Ideal models exist only in human knowledge and function according to the laws of logic. Logical models include various signed models. An essential point in creating any symbolic model is the formalization procedure (formulas, alphabet, number systems).

Currently, in a number of areas of science and technology, the concept of a model is interpreted not in the spirit of classical physics, as a visual, for example, mechanical system, but in the spirit modern stage knowledge as an abstract logical-mathematical structure.

In modern modeling, along with the increasing role of abstract logical models in cognition, there is another trend associated with the widespread use of cybernetic functional information models.

The uniqueness of cybernetic modeling is that the objective similarity of the model and the simulated object concerns only their functions, areas of application, and connection with the external environment. The basis of the information approach to the study of cybernetic processes is abstraction.

Let's consider the models that take place in CAD LSI: structural, functional, geometric, symbolic, mental, analytical, numerical and simulation.

Structural models reproduce the composition of the elements of an object or system, their location in space and relationships, i.e. the structure of the system. Structural models can be both real (layouts) and ideal (for example, mechanical engineering drawings, printed circuit board topology and IC topology).

Functional models imitate only the way the original behaves, its functional dependence on the external environment. The most typical example is models built on the “black box” concept.

In these models, it is possible to reproduce the functioning of the original, completely abstracting from its contents and structure, connecting various input and output quantities using a mathematical relationship.

Rice. 3.10. General classification of models (a), as well as full-scale (b), physical (c), real mathematical (d), visual (e), symbolic (f), ideal mathematical (g) models

Geometric models reflect only the structure of an object and are of great importance in connection with design electronic systems. These models, built on the basis of geometric similarity, allow solving problems related to the optimal placement of objects, laying out traces on printed circuit boards and integrated circuits.

Sign models are an ordered record of symbols (signs). Signs interact with each other not according to physical laws, but according to the rules established in a particular field of knowledge, or, as they say, according to the nature of signs. Iconic models are now extremely widespread. Almost every field of knowledge - linguistics, programming, electronics and many others - has developed its own symbolism to describe models. These are programs, schemes, etc.

Mental models are a product of sensory perception and the activity of abstract thinking. Mental models include the well-known planetary model of the Bohr atom. To convey these models, they are presented in the form of a verbal or symbolic description, that is, mental models can be recorded in the form of various sign systems.

Analytical models make it possible to obtain explicit dependences of the required quantities on the parameters and variables characterizing the phenomenon being studied. The analytical solution of a mathematical relationship is a generalized description of the object

Numerical models are characterized by the fact that the values ​​of the required quantities can be obtained as a result of applying the appropriate numerical methods. All numerical methods allow one to obtain only private information regarding the desired quantities, since for their implementation they require specifying specific values ​​of all parameters included in the mathematical relationship. For each desired value, one has to transform the mathematical model in its own way and apply the corresponding numerical procedure.

Simulation models are implemented on a computer in the form of modeling algorithms (programs) that allow one to calculate the values ​​of output variables and determine the new state into which the model goes for given values ​​of input variables, parameters and the initial state of the model. Simulation modeling, unlike numerical modeling, is characterized by the independence of the modeling algorithm from the type of information that needs to be obtained as a result of the modeling. A mathematical model that is represented in an abstract mathematical form through variables, parameters, equations and inequalities is quite universal, flexible and effective.

The MM includes the following elements: variables (dependent and independent); constants or fixed parameters (determining the degree of connection between variables); mathematical expressions (equations and/or inequalities that combine variables and parameters); logical expressions (defining various restrictions in the mathematical model); information (alphanumeric and graphic).

Mathematical models are classified according to the following criteria: 1) behavior of models over time; 2) types of input information, parameters and expressions that make up the mathematical model; 3) the structure of the mathematical model; 4) the type of mathematical apparatus used.

With regard to integrated circuits, the following classification can be proposed.

Depending on the nature of the properties of the integrated circuit, mathematical models are divided into functional and structural.

Functional models reflect the functioning processes of an object; these models have the form of systems of equations.

When solving a number of design problems, mathematical models that reflect only the structural properties of the designed object are widely used; such structural models can take the form of matrices, graphs, lists of vectors and express mutual arrangement elements in space, the presence of a direct connection in the form of conductors, etc. Structural models are used in the case when the problems of structural synthesis can be formalized and solved, abstracting from the peculiarities of physical processes in the object.

Rice. 3.11. Structural model of inverter = it. d.)

According to the method of obtaining, functional mathematical models are divided into theoretical and formal.

Theoretical models are obtained based on the study of physical laws, and the structure of the equations and parameters of the models have a clear physical basis.

Formal models are obtained by considering the properties of a real object as a black box.

Theoretical approach allows us to obtain more universal models that are valid for various operating modes and for wide ranges of changes in external parameters.

A number of features in the classification are associated with the features of the equations that make up the mathematical model; Depending on the linearity or nonlinearity of the equations, models are divided into linear and nonlinear.

