Drawing up mathematical models. Mathematical model in practice What type of mathematical models uses algorithms

Math modeling

1. What is mathematical modeling?

From the middle of the 20th century. Mathematical methods and computers began to be widely used in various fields of human activity. New disciplines have emerged such as “mathematical economics”, “mathematical chemistry”, “mathematical linguistics”, etc., studying mathematical models of relevant objects and phenomena, as well as methods for studying these models.

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.

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.

3. Classification of models

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.

4. Examples of mathematical models

1) Problems about the motion of a projectile.

Consider the following mechanics problem.

The projectile is launched from the Earth with an initial speed v 0 = 30 m/s at an angle a = 45° to its surface; it is required to find the trajectory of its movement and the distance S between the starting and ending points of this trajectory.

Then, as is known from a school physics course, the motion of a projectile is described by the formulas:

where t is time, g = 10 m/s 2 is the acceleration of gravity. These formulas provide a mathematical model of the problem. Expressing t through x from the first equation and substituting it into the second, we obtain the equation for the trajectory of the projectile:

This curve (parabola) intersects the x axis at two points: x 1 = 0 (beginning of the trajectory) and (place where the projectile fell). Substituting the given values ​​of v0 and a into the resulting formulas, we obtain

answer: y = x – 90x 2, S = 90 m.

Note that when constructing this model, a number of assumptions were used: for example, it is assumed that the Earth is flat, and air and the rotation of the Earth do not affect the movement of the projectile.

2) Problem about a tank with the smallest surface area.

It is required to find the height h 0 and radius r 0 of a tin tank with a volume V = 30 m 3, having the shape of a closed circular cylinder, at which its surface area S is minimal (in this case, the least amount of tin will be used for its production).

Let us write the following formulas for the volume and surface area of ​​a cylinder of height h and radius r:

V = p r 2 h, S = 2p r(r + h).

Expressing h through r and V from the first formula and substituting the resulting expression into the second, we get:

Thus, from a mathematical point of view, the problem comes down to determining the value of r at which the function S(r) reaches its minimum. Let us find those values ​​of r 0 for which the derivative

goes to zero: You can check that the second derivative of the function S(r) changes sign from minus to plus when the argument r passes through the point r 0 . Consequently, at the point r0 the function S(r) has a minimum. The corresponding value is h 0 = 2r 0 . Substituting the given value V into the expression for r 0 and h 0, we obtain the desired radius and height

3) Transport problem.

The city has two flour warehouses and two bakeries. Every day, 50 tons of flour are transported from the first warehouse, and 70 tons from the second to factories, with 40 tons to the first, and 80 tons to the second.

Let us denote by a ij is the cost of transporting 1 ton of flour from the i-th warehouse to the j-th plant (i, j = 1.2). Let

a 11 = 1.2 rubles, a 12 = 1.6 rubles, a 21 = 0.8 rub., a 22 = 1 rub.

How should transportation be planned so that its cost is minimal?

Let's give the problem a mathematical formulation. Let us denote by x 1 and x 2 the amount of flour that must be transported from the first warehouse to the first and second factories, and by x 3 and x 4 - from the second warehouse to the first and second factories, respectively. Then:

x 1 + x 2 = 50, x 3 + x 4 = 70, x 1 + x 3 = 40, x 2 + x 4 = 80. (1)

The total cost of all transportation is determined by the formula

f = 1.2x 1 + 1.6x 2 + 0.8x 3 + x 4.

From a mathematical point of view, the problem is to find four numbers x 1, x 2, x 3 and x 4 that satisfy all given conditions and give the minimum of the function f. Let us solve the system of equations (1) for xi (i = 1, 2, 3, 4) by eliminating the unknowns. We get that

x 1 = x 4 – 30, x 2 = 80 – x 4, x 3 = 70 – x 4, (2)

and x 4 cannot be determined uniquely. Since x i і 0 (i = 1, 2, 3, 4), it follows from equations (2) that 30Ј x 4 Ј 70. Substituting the expression for x 1, x 2, x 3 into the formula for f, we get

f = 148 – 0.2x 4.

It is easy to see that the minimum of this function is achieved at the maximum possible value of x 4, that is, at x 4 = 70. The corresponding values ​​of other unknowns are determined by formulas (2): x 1 = 40, x 2 = 10, x 3 = 0.

4) The problem of radioactive decay.

Let N(0) be the initial number of atoms of a radioactive substance, and N(t) be the number of undecayed atoms at time t. It has been experimentally established that the rate of change in the number of these atoms N"(t) is proportional to N(t), that is, N"(t)=–l N(t), l >0 is the radioactivity constant of a given substance. In the school course of mathematical analysis it is shown that the solution to this differential equation has the form N(t) = N(0)e –l t. The time T during which the number of initial atoms has halved is called the half-life, and is an important characteristic of the radioactivity of a substance. To determine T, we must put in the formula Then For example, for radon l = 2.084 · 10 –6, and therefore T = 3.15 days.

5) The traveling salesman problem.

