According to the textbook of Sovetov and Yakovlev: “a model (lat. modulus - measure) is an object-substitute of the original object, providing 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 the model object is called modeling.” (p. 6) “Under mathematical modeling we will understand the process of establishing correspondence to a given real object of some mathematical object, called a mathematical model, and the study of this model, which allows obtaining the characteristics of the real object under consideration. The type of mathematical model depends on both 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 built in the form of dichotomies. For example, one of the popular sets of dichotomies is:

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

Classification by the way the object is represented

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

• Structural or functional models

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

Content and formal models

Almost all authors describing the process of mathematical modeling indicate that first a special ideal construction 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 just a mathematical model obtained as a result of the formalization of this content model (pre-model). The construction of a meaningful model can be carried out using a set of ready-made idealizations, as in mechanics, where ideal springs, solid bodies, perfect 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 is dramatically more complicated.

Meaningful classification of models

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

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

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

Type 2: Phenomenological model (behave as if…)

The phenomenological model contains a mechanism for describing the phenomenon. However, this mechanism is not sufficiently convincing, cannot be sufficiently confirmed by the available data, or does not agree well with the available 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 it is necessary to continue the search for "true mechanisms". Peierls refers, for example, the caloric model and the quark model of elementary particles to the second type.

The role of the model in research may change over time, it may happen that new data and theories confirm phenomenological models and they are promoted to the status of a hypothesis. Likewise, new knowledge may gradually come into conflict with models-hypotheses of the first type, and they may be transferred to 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 passed into the first type. But the ether models have gone from type 1 to type 2, and now they are outside of science.

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

Type 3: Approximation (something is considered very large or very small)

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

And here is type 8, which is widely used in mathematical models of biological systems.

Type 8: Possibility 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 oscillations, 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 mechanical system, consisting of a spring fixed at one end, and a load of mass m attached to the free end of the spring. We will assume that the load can move only in the direction of the spring axis (for example, the movement occurs along the rod). Let us construct 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 a spring and a load using Hooke's law (F = − kx ) after which we 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 pattern is called the "harmonic oscillator".

According to the formal classification, this model is linear, deterministic, dynamic, concentrated, continuous. In the process of constructing it, 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 omit some details for clarity”), since some essential universal features (for example, dissipation) are omitted. In some approximation (say, while the deviation of the load from equilibrium is small, with little friction, for a not too long 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.

However, when the model is refined, the complexity of its mathematical study can increase significantly and make the model virtually useless. Often more simple model allows you to better and deeper explore the real system than a more complex (and, formally, “more correct”) one.

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

Hard and soft models

The harmonic oscillator is an example of a 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 are the factors that we have neglected. In other words, it is necessary to investigate 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 - some function, which can take into account the friction force or the dependence of the coefficient of stiffness of the spring on the degree of its stretching - some small parameter. Explicit form of a function f we are not interested at the moment. If we prove that the behavior of a soft model does not fundamentally differ from that of a hard model (regardless of the explicit form of the perturbing factors, if they are small enough), the problem will be reduced to studying the hard model. Otherwise, the application of the results obtained in the study of the rigid model will require additional research. For example, the solution to the equation of a harmonic oscillator are 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 an arbitrarily small friction (always present in a real system), we get damped oscillations. The behavior of the system has changed qualitatively.

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

Universality of models

The most important mathematical models usually have the important property universality: 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 liquid level in U-shaped vessel or a change in the current strength in the oscillatory circuit. Thus, studying one mathematical model, we study at once a whole class of phenomena described by it. It is this isomorphism of the laws expressed by mathematical models in various segments of scientific knowledge that led Ludwig von Bertalanffy to create the "General Systems Theory".

Direct and inverse problems of mathematical modeling

There are many problems associated with mathematical modeling. First, it is necessary to come up with the basic scheme of the object being modeled, to reproduce it within the framework of the idealizations of this science. So, 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 compiled, along the way some details are discarded as insignificant, calculations are made, compared with measurements, the model is refined, and so on. However, for the development of mathematical modeling technologies, it is useful to disassemble this process into its main constituent elements.

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

Direct problem: the structure of the model and all its parameters are considered known, the main task is to study the model to extract useful knowledge about the object. What static load can 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 task. Setting the correct direct problem (asking the correct question) requires special skill. If the right questions are not asked, the bridge may collapse even if it was built. good model for his behaviour. So, in 1879 in England, a metal bridge across the River Tey collapsed, the designers of which built a model of the bridge, calculated it for a 20-fold margin of safety for 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, it is necessary to choose a specific model 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 in additional empirical data, or in the requirements for the object ( design task). Additional data can come regardless of the process of solving the inverse problem ( passive observation) or be the result of an experiment specially planned in the course of solving ( active surveillance).

One of the first examples of a virtuoso solution of an inverse problem with the fullest possible use of available data was the method constructed by I. Newton for reconstructing friction forces from observed damped oscillations.

Additional examples

Where x s- "equilibrium" population size, at which the birth rate is exactly compensated by the death rate. The population size in such a model tends to the equilibrium value x s, and this behavior is structurally stable.

This system has an equilibrium state where the number of rabbits and foxes is constant. Deviation from this state leads to fluctuations in the number of rabbits and foxes, similar to fluctuations in the harmonic oscillator. As in the case of the harmonic oscillator, this behavior is not structurally stable: a small change in the model (for example, taking into account the limited resources needed by rabbits) can lead to a qualitative change in behavior. For example, the equilibrium state can become stable, and population fluctuations will fade. 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. To the question of which of these scenarios is realized, the Volterra-Lotka model does not give an answer: additional research is required here.