Depending on the power of the set of variable values, models are divided into continuous and discrete (Fig. 3.12).

In continuous models, the variable appearing in them is continuous or piecewise continuous.

Variables in discrete models are discrete quantities, the set of which is countable.

Rice. 3.12. Continuous and discrete variables

Based on the form of connection between output, internal and external parameters, models in the form of systems of equations and models in the form of an explicit dependence of output parameters on internal and external parameters are distinguished. The first of them are called algorithmic, and the second - analytical.

Depending on whether the model equations take into account the inertia of processes in the design object, dynamic and static models are distinguished.

The concept of model and simulation.

Model in a broad sense- this is any image, mental analogue or established image, description, diagram, drawing, map, etc. of any volume, process or phenomenon, used as its substitute or representative. The object, process or phenomenon itself is called the original of this model.

Modeling - this is the study of any object or system of objects by constructing and studying their models. This is the use of models to determine or clarify the characteristics and rationalize the methods of constructing newly constructed objects.

Any method of scientific research is based on the idea of ​​modeling, while theoretical methods use various kinds of symbolic, abstract models, and experimental methods use subject models.

During research, a complex real phenomenon is replaced by some simplified copy or diagram; sometimes such a copy serves only to remember and recognize the desired phenomenon at the next meeting. Sometimes the constructed diagram reflects some essential features, allows one to understand the mechanism of a phenomenon, and makes it possible to predict its change. Different models can correspond to the same phenomenon.

The researcher's task is to predict the nature of the phenomenon and the course of the process.

Sometimes, it happens that an object is available, but experiments with it are expensive or lead to serious environmental consequences. Knowledge about such processes is obtained using models.

An important point is that the very nature of science involves the study of not one specific phenomenon, but a wide class of related phenomena. It assumes the need to formulate some general categorical statements, which are called laws. Naturally, with such a formulation many details are neglected. In order to more clearly identify a pattern, they consciously go for coarsening, idealization, and sketchiness, that is, they study not the phenomenon itself, but a more or less accurate copy or model of it. All laws are laws about models, and therefore it is not surprising that over time some scientific theories are considered unsuitable. This does not lead to the collapse of science, since one model has been replaced by another more modern.

A special role in science is played by mathematical models, building materials and tools of these models - mathematical concepts. They accumulated and improved over thousands of years. Modern mathematics provides extremely powerful and universal means of research. Almost every concept in mathematics, every mathematical object, starting from the concept of number, is a mathematical model. When constructing a mathematical model of the object or phenomenon being studied, those of its features, features and details are identified that, on the one hand, contain more or less complete information about the object, and on the other, allow mathematical formalization. Mathematical formalization means that the features and details of an object can be associated with suitable adequate mathematical concepts: numbers, functions, matrices, and so on. Then the connections and relationships discovered and assumed in the object under study between its individual parts and components can be written using mathematical relations: equalities, inequalities, equations. The result is a mathematical description of the process or phenomenon being studied, that is, its mathematical model.

The study of a mathematical model is always associated with certain rules of action on the objects being studied. These rules reflect the relationships between causes and effects.

Building a mathematical model is the central stage of research or design of any system. All subsequent analysis of the object depends on the quality of the model. Building a model is not a formal procedure. It strongly depends on the researcher, his experience and taste, and is always based on certain experimental material. The model must be sufficiently accurate, adequate and convenient to use.

Math modeling.

Classification of mathematical models.

Mathematical models can bedeterministic And stochastic .

Determinate model and are models in which a one-to-one correspondence is established between variables describing an object or phenomenon.

This approach is based on knowledge of the functioning mechanism of objects. Often the object being modeled is complex and deciphering its mechanism can be very labor-intensive and time-consuming. In this case, proceed as follows: experiments are carried out on the original, the results are processed and, without delving into the mechanism and theory of the simulated object, using methods mathematical statistics and probability theories, establish connections between variables that describe an object. In this case you getstochastic model . IN stochastic model, the relationship between variables is random, sometimes it is fundamental. The influence of a huge number of factors, their combination leads to a random set of variables describing an object or phenomenon. According to the nature of the modes, the model isstatistical And dynamic.

Statisticalmodelincludes a description of the relationships between the main variables of the modeled object in a steady state without taking into account changes in parameters over time.

IN dynamicmodelsthe relationships between the main variables of the modeled object during the transition from one mode to another are described.

There are models discrete And continuous, and mixed type. IN continuous variables take values ​​from a certain interval, indiscretevariables take isolated values.

Linear models- all functions and relations that describe the model linearly depend on the variables andnot linearotherwise.

Math modeling.

Requirements ,p presented to the models.