A traveling salesman living in city A 1 needs to visit cities A 2 , A 3 and A 4 , each city exactly once, and then return back to A 1 . It is known that all cities are connected in pairs by roads, and the lengths of roads b ij between cities A i and A j (i, j = 1, 2, 3, 4) are as follows:

b 12 = 30, b 14 = 20, b 23 = 50, b 24 = 40, b 13 = 70, b 34 = 60.

It is necessary to determine the order of visiting cities in which the length of the corresponding path is minimal.

Let us depict each city as a point on the plane and mark it with the corresponding label Ai (i = 1, 2, 3, 4). Let's connect these points with straight lines: they will represent roads between cities. For each “road” we indicate its length in kilometers (Fig. 2). The result is a graph - a mathematical object consisting of a certain set of points on the plane (called vertices) and a certain set of lines connecting these points (called edges). Moreover, this graph is labeled, since its vertices and edges are assigned some labels - numbers (edges) or symbols (vertices). A cycle on a graph is a sequence of vertices V 1 , V 2 , ..., V k , V 1 such that the vertices V 1 , ..., V k are different, and any pair of vertices V i , V i+1 (i = 1, ..., k – 1) and the pair V 1, V k are connected by an edge. Thus, the problem under consideration is to find a cycle on the graph passing through all four vertices for which the sum of all edge weights is minimal. Let us search through all the different cycles passing through four vertices and starting at A 1:

1) A 1, A 4, A 3, A 2, A 1;
2) A 1, A 3, A 2, A 4, A 1;
3) A 1, A 3, A 4, A 2, A 1.

Let us now find the lengths of these cycles (in km): L 1 = 160, L 2 = 180, L 3 = 200. So, the route of the shortest length is the first.

Note that if there are n vertices in a graph and all vertices are connected in pairs by edges (such a graph is called complete), then the number of cycles passing through all vertices is Therefore, in our case there are exactly three cycles.

6) The problem of finding a connection between the structure and properties of substances.

Let's look at several chemical compounds called normal alkanes. They consist of n carbon atoms and n + 2 hydrogen atoms (n = 1, 2 ...), interconnected as shown in Figure 3 for n = 3. Let the experimental values ​​of the boiling points of these compounds be known:

y e (3) = – 42°, y e (4) = 0°, y e (5) = 28°, y e (6) = 69°.

It is required to find an approximate relationship between the boiling point and the number n for these compounds. Let us assume that this dependence has the form

y" a n+b,

Where a, b - constants to be determined. To find a and b we substitute into this formula sequentially n = 3, 4, 5, 6 and the corresponding values ​​of boiling points. We have:

– 42 » 3 a+ b, 0 » 4 a+ b, 28 » 5 a+ b, 69 » 6 a+ b.

To determine the best a and b there are many different methods. Let's use the simplest of them. Let's express b through a from these equations:

b » – 42 – 3 a, b " – 4 a, b » 28 – 5 a, b » 69 – 6 a.

Let us take the arithmetic mean of these values ​​as the desired b, that is, we put b » 16 – 4.5 a. Let us substitute this value of b into the original system of equations and, calculating a, we get for a the following values: a» 37, a» 28, a» 28, a" 36. Let's take as the required a the average value of these numbers, that is, let's put a" 34. So, the required equation has the form

y » 34n – 139.

Let's check the accuracy of the model on the original four compounds, for which we calculate the boiling points using the resulting formula:

y р (3) = – 37°, y р (4) = – 3°, y р (5) = 31°, y р (6) = 65°.

Thus, the error in calculating this property for these compounds does not exceed 5°. We use the resulting equation to calculate the boiling point of a compound with n = 7, which is not included in the original set, for which we substitute n = 7 into this equation: y р (7) = 99°. The result was quite accurate: it is known that the experimental value of the boiling point y e (7) = 98°.

7) The problem of determining the reliability of an electrical circuit.

Here we will look at an example of a probabilistic model. First, we present some information from probability theory - a mathematical discipline that studies the patterns of random phenomena observed during repeated repetition of experiments. Let us call a random event A a possible outcome of some experiment. Events A 1, ..., A k form a complete group if one of them necessarily occurs as a result of the experiment. Events are called incompatible if they cannot occur simultaneously in one experience. Let the event A occur m times during an n-fold repetition of the experiment. The frequency of event A is the number W = . Obviously, the value of W cannot be predicted accurately until a series of n experiments is carried out. However, the nature of random events is such that in practice the following effect is sometimes observed: as the number of experiments increases, the value practically ceases to be random and stabilizes around some non-random number P(A), called the probability of the event A. For an impossible event (which never occurs in an experiment) P(A)=0, and for a reliable event (which always occurs in experience) P(A)=1. If the events A 1 , ..., A k form a complete group of incompatible events, then P(A 1)+...+P(A k)=1.

Let, for example, the experiment consist of tossing a dice and observing the number of points X rolled out. Then we can introduce the following random events A i = (X = i), i = 1, ..., 6. They form a complete group of incompatible equally probable events, therefore P(A i) = (i = 1, ..., 6).

The sum of events A and B is the event A + B, which consists in the fact that at least one of them occurs in experience. The product of events A and B is the event AB, which consists of the simultaneous occurrence of these events. For independent events A and B, the following formulas are true:

P(AB) = P(A) P(B), P(A + B) = P(A) + P(B).

