In recent decades, with the development of science and technology and social progress, mathematics has been widely and deeply applied in natural sciences, social sciences, engineering technology and modern management, which has gradually made people realize the importance of establishing mathematical models.
A mathematical model is an (approximate) description of a class of practical problems or phenomena that occur in real systems using a mathematical symbol system. However, real-world problems are often more complex, and in the process of abstracting mathematical problems from reality, we must grasp the main factors, ignore the secondary factors, and make necessary simplifications, so that the mathematical problems obtained by abstraction can be solved by appropriate methods.
Generally speaking, a mathematical model is a mathematical structure obtained by making some necessary simplifications and assumptions for a specific object in the real world for a specific purpose and according to its unique internal laws, and using appropriate mathematical tools. The specific object set up refers to a specific problem that we want to study and solve;The specific purpose here refers to the purpose to be achieved when a specific object is studied, such as analysis, control, decision-making, etc.;Mathematical tools here refer to the theories and methods of the various branches of mathematics and certain software systems of mathematics;The mathematical structure here includes mathematical relations, mathematical propositions, graphs and charts, etc.
There are many ways to classify mathematical models, and the following are some commonly used classifications.
According to the different application fields of mathematical models, they can be divided into population models, transportation models, economic models, financial models, environmental models, ecological models, enterprise management models, urban planning models, etc.
According to the different mathematical methods used to establish mathematical models, they can be divided into elementary models, geometric models, operations research models, differential equation models, probability and statistical models, analytic hierarchy process models, cybernetic models, grey system models, etc.
According to the different purposes of establishing mathematical models, they can be divided into analysis models, optimization models, decision-making models, control models, etc.
According to the performance characteristics of the model, it can be divided into deterministic model and random model (the former does not consider the influence of random factors, and the latter considers the influence of random factors).Discrete vs. continuous models (the difference between the two is whether the variables that describe the state of the system are discrete or continuous);Static model vs. dynamic model (the difference between the two is: whether or not to take into account changes caused by the time factor).
Mathematical models are a powerful tool for understanding the external world and control, and the analysis of mathematical models can gain insight into the nature of the object being studied. Mathematical modeling, on the other hand, is the whole process of obtaining the model, solving the model, obtaining conclusions, and verifying whether the conclusions are correct. Mathematical modeling is not only a powerful tool for the study of practical problems with the help of mathematical models, but also a powerful tool for controlling the behavior of the established model system from an applied point of view.
In particular, with the rapid development of modern computing technology and other technologies, the application field of mathematical modeling has been greatly expanded, and people have become more and more aware of the importance of mathematics and mathematical modeling. Learning Xi and initially mastering the ideas and methods of mathematical modeling has become an important content that everyone living in modern society should learn Xi.
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