Notes

1. "A mathematical representation of reality" (Encyclopaedia Britanica)
2. Novik I. B., On philosophical questions of cybernetic modeling. M., Knowledge, 1964.
3. Sovetov B. Ya., Yakovlev S. A., Systems Modeling: 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. Wiktionary: mathematical models
7. Cliffs Notes
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 non-linear, depending on what - linear or non-linear - mathematical apparatus, what - linear or non-linear - mathematical models it uses. ... without denying the latter. A modern physicist, if he happened to redefine such an important entity as non-linearity, would most likely act differently, and, preferring non-linearity as the more important and common of the two opposites, would define linearity as "non-non-linearity". Danilov Yu. A., Lectures on nonlinear dynamics. Elementary introduction. Synergetics: from the past to the future series. Ed.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 in various conditions can be considered either concentrated or distributed. Mathematical models of distributed systems are partial differential equations, integral equations or ordinary equations with a delayed 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 studied processes 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 displays deterministic processes, that is, processes in which the absence of any random influences is assumed; stochastic modeling displays probabilistic processes and events. … Static modeling is used to describe the behavior of an object at any point in time, while dynamic modeling reflects the behavior of an object over time. Discrete modeling serves to describe processes that are assumed to be discrete, respectively, continuous modeling allows you to reflect continuous processes in systems, and discrete-continuous modeling is used for cases where you want to highlight the presence of both discrete and continuous processes. Sovetov B. Ya., Yakovlev S. A., Systems Modeling: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2
12. Usually, the mathematical model reflects the structure (arrangement) of the object being modeled, the properties and interconnections of the components of this object that are essential for the purposes of the study; 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 a 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 the most important initial stage of constructing or choosing a mathematical model is to get as clear as possible about the object being modeled and to refine its content model based on informal discussions. Time and efforts should not be spared at this stage; the success of the entire study largely depends on it. More than once it happened that considerable 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 sub-stage 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 typical mathematical schemes; c) hypotheses and assumptions are finally accepted; d) the choice of a procedure for approximating real processes when building a model is substantiated.” Sovetov B. Ya., Yakovlev S. A., Systems Modeling: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2, p. 93.

A mathematical model of a technical object is a set of mathematical objects and relations between them that adequately reflects the properties of the object under study that are of interest to the researcher (engineer).

The model can be represented in various ways.

Model representation forms:

invariant - recording model relations using a traditional mathematical language, regardless of the method for solving model equations;

analytical - recording the model in the form of the result of an analytical solution of the initial equations of the model;

algorithmic - recording the relations of the model and the selected numerical method of solution in the form of an algorithm.

schematic (graphic) - representation of the model in some graphic language (for example, the language of graphs, equivalent circuits, diagrams, etc.);

physical

analog

The most universal is the mathematical description of processes - mathematical modeling.

The concept of mathematical modeling also includes the process of solving a problem on a computer.

Generalized mathematical model

The mathematical model describes the relationship between the initial data and the desired values.

The elements of the generalized mathematical model are (Fig. 1): a set of input data (variables) X,Y;

X - set of variable variables; Y - independent variables (constant);

mathematical operator L that defines operations on these data; which is understood as a complete system of mathematical operations that describe numerical or logical relationships between sets of input and output data (variables);

set of output data (variables) G(X,Y); is a set of criterion functions, including (if necessary) the objective function.

The mathematical model is a mathematical analogue of the designed object. The degree of adequacy of its object is determined by the formulation and correctness of solutions to the design problem.

The set of variable parameters (variables) X forms the space of variable parameters Rx (search space), which is metric with dimension n equal to the number of variable parameters.

The set of independent variables Y form the metric space of input data Ry. In the case when each component of the space Ry is given by a range of possible values, the set of independent variables is mapped to some limited subspace of the space Ry.

The set of independent variables Y determines the environment for the operation of the object, i.e. external conditions in which the designed object will operate

It can be:

• - technical specifications an object that is not subject to change during the design process;
• - physical perturbations of the environment with which the design object interacts;
• - tactical parameters that the design object should achieve.

The output data of the considered generalized model form a metric space of criteria indicators RG.

The scheme of using a mathematical model in a computer-aided design system is shown in Fig.2.

Requirements for the mathematical model

The main requirements for mathematical models are the requirements of adequacy, universality and economy.

Adequacy. The model is considered adequate if it reflects the given properties with acceptable accuracy. Accuracy is defined as the degree of agreement between the values ​​of the output parameters of the model and the object.

The accuracy of the model is different in different conditions functioning of the object. These conditions are characterized by external parameters. In the space of external parameters, select the region of model adequacy, where the error is less than the specified maximum permissible error. Determining the domain of model adequacy is a complex procedure that requires large computational costs, which grow rapidly with an increase in the dimension of the space of external parameters. This task can significantly exceed the task of parametric optimization of the model itself in volume, therefore, it may not be solved for newly designed objects.

Universality - is determined mainly by the number and composition of external and output parameters taken into account in the model.

The economy of the model is characterized by the cost of computing resources for its implementation - the cost of computer time and memory.

The contradictory requirements for a model to have a wide range of adequacy, a high degree of universality and high efficiency determine the use of a number of models for objects of the same type.

Model Retrieval Methods

Get models in general case- unformalized procedure. The main decisions regarding the choice of the type of mathematical relationships, the nature of the variables and parameters used, are made by the designer. At the same time, such operations as the calculation of the numerical values ​​of the model parameters, the determination of adequacy areas, and others are algorithmized and solved on a computer. Therefore, the modeling of the elements of the designed system is usually performed by specialists in specific technical fields using traditional experimental studies.

Methods for obtaining functional models of elements are divided into theoretical and experimental.

Theoretical methods are based on the study of the physical regularities of the processes occurring in the object, determining the mathematical description corresponding to these regularities, substantiating and accepting simplifying assumptions, performing the necessary calculations and bringing the result to the accepted form of the model representation.

Experimental methods are based on the use external manifestations properties of the object, recorded during the operation of the same type of objects or during targeted experiments.

Despite the heuristic nature of many operations, modeling has a number of provisions and techniques common to obtaining models of various objects. They are quite general in nature.

macro modeling technique,

mathematical methods for planning experiments,

algorithms for formalized operations for calculating the numerical values ​​of parameters and determining the areas of adequacy.

Using Mathematical Models

The computing power of modern computers, combined with the provision of all system resources to the user, the possibility of an interactive mode when solving a problem and analyzing the results, make it possible to minimize the time for solving a problem.

When compiling a mathematical model, the researcher is required to:

study the properties of the object under study;

the ability to separate the main properties of the object from the secondary ones;

evaluate the assumptions made.

The model describes the relationship between the input data and the desired values. The sequence of actions that must be performed in order to move from the initial data to the desired values ​​is called an algorithm.

The algorithm for solving the problem on a computer is associated with the choice of a numerical method. Depending on the form of representation of the mathematical model (algebraic or differential form), various numerical methods are used.

The essence of economic and mathematical modeling lies in the description of socio-economic systems and processes in the form of economic and mathematical models.

Let's consider questions of classification of economic and mathematical methods. These methods, as noted above, are a complex of economic and mathematical disciplines that are an alloy of economics, mathematics and cybernetics.