8) Let us now consider the following task. Let us assume that three elements are connected in series to an electrical circuit and operate independently of each other. The failure probabilities of the 1st, 2nd and 3rd elements are respectively equal to P1 = 0.1, P2 = 0.15, P3 = 0.2. We will consider a circuit reliable if the probability that there will be no current in the circuit is no more than 0.4. It is necessary to determine whether a given circuit is reliable.

Since the elements are connected in series, there will be no current in the circuit (event A) if at least one of the elements fails. Let A i be the event that the i-th element works (i = 1, 2, 3). Then P(A1) = 0.9, P(A2) = 0.85, P(A3) = 0.8. Obviously, A 1 A 2 A 3 is an event in which all three elements work simultaneously, and

P(A 1 A 2 A 3) = P(A 1) P(A 2) P(A 3) = 0.612.

Then P(A) + P(A 1 A 2 A 3) = 1, so P(A) = 0.388< 0,4. Следовательно, цепь является надежной.

In conclusion, we note that the given examples of mathematical models (including functional and structural, deterministic and probabilistic) are illustrative in nature and, obviously, do not exhaust the variety of mathematical models that arise in the natural sciences and humanities.

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.

In simple terms, 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 to the source 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 the additional information that 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 a percentage is will never reveal it, yes... But whoever knows will immediately say that the percentages here are based on the total number of fish. 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 paragraph.

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-related and other life tasks, these rules are not strictly observed. 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.

According to the textbook by Sovetov and Yakovlev: “a model (lat. modulus - measure) is a substitute object for the original object, which ensures the study of some properties of the original.” (p. 6) “Replacing one object with another in order to obtain information about the most important properties of the original object using a model object is called modeling.” (p. 6) “By mathematical modeling we understand the process of establishing a correspondence to a given real object with a certain mathematical object, called a mathematical model, and the study of this model, which allows us to obtain the characteristics of the real object under consideration. The type of mathematical model depends both on the nature of the real object and the tasks of studying the object and the required reliability and accuracy of solving this problem.”

Finally, the most concise definition of a mathematical model: "An equation expressing an idea."

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

Structural models represent an object as a system with its own structure and functioning mechanism. Functional models do not use such representations and reflect only the externally perceived behavior (functioning) of an object. In their extreme expression, they are also called “black box” models. Combined types of models are also possible, which are sometimes called “gray box” models.

Content and formal models

Almost all authors describing the process of mathematical modeling indicate that first a special ideal structure is built, content model. There is no established terminology here, and other authors call this ideal object conceptual model , speculative model or premodel. In this case, the final mathematical construction is called formal model or simply a mathematical model obtained as a result of the formalization of a given meaningful model (pre-model). The construction of a meaningful model can be done using a set of ready-made idealizations, as in mechanics, where ideal springs, rigid bodies, ideal pendulums, elastic media, etc. provide ready-made structural elements for meaningful modeling. However, in areas of knowledge where there are no fully completed formalized theories (the cutting edge of physics, biology, economics, sociology, psychology, and most other areas), the creation of meaningful models becomes dramatically more difficult.

Content classification of models

No hypothesis in science can be proven once and for all. Richard Feynman formulated this very clearly:

“We always have the opportunity to disprove a theory, but note that we can never prove that it is correct. Let's assume that you have put forward a successful hypothesis, calculated where it leads, and found that all its consequences are confirmed experimentally. Does this mean your theory is correct? No, it simply means that you failed to refute it.”

If a model of the first type is built, this means that it is temporarily recognized 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.

Type 2: Phenomenological model (we behave as if…)

A phenomenological model contains a mechanism for describing a phenomenon. However, this mechanism is not convincing enough, cannot be sufficiently confirmed by the 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. Likewise, new knowledge can gradually come into conflict with models-hypotheses 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 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.

Type 3: Approximation (we consider something very big 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 common 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.

Here comes Type 8, which is widespread in mathematical models of biological systems.

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

These are also thought experiments with imaginary entities, demonstrating that supposed phenomenon consistent with basic principles and internally consistent. This is the main difference from models of type 7, which reveal hidden contradictions.

One of the most famous of these experiments is Lobachevsky’s geometry (Lobachevsky called it “imaginary geometry”). Another example is the mass production of formally kinetic models of chemical and biological vibrations, autowaves, etc. The Einstein-Podolsky-Rosen paradox was conceived as a type 7 model to demonstrate the inconsistency of quantum mechanics. In a completely unplanned way, it eventually turned into a type 8 model - a demonstration of the possibility of quantum teleportation of information.

Example

Consider a mechanical system consisting of a spring attached at one end and a mass of mass m attached to the free end of the spring. We will assume that the load can only move in the direction of the spring axis (for example, movement occurs along the rod). Let's build a mathematical model of this system. We will describe the state of the system by the distance x from the center of the load to its equilibrium position. Let us describe the interaction of the spring and the load using Hooke's law (F = − kx ) and then use Newton's second law to express it in the form of a differential equation:

where means the second derivative of x by time: .