1. Versatility- characterizes the completeness of the model’s representation of the studied properties of a real object.

1. Adequacy is the ability to reflect the desired properties of an object with an error no higher than a given one.
2. Accuracy is assessed by the degree of agreement between the values ​​of the characteristics of a real object and the values ​​of these characteristics obtained using models.
3. Economical - determined by the expenditure of computer memory resources and time for its implementation and operation.

Math modeling.

Main stages of modeling.

1. Statement of the problem.

Determining the purpose of the analysis and the way to achieve it and developing a general approach to the problem under study. At this stage, a deep understanding of the essence of the task is required. Sometimes, setting a problem correctly is no less difficult than solving it. Staging is not a formal process; there are no general rules.

2. Studying the theoretical foundations and collecting information about the original object.

At this stage, a suitable theory is selected or developed. If it is not there, cause-and-effect relationships are established between the variables describing the object. Input and output data are determined, and simplifying assumptions are made.

3. Formalization.

It consists in choosing a system of symbols and using them to write down the relationships between the components of an object in the form of mathematical expressions. The class of problems to which the resulting mathematical model of the object can be classified is established. The values ​​of some parameters may not yet be specified at this stage.

4. Choosing a solution method.

At this stage, the final parameters of the models are established taking into account the operating conditions of the object. For the resulting mathematical problem, a solution method is selected or a special method is developed. When choosing a method, the user's knowledge, his preferences, and the developer's preferences are taken into account.

5. Implementation of the model.

Having developed an algorithm, a program is written, which is debugged, tested, and a solution to the desired problem is obtained.

6. Analysis of the information received.

The obtained and expected solutions are compared, and the modeling error is monitored.

7. Checking the adequacy of the real object.

The results obtained from the model are comparedeither with the information available about the object, or an experiment is carried out and its results are compared with the calculated ones.

The modeling process is iterative. In case of unsatisfactory results of the stages 6. or 7. a return is made to one of the earlier stages, which could have led to the development of an unsuccessful model. This stage and all subsequent ones are refined and such refinement of the model occurs until acceptable results are obtained.

A mathematical model is an approximate description of any class of phenomena or objects of the real world in the language of mathematics. The main purpose of modeling is to explore these objects and predict the results of future observations. However, modeling is also a method of understanding the world around us, making it possible to control it.

Mathematical modeling and the associated computer experiment are indispensable in cases where a full-scale experiment is impossible or difficult for one reason or another. For example, it is impossible to set up a natural experiment in history to check “what would have happened if...” It is impossible to check the correctness of one or another cosmological theory. It is possible, but unlikely to be reasonable, to experiment with the spread of a disease, such as the plague, or carry out a nuclear explosion to study its consequences. However, all this can be done on a computer by first constructing mathematical models of the phenomena being studied.

1.1.2 2. Main stages of mathematical modeling

1) Model building. At this stage, some “non-mathematical” object is specified - a natural phenomenon, design, economic plan, production process, etc. In this case, as a rule, a clear description of the situation is difficult. First, the main features of the phenomenon and the connections 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 stage of modeling.

2) Solving the mathematical problem to which the model leads. At this stage, much attention is paid to the development of algorithms and numerical methods for solving the 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 the field.

4) Checking the adequacy of the model.At this stage, it is determined whether the experimental results agree with the theoretical consequences of the model within a certain accuracy.

5) Modification of the model.At this stage, either the model is complicated so that it is more adequate to reality, or it is simplified in order to achieve a practically acceptable solution.

1.1.3 3. Model classification

Models can be classified according to different criteria. For example, according to the nature of the problems being solved, models can be divided into functional and structural. In the first case, all quantities characterizing a phenomenon or object are expressed quantitatively. Moreover, some of them are considered as independent variables, while others are considered as functions of these quantities. A mathematical model is usually a system of equations of various types (differential, algebraic, etc.) that establish quantitative relationships between the quantities under consideration. In the second case, the model characterizes the structure of a complex object consisting of individual parts, between which there are certain connections. Typically, these connections are not quantifiable. To construct such models, it is convenient to use graph theory. A graph is a mathematical object that represents a set of points (vertices) on a plane or in space, some of which are connected by lines (edges).

Based on the nature of the initial data and results, prediction models can be divided into deterministic and probabilistic-statistical. Models of the first type make certain, unambiguous predictions. Models of the second type are based on statistical information, and the predictions obtained with their help are probabilistic in nature.

MATHEMATICAL MODELING AND GENERAL COMPUTERIZATION OR SIMULATION MODELS

Now, when almost universal computerization is taking place in the country, we hear statements from specialists in various professions: “If we introduce a computer, then all problems will be solved immediately.” This point of view is completely incorrect; computers themselves, without mathematical models of certain processes, will not be able to do anything, and one can only dream of universal computerization.

In support of the above, we will try to substantiate the need for modeling, including mathematical modeling, and reveal its advantages in human cognition and transformation outside world, let's identify existing shortcomings and go... to simulation modeling, i.e. modeling using a computer. But everything is in order.