Therefore, the classification of economic and mathematical methods is reduced to the classification of the scientific disciplines included in their composition. Although the generally accepted classification of these disciplines has not yet been developed, with a certain degree of approximation, the following sections can be distinguished in the composition of economic and mathematical methods:

• * economic cybernetics: system analysis of economics, theory of economic information and theory of control systems;
• * mathematical statistics: economic applications of this discipline -- sampling method, analysis of variance, correlation analysis, regression analysis, multivariate statistical analysis, factor analysis, index theory, etc.;
• * mathematical economy and econometrics studying the same questions from the quantitative side: the theory of economic growth, the theory of production functions, intersectoral balances, national accounts, analysis of demand and consumption, regional and spatial analysis, global modeling, etc.;
• * methods for making optimal decisions, including the study of operations in the economy. This is the most voluminous section, which includes the following disciplines and methods: optimal (mathematical) programming, including branch and bound methods, network planning and control methods, program-target planning and control methods, inventory management theory and methods, queuing theory , game theory, decision theory and methods, scheduling theory. Optimal (mathematical) programming includes, in turn, linear programming, non-linear programming, dynamic programming, discrete (integer) programming, linear-fractional programming, parametric programming, separable programming, stochastic programming, geometric programming;
• * Methods and disciplines that are specific to both a centrally planned economy and a market (competitive) economy. The first include the theory of optimal functioning of the economy, optimal planning, the theory of optimal pricing, models of logistics, etc. The second are methods that allow developing models of free competition, models of the capitalist cycle, models of monopoly, models of indicative planning, models of the theory of the firm etc.

Many of the methods developed for a centrally planned economy can also be useful in economic and mathematical modeling in a market economy;

* methods of experimental study of economic phenomena. These include, as a rule, mathematical methods of analysis and planning of economic experiments, methods of machine simulation (simulation modeling), business games. This also includes methods of expert assessments developed to evaluate phenomena that cannot be directly measured.

Let us now turn to the questions of classifying economic and mathematical models, in other words, mathematical models of socio-economic systems and processes.

A unified classification system for such models currently does not exist either, however, more than ten main features of their classification, or classification headings, are usually distinguished. Let's take a look at some of these sections.

According to the general purpose, economic and mathematical models are divided into theoretical and analytical, used in the study common properties and laws of economic processes, and applied, used in solving specific economic problems of analysis, forecasting and management. Various types applied economic and mathematical models are just considered in this tutorial.

According to the degree of aggregation of modeling objects, the models are divided into macroeconomic and microeconomic. Although there is no clear distinction between them, the first of them include models that reflect the functioning of the economy as a whole, while microeconomic models are associated, as a rule, with such parts of the economy as enterprises and firms.

According to a specific purpose, i.e., according to the purpose of creation and application, balance models are distinguished, expressing the requirement that the availability of resources correspond to their use; trend models, in which the development of the modeled economic system is reflected through the trend (long-term trend) of its main indicators; optimization models designed for selection the best option from a certain number of options for production, distribution or consumption; simulation models intended for use in the process of machine simulation of the systems or processes under study, etc.

According to the type of information used in the model, economic-mathematical models are divided into analytical, built on a priori information, and identifiable, built on a posteriori information.

By taking into account the time factor, the models are divided into static, in which all dependencies are related to one point in time, and dynamic, which describe economic systems in development.

By taking into account the uncertainty factor, the models are divided into deterministic ones, if the output results in them are uniquely determined by control actions, and stochastic (probabilistic), if when a certain set of values ​​is specified at the model input, its output can produce different results depending on the action of a random factor.

Economic and mathematical models can also be classified according to the characteristics of the mathematical objects included in the model, in other words, according to the type of mathematical apparatus used in the model. On this basis, matrix models, linear and non-linear programming models, correlation-regression models,

Basic concepts of mathematical modeling of the queuing theory model, network planning and control model, game theory model, etc.

Finally, according to the type of approach to the studied socio-economic systems, descriptive and normative models are distinguished. With a descriptive (descriptive) approach, models are obtained that are designed to describe and explain actually observed phenomena or to predict these phenomena; As an example of descriptive models, we can cite the previously named balance and trend models. With a normative approach, they are not interested in how the economy is organized and develops. economic system, but how it should be arranged and how it should act in the sense of certain criteria. In particular, all optimization models are of the normative type; normative models of standard of living can serve as another example.

Let us consider as an example the economic-mathematical model of the input-output balance (EMM IOB). Taking into account the above classification headings, this is an applied, macroeconomic, analytical, descriptive, deterministic, balance, matrix model; while they exist as static methods as well as dynamic

Linear programming is a particular branch of optimal programming. In turn, optimal (mathematical) programming is a branch of applied mathematics that studies problems of conditional optimization. In economics, such problems arise in the practical implementation of the principle of optimality in planning and management.

A necessary condition for using the optimal approach to planning and management (the principle of optimality) is flexibility, alternativeness of production and economic situations in which planning and management decisions have to be made. It is these situations, as a rule, that make up the daily practice of an economic entity (choosing a production program, attaching to suppliers, routing, cutting materials, preparing mixtures, etc.).

The essence of the principle of optimality lies in the desire to choose such a planning and management decision the best way would take into account the internal capabilities and external conditions of the production activity of an economic entity.

The words "in the best way" here mean the choice of some criterion of optimality, i.e. some economic indicator that allows you to compare the effectiveness of certain planning and management decisions. Traditional optimality criteria: “maximum profit”, “minimum costs”, “maximum profitability”, etc. The words “would take into account the internal capabilities and external conditions of production activity” mean that a number of conditions are imposed on the choice of a planning and management decision (behavior), t .e. the choice of X is carried out from a certain region of possible (admissible) solutions D; this area is also called the problem definition area. a general problem of optimal (mathematical) programming, otherwise, a mathematical model of an optimal programming problem, the construction (development) of which is based on the principles of optimality and consistency.

A vector X (a set of control variables Xj, j = 1, n) is called a feasible solution, or an optimal programming problem plan, if it satisfies the system of constraints. And that plan X (admissible solution) that delivers the maximum or minimum of the objective function f(xi, *2, ..., xn) is called the optimal plan (optimal behavior, or simply solution) of the optimal programming problem.

Thus, the choice of optimal managerial behavior in a specific production situation is associated with conducting economic and mathematical modeling from the standpoint of consistency and optimality and solving the problem of optimal programming. Optimal programming problems in the most general form are classified according to the following criteria.

• 1. By the nature of the relationship between variables --
• a) linear
• b) non-linear.

In case a) all functional connections in the system of restrictions and the goal function are linear functions; the presence of a nonlinearity in at least one of the mentioned elements leads to case b).

• 2. By the nature of the change in variables --
• a) continuous
• b) discrete.