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 fulfilled.

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. To resolve the issue of its applicability, it is necessary to understand how significant the factors that we have neglected are. In other words, it is necessary to study the “soft” model, which is obtained by a small perturbation of the “hard” one. It can be given, for example, by the following equation:

Here is some function that can take into account the friction force or the dependence of the spring stiffness coefficient on the degree of its stretching - some small parameter. Explicit function form 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. For example, the solution to the equation of a harmonic oscillator is functions of the form , that is, oscillations with a constant amplitude. Does it follow from this that a real oscillator will oscillate indefinitely with a constant amplitude? No, because considering a system with arbitrarily small friction (always present in a real system), we get damped oscillations. The behavior of the system has changed qualitatively.

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-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 the “General Theory of Systems”.

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, and 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. So, in 1879, a metal bridge across the River Tay collapsed in England, the designers of which built a model of the bridge, calculated it to have 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 the method constructed by I. Newton for reconstructing friction forces from observed damped oscillations.

Additional examples

Where x s- 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, and this behavior is structurally stable.

This system has an equilibrium state when the number of rabbits and foxes is constant. Deviation from this state results in fluctuations in the numbers of rabbits and foxes, similar to the fluctuations of a harmonic oscillator. As with the harmonic oscillator, this behavior is not structurally stable: a small change in the model (for example, taking into account the limited resources required by rabbits) can lead to a qualitative change in behavior. For example, the equilibrium state may become stable, and fluctuations in numbers will die out. 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-Lotka model does not answer the question of which of these scenarios is being realized: additional research is required here.

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., revised. - 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. Wiktionary: mathematical model
7. CliffsNotes
8. Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, Springer, Complexity series, Berlin-Heidelberg-New York, 2006. XII+562 pp. ISBN 3-540-35885-4
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 partial differential equations, integral equations, or ordinary delay equations. 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 with ISBN 978-5-484-00953-4
13. “The obvious, but most important initial stage of constructing or selecting a mathematical model is obtaining as clear a picture as possible about the object being modeled and refining its meaningful model, based on informal discussions. You should not spare time and effort at this stage; the success of the entire study largely depends on it. It has happened more than once that significant work spent on solving a mathematical problem turned out to be ineffective or even wasted due to insufficient attention to this side of the matter.” Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4, p. 35.
14. « Description of the conceptual model of the system. At this substage of building a system model: a) the conceptual model M is described in abstract terms and concepts; b) a description of the model is given using standard mathematical schemes; c) hypotheses and assumptions are finally accepted; d) the choice of procedure for approximating real processes when constructing a model is justified.” 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, p. 93.
15. Blekhman I. I., Myshkis A. D., Panovko N. G., Applied mathematics: Subject, logic, features of approaches. With examples from mechanics: Textbook. - 3rd ed., rev. and additional - M.: URSS, 2006. - 376 p. ISBN 5-484-00163-3, Chapter 2.

Lecture 1.

METHODOLOGICAL BASICS OF MODELING

Current state of the problem of system modeling

Modeling and Simulation Concepts

Modeling can be considered as the replacement of the object under study (original) with its conventional image, description or other object called model and providing behavior close to the original within the framework of certain assumptions and acceptable errors. Modeling is usually performed with the goal of understanding the properties of the original by studying its model, and not the object itself. Of course, modeling is justified when it is simpler than creating the original itself, or when for some reason it is better not to create the original at all.

Under model is understood as a physical or abstract object, the properties of which are in a certain sense similar to the properties of the object under study. In this case, the requirements for the model are determined by the problem being solved and the available means. There are a number of general requirements for models:

2) completeness – providing the recipient with all the necessary information

about the object;

3) flexibility - the ability to reproduce different situations in everything

range of changes in conditions and parameters;

4) the complexity of development must be acceptable for the existing

time and software.

Modeling is the process of constructing a model of an object and studying its properties by examining the model.

Thus, modeling involves 2 main stages:

1) development of a model;

2) study of the model and drawing conclusions.

At the same time, at each stage different tasks are solved and

essentially different methods and means.

In practice, various modeling methods are used. Depending on the method of implementation, all models can be divided into two large classes: physical and mathematical.

Math modeling It is usually considered as a means of studying processes or phenomena using their mathematical models.

Under physical modeling refers to the study of objects and phenomena on physical models, when the process being studied is reproduced while preserving its physical nature or another physical phenomenon similar to the one being studied is used. Wherein physical models As a rule, they assume the real embodiment of those physical properties of the original that are significant in a particular situation. For example, when designing a new aircraft, a mock-up is created that has the same aerodynamic properties; When planning a development, architects make a model that reflects the spatial arrangement of its elements. In this regard, physical modeling is also called prototyping.

Half-life modeling is a study of controllable systems on modeling complexes with the inclusion of real equipment in the model. Along with real equipment, the closed model includes simulators of influences and interference, mathematical models of the external environment and processes for which a sufficiently accurate mathematical description is unknown. The inclusion of real equipment or real systems in the circuit of modeling complex processes makes it possible to reduce a priori uncertainty and explore processes for which there is no exact mathematical description. Using semi-natural modeling, research is carried out taking into account small time constants and linearities inherent in real equipment. When studying models using real equipment, the concept is used dynamic simulation, when studying complex systems and phenomena - evolutionary, imitation And cybernetic modeling.