First of all, let's answer the question: what is a model?

A model is a material or mentally represented object, which in the process of cognition (study) replaces the original, preserving some typical properties that are important for this study.

A well-built model is more accessible for research than a real object. For example, experiments with the country’s economy in educational purposes, you can’t do without a model here.

Summarizing what has been said, we can answer the question: what are models for? In order to

• understand how an object works (its structure, properties, laws of development, interaction with the outside world).
• learn to manage an object (process) and determine the best strategies
• predict the consequences of impact on the object.

What's positive about any model? It allows you to gain new knowledge about the object, but, unfortunately, it is incomplete to one degree or another.

Modelformulated in the language of mathematics using mathematical methods is called a mathematical model.

The starting point for its construction is usually some problem, for example an economic one. Both descriptive and optimization mathematical ones are widespread, characterizing various economic processes and phenomena, for example:

• resource allocation
• rational cutting
• transportation
• consolidation of enterprises
• network planning.

How is a mathematical model constructed?

• Firstly, the purpose and subject of the study are formulated.
• Secondly, the most important characteristics corresponding to this goal are highlighted.
• Thirdly, the relationships between the elements of the model are described verbally.
• Next, the relationship is formalized.
• And a calculation is made using a mathematical model and the resulting solution is analyzed.

Using this algorithm, you can solve any optimization problem, including multicriteria, i.e. one in which not one, but several goals are pursued, including contradictory ones.

Let's give an example. Theory queuing– the problem of queuing. It is necessary to balance two factors - the cost of maintaining service devices and the cost of staying in line. Having constructed a formal description of the model, calculations are made using analytical and computational methods. If the model is good, then the answers found with its help are adequate to the modeling system; if it is bad, then it must be improved and replaced. The criterion of adequacy is practice.

Optimization models, including multicriteria ones, have a common property - a goal (or several goals) is known, to achieve which one often has to deal with complex systems, where it is not so much about solving optimization problems, but about studying and predicting states depending on selected management strategies. And here we are faced with the difficulties of implementing the previous plan. They are as follows:

• a complex system contains many connections between elements
• a real system is influenced by random factors, taking them into account analytically is impossible
• the possibility of comparing the original with the model exists only at the beginning and after using the mathematical apparatus, because intermediate results may have no analogues in the real system.

In connection with the listed difficulties that arise when studying complex systems, practice required a more flexible method, and it appeared - “Simujation modeling”.

Typically, a simulation model is understood as a set of computer programs that describes the functioning of individual system blocks and the rules of interaction between them. Usage random variables makes it necessary to carry out repeated experiments with the simulation system (on a computer) and subsequent statistical analysis obtained results. A very common example of using simulation models is solving the queuing problem using the MONTE CARLO method.

Thus, working with a simulation system is an experiment carried out on a computer. What are the advantages?

– Greater proximity to the real system than mathematical models;

–The block principle makes it possible to verify each block before its inclusion in the overall system;

–The use of dependencies of a more complex nature that cannot be described by simple mathematical relationships.

The listed advantages determine the disadvantages

– building a simulation model takes longer, is more difficult and more expensive;

– to work with the simulation system, you must have a computer suitable for the class;

– the interaction between the user and the simulation model (interface) should not be too complex, convenient and well known;

-building a simulation model requires a more in-depth study of the real process than mathematical modeling.

The question arises: can simulation modeling replace optimization methods? No, but it conveniently complements them. A simulation model is a program that implements a certain algorithm, to optimize the control of which an optimization problem is first solved.

So, neither a computer, nor a mathematical model, nor an algorithm for its study alone can solve a sufficiently complex problem. But together they represent the force that allows us to understand the world around us and manage it in the interests of man.

1.2 Model classification

1.2.1 Classification taking into account the time factor and area of ​​use (Makarova N.A.) Static model - it’s like a one-time snapshot of information on an object (the result of one survey) Dynamic model-allows see changes in an object over time (Card in the clinic) Models can also be classified according to what area of ​​knowledge do they belong to?(biological, historical, environmental, etc.) Return to top

1.2.2 Classification by area of ​​use (Makarova N.A.) Educational- visual manuals, simulators oh, howling ones programs Experienced models-reduced copies (car in a wind tunnel) Scientific and technical synchrophasotron, stand for testing electronic equipment Gaming- economic, sports, business games Imitation- Not They simply reflect reality, but imitate it (medicines are tested on mice, experiments are conducted in schools, etc. This modeling method is called trial and error Return to top

1.2.3 Classification according to the method of presentation Makarov N.A.) Material models- otherwise can be called subject. They perceive geometric and physical properties original and always have a real embodiment Information models are not allowed touch or see. They are based only on information .And informational model is a set of information that characterizes the properties and states of an object, process, phenomenon, as well as the relationship with the outside world. Verbal model - information model in mental or spoken form. Iconic model-information model expressed by signs ,i.e.. by means of any formal language. Computer model - m A model implemented by means of a software environment.