In case a) the values ​​of each of the control variables can completely fill a certain area of ​​real numbers; in case b) all or at least one variable can take only integer values.

• 3. By taking into account the time factor -
• a) static
• b) dynamic.

In tasks a), modeling and decision-making are carried out under the assumption that the elements of the model are independent of time during the period of time for which a planning and management decision is made. In case b), such an assumption cannot be accepted with sufficient reason and the time factor must be taken into account.

• 4. According to the availability of information about variables --
• a) tasks under conditions of complete certainty (deterministic),
• b) tasks in conditions of incomplete information,
• c) tasks under conditions of uncertainty.

In tasks b), individual elements are probabilistic quantities, however, their distribution laws are known or additional statistical studies can be established. In case c), one can make an assumption about the possible outcomes of random elements, but it is not possible to draw a conclusion about the probabilities of the outcomes.

• 5. According to the number of criteria for evaluating alternatives -
• a) simple, single-criteria tasks,
• b) complex, multicriteria tasks.

In tasks a) it is economically acceptable to use one optimality criterion or it is possible by special procedures (for example, “priority weighting”)

INTRODUCTION

It is impossible to imagine modern science without wide application mathematical modeling. The essence of this methodology is to replace the original object with its "image" - a mathematical model - and further study of the model using computational logic algorithms implemented on computers. This "third method" of cognition, design, design combines many advantages of both theory and experiment. Working not with the object itself (phenomenon, process), but with its model makes it possible to painlessly, relatively quickly and without significant costs to investigate its properties and behavior in any conceivable situations (advantages of the theory). At the same time, computational (computer, simulation, simulation) experiments with object models make it possible, relying on the power of modern computational methods and technical tools of informatics, to study objects in sufficient detail and in depth, in sufficient completeness, inaccessible to purely theoretical approaches (experimental advantages). It is not surprising that the methodology of mathematical modeling is rapidly developing, covering all new areas - from the development technical systems and their management to the analysis of the most complex economic and social processes.

Elements of mathematical modeling have been used since the very beginning of the appearance of the exact sciences, and it is no coincidence that some calculation methods bear the names of such luminaries of science as Newton and Euler, and the word "algorithm" comes from the name of the medieval Arab scientist Al-Khwarizmi. The second “birth” of this methodology took place in the late 1940s and early 1950s and was due to at least two reasons. The first of these is the emergence of computers (computers), although modest by today's standards, but nevertheless saved scientists from a huge amount of routine computational work. The second is an unprecedented social order - the implementation of the national programs of the USSR and the USA to create a nuclear missile shield, which could not be implemented by traditional methods. Mathematical modeling coped with this task: nuclear explosions and flights of rockets and satellites were previously “carried out” in the depths of computers using mathematical models and only then put into practice. This success largely determined the further achievements of the methodology, without which no large-scale technological, environmental or economic project is now seriously considered in developed countries (this is also true in relation to some socio-political projects).

Now mathematical modeling is entering the third fundamentally important stage of its development, "integrating" into the structures of the so-called information society. Impressive progress in the means of processing, transmitting and storing information corresponds to global trends towards complication and mutual penetration various areas human activity. Without the possession of information "resources" it is impossible to even think about solving the increasingly larger and more diverse problems facing the world community. However, information as such often does little for analysis and forecasting, for making decisions and monitoring their implementation. We need reliable ways of processing information "raw materials" into a finished "product", i.e., into accurate knowledge. The history of the methodology of mathematical modeling convinces: it can and should be the intellectual core information technologies, the whole process of informatization of society.

Technical, ecological, economic and other systems studied modern science, are no longer amenable to investigation (in the required completeness and accuracy) by conventional theoretical methods. A direct full-scale experiment on them is long, expensive, often either dangerous or simply impossible, since many of these systems exist in a "single copy". The price of mistakes and miscalculations in handling them is unacceptably high. Therefore, mathematical (more broadly - informational) modeling is an inevitable component of scientific and technological progress.

Considering the issue more broadly, we recall that modeling is present in almost all types of creative activity of people of various "specialties" - researchers and entrepreneurs, politicians and military leaders. The introduction of exact knowledge into these spheres helps to limit the intuitive speculative "modeling", expands the field of application of rational methods. Of course, mathematical modeling is fruitful only when well-known professional requirements are met: a clear formulation of basic concepts and assumptions, a posteriori analysis of the adequacy of the models used, guaranteed accuracy of computational algorithms, etc. If we talk about modeling systems with the participation of the "human factor", then i.e. objects that are difficult to formalize, then to these requirements it is necessary to add an accurate distinction between mathematical and everyday terms (sounding the same, but having a different meaning), careful application of a ready-made mathematical apparatus to the study of phenomena and processes (the path “from problem to method” is preferable, and not vice versa) and a number of others.

Solving the problems of the information society, it would be naive to rely only on the power of computers and other informatics tools. Continuous improvement of the triad of mathematical modeling and its implementation in modern information modeling systems is a methodological imperative. Only its implementation makes it possible to obtain the high-tech, competitive and diverse material and intellectual products that we so desperately need.

The topic I have chosen is relevant in modern mathematics and its applications. In the modern scientific approach in the study of natural, technical and socio-economic objects, the importance of mathematical modeling of the processes occurring in them is increasing. Natural study of the behavior of objects and systems in such modes and conditions is impossible or difficult, which forces the use of mathematical modeling methods.

The purpose of this course work is to learn how to use the methods of mathematical modeling to study various natural social processes.

Tasks set to achieve the goal:

n To study theoretical questions of mathematical modeling, classification of models.

BASIC CONCEPTS OF MATHEMATICAL MODELING

Modeling- a method of scientific research of phenomena, processes, objects, devices or systems (generally - research objects), based on the construction and study of models in order to obtain new knowledge, improve the characteristics of research objects or manage them.

Model- a material object or image (mental or conditional: hypothesis, idea, abstraction, image, description, diagram, formula, drawing, plan, map, algorithm flowchart, notes, etc.), which simply display the most essential properties of the object research.

Any model is always simpler than a real object and displays only a part of its most essential features, main elements and connections. For this reason, for one object of study, there are many different models. The type of model depends on the chosen purpose of modeling.

The term "model" is based on the Latin word modulus - measure, sample. The model is a substitute for the real object of study. The model is always simpler than the object under study. When studying complex phenomena, processes, objects, it is not possible to take into account the totality of all elements and relationships that determine their properties.