Obviously, the real benefit of modeling can only be obtained if two conditions are met:

1) the model provides a correct (adequate) display of properties

the original, significant from the point of view of the operation under study;

2) the model allows you to eliminate the problems listed above inherent

conducting research on real objects.

2. Basic concepts of mathematical modeling

Solving practical problems using mathematical methods is consistently carried out by formulating the problem (developing a mathematical model), choosing a method for studying the resulting mathematical model, and analyzing the obtained mathematical result. The mathematical formulation of the problem is usually presented in the form of geometric images, functions, systems of equations, etc. The description of an object (phenomenon) can be represented using continuous or discrete, deterministic or stochastic and other mathematical forms.

Theory of mathematical modeling ensures the identification of patterns of occurrence of various phenomena in the surrounding world or the operation of systems and devices by means of their mathematical description and modeling without carrying out full-scale tests. In this case, the provisions and laws of mathematics are used that describe the simulated phenomena, systems or devices at some level of their idealization.

Mathematical model (MM) is a formalized description of a system (or operation) in some abstract language, for example, in the form of a set of mathematical relationships or an algorithm diagram, i.e. i.e. such a mathematical description that provides simulation of the operation of systems or devices at a level sufficiently close to their real behavior obtained during full-scale testing of systems or devices.

Any MM describes a real object, phenomenon or process with some degree of approximation to reality. The type of MM depends both on the nature of the real object and on the objectives of the study.

Math modeling social, economic, biological and physical phenomena, objects, systems and various devices is one of the most important means of understanding nature and designing a wide variety of systems and devices. There are known examples of the effective use of modeling in the creation of nuclear technologies, aviation and aerospace systems, in forecasting atmospheric and oceanic phenomena, weather, etc.

However, such serious areas of modeling often require supercomputers and years of work by large teams of scientists to prepare data for modeling and its debugging. However, in this case, mathematical modeling of complex systems and devices not only saves money on research and testing, but can also eliminate environmental disasters - for example, it allows you to abandon the testing of nuclear and thermonuclear weapons in favor of their mathematical modeling or testing of aerospace systems before their actual flights. Between Therefore, mathematical modeling at the level of solving simpler problems, for example, from the field of mechanics, electrical engineering, electronics, radio engineering and many other areas of science and technology has now become available to perform on modern PCs. And when using generalized models, it becomes possible to simulate fairly complex systems, for example, telecommunication systems and networks, radar or radio navigation systems.

The purpose of mathematical modeling is the analysis of real processes (in nature or technology) using mathematical methods. In turn, this requires the formalization of the MM process to be studied. The model can be a mathematical expression containing variables whose behavior is similar to the behavior of a real system. The model can include elements of randomness that take into account the probabilities of possible actions of two or more “players”, as, for example, in the theory games; or it may represent real variables of interconnected parts of the operating system.

Mathematical modeling for studying the characteristics of systems can be divided into analytical, simulation and combined. In turn, MMs are divided into simulation and analytical.

Analytical Modeling

For analytical modeling It is characteristic that the processes of the functioning of the system are written in the form of certain functional relationships (algebraic, differential, integral equations). The analytical model can be studied using the following methods:

1) analytical, when they strive to obtain, in a general form, explicit dependencies for the characteristics of systems;

2) numerical, when it is not possible to find a solution to the equations in general form and they are solved for specific initial data;

3) qualitative, when in the absence of a solution some of its properties are found.

Analytical models can only be obtained for relatively simple systems. For complex systems, large mathematical problems often arise. To apply the analytical method, they go to a significant simplification of the original model. However, research using a simplified model helps to obtain only indicative results. Analytical models mathematically correctly reflect the relationship between input and output variables and parameters. But their structure does not reflect the internal structure of the object.

During analytical modeling, its results are presented in the form of analytical expressions. For example, by connecting R.C.- circuit to a constant voltage source E(R, C And E- components of this model), we can create an analytical expression for the time dependence of voltage u(t) on the capacitor C:

This linear differential equation (DE) is the analytical model of this simple linear circuit. Its analytical solution, under the initial condition u(0) = 0, meaning a discharged capacitor C at the start of modeling, allows you to find the desired dependence - in the form of a formula:

u(t) = E(1− exp(- t/RC)). (2)

However, even in this simplest example, certain efforts are required to solve DE (1) or to apply computer mathematics systems(SCM) with symbolic calculations – computer algebra systems. For this completely trivial case, solving the problem of modeling a linear R.C.-circuit gives analytical expression (2) of a fairly general form - it is suitable for describing the operation of the circuit for any component ratings R, C And E, and describes the exponential charge of the capacitor C through a resistor R from a constant voltage source E.

