Mathematical model: design stages

From the middle of the last century, computer and mathematical methods began to enter into various fields of human activity. New disciplines such as mathematical economics, mathematical linguistics, mathematical chemistry and others began to appear, the subject of study of which were mathematical models of phenomena and objects, as well as methods for their investigation.

A mathematical model is an approximate description in a mathematical language of objects or phenomena of the real world. The main purpose of the simulation is to study these objects and predict the results of future observations. In addition, modeling is also a method of cognition of the environment, a world that enables management.

The use of mathematical modeling is indispensable in cases where, for various reasons, it is difficult or impossible to make a natural experiment. For example, it is difficult to verify whether a particular cosmological theory is correct or to study the consequences of a nuclear explosion. But all this can be seen on the computer, having previously constructed a mathematical model.

Mathematical model: design stages

First, the model is constructed. For this purpose, consider a certain phenomenon of nature, an economic plan, a design, a production process or another non-mathematical object. First determine the features of the phenomena and the relationship between them at a qualitative level. Further, the obtained dependencies are transformed into a formula form or a mathematical model is constructed. This stage is the most difficult.

At the second stage, the mathematical problem formulated on the basis of the model is solved. Here, special attention is paid to the development of numerical methods and algorithms for solving the problem with the help of computers, which allow one to obtain the result with the necessary accuracy for a permissible time.

At the next stage, an interpretation of the consequences arising from the model is necessary, translation of the results from the mathematical language into the form adopted in the field under study.

Then, the adequacy of the obtained model is checked, it is determined whether the results of the consequences are within the specified accuracy.

At the final stage, the model is modified. It is either complicated for greater adequacy of reality or simplified to achieve an acceptable practical solution.

Classification of mathematical models

There are different criteria for separating mathematical models into groups. So, by the nature of the problems being solved, division into structural and functional models is made. In this case, the quantities characterizing an object or phenomenon are expressed quantitatively.

The structural mathematical model is represented in the form of a system of different types of equations (algebraic, differential), which establish quantitative relationships between the quantities studied. In this case, both independent variables and functions formed from them are considered as quantities.

Functional models characterize complex objects, consisting of several separate elements, between which some connections are established. Usually, these connections are difficult or impossible to measure quantitatively. For their study use the theory of graphs, mathematical objects, which represent a set of points in space or on a plane.

According to the nature of the forecasting results and the initial data, the models are divided into probabilistic static and deterministic. The first type is based on the collected statistical data, and the forecasts obtained with their help are of a probabilistic nature.

Examples of mathematical models include tasks for the flight of a projectile, transport and other tasks.