Mathematical model- this is a set of mathematical objects and relationships between them, adequately reflecting the properties and behavior of the object under study.
Mathematics in the most general sense deals with the definition and use of symbolic models. A mathematical model covers a class of undefined (abstract, symbolic) mathematical objects such as numbers or vectors, and the relationships between these objects.
A mathematical relation is a hypothetical rule relating two or more symbolic objects. Many relationships can be described using mathematical operations that relate one or more objects to another object or to a set of objects (the result of an operation). The abstract model, with its objects of an arbitrary nature, relations and operations, is defined by a consistent set of rules that introduce operations that can be used and establish general relationships between their results. The constructive definition introduces a new mathematical model, using already known mathematical concepts (for example, the definition of addition and multiplication of matrices in terms of addition and multiplication of numbers).
A mathematical model will reproduce appropriately selected aspects of a physical situation if a correspondence rule can be established linking specific physical objects and relations to certain mathematical objects and relations. It can also be instructive and/or interesting to build mathematical models for which physical world analogues do not exist. The most commonly known mathematical models are systems of integers and real numbers and Euclidean geometry; the defining properties of these models are more or less direct abstractions of physical processes (counting, ordering, comparison, measurement).
Objects and operations of more general mathematical models are often associated with sets of real numbers, which can be correlated with the results of physical measurements.
Mathematical modeling is a method of qualitative and (or) quantitative description process through the so-called mathematical model, in the construction of which the real process or phenomenon is described using one or another adequate mathematical apparatus. Mathematical modeling is an integral part of modern research.
Mathematical modeling is a typical discipline located, as it is now often said, at the "junction" of several sciences. An adequate mathematical model cannot be built without deep knowledge of the object that is “served” by the mathematical model. Sometimes an illusory hope is expressed that a mathematical model can be created jointly by a mathematician who does not know the object of modeling, and a specialist in the “object” who does not know mathematics. For successful activity in the field mathematical modeling it is necessary to know both mathematical methods and the object of modeling. This is connected, for example, with the presence of such a specialty as a theoretical physicist, whose main activity is mathematical modeling in physics. The division of specialists into theoreticians and experimenters, which has been established in physics, will undoubtedly occur in other sciences, both fundamental and applied.
Due to the variety of applied mathematical models, their general classification difficult. In the literature, classifications are usually given, which are based on various approaches. One of these approaches is related to the nature of the process being modeled, when deterministic and probabilistic models are distinguished. Along with such a widespread classification of mathematical models, there are others.
Classification of mathematical models based on the features of the applied mathematical apparatus . It includes the following varieties.
Usually, such models are used to describe the dynamics of systems consisting of discrete elements. From the mathematical side, these are systems of ordinary linear or non-linear differential equations.
Mathematical models with lumped parameters are widely used to describe systems consisting of discrete objects or sets of identical objects. For example, the dynamic model of a semiconductor laser is widely used. In this model, two dynamic variables appear - the concentrations of minor charge carriers and photons in the active zone of the laser.
When complex systems the number of dynamic variables and, consequently, differential equations can be large (up to 102 ... 103). In these cases, useful various methods system reductions based on the temporal hierarchy of processes, assessment of the influence of various factors and neglect of insignificant ones, etc.
The method of successive extension of the model can lead to the creation of an adequate model of a complex system.
Models of this type describe the processes of diffusion, heat conduction, propagation of waves of various natures, etc. These processes can be not only of a physical nature. Mathematical models with distributed parameters are widely used in biology, physiology and other sciences. Most often, the equations of mathematical physics, including nonlinear ones, are used as the basis of a mathematical model.
The fundamental role of the principle of greatest action in physics is well known. For example, all known systems of equations describing physical processes can be derived from extremal principles. However, in other sciences extremal principles play an essential role.
The extremal principle is used when approximating empirical dependencies by an analytical expression. The graphic representation of such a dependence and the specific form of the analytical expression describing this dependence is determined using the extremal principle, called the least squares method (Gauss method), the essence of which is as follows.