1.2.4 Classification of models given in the book "Earth Informatics" (Gein A.G.)) "...here is a seemingly simple task: how long will it take to cross the Karakum Desert? The answer is of course depends on the mode of transportation. If travel on camels, then it will take one term, another if you go by car, a third if you fly by plane. And most importantly, different models are required to plan a trip. For the first case, the required model can be found in the memoirs famous explorers deserts: after all, here you cannot do without information about oases and camel trails. In the second case, the information contained in the road atlas is irreplaceable. In the third, you can use the flight schedule. These three models differ - memoirs, atlas and schedule - and the nature of the presentation of information. In the first case, the model is represented by a verbal description of information (descriptive model), in the second - as if a photograph from life (full-scale model), in the third - a table containing symbols: departure and arrival times, day of the week, ticket price (the so-called sign model) However, this division is very arbitrary - in memoirs you may find maps and diagrams (elements of a full-scale model), on maps there are symbols (elements of a symbolic model), in the schedule there is a decoding of symbols (elements of a descriptive model). So this classification of models... in our opinion, is unproductive" In my opinion, this fragment demonstrates the descriptive (wonderful language and style of presentation) and, as it were, Socratic teaching style common to all of Hein’s books (Everyone thinks it’s like this. I completely agree with you, but if you look closely...). In such books it is quite difficult to find a clear system of definitions (it is not intended by the author). In the textbook edited by N.A. Makarova demonstrates a different approach - the definitions of concepts are clearly highlighted and somewhat static.

1.2.5 Classification of models given in the manual by A.I. Bochkin There are an unusually large number of classification methods .P bring just some of the most well-known grounds and signs: discreteness And continuity,matrix and scalar models, static and dynamic models, analytical and information models, subject and figurative-sign models, large-scale and non-scale... Every sign gives a certain knowledge about the properties of both the model and the simulated reality. The sign can serve as a hint about the method of the completed or upcoming modeling. Discreteness and continuity Discreteness - a characteristic feature of computer models .After all a computer can be in a finite, albeit very large number of states. Therefore, even if the object is continuous (time), in the model it will change in jumps. It could be considered continuity a sign of non-computer type models. Chance and determinism . Uncertainty, accident initially opposed computer world: The algorithm launched again should repeat itself and give the same results. But to simulate random processes, pseudorandom number sensors are used. Introducing randomness into deterministic problems leads to powerful and interesting models (Calculation of area by random toss). Matrixity - scalarity. Availability of parameters matrix model indicates its greater complexity and, possibly, accuracy compared to scalar. For example, if we do not identify all age groups in the country’s population, considering its change as a whole, we will obtain a scalar model (for example, the Malthus model); if we isolate it, we will obtain a matrix (sex-age) model. It was the matrix model that made it possible to explain the fluctuations in fertility after the war. Static dynamic. These properties of the model are usually predetermined by the properties of the real object. There is no freedom of choice here. Just static the model could be a step towards dynamic, or some of the model variables can be considered unchanged for now. For example, a satellite moves around the Earth, its movement is influenced by the Moon. If we consider the Moon stationary during the satellite's revolution, we obtain a simpler model. Analytical models. Description of processes analytically, formulas and equations. But when trying to build a graph, it is more convenient to have tables of function values ​​and arguments. Simulation models. Imitation models appeared a long time ago in the form of scale copies of ships, bridges, etc. appeared a long time ago, but are being considered recently in connection with computers. Knowing how connected elements of the model analytically and logically, it is easier not to solve a system of certain relationships and equations, but to display the real system in the computer memory, taking into account the connections between the memory elements. Information models. Information Models are usually contrasted with mathematical ones, or rather algorithmic ones. The ratio of data volumes to algorithms is important here. If there is more data or it is more important, we have an information model, otherwise - mathematical. Subject models. This is primarily a children's model - a toy. Iconic models. This is primarily a model in the human mind: figurative, if graphic images predominate, and iconic, if there are more words and/or numbers. Figurative-sign models are built on a computer. Scale models. TO large-scale models are those of subject or figurative models that repeat the shape of an object (map).

It is possible to trace the dynamics of the development of an object, the internal essence of the relationships of its elements and various states in the design process only with the help of models that use the principle of dynamic analogy, i.e. with the help of mathematical models.

Mathematical model is a system of mathematical relationships that describe the process or phenomenon being studied. To compile a mathematical model, you can use any mathematical means - set theory, mathematical logic, the language of differential or integral equations. The process of compiling a mathematical model is called mathematical modeling. Like other types of models, a mathematical model represents a problem in a simplified form and describes only the properties and patterns that are most important for a given object or process. The mathematical model allows for multilateral quantitative analysis. By changing the initial data, criteria, restrictions, each time you can obtain an optimal solution under given conditions and determine further direction search.

