Mathematical modeling is a mathematical thinking method, and it is a powerful mathematical means to describe and "solve" practical problems by using mathematical language and methods through abstraction and simplification.

Mathematical modeling is a process of describing actual phenomena with mathematical language. The actual phenomena here include both concrete natural phenomena, such as free fall, and abstract phenomena, such as customers' value tendency to a certain commodity. The description here includes not only the description of external form and internal mechanism, but also the prediction, experiment and explanation of actual phenomena.

We can also intuitively understand this concept: mathematical modeling is a process that makes pure mathematicians (mathematicians who only know mathematics but don't know its application in practice) become physicists, biologists, economists and even psychologists.

A mathematical model is generally a mathematical simplification of actual things. It often exists in an abstract form close to the real thing in a sense, but it is essentially different from the real thing. There are many ways to describe an actual phenomenon, such as recording, video recording, metaphor, rumors and so on. In order to make the description more scientific, reasonable, objective and repeatable, people use a generally accepted strict language to describe various phenomena, which is mathematics. What is described in mathematical language is called a mathematical model. Sometimes we need to do some experiments, but these experiments often use abstract mathematical models as substitutes for actual objects and carry out corresponding experiments. The experiment itself is also a theoretical substitute for the actual operation.

Second, several processes of mathematical modeling

Model preparation: understand the actual background of the problem, clarify its practical significance, and master all kinds of information of the object. Describe the problem in mathematical language.

Model hypothesis: according to the characteristics of the actual object and the purpose of modeling, simplify the problem and put forward some appropriate assumptions in accurate language.

Modeling: On the basis of hypothesis, use appropriate mathematical tools to describe the mathematical relationship between variables and establish the corresponding mathematical structure.

Model solution: calculate (estimate) all parameters of the model by using the obtained data.

Model analysis: analyze the results by mathematical methods.

Model test: compare the results of model analysis with the actual situation to verify the accuracy, rationality and applicability of the model. If the model is in good agreement with the actual situation, the practical significance of the calculation results should be given and explained. If the model is not consistent with the actual situation, it is necessary to modify the assumptions and repeat the modeling process.

Model application: the application method varies with the nature of the problem and the purpose of modeling.