Let an experiment be carried out, the purpose of which is to study the dependence of some physical quantity Y from physical quantity x. It is assumed that the values x and y linked by functional dependency
The form of this dependence needs to be determined from experience. Let us assume that as a result of the experiment, we obtained a number of experimental points and built a dependence graph at from X. Usually, the experimental points on such a graph are not located quite correctly, they give some spread, i.e., they reveal random deviations from the visible general pattern. These deviations are associated with measurement errors that are inevitable in any experiment. Then the problem of smoothing the experimental dependence, which is typical for practice, arises.
To solve this problem, a calculation method is usually used, known as the method of least squares (or the Gauss method).
Of course, the listed varieties of mathematical models do not exhaust the entire mathematical apparatus used in mathematical modeling. The mathematical apparatus of theoretical physics and, in particular, its most important section, elementary particle physics, is especially diverse.
The areas of their application are often used as the main principle of the classification of mathematical models. With this approach, the following areas of application are distinguished:
physical processes;
technical applications, including managed systems, artificial intelligence;
life processes(biology, physiology, medicine);
large systems associated with the interaction of people (social, economic, environmental);
humanities (linguistics, art).
(The areas of application are listed in descending order according to the level of adequacy of the models).
Types of mathematical models: deterministic and probabilistic, theoretical and experimental factorial. Linear and non-linear, dynamic and static. continuous and discrete, functional and structural.
Classification of mathematical models (TO - technical object)
The structure of a model is an ordered set of elements and their relationships. A parameter is a value that characterizes a property or mode of operation of an object. The output parameters characterize the properties of the technical object, and the internal parameters characterize the properties of its elements. External parameters are parameters External Environment, which affects the functioning of the technical object.
Mathematical models are subject to the requirements of adequacy, economy, universality. These claims are contradictory.
Depending on the degree of abstraction in the description physical properties The technical system distinguishes three main hierarchical levels: the upper or meta level, the middle or macro level, the lower or micro level.
The meta level corresponds to the initial stages of design, at which scientific and technical1 search and forecasting, development of a concept and technical solution, and development of a technical proposal are carried out. To build mathematical models of the metalevel, the methods of morphological synthesis, graph theory, mathematical logic, theory automatic control, the theory of queuing, the theory of finite automata.
At the macrolevel, an object is considered as a dynamic system with lumped parameters. Mathematical models of the macrolevel are systems of ordinary differential equations. These models are used in determining the parameters of a technical object and its functional elements.
At the micro level, an object is represented as a continuous medium with distributed parameters. To describe the processes of functioning of such objects, partial differential equations are used. At the micro level, elements of a technical system that are indivisible in terms of functional characteristics, called basic elements, are designed. At the same time, the base element is considered as a system consisting of a set of similar functional elements of the same physical nature, interacting with each other and being influenced by the external environment and other elements of the technical object, which are the external environment in relation to the base element.
According to the form of representation of mathematical models, invariant, algorithmic, analytical and graphical models of the design object are distinguished.
AT invariant form, the mathematical model is represented by a system of equations without regard to the method of solving these equations.
AT algorithmic in the form of model relations are associated with the chosen numerical solution method and are written in the form of an algorithm - a sequence of calculations. Algorithmic models include imitation, models designed to simulate the physical and information processes occurring in the object during its operation under the influence of various environmental factors.
Analytical the model represents the explicit dependences of the desired variables on the given values (usually, the dependences of the output parameters of the object on internal and external parameters). Such models are obtained on the basis of physical laws, or as a result of direct integration of the original differential equations. Analytical mathematical models make it easy and simple to solve the problem of determining the optimal parameters. Therefore, if it is possible to obtain a model in this form, it is always advisable to implement it, even if it requires performing a number of auxiliary procedures. Such models are usually obtained by experimental design (computational or physical).
Graphic(circuit) model is represented in the form of graphs, equivalent circuits, dynamic models, diagrams, etc. To use graphical models, there must be a rule of one-to-one correspondence between conditional images of graphical elements and components of invariant mathematical models.
The division of mathematical models into functional and structural ones is determined by the nature of the displayed properties of the technical object.
Structural models display only the structure of objects and are used only in solving problems of structural synthesis. Parameters of structural models are signs of functional or structural elements that make up a technical object and in which one version of the object structure differs from another. These parameters are called morphological variables. Structural models take the form of tables, matrices and graphs. The most promising is the use of tree-like graphs of the AND-OR-tree type. Such models are widely used at the meta level when choosing a technical solution.