The creation of mathematical models requires from their developers, in addition to knowledge of formal logical methods, a thorough analysis of the object being studied in order to strictly formulate the main ideas and rules, as well as to identify a sufficient amount of reliable factual, statistical and regulatory data.

It should be noted that all currently used mathematical models relate to prescriptive. The purpose of developing prescriptive models is to indicate the direction of finding a solution, while the purpose of developing describing models are a reflection of actual human thinking processes.

There is a fairly widespread point of view that with the help of mathematics it is possible to obtain only some numerical data on the object or process being studied. “Of course, many mathematical disciplines are aimed at obtaining a final numerical result. But to reduce mathematical methods only to the problem of obtaining a number means to endlessly impoverish mathematics, to impoverish the possibility of that powerful weapon that today is in the hands of researchers...

A mathematical model written in one or another private language (for example, differential equations) reflects certain properties real physical processes. As a result of the analysis of mathematical models, we obtain, first of all, qualitative ideas about the features of the processes under study, establish patterns that determine the dynamic series of successive states, and gain the opportunity to predict the course of the process and determine its quantitative characteristics.”

Mathematical models are used in many well-known modeling methods. Among them are the development of models that describe the static and dynamic state of an object, optimization models.

An example of mathematical models that describe the static and dynamic state of an object can be various methods of traditional structural calculations. The calculation process, presented in the form of a sequence of mathematical operations (algorithm), allows us to say that a mathematical model has been compiled for calculating a certain structure.

IN optimization models contain three elements:

Objective function reflecting the accepted quality criterion;

Adjustable parameters;

Imposed restrictions.

All these elements must be described mathematically in the form of equations, logical conditions, etc. Solving an optimization problem is the process of finding the minimum (maximum) value of the objective function while complying with specified restrictions. The solution result is considered optimal if the objective function reaches its extreme value.

An example of an optimization model is a mathematical description of the “connection length” criterion in the method of alternative design of industrial buildings.

The objective function reflects the total weighted length of all functional connections, which should tend to a minimum:

where is the weight value of the element’s connection with ;

– length of connection between and elements;

– the total number of placed elements.

Since the areas of the placed elements of the premises are equal in all variants of the design solution, the variants differ from one another only in the different distances between the elements and their location relative to each other. Consequently, the adjustable parameters in this case are the coordinates of the elements placed on the floor plans.

Imposed restrictions on the location of elements (in a pre-fixed place on the plan, at the outer perimeter, on top of each other, etc.) and on the length of connections (the lengths of connections between elements are rigidly specified, minimum or maximum limits of values ​​are specified, boundaries of change are specified values) are written formally.

An option is considered optimal (according to this criterion) if the value of the objective function calculated for this option is minimal.

A variety of mathematical models – economic-mathematical model– represents a communication model economic characteristics and system parameters.

An example of economic-mathematical models is the mathematical description of cost criteria in the above-mentioned method of alternative design of industrial buildings. Mathematical models obtained based on the use of mathematical statistics methods reflect the dependence of the cost of the frame, foundations, earthworks of one-story and multi-story industrial buildings and their height, span and pitch of load-bearing structures.

Based on the method of taking into account the influence of random factors on decision-making, mathematical models are divided into deterministic and probabilistic. Deterministic the model does not take into account the influence of random factors in the process of system operation and is based on an analytical representation of the functioning patterns. Probabilistic (stochastic) the model takes into account the influence of random factors during the operation of the system and is based on statistical, i.e. quantitative assessment of mass phenomena, allowing to take into account their nonlinearity, dynamics, random disturbances described by different distribution laws.

Using the above examples, we can say that the mathematical model that describes the criterion “length of connections” refers to deterministic models, and the mathematical models that describe the group of criteria “costs” refer to probabilistic models.

Linguistic, semantic and information models

Mathematical models have obvious advantages because quantifying aspects of a problem provides a clear picture of the priorities of goals. It is important that a specialist can always justify the adoption of a particular decision by presenting the relevant numerical data. However, the full mathematical description project activities impossible, therefore most of the problems solved at the initial stage of architectural and construction design relate to poorly structured.

One of the features of semi-structured problems is a verbal description of the criteria used in them. Introduction of criteria described in natural language (such criteria are called linguistic), allows you to use less complex methods to find optimal design solutions. Given such criteria, the designer makes a decision based on familiar, unquestionable expressions of goals.

A meaningful description of all aspects of the problem introduces systematization into the process of solving it, on the one hand, and on the other, greatly facilitates the work of specialists who, without studying the relevant branches of mathematics, can solve their professional problems more rationally. In Fig. 5.2 is given linguistic model, describing the possibilities of creating conditions for natural ventilation in various layout options for a bakery.