But all the elements and connections in the created model should not be taken into account. It is only necessary to single out the most characteristic, dominant components, which overwhelmingly determine the main properties of the object of study. As a result, the object of study is replaced by some simplified similarity, but with characteristic, main properties similar to those of the object of study. A new object (or abstraction) that appeared as a result of the substitution is usually called a model of the object of study.

To compile mathematical models, you can use any mathematical means - differential and integral calculus, regression analysis, probability theory, mathematical statistics, etc. A mathematical model is a set of formulas, equations, inequalities, logical conditions, etc. The mathematical relations used in mathematical modeling determine the process of changing the state of the object of study depending on its parameters, input signals, initial conditions and time. Essentially, all mathematics is designed to form mathematical models.

ABOUT great importance mathematics for all other sciences (including modeling) says the following fact. The great English physicist I. Newton (1643-1727) in the middle of the 17th century got acquainted with the works of Rene Descartes and Pierre Gassendi. These works argued that the entire structure of the world can be described mathematical formulas. Under the influence of these works, I. Newton began to intensively study mathematics. His contribution to physics and mathematics is widely known.

Mathematical modeling is a method of studying an object of study based on creating its mathematical model and using it to obtain new knowledge, improve the object of study or control the object.

For mathematical modeling, it is characteristic that the processes of the object's functioning are written in the form of mathematical relations (algebraic, integral), written in the form of logical conditions.

Differential equations are one of the main means of compiling mathematical models that are most widely used in solving mathematical problems. When studying physical processes, solving various applied problems, as a rule, it is not possible to directly find the laws that connect the quantities that characterize the phenomena under study. It is usually easier to establish relationships between the same quantities and their derivatives or differentials. Relations of this kind are called differential equations. The possibilities and rules for compiling differential equations are determined by knowledge of the laws of the field of science with which the nature of the problem under study is associated. So, for example, Newton's laws can be used in mechanics, in the theory of velocities chemical reactions- the law of mass action, etc. However, in practice there are often cases when the laws that could make it possible to draw up a differential equation are not known. Then resort to various simplifying assumptions concerning the course of the process with small changes in the parameters-variables. In this case, the passage to the limit leads to differential equations. The question of the correspondence of the mathematical model and the real phenomenon is solved on the basis of the analysis of the results, experiments and their comparison with the behavior of the solution of the obtained differential equation

Mathematical models

Mathematical model - approximate opidescription of the object of modeling, expressed usingschyu mathematical symbolism.

Mathematical models appeared along with mathematics many centuries ago. A huge impetus to the development of mathematical modeling was given by the appearance of computers. The use of computers made it possible to analyze and put into practice many mathematical models that had not previously been amenable to analytical research. Computer-implemented mathematicalsky model called computer mathematical model, A carrying out targeted calculations using a computer model called computational experiment.

Stages of computer mathematical modeletion shown in the figure. Firststage - definition of modeling goals. These goals can be different:

1. a model is needed in order to understand how a particular object works, what is its structure, basic properties, laws of development and interaction
   with the outside world (understanding);
2. a model is needed in order to learn how to control an object (or process) and determine best ways management with given goals and criteria (management);
3. the model is needed in order to predict the direct and indirect consequences of the implementation of the specified methods and forms of impact on the object (forecasting).

Let's explain with examples. Let the object of study be the interaction of a liquid or gas flow with a body that is an obstacle to this flow. Experience shows that the force of resistance to flow from the side of the body increases with increasing flow velocity, but at some sufficiently high speed, this force decreases abruptly in order to increase again with a further increase in speed. What caused the decrease in resistance force? Mathematical modeling allows us to get a clear answer: at the moment of an abrupt decrease in resistance, the vortices formed in the flow of liquid or gas behind the streamlined body begin to break away from it and are carried away by the flow.

An example from a completely different area: peacefully coexisting with stable populations of two species of individuals with a common food base, "suddenly" begin to dramatically change their numbers. And here mathematical modeling allows (with a certain degree of certainty) to establish the cause (or at least to refute a certain hypothesis).

Development of the concept of object management is another possible goal of modeling. Which aircraft flight mode should be chosen in order for the flight to be safe and most economically advantageous? How to schedule hundreds of types of work on the construction of a large facility so that it ends as soon as possible? Many such problems systematically arise before economists, designers, and scientists.

Finally, predicting the consequences of certain impacts on an object can be both a relatively simple matter in simple physical systems, and extremely complex - on the verge of feasibility - in biological, economic, social systems. If it is relatively easy to answer the question about the change in the mode of heat propagation in a thin rod with changes in its constituent alloy, then it is incomparably more difficult to trace (predict) the environmental and climatic consequences of the construction of a large hydroelectric power station or the social consequences of changes in tax legislation. Perhaps, here, too, mathematical modeling methods will provide more significant assistance in the future.

Second phase: definition of input and output parameters of the model; division of input parameters according to the degree of importance of the impact of their changes on the output. This process is called ranking, or division by rank (see below). "Formalisation and modeling").

Third stage: construction of a mathematical model. At this stage, there is a transition from the abstract formulation of the model to a formulation that has a specific mathematical representation. A mathematical model is equations, systems of equations, systems of inequalities, differential equations or systems of such equations, etc.

Fourth stage: choice of method for studying the mathematical model. Most often, numerical methods are used here, which lend themselves well to programming. As a rule, several methods are suitable for solving the same problem, differing in accuracy, stability, etc. The success of the entire modeling process often depends on the correct choice of method.

Fifth stage: the development of an algorithm, the compilation and debugging of a computer program is a process that is difficult to formalize. Of the programming languages, many professionals for mathematical modeling prefer FORTRAN: both due to tradition, and due to the unsurpassed efficiency of compilers (for computational work) and the presence of huge, carefully debugged and optimized libraries of standard programs of mathematical methods written in it. Languages ​​such as PASCAL, BASIC, C are also in use, depending on the nature of the task and the inclinations of the programmer.

Sixth stage: program testing. The operation of the program is checked for test task with a known answer. This is just the beginning of a testing procedure that is difficult to describe in a formally exhaustive way. Usually, testing ends when the user, according to his professional characteristics, considers the program correct.

Seventh stage: actual computational experiment, during which it becomes clear whether the model corresponds to a real object (process). The model is sufficiently adequate to the real process if some characteristics of the process obtained on a computer coincide with the experimentally obtained characteristics with a given degree of accuracy. If the model does not correspond to the real process, we return to one of the previous stages.