Functional models describe the processes of functioning technical objects and have the form of systems of equations. They take into account the structural and functional properties of the object and allow solving problems of both parametric and structural synthesis. They are widely used at all levels of design. At the meta level, functional tasks allow solving forecasting problems, at the macro level - choosing the structure and optimizing the internal parameters of a technical object, at the micro level - optimizing parameters basic elements.
According to the methods of obtaining functional mathematical models are divided into theoretical and experimental.
Theoretical models are obtained based on the description of the physical processes of the object's functioning, and experimental- based on the behavior of the object in the external environment, considering it as a "black box". Experiments in this case can be physical (on a technical object or its physical model) or computational (on a theoretical mathematical model).
When constructing theoretical models, physical and formal approaches are used.
The physical approach is reduced to the direct application of physical laws to describe objects, for example, the laws of Newton, Hooke, Kirchhoff, etc.
The formal approach uses general mathematical principles and is used in the construction of both theoretical and experimental models. Experimental models are formal. They do not take into account the entire complex of physical properties of the elements of the technical system under study, but only establish a connection found during the experiment between the individual parameters of the system, which can be varied and (or) measured. Such models provide an adequate description of the processes under study only in a limited region of the parameter space, in which the parameters were varied in the experiment. Therefore, experimental mathematical models are of a particular nature, while physical laws reflect the general patterns of phenomena and processes that occur throughout technical system, as well as in each of its elements separately. Consequently, experimental mathematical models cannot be accepted as physical laws. However, the methods used to build these models are widely used in testing scientific hypotheses.
Functional mathematical models can be linear and non-linear. Linear models contain only linear functions of quantities characterizing the state of the object during its operation, and their derivatives. The characteristics of many elements of real objects are non-linear. Mathematical models of such objects include non-linear functions of these quantities and their derivatives and refer to non-linear .
If the modeling takes into account the inertial properties of the object and (or) the change in time of the object or the external environment, then the model is called dynamic. Otherwise the model is static. The mathematical representation of a dynamic model in the general case can be expressed by a system of differential equations, and a static one - by a system of algebraic equations.
If the impact of the environment on the object is of a random nature and is described by random functions. In this case, it is necessary to build probabilistic mathematical model. However, such a model is very complex and its use in the design of technical objects requires a lot of computer time. Therefore, it is used for final stage design.
Most design procedures are performed on deterministic models. A deterministic mathematical model is characterized by a one-to-one correspondence between external influence on the dynamic system and its response to this impact. In a computational experiment, when designing, some standard typical actions on an object are usually set: stepped, impulse, harmonic, piecewise linear, exponential, etc. They are called test actions.
Continuation of the Table “Classification of mathematical models
Types of mathematical models of technical objects |
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By taking into account the physical properties of TO | By the ability to predict results |
dynamic | deterministic |
Static | Probabilistic |
continuous |
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Discrete |
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Linear |
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| At this stage, the following steps are performed. A plan is drawn up for creating and using a software model. As a rule, the model program is created using computer simulation automation tools. Therefore, the plan indicates: type of computer; simulation automation tool; approximate costs of computer memory for creating a model program and its working arrays; the cost of machine time for one cycle of the model; estimates of costs for programming and debugging the model program. Then the researcher starts programming the model. As terms of reference description for programming simulation model. The specifics of model programming work depends on the modeling automation tools that are available to the researcher. There are no significant differences between the creation of a model program and the usual offline debugging of program modules of a large program or software package. In accordance with the text, the model is divided into blocks and subblocks. In contrast to the usual offline debugging of program modules, when debugging blocks and subblocks of a program model, the amount of work increases significantly, since for each module it is necessary to create and debug an external environment simulator. It is very important to verify the implementation of the module functions in the model time t and estimate the cost of computer time for one cycle of the model as a function of the values of the model parameters. The work is completed during autonomous debugging of the model components by preparing the forms for representing the input and output data of the simulation. Next, proceed to the second verification of the reliability of the system model