Other benefits of meaningful problem descriptions include:

The ability to describe all the criteria that determine the effectiveness of a design solution. At the same time, it is important that complex concepts can be introduced into the description and the specialist’s field of view, along with quantitative, measurable factors, will also include qualitative, non-measurable ones. Thus, at the time of decision making, all subjective and objective information will be used;

Rice. 5.2 Description of the content of the “ventilation” criterion in the form of a linguistic model

The ability to unambiguously assess the degree of achievement of the goal in the options for this criterion based on the formulations accepted by specialists, which ensures the reliability of the information received;

The ability to take into account the uncertainty associated with incomplete knowledge of all the consequences of decisions made, as well as predictive information.

Models that use natural language to describe the object of study also include semantic models.

Semantic model- there is such a representation of an object that reflects the degree of interconnectedness (proximity) between the various components, aspects, properties of the object. Interconnectedness does not mean a relative spatial arrangement, but a connection in meaning.

Thus, in a semantic sense, the relationship between the coefficient of natural illumination and the light area of ​​transparent fences will be presented as closer than the relationship between window openings and adjacent blind sections of the wall.

The set of connectivity relationships shows what each element selected in an object and the object as a whole represents. At the same time, the semantic model reflects, in addition to the degree of connectedness of various aspects in an object, the content of concepts. Elementary models are concepts expressed in natural language.

The construction of semantic models is based on the principles according to which concepts and connections do not change throughout the entire time the model is used; the content of one concept does not transfer to another; connections between two concepts have an equal and non-oriented interaction in relation to them.

Each model analysis aims to select elements of the model that have a certain quality in common. This gives grounds for constructing an algorithm that takes into account only direct connections. When converting a model to an undirected graph, a path is found between two elements that traces the movement from one element to another, using each element only once. The order in which the elements appear is called the sequence of the two elements. Sequences can have different lengths. The shortest of them are called element relations. A sequence of two elements exists even if there is a direct connection between them, but in this case there is no relationship.

As an example of a semantic model, we give a description of the layout of an apartment along with communication connections. The concept is the premises of an apartment. Direct connection means the functional connection of two rooms, for example by a door (see Table 5.1).

Transforming the model into the form of an undirected graph allows us to obtain a sequence of elements (Fig. 5.3).

Examples of the sequence formed between element 2 (bathroom) and element 6 (pantry) are given in table. 5.2. As can be seen from the table, sequence 3 represents the relationship of these two elements.

Table 5.1

Description of the apartment layout

Rice. 5.3 Description of the planning solution in the form of an undirected graph

What is a mathematical model?

The concept of a mathematical model.

A mathematical model is a very simple concept. And very important. It is mathematical models that connect mathematics and real life.

Speaking in simple language, a mathematical model is a mathematical description of any situation. That's all. The model can be primitive, or it can be super complex. Whatever the situation, such is the model.)

In any (I repeat - in any!) in a case where you need to count and calculate something - we are engaged in mathematical modeling. Even if we don’t suspect it.)

P = 2 CB + 3 CM

This entry will be a mathematical model of the costs of our purchases. The model does not take into account the color of the packaging, expiration date, politeness of cashiers, etc. That's why she model, not an actual purchase. But expenses, i.e. what we need- we will find out for sure. If the model is correct, of course.

It is useful to imagine what a mathematical model is, but it is not enough. The most important thing is to be able to build these models.

Drawing up (construction) of a mathematical model of the problem.

To create a mathematical model means to translate the conditions of the problem into mathematical form. Those. turn words into an equation, formula, inequality, etc. Moreover, transform it so that this mathematics strictly corresponds original text. Otherwise, we will end up with a mathematical model of some other problem unknown to us.)

More specifically, you need

There are an endless number of tasks in the world. Therefore, offer clear step-by-step instructions for drawing up a mathematical model any tasks are impossible.

But there are three main points that you need to pay attention to.

1. Any problem contains text, oddly enough.) This text, as a rule, contains explicit, open information. Numbers, values, etc.

2. Any problem has hidden information. This is a text that assumes additional knowledge in your head. There is no way without them. In addition, mathematical information is often hidden behind simple words and... slips past attention.

3. Any task must be given connection of data with each other. This connection can be given in plain text (something equals something), or it can be hidden behind simple words. But simple and clear facts are often overlooked. And the model is not compiled in any way.

I’ll say right away: in order to apply these three points, you have to read the problem (and carefully!) several times. The usual thing.

And now - examples.

Let's start with a simple problem:

Petrovich returned from fishing and proudly presented his catch to the family. Upon closer examination, it turned out that 8 fish came from the northern seas, 20% of all fish came from the southern seas, and not a single one came from the local river where Petrovich was fishing. How many fish did Petrovich buy in the Seafood store?

All these words need to be turned into some kind of equation. To do this you need, I repeat, establish a mathematical connection between all the data in the problem.

Where to start? First, let's extract all the data from the task. Let's start in order:

Let's pay attention to the first point.