Classification of mathematical models

The classification of mathematical models can be based on various principles. It is possible to classify models by branches of science (mathematical models in physics, biology, sociology, etc.). It can be classified according to the applied mathematical apparatus (models based on the use of ordinary differential equations, partial differential equations, stochastic methods, discrete algebraic transformations, etc.). Finally, based on common tasks modeling in different sciences, regardless of the mathematical apparatus, the following classification is most natural:

• descriptive (descriptive) models;
• optimization models;
• multicriteria models;
• game models.

Let's explain this with examples.

Descriptive (descriptive) models. For example, modeling the motion of a comet that invaded solar system, is made in order to predict the trajectory of its flight, the distance at which it will pass from the Earth, etc. In this case, the goals of modeling are descriptive, since there is no way to influence the motion of the comet, to change something in it.

Optimization Models are used to describe the processes that can be influenced in an attempt to achieve a given goal. In this case, the model includes one or more parameters that can be influenced. For example, by changing the thermal regime in a granary, one can set a goal to choose such a regime in order to achieve maximum grain preservation, i.e. optimize the storage process.

Multicriteria models. Often it is necessary to optimize the process in several parameters at the same time, and the goals can be very contradictory. For example, knowing food prices and a person's need for food, it is necessary to organize the nutrition of large groups of people (in the army, children's summer camp, etc.) physiologically correctly and, at the same time, as cheaply as possible. It is clear that these goals do not coincide at all; when modeling, several criteria will be used, between which a balance must be sought.

Game models may be related not only to computer games but also to very serious things. For example, before a battle, if there is incomplete information about the opposing army, a commander must develop a plan: in what order to bring certain units into battle, etc., taking into account the possible reaction of the enemy. There is a special section of modern mathematics - game theory - that studies the methods of decision making under conditions of incomplete information.

In the school course of computer science, students receive an initial idea of ​​​​computer mathematical modeling within the framework of basic course. In high school, mathematical modeling can be studied in depth in a general education course for classes in physics and mathematics, as well as within a specialized elective course.

The main forms of teaching computer mathematical modeling in high school are lectures, laboratory and credit classes. Usually, the work on creating and preparing for the study of each new model takes 3-4 lessons. In the course of the presentation of the material, tasks are set, which in the future should be solved by students independently, in in general terms ways to solve them are outlined. Questions are formulated, the answers to which should be obtained when performing tasks. Additional literature is indicated, which allows obtaining auxiliary information for more successful completion of tasks.

The form of organizing classes in the study of new material is usually a lecture. After the completion of the discussion of the next model students have at their disposal the necessary theoretical information and a set of tasks for further work. In preparation for the task, students choose the appropriate solution method, using some known private solution, they test the developed program. In case of quite possible difficulties in the performance of tasks, consultation is given, a proposal is made to work out these sections in more detail in the literature.

The most relevant to the practical part of teaching computer modeling is the method of projects. The task is formulated for the student in the form of an educational project and is completed over several lessons, with the main organizational form while doing computer lab work. Learning to model using the learning project method can be implemented at different levels. The first is a problem statement of the project implementation process, which is led by the teacher. The second is the implementation of the project by students under the guidance of a teacher. The third is the independent implementation by students of an educational research project.

The results of the work should be presented in numerical form, in the form of graphs, diagrams. If possible, the process is presented on the computer screen in dynamics. Upon completion of the calculations and the receipt of the results, they are analyzed, compared with known facts from the theory, the reliability is confirmed and a meaningful interpretation is carried out, which is subsequently reflected in a written report.

If the results satisfy the student and the teacher, then the work counts completed, and its final stage is the preparation of a report. The report includes brief theoretical information on the topic under study, the mathematical formulation of the problem, the solution algorithm and its justification, a computer program, the results of the program, analysis of the results and conclusions, a list of references.

When all the reports have been drawn up, at the test session, students make brief reports on the work done, defend their project. This is an effective form of reporting to the class by the project team, including setting the problem, building a formal model, choosing methods for working with the model, implementing the model on a computer, working with the finished model, interpreting the results, and forecasting. As a result, students can receive two grades: the first - for the elaboration of the project and the success of its defense, the second - for the program, the optimality of its algorithm, interface, etc. Students also receive marks in the course of surveys on theory.

An essential question is what kind of tools to use in the school informatics course for mathematical modeling? Computer implementation of models can be carried out:

• using a spreadsheet (usually MS Excel);
• by creating programs in traditional programming languages ​​(Pascal, BASIC, etc.), as well as in their modern versions (Delphi, Visual
  Basic for Application, etc.);
• using special software packages for solving mathematical problems (MathCAD, etc.).

At the elementary school level, the first remedy appears to be the preferred one. However, in high school, when programming is, along with modeling, a key topic of computer science, it is desirable to involve it as a modeling tool. In the process of programming, the details of mathematical procedures become available to students; moreover, they are simply forced to master them, and this also contributes to mathematical education. As for the use of special software packages, this is appropriate in a profile computer science course as a supplement to other tools.

Exercise :

• Outline key concepts.

LECTURE 4

Definition and purpose of mathematical modeling

Under model(from the Latin modulus - measure, sample, norm) we will understand such a materially or mentally represented object that, in the process of cognition (study), replaces the original object, retaining some of its typical features that are important for this study. The process of building and using a model is called modeling.

essence mathematical modeling (MM) is to replace the studied object (process) with an adequate mathematical model and then study the properties of this model using either analytical methods or computational experiments.

Sometimes it is more useful, instead of giving strict definitions, to describe a particular concept with a specific example. Therefore, we illustrate the above definitions of MM using the example of the problem of calculating the specific impulse. In the early 1960s, scientists were faced with the task of developing rocket fuel with the highest specific impulse. The principle of rocket movement is as follows: liquid fuel and oxidizer from the rocket tanks are fed into the engine, where they are burned, and the combustion products are released into the atmosphere. From the law of conservation of momentum, it follows that in this case the rocket will move with speed.

The specific impulse of a fuel is the resulting impulse divided by the mass of the fuel. The experiments were very expensive and led to systematic damage to the equipment. It turned out that it is easier and cheaper to calculate the thermodynamic functions of ideal gases, to calculate with their help the composition of the emitted gases and the plasma temperature, and then the specific impulse. That is, to carry out the MM of the fuel combustion process.

The concept of mathematical modeling (MM) is one of the most common in the scientific literature today. The vast majority of modern theses and dissertations are associated with the development and use of appropriate mathematical models. Computer MM today is an integral part of many areas of human activity (science, technology, economics, sociology, etc.). This is one of the reasons for today's shortage of specialists in the field of information technology.