program. During this check, the correspondence of operations in the program and the description of the model is established. For this, it is produced reverse translation programs into the model diagram (manual "scrolling" allows you to find gross errors in the model statics). After eliminating gross errors, a number of blocks are combined and complex debugging of the model begins using tests. Debugging for tests starts with a few blocks, then an increasing number of model blocks are involved in this process. Note that complex debugging of the model program is much more difficult than debugging application packages, since simulation dynamics errors in this case are much more difficult to find due to the quasi-parallel operation of various model components. Upon completion of the complex debugging of the model program, it is necessary to re-estimate the costs of computer time for one cycle of calculations on the model. In this case, it is useful to obtain an approximation of the simulation time for one simulation cycle. The next step is to draw up technical documentation for a model of a complex system. By the end of the complex debugging of the model program, the result of the stage should be the following documents:
To check the adequacy of the model to the object of study, after compiling a formal description of the system, the researcher draws up a plan for conducting full-scale experiments with a system prototype. If there is no prototype of the system, then a system of nested IMs can be used, differing from each other in the degree of detail of imitation of the same phenomena. Then the more detailed model serves as a prototype for the generalized IM. If it is impossible to build such a sequence, either because of the lack of resources to perform this work, or because of insufficient information, then they do without checking the adequacy of the IM. According to this plan, in parallel with the debugging of the IM, a series of full-scale experiments on a real system is carried out, during which control results. Having control results and MI test results at his disposal, the researcher checks the adequacy of the model to the object. If errors are found during the debugging phase that can only be corrected in the previous phases, a return to the previous phase can take place. In addition to the technical documentation, the results of the stage are accompanied by a machine implementation of the model (a program translated in the machine code of the computer on which the simulation will take place). This is an important step in the creation of the model. In this case, you must do the following. First, make sure that the dynamics of the development of the algorithm for modeling the object of study is correct in the course of simulating its functioning (to verify the model). Secondly, to determine the degree of adequacy of the model and the object of study. The adequacy of a software simulation model to a real object is understood as the coincidence with a given accuracy of the vectors of the characteristics of the behavior of the object and the model. In the absence of adequacy, the simulation model is calibrated (“correct” the characteristics of the model component algorithms). The presence of errors in the interaction of model components returns the researcher to the stage of creating a simulation model. It is possible that in the course of formalization, the researcher oversimplified the physical phenomena, excluded from consideration a number of important aspects of the functioning of the system, which led to the inadequacy of the model to the object. In this case, the researcher must return to the stage of system formalization. In cases where the choice of the formalization method turned out to be unsuccessful, the researcher needs to repeat the stage of compiling a conceptual model, taking into account new information and experience. Finally, when the researcher has insufficient information about the object, he must return to the stage of compiling a meaningful description of the system and refine it, taking into account the results of testing the previous system model. At the same time, the accuracy of the simulation of phenomena, the stability of the simulation results, the sensitivity of the quality criteria to changes in the model parameters are evaluated. It is very difficult to obtain these estimates in some cases. However, without the successful results of this work, neither the developer nor the IM customer will have confidence in the model. Different researchers, depending on the type of IM, have developed different interpretations of the concepts of accuracy, stability, stationarity, sensitivity of IM. So far, there is no generally accepted theory of imitation of phenomena on a computer. Each researcher has to rely on his experience in organizing the simulation and on his understanding of the features of the simulation object. The simulation accuracy of phenomena is an assessment of the influence of stochastic elements on the functioning of a complex system model. The stability of the simulation results is characterized by the convergence of the controlled simulation parameter to a certain value with an increase in the simulation time of a variant of a complex system. The stationarity of the simulation mode characterizes a certain equilibrium of processes in the system model, when further simulation is meaningless, since the researcher will not receive new information from the model, and continuing the simulation practically only leads to an increase in computer time. It is necessary to provide for such a possibility and develop a method for determining the moment when the stationary simulation mode is reached. The MI sensitivity is represented by the value of the minimum increment of the selected quality criterion, calculated from the simulation statistics, with sequential variation of the simulation parameters over the entire range of their changes. This stage begins with the design of an experiment that allows the researcher to