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Stages of modeling in mathematics, economics and informatics

In the scale version, the model represents a certain image, diagram, map, description, image of a certain phenomenon or process. The phenomenon itself is called the original mathematical or economic model.

What is modeling?

Modeling is the study of an object, a system. To implement it, a model is constructed and analyzed.

All stages of modeling involve a scientific experiment, the object of which is an abstract or objective model. In carrying out the experiment, a specific phenomenon is substituted by a scheme or a simplified model (a copy). In some cases, a working model is collected to understand the mechanism of work on its example, to analyze the economic feasibility of introducing the results of experience in the market economy. The same phenomenon can be considered by different models.

The researcher should choose the necessary stages of modeling, optimally use them. The application of models is relevant in cases where a real object is not available, or experiments with it are associated with serious environmental problems. The current model is also used in those situations where a real experiment involves substantial material costs.

Features of mathematical modeling

In science, mathematical models are irreplaceable, and also tools for them - mathematical concepts. For several millennia, they accumulated, modernized. In modern mathematics there are universal and powerful methods of investigation. Any objects considered by the "queen of sciences" are a mathematical model. For a detailed analysis of the selected object, the stages of mathematical modeling are selected. With their help, the details, features, characteristics, systematize the information obtained, make a complete description of the object.

Mathematical formalization involves operating during the study with special concepts: a matrix, a function, a derivative, a primitive, numbers. Those relations and connections that can be found in the object under study between composite elements and details are written down by mathematical relations: equations, inequalities, and equalities. As a result, a mathematical description of the phenomenon or process, and therefore its mathematical model, is obtained.

Rules for Studying the Mathematical Model

There is a certain order of the stages of modeling, which allows you to establish the relationship between the consequences and causes. The central stage of designing or researching the system is the construction of a full-fledged mathematical model. The further analysis of this object directly depends on the quality of the performed actions. The construction of a mathematical or economic model is not a formal procedure. It should be easy to use, accurate so that there is no distortion in the results of the analysis.

On the classification of mathematical models

There are two types: deterministic and stochastic models. Deterministic models assume the establishment of a one-to-one correspondence between the variables used to describe the phenomenon or object.

This approach is based on information about the principle of the operation of the object. In many cases, the simulated phenomenon has a complex structure, it takes a lot of time and knowledge to decode it. In such situations, we select the stages of modeling that allow us to carry out experiments on the original, perform the processing of the results obtained, without going into the theoretical features of the object. Most often use statistics and probability theory. The result is a stochastic model. There is a random connection between the variables. A huge number of different factors cause a random set of variables through which a phenomenon or object is characterized.

Modern stages of modeling apply to static and dynamic models. In static views, the description of the relationships between the variables of the created phenomenon does not involve accounting for the time variation of the main parameters. For dynamic models, the description of relationships between variables is made taking into account temporal changes.

Varieties of models:

  • Continuous;
  • Discrete;
  • Mixed

Different stages of mathematical modeling allow us to describe relations and functions in linear models using a direct relationship of variables.

What are the requirements for the models?

  • Versatility. The model must be a complete mapping of all properties inherent in the real object.
  • Adequacy. Important characteristics of the object should not exceed the specified error value.
  • Accuracy. Characterizes the degree of coincidence of the characteristics of an existing in reality object with similar parameters obtained in the study of the model.
  • Economical. The model should be minimal in material costs.

Stages of modeling

Let's consider the basic stages of mathematical modeling.

  • Select a task. The aim of the research is chosen, the methods of its implementation are selected, the strategy of the experiment is developed. This stage involves serious work. It is from the correctness of the task posed that the final result of the simulation depends.

  • Analysis of theoretical foundations, summation of information received about the object. Such a stage implies the selection or creation of a theory. In the absence of theoretical knowledge of the object, cause-effect relationships are established between all the variables chosen to describe the phenomenon or object. At this stage, determine the initial and final data, put forward a hypothesis.
  • Formalization. A selection of a system of special designations is made that will help to write in the form of mathematical expressions the relationships between the components of the object under consideration.

Additions to the algorithm

After setting the model parameters, you select a particular method or solution method.

  • Implementation of the created model. After the stages of system modeling are selected, a program is created that passes the testing and is applied to solve the task.
  • Analysis of the collected information. An analogy is made between the task and the solution obtained, and the simulation error is determined.
  • Verify that the model matches the real object. If there is a significant difference between them, a new model is being developed. Until the ideal correspondence of the model to its real analogue is obtained, refinement and change of details are carried out.

Characteristic of modeling

In the middle of the last century in the life of modern man appeared computer technology, the relevance of mathematical methods of research of objects and phenomena has increased. There appeared such sections as "mathematical chemistry", "mathematical linguistics", "mathematical economics", dealing with the study of phenomena and objects, the main stages of modeling were created.