Of course, finding analytical solutions during analytical modeling turns out to be extremely valuable for identifying general theoretical patterns of simple linear circuits, systems and devices. However, its complexity increases sharply as the influences on the model become more complex and the order and number of state equations describing the modeled object increase. You can get more or less visible results when modeling objects of the second or third order, but with a higher order, analytical expressions become overly cumbersome, complex and difficult to comprehend. For example, even a simple electronic amplifier often contains dozens of components. However, many modern SCMs, for example, systems of symbolic mathematics Maple, Mathematica or environment MATLAB, are capable of largely automating the solution of complex analytical modeling problems.

One type of modeling is numerical modeling, which consists in obtaining the necessary quantitative data on the behavior of systems or devices by any suitable numerical method, such as the Euler or Runge-Kutta methods. In practice, modeling nonlinear systems and devices using numerical methods turns out to be much more effective than analytical modeling of individual private linear circuits, systems or devices. For example, for solving DE (1) or DE systems in more complex cases, a solution in analytical form cannot be obtained, but using numerical simulation data, you can obtain fairly complete data on the behavior of the simulated systems and devices, as well as construct graphs of dependencies describing this behavior.

Simulation modeling

At imitation 10and modeling, the algorithm that implements the model reproduces the process of system functioning over time. The elementary phenomena that make up the process are simulated, preserving their logical structure and sequence of events over time.

The main advantage of simulation models compared to analytical ones is the ability to solve more complex problems.

Simulation models make it easy to take into account the presence of discrete or continuous elements, nonlinear characteristics, random influences, etc. Therefore, this method is widely used at the design stage of complex systems. The main means of implementing simulation modeling is a computer, which allows for digital modeling of systems and signals.

In this regard, let us define the phrase “ computer modelling”, which is increasingly used in the literature. Let's assume that computer modelling is mathematical modeling using computer technology. Accordingly, computer modeling technology involves performing the following actions:

1) determining the purpose of the modeling;

2) development of a conceptual model;

3) formalization of the model;

4) software implementation of the model;

5) planning model experiments;

6) implementation of the experimental plan;

7) analysis and interpretation of modeling results.

At simulation modeling the MM used reproduces the algorithm (“logic”) of the functioning of the system under study over time for various combinations of values ​​of system parameters and the external environment.

An example of the simplest analytical model is the equation of rectilinear uniform motion. When studying such a process using a simulation model, observation of changes in the path traveled over time should be implemented. Obviously, in some cases analytical modeling is more preferable, in others - simulation (or a combination of both). To make a successful choice, you need to answer two questions.

What is the purpose of modeling?

To what class can the modeled phenomenon be classified?

Answers to both of these questions can be obtained during the first two stages of modeling.

Simulation models not only in properties, but also in structure correspond to the modeled object. In this case, there is an unambiguous and obvious correspondence between the processes obtained on the model and the processes occurring at the object. The disadvantage of simulation is that it takes a long time to solve the problem to obtain good accuracy.

The results of simulation modeling of the operation of a stochastic system are realizations of random variables or processes. Therefore, to find the characteristics of the system, multiple repetitions and subsequent data processing are required. Most often in this case, a type of simulation is used - statistical

modeling(or Monte Carlo method), i.e. reproduction of random factors, events, quantities, processes, fields in models.

Based on the results of statistical modeling, estimates of probabilistic quality criteria, general and specific, characterizing the functioning and efficiency of the managed system are determined. Statistical modeling is widely used to solve scientific and applied problems in various fields of science and technology. Statistical modeling methods are widely used in the study of complex dynamic systems, assessing their functioning and efficiency.

The final stage of statistical modeling is based on mathematical processing of the results obtained. Here, methods of mathematical statistics are used (parametric and nonparametric estimation, hypothesis testing). An example of a parametric estimator is the sample mean of a performance measure. Among nonparametric methods, widespread histogram method.

The considered scheme is based on repeated statistical tests of the system and methods of statistics of independent random variables. This scheme is not always natural in practice and optimal in terms of costs. Reducing system testing time can be achieved through the use of more accurate evaluation methods. As is known from mathematical statistics, effective estimates have the greatest accuracy for a given sample size. Optimal filtering and the maximum likelihood method provide a general method for obtaining such estimates. In statistical modeling problems, processing implementations of random processes is necessary not only for analyzing output processes.

Control of the characteristics of input random influences is also very important. Control consists of checking the compliance of the distributions of generated processes with the given distributions. This problem is often formulated as hypothesis testing problem.

The general trend in computer modeling of complex controlled systems is the desire to reduce modeling time, as well as conduct research in real time. It is convenient to represent computational algorithms in a recurrent form, allowing their implementation at the rate of receipt of current information.

PRINCIPLES OF A SYSTEM APPROACH IN MODELING

Basic principles of systems theory

The basic principles of systems theory arose during the study of dynamic systems and their functional elements. A system is understood as a group of interconnected elements that act together to accomplish a predetermined task. Analysis of systems allows us to determine the most realistic ways to perform a given task, ensuring maximum satisfaction of the stated requirements.

The elements that form the basis of systems theory are not created through hypotheses, but are discovered experimentally. In order to begin building a system, it is necessary to have general characteristics of technological processes. The same is true with regard to the principles of creating mathematically formulated criteria that a process or its theoretical description must satisfy. Modeling is one of the most important methods of scientific research and experimentation.