obtain maximum information with minimum computational effort. Statistical substantiation of the experimental plan is required. Experiment planning is a procedure for choosing the number and conditions of experiments that are necessary and sufficient to solve the problem with the required accuracy. At the same time, the following is essential: striving to minimize the total number of experiments, ensuring the possibility of simultaneous variation of all variables; the use of a mathematical apparatus that formalizes many of the actions of experimenters; choosing a clear strategy that allows you to make informed decisions after each series of experiments on the model. Then the researcher proceeds to carry out working calculations on the model. This is a very time-consuming process that requires a large computer resource and an abundance of clerical work. It should be noted that already at the early stages of creating an IM, it is necessary to carefully consider the composition and volume of modeling information in order to significantly facilitate further analysis of the simulation results. The result of the work are the simulation results. This stage completes the technological chain of stages of creating and using simulation models. Having received the simulation results, the researcher proceeds to interpret the results. The following simulation cycles are possible here. In the first cycle of the simulation experiment in the IM, the choice of options for the system under study is provided in advance by setting the initial conditions for the simulation for the computer program of the model. In the second cycle of the simulation experiment, the model is modified in the modeling language, and therefore re-translation and editing of the program is required. It is possible that in the course of interpreting the results, the researcher found the presence of errors either when creating the model or when formalizing the modeling object. In these cases, a return is made to the stages of constructing a description of the simulation model or to compiling a conceptual model of the system, respectively. The result of the stage of interpretation of the simulation results are recommendations for the design of the system or its modification. With the recommendations at their disposal, the researchers begin to make design decisions. The interpretation of the simulation results is significantly influenced by the imaging capabilities of the computer used and the simulation system implemented on it. 1. How is the classification of mathematical models based on the features of the applied mathematical apparatus. Mathematics abstract Development of an economic and mathematical model for optimizing the sectoral structure of production in the agricultural sector |
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A mathematical model is a mathematical representation of reality, one of the variants of a model, as a system, the study of which allows obtaining information about some other system. The process of building and studying mathematical models is called mathematical modeling. All natural and social sciences that use the mathematical apparatus are, in fact, engaged in mathematical modeling: they replace the object of study with its mathematical model and then study the latter. The connection of a mathematical model with reality is carried out with the help of a chain of hypotheses, idealizations and simplifications. By using mathematical methods describes, as a rule, an ideal object built at the stage of meaningful modeling. General information
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No definition can fully cover the real-life activity of mathematical modeling. Despite this, definitions are useful in that they attempt to highlight the most significant features. According to Lyapunov, mathematical modeling is an indirect practical or theoretical study of an object, in which not the object of interest to us is directly studied, but some auxiliary artificial or natural system (model) that is in some objective correspondence with the object being known, capable of replacing it in certain respects. and giving, in its study, ultimately, information about the modeled object itself. In other versions, the mathematical model is defined as an object-substitute of the original object, providing the study of some properties of the original, as "an" equivalent "of the object, reflecting in mathematical form its most important properties - the laws to which it obeys, the connections inherent in its constituent parts", as a system of equations, or arithmetic relations, or geometric figures, or a combination of both, the study of which by means of mathematics should answer the questions posed about the properties of a certain set of properties of a real world object, as a set of mathematical relations, equations, inequalities that describe the basic patterns inherent the process, object or system being studied. Definitions
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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: Linear versus non-linear models; Concentrated or distributed systems; Deterministic or stochastic; Static or dynamic; Discrete or continuous and so on. Each constructed model is linear or non-linear, deterministic or stochastic, ... Naturally, mixed types are also possible: in one respect concentrated (in terms of parameters), in another - distributed models, etc. Formal classification of models
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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. Also possible combined types models, sometimes referred to as gray box models. Mathematical models of complex systems can be divided into three types: Black box models (phenomenological), Gray box models (a mixture of phenomenological and mechanistic models), Type models white box(mechanistic, axiomatic). Schematic representation of black box, gray box and white box models