Which one is here? explicit mathematical information? 8 fish and 20%. Not a lot, but we don’t need a lot.)

Let us pay attention to the second point.

Are looking for hidden information. It's here. These are the words: "20% of all fish". Here you need to understand what percentages are and how they are calculated. Otherwise, the problem cannot be solved. This is exactly what Additional Information, which should be in your head.

There is also mathematical information that is completely invisible. This task question: "How many fish did I buy..." This is also a number. And without it, no model will be formed. Therefore, let's denote this number by the letter "X". We don’t yet know what x is equal to, but this designation will be very useful to us. More details on what to take for X and how to handle it are written in the lesson How to solve problems in mathematics? Let’s write it down right away:

x pieces - total number of fish.

In our problem, southern fish are given as percentages. We need to convert them into pieces. For what? Then what in any the problem of the model must be drawn up in the same type of quantities. Pieces - so everything is in pieces. If given, say, hours and minutes, we translate everything into one thing - either only hours, or only minutes. It doesn't matter what it is. It is important that all values ​​were of the same type.

Let's return to information disclosure. Whoever doesn’t know what interest is will never reveal it, yes... But whoever knows will immediately say that interest here is from total number fish are given. And we don’t know this number. Nothing will work!

It’s not for nothing that we letter the total number of fish (in pieces!) "X" designated. It won't be possible to count the number of southern fish, but we can write them down? Like this:

0.2 x pieces - the number of fish from the southern seas.

Now we have downloaded all the information from the task. Both obvious and hidden.

Let us pay attention to the third point.

Are looking for mathematical connection between task data. This connection is so simple that many do not notice it... This often happens. Here it’s useful to simply write down the collected data in a pile and see what’s what.

What do we have? Eat 8 pieces northern fish, 0.2 x pieces- southern fish and x fish- total amount. Is it possible to link this data together somehow? Yes Easy! Total number of fish equals the sum of southern and northern! Well, who would have thought...) So we write it down:

x = 8 + 0.2x

This is the equation mathematical model of our problem.

Please note that in this problem We are not asked to fold anything! It was we ourselves, out of our heads, who realized that the sum of the southern and northern fish would give us the total number. The thing is so obvious that it goes unnoticed. But without this evidence, a mathematical model cannot be created. Like this.

Now you can use the full power of mathematics to solve this equation). This is precisely why the mathematical model was compiled. We solve this linear equation and get the answer.

Answer: x=10

Let's create a mathematical model of another problem:

They asked Petrovich: “Do you have a lot of money?” Petrovich began to cry and answered: “Yes, just a little. If I spend half of all the money, and half of the rest, then I’ll only have one bag of money left...” How much money does Petrovich have?

Again we work point by point.

1. We are looking for explicit information. You won’t find it right away! Explicit information is one money bag. There are some other halves... Well, we’ll look into that in the second point.

2. We are looking for hidden information. These are halves. What? Not very clear. We are looking further. There is one more question: "How much money does Petrovich have?" Let us denote the amount of money by the letter "X":

X- all the money

And again we read the problem. Already knowing that Petrovich X money. This is where halves will work! We write down:

0.5 x- half of all money.

The remainder will also be half, i.e. 0.5 x. And half of half can be written like this:

0.5 0.5 x = 0.25x- half of the remainder.

Now all hidden information has been revealed and recorded.

3. We are looking for a connection between the recorded data. Here you can simply read Petrovich’s suffering and write it down mathematically):

If I spend half of all the money...

Let's record this process. All the money - X. Half - 0.5 x. To spend is to take away. The phrase turns into a recording:

x - 0.5 x

yes half the rest...

Let's subtract another half of the remainder:

x - 0.5 x - 0.25x

then I'll only have one bag of money left...

And here we have found equality! After all the subtractions, one bag of money remains:

x - 0.5 x - 0.25x = 1

Here it is, a mathematical model! This is again a linear equation, we solve it, we get:

Question for consideration. What is four? Ruble, dollar, yuan? And in what units is money written in our mathematical model? In bags! That means four bag money from Petrovich. Good too.)

The tasks are, of course, elementary. This is specifically to capture the essence of drawing up a mathematical model. Some tasks may contain much more data, which can be easy to get lost in. This often happens in the so-called. competency tasks. How to extract mathematical content from a pile of words and numbers is shown with examples

One more note. In classic school problems (pipes filling a pool, boats floating somewhere, etc.), all data, as a rule, is selected very carefully. There are two rules:
- there is enough information in the problem to solve it,
- There is no unnecessary information in a problem.

This is a hint. If there is some value left unused in the mathematical model, think about whether there is an error. If there is not enough data, most likely, not all hidden information has been identified and recorded.

In competence and other life tasks these rules are not strictly followed. No clue. But such problems can also be solved. If, of course, you practice on the classic ones.)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.