The rapid growth of mathematical modeling is due to the rapid improvement of computer technology. If 20 years ago only a small number of programmers were engaged in numerical calculations, now the amount of memory and speed of modern computers, which make it possible to solve problems of mathematical modeling, are available to all specialists, including university students.

In any discipline, a qualitative description of the phenomena is first given. And then - quantitative, formulated in the form of laws that establish relationships between various quantities (field strength, scattering intensity, electron charge, ...) in the form of mathematical equations. Therefore, we can say that in each discipline there is as much science as there are mathematicians in it, and this fact allows us to successfully solve many problems using mathematical modeling methods.

This course is designed for students majoring in applied mathematics who are completing their theses under the supervision of leading scientists working in various fields. Therefore, this course is necessary not only as educational material but also as a preparation for thesis. For studying this course we will need the following sections of mathematics:

1. Equations of mathematical physics (Kantian mechanics, gas and hydrodynamics)

2. Linear algebra (the theory of elasticity)

3. Scalar and vector fields (field theory)

4. Probability theory (quantum mechanics, statistical physics, physical kinetics)

5. Special features.

6. Tensor analysis (theory of elasticity)

7. Mathematical analysis

MM in natural science, engineering, and economics

Let us first consider the various branches of natural science, technology, economics, in which mathematical models are used.

natural science

Physics, which establishes the basic laws of natural science, has long been divided into theoretical and experimental. Theoretical physics deals with the derivation of equations describing physical phenomena. Thus, theoretical physics can also be considered one of the areas of mathematical modeling. (Recall that the title of the first book on physics - "The Mathematical Principles of Natural Philosophy" by I. Newton can be translated into modern language as "Mathematical models of natural sciences.") Based on the obtained laws, engineering calculations are carried out, which are carried out in various institutes, firms, design bureaus. These organizations develop technologies for the manufacture of modern products that are science-intensive. Thus, the concept of science-intensive technologies includes calculations using appropriate mathematical models.

One of the most extensive branches of physics - classical mechanics(sometimes this section is called theoretical or analytical mechanics). This section of theoretical physics studies the motion and interaction of bodies. Calculations using the formulas of theoretical mechanics are necessary when studying the rotation of bodies (calculating the moments of inertia, gyrostats - devices that keep the axes of rotation stationary), analyzing the movement of a body in a vacuum, etc. One of the sections of theoretical mechanics is called the theory of stability and underlies many mathematical models describing the movement of aircraft, ships, rockets. Sections of practical mechanics - courses "Theory of machines and mechanisms", "Machine parts", are studied by students of almost all technical universities (including MGIU).

Theory of elasticity- part of a section continuum mechanics, which assumes that the material of the elastic body is homogeneous and continuously distributed over the entire volume of the body, so that the smallest element cut out of the body has the same physical properties, which is the whole body. The application of the theory of elasticity - the course "strength of materials", is studied by students of all technical universities (including MGIU). This section is required for all strength calculations. Here is the calculation of the strength of the hulls of ships, aircraft, missiles, the calculation of the strength of steel and reinforced concrete structures of buildings, and much more.

Gas and hydrodynamics, as well as the theory of elasticity - part of the section continuum mechanics, considers the laws of motion of liquid and gas. The equations of gas and hydrodynamics are necessary when analyzing the movement of bodies in a liquid and gaseous medium (satellites, submarines, rockets, shells, cars), when calculating the outflow of gas from the nozzles of rocket and aircraft engines. Practical Application of Fluid Dynamics – Hydraulics (Brake, Rudder,…)

The previous sections of mechanics considered the movement of bodies in the macrocosm, and the physical laws of the macrocosm are not applicable in the microcosm, in which particles of matter move - protons, neutrons, electrons. Here, completely different principles operate, and to describe the microworld, it is necessary to quantum mechanics. The basic equation describing the behavior of microparticles is the Schrödinger equation: . Here, is the Hamiltonian operator (Hamiltonian). For a one-dimensional particle motion equation https://pandia.ru/text/78/009/images/image005_136.gif" width="35" height="21 src=">-potential energy. The solution of this equation is a set of energy eigenvalues and eigenfunctions..gif" width="55" height="24 src=">– probability density. Quantum mechanical calculations are needed for the development of new materials (microcircuits), the creation of lasers, the development of spectral analysis methods, etc.

A large number of tasks are solved kinetics describing the motion and interaction of particles. Here and diffusion, heat transfer, the theory of plasma - the fourth state of matter.

statistical physics considers ensembles of particles, allows you to say about the parameters of the ensemble, based on the properties of individual particles. If the ensemble consists of gas molecules, then the properties of the ensemble derived by the methods of statistical physics are the equations of the gas state well known from high school: https://pandia.ru/text/78/009/images/image009_85.gif" width="16" height="17 src=">.gif" width="16" height="17">-molecular weight of the gas. K is the Rydberg constant. Statistical methods are also used to calculate the properties of solutions, crystals, and electrons in metals. MM statistical physics - theoretical background thermodynamics, which underlies the calculation of engines, heating networks and stations.

Field theory describes by MM methods one of the main forms of matter - the field. In this case, electromagnetic fields are of primary interest. The equations of the electromagnetic field (electrodynamics) were derived by Maxwell: , , , . Here and https://pandia.ru/text/78/009/images/image018_44.gif" width="16" height="17"> - charge density, - current density. The equations of electrodynamics underlie the calculations of the propagation of electromagnetic waves necessary to describe the propagation of radio waves (radio, television, cellular communications), explain the operation of radar stations.

Chemistry can be represented in two aspects, highlighting descriptive chemistry - the discovery of chemical factors and their description - and theoretical chemistry - the development of theories that allow generalizing the established factors and presenting them in the form of a specific system (L. Pauling). Theoretical chemistry is also called physical chemistry and is, in essence, a branch of physics that studies substances and their interactions. Therefore, everything that has been said about physics fully applies to chemistry. Sections of physical chemistry will be thermochemistry, which studies the thermal effects of reactions, chemical kinetics (reaction rates), quantum chemistry (the structure of molecules). At the same time, the problems of chemistry are extremely complex. So, for example, to solve the problems of quantum chemistry - the science of the structure of atoms and molecules, programs are used that are comparable in volume to the air defense programs of the country. For example, in order to describe a UCl4 molecule, consisting of 5 atomic nuclei and +17 * 4) electrons, you need to write down the equation of motion - equations in partial derivatives.