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Almost all authors describing the process of mathematical modeling indicate that first a special ideal construction, a meaningful model, is built. There is no established terminology here, and other authors call this ideal object a conceptual model, a speculative model, or a premodel. In this case, the final mathematical construction is called a formal model or simply a mathematical model obtained as a result of the formalization of this content model (pre-model). A meaningful model can be built 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 much more complicated. Content and formal models
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Peierls' work gives a classification of mathematical models used in physics and, more broadly, in the natural sciences. In the book by A. N. Gorban and R. G. Khlebopros, this classification is analyzed and expanded. This classification is focused primarily on the stage of constructing a meaningful model. Hypothesis Models of the first type - hypotheses ("this could be"), "represent a trial description of the phenomenon, and the author either believes in its possibility, or even considers it to be true." According to Peierls, these are, for example, the Ptolemy model of the solar system and the Copernican model (improved by Kepler), the Rutherford model of the atom and the Big Bang model. Model-hypotheses in science cannot be proven once and for all, one can only talk about their refutation or non-refutation as a result of the experiment. 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. Phenomenological model The second type, the phenomenological model (“behave as if…”), contains a mechanism for describing the phenomenon, although this mechanism is not convincing enough, cannot be sufficiently confirmed by the available data, or is poorly consistent 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 the search for "true mechanisms" must continue. 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. Similarly, new knowledge can gradually come into conflict with models-hypotheses of the first type, and they can be transferred to the second. Meaningful classification of models
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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. Approximation The third type of models is approximations (“we consider something very large or very small”). If it is possible to construct equations describing the system under study, this does not mean that they can be solved even with the help of a computer. A common technique in this case is the use of approximations (models of type 3). Among them are linear response models. The equations are replaced by linear ones. The standard example is Ohm's law. If we use the ideal gas model to describe sufficiently rarefied gases, then this is a type 3 model (approximation). At higher gas densities, it is also useful to imagine a simpler situation with an ideal gas for qualitative understanding and evaluation, but then this is already type 4. Simplification noticeable and not always controllable effect on the result. The same equations can serve as a model of type 3 (approximation) or 4 (we omit some details for clarity) - this depends on the phenomenon for which the model is used to study. So, if linear response models are used in the absence of more complex models (that is, nonlinear equations are not linearized, but linear equations describing the object are simply searched), then these are already phenomenological linear models, and they belong to the following type 4 (all non-linear details " omitted for clarity). Examples: application of an ideal gas model to a non-ideal one, the van der Waals equation of state, most models of solid state, liquid and nuclear physics. The path from microdescription to the properties of bodies (or media) consisting of a large number of particles, Meaningful classification of models (continued)
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very long. Many details have to be left out. This leads to models of the fourth type. Heuristic model The fifth type is the heuristic model (“there is no quantitative confirmation, but the model contributes to a deeper insight into the essence of the matter”), such a model preserves only a qualitative similarity of reality and gives predictions only “in order of magnitude”. A typical example is the mean free path approximation in kinetic theory. It gives simple formulas for the coefficients of viscosity, diffusion, thermal conductivity, consistent with reality in order of magnitude. But when building a new physics, it is far from immediately obtained a model that gives at least a qualitative description of an object - a model of the fifth type. In this case, a model is often used by analogy, reflecting reality at least in some way. Analogy The sixth type is an analogy model (“let's take into account only some features”). Peierls gives a history of the use of analogies in Heisenberg's first paper on nature. nuclear forces. The Thought Experiment The seventh type of model is the thought experiment (“the main thing is to disprove the possibility”). This type of simulation was often used by Einstein, in particular, one of these experiments led to the construction of the special theory of relativity. Suppose that in classical physics we follow a light wave at the speed of light. We will observe an electromagnetic field periodically changing in space and constant in time. According to Maxwell's equations, this cannot be. From here, Einstein concluded: either the laws of nature change when the frame of reference changes, or the speed of light does not depend on the frame of reference, and chose the second option. Possibility demonstration The eighth type is the possibility demonstration (“the main thing is to show the internal consistency of the possibility”), such models are also thought experiments with imaginary entities, demonstrating that the alleged phenomenon is consistent with the basic principles and Meaningful classification of models (continued)