When constructing models of objects, a systems approach is used, which is a methodology for solving complex problems, which is based on considering the object as a system operating in a certain environment. A systematic approach involves revealing the integrity of an object, identifying and studying its internal structure, as well as connections with the external environment. In this case, the object is presented as a part of the real world, which is isolated and studied in connection with the problem of constructing a model. In addition, the systems approach involves a consistent transition from the general to the specific, when the design goal is the basis of consideration, and the object is considered in relation to the environment.

A complex object can be divided into subsystems, which are parts of the object that meet the following requirements:

1) a subsystem is a functionally independent part of an object. It is connected with other subsystems, exchanges information and energy with them;

2) for each subsystem functions or properties that do not coincide with the properties of the entire system can be defined;

3) each of the subsystems can be subjected to further division to the level of elements.

In this case, an element is understood as a lower-level subsystem, the further division of which is inappropriate from the standpoint of the problem being solved.

Thus, a system can be defined as a representation of an object in the form of a set of subsystems, elements and connections for the purpose of its creation, research or improvement. In this case, an enlarged representation of the system, including the main subsystems and connections between them, is called macrostructure, and a detailed disclosure of the internal structure of the system down to the level of elements is called microstructure.

Along with the system, there is usually a supersystem - a system of a higher level, which includes the object in question, and the function of any system can be determined only through the supersystem.

It is necessary to highlight the concept of the environment as a set of objects of the external world that significantly influence the efficiency of the system, but are not part of the system and its supersystem.

In connection with the systems approach to building models, the concept of infrastructure is used, which describes the relationship of the system with its environment (environment). In this case, the identification, description and study of the properties of an object that are essential within the framework of a specific task is called stratification of the object, and any model of the object is its stratified description.

For a systems approach, it is important to determine the structure of the system, i.e. a set of connections between elements of the system, reflecting their interaction. To do this, we first consider the structural and functional approaches to modeling.

With a structural approach, the composition of the selected elements of the system and the connections between them are revealed. The set of elements and connections allows us to judge the structure of the system. The most general description of a structure is a topological description. It allows you to determine the components of the system and their connections using graphs. Less general is the functional description, when individual functions are considered, i.e., algorithms for the behavior of the system. In this case, a functional approach is implemented that defines the functions that the system performs.

Based on the systems approach, a sequence of model development can be proposed, when two main design stages are distinguished: macrodesign and microdesign.

At the macro-design stage, a model of the external environment is built, resources and limitations are identified, a system model and criteria are selected for assessing adequacy.

The micro-design stage depends largely on the specific type of model chosen. In general, it involves the creation of information, mathematical, technical and software modeling systems. At this stage, the main technical characteristics of the created model are established, the time required to work with it and the cost of resources to obtain the specified quality of the model are estimated.

Regardless of the type of model, when constructing it, it is necessary to be guided by a number of principles of a systematic approach:

1) consistent progression through the stages of creating a model;

2) coordination of information, resource, reliability and other characteristics;

3) the correct relationship between the different levels of model construction;

4) the integrity of the individual stages of model design.

In this article, we offer examples of mathematical models. In addition, we will pay attention to the stages of creating models and analyze some problems associated with mathematical modeling.

Another question we have is mathematical models in economics, examples of which we will look at the definition a little later. We propose to start our conversation with the very concept of “model”, briefly consider their classification and move on to our main questions.

The concept of "model"

We often hear the word “model”. What is it? This term has many definitions, here are just three of them:

• a specific object that is created to receive and store information, reflecting some properties or characteristics, and so on, of the original of this object (this specific object can be expressed in different forms: mental, description using signs, and so on);
• A model also means a representation of a specific situation, life or management;
• a model can be a reduced copy of an object (they are created for more detailed study and analysis, since the model reflects the structure and relationships).

Based on everything that was said earlier, we can draw a small conclusion: the model allows you to study a complex system or object in detail.

All models can be classified according to a number of characteristics:

• by area of ​​use (educational, experimental, scientific and technical, gaming, simulation);
• by dynamics (static and dynamic);
• by branch of knowledge (physical, chemical, geographical, historical, sociological, economic, mathematical);
• by the method of presentation (material and informational).

Information models, in turn, are divided into symbolic and verbal. And symbolic ones - into computer and non-computer ones. Now let's move on to a detailed consideration of examples of the mathematical model.

Mathematical model

As you might guess, a mathematical model reflects any features of an object or phenomenon using special mathematical symbols. Mathematics is needed in order to model the patterns of the surrounding world in its own specific language.

The method of mathematical modeling originated quite a long time ago, thousands of years ago, along with the advent of this science. However, the impetus for the development of this modeling method was given by the emergence of computers (electronic computers).

Now let's move on to classification. It can also be carried out according to some signs. They are presented in the table below.

We propose to stop and take a closer look at the latest classification, since it reflects the general patterns of modeling and the goals of the models being created.

Descriptive models

In this chapter, we propose to dwell in more detail on descriptive mathematical models. To make everything very clear, an example will be given.

