From the middle of the last century, computers and mathematical methods began to enter various fields of human activity. New disciplines began to appear, such as mathematical economics, mathematical linguistics, mathematical chemistry and others, the subject of study of which were mathematical models of phenomena and objects, as well as methods for their study.
A mathematical model is an approximate description in a mathematical language of objects or phenomena of the real world. The main goal of modeling is to study these objects and predict the results of future observations. In addition, modeling is a method of understanding the environment, the world, which makes it possible to control.
The use of mathematical modeling is indispensable in cases where for various reasons it is difficult or impossible to conduct a natural experiment. For example, it is difficult to verify whether one or another cosmological theory is true, or to study the consequences of a nuclear explosion. But all this can be seen on the computer, having previously built a mathematical model.
Mathematical Model: Design Steps
Firstly, the model is being built. To do this, consider some natural phenomenon, economic plan, design, production process or other non-mathematical object. First, they determine the features of phenomena and the relationship between them at a qualitative level. Further, the obtained dependencies are translated into a formulaic form or a mathematical model is constructed. This stage is the most difficult.
At the second stage, they solve the mathematical problem formulated on the basis of the model. Here, increased attention is paid to the development of numerical methods and algorithms for solving the problem with the help of computers, which make it possible to obtain a result with the necessary accuracy in a reasonable time.
At the next stage, it is necessary to interpret the consequences arising from the model, translate the results from the mathematical language into the form adopted in the study area.
Then, the adequacy of the obtained model is checked, and it is determined whether the results correspond to the consequences within the given accuracy.
At the final stage, the model is modified. It is either complicated for greater adequacy of reality or simplified in order to reach an acceptable practical solution.
Classification of mathematical models
There are various criteria for dividing mathematical models into groups. So, by the nature of the problems being solved, they are divided into structural and functional models. Moreover, the values characterizing the 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 studied quantities. 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. Typically, these relationships are difficult or impossible to quantify. To study them, they use the theory of graphs, mathematical objects that represent many points in space or on a plane.
By 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 probabilistic in nature.
Examples of mathematical models include missile, transport, and other tasks.