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internally consistent. This is the main difference from models of type 7, which reveal hidden contradictions. One of the most famous such experiments is Lobachevsky's geometry. (Lobachevsky called it "imaginary geometry.") Another example is mass production formal-kinetic models of chemical and biological oscillations, autowaves. The Einstein - Podolsky - Rosen paradox was conceived as a thought experiment to demonstrate the inconsistency of quantum mechanics, but in an unplanned way over time turned into a type 8 model - a demonstration of the possibility of quantum teleportation of information. The substantive classification is based on the stages preceding mathematical analysis and calculations. Eight types of models according to Peierls are eight types of research positions in modeling. Meaningful classification of models (continued)
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actually useless. Often, a simpler 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”). Example (continued)
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The most important mathematical models usually have an important property of 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 a U-shaped vessel, or a change in the current strength in an 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 a “general systems theory”. Universality of models
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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 task: the structure of the model and all its parameters are considered known, the main task is to study the model in order to extract useful knowledge about the object. What static load can the bridge withstand? How will it 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 will the aircraft overcome sound barrier, whether it will fall apart from flutter - these are typical examples of the direct problem. Setting the correct direct problem (asking the correct question) requires special skill. If not set the right questions, then the bridge may collapse, even if it was built good model for his behaviour. So, in 1879 in the UK, a metal railway bridge across the River Tay collapsed, the designers of which built a model of the bridge, calculated it for a 20-fold margin of safety against 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 Direct and inverse problems of mathematical modeling
"System approach in modeling" - Process - dynamic change of the system in time. System - a set of interrelated elements that form integrity or unity. Peter Ferdinand Drucker. Systems approach in organizations. A systematic approach as the basis for the introduction of specialized education. Founders systems approach: Structure - a way of interaction of elements of the system through certain connections.
"ISO 20022" - Elements of the methodology of the international standard. Comparison of composition and properties. Appointment. Modeling process. Features of the methodology. Simulation results. openness and development. Migration. Name international standard. Aspects of universality. Tools. Activity. The composition of documents.
"The concept of model and modeling" - Types of models by branches of knowledge. Types of models. Basic concepts. Types of models depending on the time. Types of models depending on the external dimensions. Model adequacy. Figurative-sign models. The need to create models. Modeling. Models modeling.
"Models and Modeling" - Changing the size and proportions. A mathematical model is a model presented in the language of mathematical relations. A block diagram is one of the special varieties of a graph. Analysis of an object. Structural model - representation of the information sign model in the form of a structure. real phenomenon. Abstract. Verbal.
"Model development steps" - Descriptive information models are usually built using natural languages and drawings. Building a descriptive information model. The main stages of development and research of models on the computer. Stage 4. Stage 1. Stage 5 Model solar system. Practical task. Stage 3. Stage 2.
"Modeling as a method of cognition" - In biology - the classification of the animal world. Definitions. Definition. In physics, it is an information model of simple mechanisms. Modeling as a method of cognition. Forms of representation of information models. Tabular model. The process of building information models using formal languages is called formalization.
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Algorithm drawing up a mathematical model:
- Make a brief statement of the problem statement:
A) find out how many quantities are involved in the task;
B) identify the relationship between these quantities.
2. Make a drawing for the problem (in problems of movement or problems of geometric content) or a table.
3. Designate one of the values for X (better, a smaller value).
4. Taking into account the connections, make a mathematical model.
Problem 1. (No. 86 (1)).
The apartment consists of 3 rooms with a total area of 42 sq.m. The first room is 2 times smaller than the second, and the second is 3 square meters. m more than a third. What is the area of each room in this apartment?
Task 2. (No. 86 (2)).
Sasha paid 11200 rubles for the book, pen and notebook. A pen is 3 times more expensive than a notebook and 700 r. cheaper than a book. How much does a notebook cost?
Problem 3. (No. 86 (3)).
A motorcyclist traveled a distance between two cities equal to
980 km, in 4 days. On the first day he covered 80 km less than on the second day, on the third day he covered half the distance covered in the first two days, and on the fourth day he covered the remaining 140 km. How far did the motorcyclist travel on the third day?
Problem 4. (No. 86 (4))
The perimeter of a quadrilateral is 46 in. Its first side is 2 times smaller than the second and 3 times smaller than the third side, and the fourth side is 4 cm larger than the first side. What are the lengths of the sides of this quadrilateral?