Their main goal was the prediction of planned observations, the study of certain objects. In addition, with the help of modeling you can learn the world around you, look for ways to manage it. Carrying out of computer experiment is supposed in those cases when to spend the present it is impossible. After constructing a mathematical model of the phenomenon under study, computer graphics can be used to study nuclear explosions, plague epidemics, and so on.

Specialists distinguish three stages of mathematical modeling, and each has its own peculiarities:

  • Building a model. This stage involves setting the economic plan, the phenomenon of nature, design, production process. Clearly describe the situation in this case is difficult. First, we need to identify the specifics of the phenomenon, determine the relationship between it and other objects. Then all qualitative characteristics are translated into mathematical language, a mathematical model is built. This stage is the most difficult in the entire modeling process.
  • The stage of solving a mathematical problem associated with the development of algorithms, methods of solving the problem on computer technology, and the detection of measurement errors.
  • Translation of information obtained during the research into the language of the area for which the experiment was conducted.

These three stages of mathematical modeling are supplemented by checking the adequacy of the model obtained. The correspondence between the results obtained in the experiment and the theoretical knowledge is checked. If necessary, modify the created model. It is complicated or simplified, depending on the results obtained.

Features of economic modeling

3 stages of mathematical modeling assume the use of algebraic, differential systems of equations. Complex objects are built using the theory of graphs. It assumes a set of points in space or on a plane, partially connected by edges. The main stages of economic modeling involve the selection of resources, their distribution, accounting of transportation, network planning. What action is not a simulation step? It is difficult to answer this question unequivocally, it all depends on the specific situation. The main stages of the modeling process involve the formulation of the goal and the subject of the study, the identification of the main characteristics to achieve the goal, a description of the relationship between the fragments of the model. Further, calculations are performed using mathematical formulas.

For example, the theory of service is the problem of queue formation. It is important to find a balance between the costs of maintaining devices and the cost of being in line. After constructing a formal description of the model, calculations are performed using computational and analytical technologies. With a qualitative drawing up of the model, you can find answers to all the questions. If the model is bad, it is impossible to understand what action is not a simulation step.

Practicality is an authentic criterion for assessing the adequacy of a phenomenon or model. Multi-criteria models, including optimization options, presuppose a goal setting. But the way to achieve this goal is different. Among the difficulties that are possible in the process, it should be noted:

  • In a complex system, there are several links between the elements;
  • It is difficult to take into account all random factors, analyzing the real system;
  • It is problematic to compare the mathematical apparatus with the results that you want to obtain

Because of the many complexities that arise in the process of studying multifaceted systems, simulation modeling has been developed. It is understood as a set of special programs for computer technology, which describes the operation of individual elements of the system and the relationship between them. The use of random variables implies repeated repetition of experiments, statistical processing of the results. Work with the simulation system is an experiment that is carried out with the help of computer technology. What are the advantages of this system? Similarly, one can achieve greater affinity for the genuine system, which is impossible in the case of a mathematical model. Using the block principle, you can analyze individual blocks before they are included in a single system. Such a variant allows us to use complex dependencies, which can not be described using ordinary mathematical relationships.

Among the disadvantages of building an imitation system, we will allocate the costs of time and resources, as well as the need to use modern computer technology.

Stages of development of modeling are comparable with the changes occurring in a society. In the field of use, all models are divided into training programs, simulators, educational-visual aids. Experienced models can be reduced copies of real objects (cars). Scientific and technical options are the stands created for the analysis of electronic equipment. Simulation models not only reflect the real reality, they imply approbation on laboratory mice, experiments in the education system. Simulation is seen as a method of errors and trials.

There is a subdivision of all models according to the variant of representation. Material models are called objective models. Such variants are endowed with the geometric and physical characteristics of the original, they can be translated into reality. Information models can not be touched. They characterize the state and properties of the studied object, phenomenon, process, and their connection with the real world. Verbal variants assume information models that are implemented in a colloquial or mental form. Signed species are expressed by applying certain signs of a multifaceted mathematical language.

Conclusion

Mathematical modeling in the form of a method of scientific cognition appeared simultaneously with the foundations of higher mathematics. An important role in this process was played by I. Newton, R. Descartes, G. Leibniz. Mathematical models were first built by P. Fermat, B. Pascal. Mathematical modeling in production, economics was paid attention to VV Leontiev, VV Novozhilov, AL Lurie. Today, such a variant of studying an object or phenomenon is applied in various fields of activity. With the help of projected systems, engineers investigate phenomena and processes that can not be analyzed in real conditions.

Scientific research by modeling was applied in the ancient times, capturing with time a variety of types of scientific knowledge: architecture, construction, chemistry, construction, physics, biology, ecology, geography, and social sciences. In any modeling process, three components are used: subject, object, model. Of course, by modeling, the study of an object or phenomenon is not limited, there are other ways of obtaining the necessary information.

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