Biology

Mathematics really came into biology only in the second half of the 20th century. The first attempts to mathematically describe biological processes refer to models of population dynamics. A population is a community of individuals of the same species occupying a certain area of ​​space on Earth. This area of ​​mathematical biology, which studies the change in population size under various conditions (presence of competing species, predators, diseases, etc.), further served as a mathematical testing ground on which mathematical models in various fields of biology were "performed". Including models of evolution, microbiology, immunology and other areas related to cell populations.
The very first known model formulated in a biological setting is the famous Fibonacci series (each subsequent number is the sum of the previous two), which is cited in his work by Leonardo of Pisa in the 13th century. This is a series of numbers describing the number of pairs of rabbits that are born each month, if the rabbits start breeding from the second month and produce a pair of rabbits each month. The row represents a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, ...

1,

2 ,

3,

5,

8, 13, …

Another example is the study of ionic transmembrane transport processes on an artificial bilayer membrane. Here, in order to study the laws of formation of a pore through which an ion passes through the membrane into the cell, it is necessary to create a model system that can be studied experimentally, and for which a well-developed physical description can be used.

A classic example of MM is also the Drosophila population. An even more convenient model is viruses, which can be propagated in a test tube. The methods of modeling in biology are the methods of dynamic systems theory, and the means are differential and difference equations, methods of the qualitative theory of differential equations, simulation modeling.
Goals of modeling in biology:
3. Elucidation of the mechanisms of interaction between the elements of the system
4. Identification and verification of model parameters using experimental data.
5. Assessment of the stability of the system (model).

6. Prediction of system behavior under various external influences, various ways management and so on.
7. Optimal control of the system in accordance with the chosen optimality criterion.

Technique

A large number of specialists are engaged in the improvement of technology, who in their work rely on the results scientific research. Therefore, the MM in technology are the same as the MM in natural science, which were discussed above.

Economy and social processes

It is generally accepted that mathematical modeling as a method of analyzing macroeconomic processes was first used by the physician of King Louis XV, Dr. François Quesnay, who in 1758 published the work "Economic Table". In this work, the first attempt was made to quantitatively describe the national economy. And in 1838 in the book O. Cournot"Investigation of the mathematical principles of the theory of wealth" quantitative methods were first used to analyze competition in the market for goods in various market situations.

Malthus's theory of population is also widely known, in which he proposed the idea that population growth is far from always desirable, and this growth is faster than the growing possibilities of providing the population with food. The mathematical model of such a process is quite simple: Let - population growth over time https://pandia.ru/text/78/009/images/image027_26.gif" width="15" height="24"> the population was equal to . and are the coefficients taking into account the birth and death rates (persons/year).

https://pandia.ru/text/78/009/images/image032_23.gif" width="151" height="41 src=">Instrumental and mathematical methods" href="/text/category/instrumentalmznie_i_matematicheskie_metodi/" rel ="bookmark">mathematical methods of analysis (for example, in recent decades, mathematical theories of cultural development have appeared in the humanities, mathematical models of mobilization, cyclic development of sociocultural processes, a model of interaction between the people and the government, an arms race model, etc.) have been constructed and studied.

In the most general terms, the MM process of socio-economic processes can be conditionally divided into four stages:

formulating a system of hypotheses and developing a conceptual model; development of a mathematical model; analysis of the results of model calculations, which includes their comparison with practice; formulation of new hypotheses and refinement of the model in case of discrepancy between the results of calculations and practical data.

Note that, as a rule, the process of mathematical modeling is cyclical, since even when studying relatively simple processes, it is rarely possible to build an adequate mathematical model from the first step and select its exact parameters.

At present, the economy is considered as a complex developing system, for the quantitative description of which dynamic mathematical models of varying degrees of complexity are used. One of the areas of research of macroeconomic dynamics is associated with the construction and analysis of relatively simple nonlinear simulation models that reflect the interaction of various subsystems - the labor market, the goods market, the financial system, the natural environment, etc.

The theory of catastrophes is successfully developing. This theory considers the question of the conditions under which a change in the parameters of a nonlinear system causes a point in the phase space, characterizing the state of the system, to move from the region of attraction to the initial equilibrium position to the region of attraction to another equilibrium position. The latter is very important not only for the analysis of technical systems, but also for understanding the sustainability of socio-economic processes. In this regard, the findings about the significance of the study of nonlinear models for management. In the book "The Theory of Catastrophes", published in 1990, he, in particular, writes: "... the current restructuring is largely due to the fact that at least some feedback mechanisms (fear of personal destruction) have begun to operate."

(model parameters)

When building models of real objects and phenomena, one often encounters a lack of information. For the object under study, the distribution of properties, the parameters of the impact and the initial state are known with varying degrees of uncertainty. When building a model, the following options for describing uncertain parameters are possible:

Classification of mathematical models

(implementation methods)

MM implementation methods can be classified according to the table below.

MM Implementation Methods Very often, the analytical solution for the model is presented in the form of functions. To obtain the values ​​of these functions for specific values ​​of the input parameters, their expansion into series (for example, Taylor) is used, and the value of the function for each value of the argument is determined approximately. Models that use this technique are called approximate. At numerical approach the set of mathematical relations of the model is replaced by a finite-dimensional analogue. This is most often achieved by discretizing the initial relations, i.e., by passing from functions of a continuous argument to functions of a discrete argument (grid methods). The solution found after calculations on a computer is taken as an approximate solution of the original problem. Most existing systems are very complex, and it is impossible to create a real model for them, described analytically. Such systems should be studied using simulation modeling. One of the main methods of simulation modeling is associated with the use of a random number generator. Since a huge number of problems are solved by MM methods, the methods for implementing MM are studied in more than one training course. Here are partial differential equations, numerical methods for solving these equations, computational mathematics, computer simulation, etc. PAULING, Linus Carl (Pauling, Linus Carl) (), American chemist and physicist, awarded in 1954 Nobel Prize in chemistry for nature studies chemical bond and determining the structure of proteins. Born February 28, 1901 in Portland, Oregon. He developed a quantum mechanical method for studying the structure of molecules (along with the American physicist J. Slayer) - the method of valence bonds, as well as the theory of resonance, which makes it possible to explain the structure of carbon-containing compounds, primarily compounds of the aromatic series. During the period of the personality cult of the USSR, scientists involved in quantum chemistry were persecuted and accused of "polingism". MALTHUS, THOMAS ROBERT (Malthus, Thomas Robert) (), English economist. Born at Rookery near Dorking in Surrey on February 15 or 17, 1766. In 1798 he published anonymously An experiment on the law of population. In 1819 Malthus was elected a Fellow of the Royal Society.