Problem 5. (No. 87)
One of the numbers is 17 less than the second, and their sum is 75. Find the largest of these numbers.
Problem 6. (No. 99)
20 participants performed in three parts of the concert. In the second section, there were 3 times fewer participants than in the first, and in the third section - 5 participants more than in the second. How many participants in the concert performed in each section?
I can (or not):
Skills
Points
0 or 1
Reveal the number of quantities involved in the task
Reveal relationships between quantities
I understand what it means
B) "everything"
I can make a mathematical model
I can compose new task according to a given mathematical model
Homework:
1) № 87 (2, 3, 4), № 102 (1);
2) Compose a problem for the mathematical model of the problem
Literature 1. Samarsky A. A., Mikhailov A. P. Mathematical modeling: Ideas. Methods. Examples. - M.: Nauka, Volkov E. A. Numerical methods. - M.: Nauka, Turchak L. I. Fundamentals of numerical methods. - M.: Science, Kopchenova N. V., Maron I. A. Computational mathematics in examples and problems. – M.: Nauka, 1972.
A bit of history from manipulation of objects to manipulation of concepts about objects replacement of the object, process or phenomenon under study with a simpler and more accessible equivalent for research the inability to take into account the entire set of factors that determine the properties and behavior of the object
The role of models The building is ugly, fragile or does not fit into the surrounding landscape Demonstration of circulatory systems in nature is inhumane Voltages, for example, in wings, may be too high It is uneconomical to assemble electrical circuits for measurements
Communication of the model with the original Creating a model involves the preservation of some properties of the original, and in different models these properties may be different. The cardboard building is much smaller than the real one, but allows us to judge its appearance; the poster makes the circulatory system understandable, although it has nothing to do with organs and tissues; the model aircraft does not fly, but the voltages in its body correspond to the flight conditions.
Why are models used? 1. A model is more accessible for research than a real object, 2. It is easier and cheaper to study a model than real objects, 3. Some objects cannot be studied directly: it is not yet possible, for example, to build a device for thermonuclear fusion or conduct experiments in the interior of stars, 4. experiments with the past are impossible, experiments with the economy or social experiments are unacceptable
Appointment of models 1. With the help of the model, it is possible to identify the most significant factors that form the properties of an object. Since the model reflects only some of the characteristics of the object - the original, then by varying the set of these characteristics in the model, it is possible to determine the degree of influence of certain factors on the adequacy of the behavior of the model
The model is needed: 1. In order to understand how a particular object is arranged: what is its structure, properties, laws of development and interaction with the surrounding world. 2. In order to learn how to manage an object or process and determine best ways management under given goals and criteria. 3. In order to predict the behavior of the object and evaluate the consequences of various methods and forms of impact on the object (meteorological models, models of the development of the biosphere).
The property of a correct model A well-built, good model has a remarkable property: its study allows you to gain new knowledge about the object - the original, despite the fact that only some of the main characteristics of the original were used when creating the model.
Material modeling The model reproduces the main geometric, physical, dynamic and functional characteristics of the object under study, when a real object is compared with its enlarged or reduced copy, which allows research in the laboratory with the subsequent transfer of the properties of the studied processes and phenomena from the model to the object based on the theory of similarity (planetarium, models of buildings and devices, etc.). The research process in this case is closely related to the material impact on the model, i.e., it consists in a full-scale experiment. Thus, material modeling is, by its nature, an experimental method.
Types of ideal modeling Intuitive - modeling of objects that are not amenable to formalization or do not need it. A person's life experience can be considered as his intuitive model of the surrounding world. Sign - modeling that uses sign transformations as models different kind: diagrams, graphs, drawings, formulas, etc. and containing a set of laws by which you can operate with model elements
Mathematical modeling The study of an object is carried out on the basis of a model formulated in the language of mathematics and studied using certain mathematical methods. Mathematical modeling is a field of science that deals with modeling natural phenomena, technology, economic and public life with the help of a mathematical apparatus and, at present, implementing these models using a computer
Classification mat. models By purpose: descriptive optimization simulation By the nature of the equations: linear nonlinear By taking into account changes in the system over time: dynamic static By the property of the domain of definition of the arguments: continuous discrete By the nature of the process: deterministic stochastic
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