Let's start with the fact that this type can be called descriptive. This is due to the fact that we simply make calculations and forecasts, but cannot in any way influence the outcome of the event.

A striking example of a descriptive mathematical model is the calculation of the flight path, speed, and distance from the Earth of a comet that invaded the expanses of our Solar System. This model is descriptive, since all the results obtained can only warn us of any danger. Unfortunately, we cannot influence the outcome of the event. However, based on the calculations obtained, it is possible to take any measures to preserve life on Earth.

Optimization models

Now we will talk a little about economic and mathematical models, examples of which can serve as different current situations. In this case, we are talking about models that help find the correct answer under certain conditions. They definitely have some parameters. To make it completely clear, let’s look at an example from the agricultural sector.

We have a granary, but the grain spoils very quickly. In this case, we need to choose the right temperature conditions and optimize the storage process.

Thus, we can define the concept of “optimization model”. In a mathematical sense, it is a system of equations (both linear and not), the solution of which helps to find the optimal solution in a specific economic situation. We looked at an example of a mathematical model (optimization), but I would like to add: this type belongs to the class of extreme problems, they help describe the functioning of the economic system.

Let us note one more nuance: models can be of different nature (see table below).

Multicriteria models

Now we invite you to talk a little about the mathematical model of multicriteria optimization. Before this, we gave an example of a mathematical model for optimizing a process according to any one criterion, but what if there are many of them?

A striking example of a multi-criteria task is the organization of proper, healthy and at the same time economical nutrition for large groups of people. Such tasks are often encountered in the army, school canteens, summer camps, hospitals, and so on.

What criteria are given to us in this task?

1. Nutrition should be healthy.
2. Food expenses should be minimal.

As you can see, these goals do not coincide at all. This means that when solving a problem, it is necessary to look for an optimal solution, a balance between two criteria.

Game models

When talking about game models, it is necessary to understand the concept of “game theory”. Simply put, these models reflect mathematical models of real conflicts. You just have to understand that, unlike a real conflict, the game mathematical model has its own specific rules.

Now we will provide a minimum of information from game theory that will help you understand what a game model is. And so, the model necessarily contains parties (two or more), which are usually called players.

All models have certain characteristics.

The game model can be paired or multiple. If we have two subjects, then the conflict is paired; if there are more, it is multiple. You can also distinguish an antagonistic game, it is also called a zero-sum game. This is a model in which the gain of one of the participants is equal to the loss of the other.

Simulation models

In this section we will pay attention to simulation mathematical models. Examples of tasks include:

• model of microorganism population dynamics;
• model of molecular movement, and so on.

In this case, we are talking about models that are as close as possible to real processes. By and large, they imitate some manifestation in nature. In the first case, for example, we can simulate the dynamics of the number of ants in one colony. At the same time, you can observe the fate of each individual individual. In this case, a mathematical description is rarely used; written conditions are more often present:

• after five days the female lays eggs;
• after twenty days the ant dies, and so on.

Thus, they are used to describe a large system. A mathematical conclusion is the processing of the obtained statistical data.

Requirements

It is very important to know that this type of model has some requirements, including those listed in the table below.

Modeling stages

In total, mathematical modeling is usually divided into four stages.

1. Formulation of laws connecting parts of the model.
2. Study of mathematical problems.
3. Determining the coincidence of practical and theoretical results.
4. Analysis and modernization of the model.

Economic and mathematical model

In this section we will briefly highlight the issue. Examples of tasks include:

• formation of a production program for the production of meat products that ensures maximum production profits;
• maximizing the organization's profit by calculating the optimal quantity of tables and chairs produced at a furniture factory, and so on.

The economic-mathematical model reflects economic abstraction, which is expressed using mathematical terms and symbols.

Computer mathematical model

Examples of a computer mathematical model are:

• hydraulic problems using flowcharts, diagrams, tables, etc.;
• problems on solid mechanics, and so on.

A computer model is an image of an object or system, presented in the form:

• tables;
• block diagrams;
• diagrams;
• graphics, and so on.

Moreover, this model reflects the structure and interconnections of the system.

Construction of an economic and mathematical model

We have already talked about what an economic-mathematical model is. An example of solving the problem will be considered right now. We need to analyze the production program to identify a reserve for increasing profits with a shift in the assortment.

We will not fully consider the problem, but will only build an economic and mathematical model. The criterion of our task is profit maximization. Then the function has the form: А=р1*х1+р2*х2..., tending to the maximum. In this model, p is the profit per unit and x is the number of units produced. Next, based on the constructed model, it is necessary to make calculations and summarize.

An example of building a simple mathematical model

Task. The fisherman returned with the following catch:

• 8 fish - inhabitants of the northern seas;
• 20% of the catch is inhabitants of the southern seas;
• Not a single fish was found from the local river.

How many fish did he buy at the store?

So, an example of constructing a mathematical model of this problem looks like this. We denote the total number of fish by x. Following the condition, 0.2x is the number of fish living in southern latitudes. Now we combine all the available information and get a mathematical model of the problem: x=0.2x+8. We solve the equation and get the answer to the main question: he bought 10 fish in the store.