The essence of mathematical model is to describe the simplified practical problems approximately with mathematical language.

I. Mechanism model

For a practical problem, if the modeler's attention is focused on describing the interrelationships and mutual constraints among the main factors in the problem with mathematical language, the model thus constructed is generally called a mechanism model.

This model describes the interaction mechanism between the main factors in practical problems, so it can widely accommodate all kinds of mathematical knowledge and methods. The mathematical analysis through this model can often make people have a deep understanding of the practical problems studied from the action mechanism.

Mechanism model is a widely used model in mathematical research of related disciplines.

Second, the appropriate "hypothesis"

Because the practical problems we face are often complicated, we need to give appropriate "assumptions" to simplify the studied problems when establishing the mechanism model of the practical problems.

"Hypothesis" is a means to remove the rough and extract the essence, and simplify the complex. The process of putting forward "hypothesis" is the process of abstracting and simplifying practical problems.

The content of "hypothesis" determines the simplification of practical problems, and has a great influence on the operability of the model and the consistency with the actual situation. If the "hypothesis" is too strong, the model will generally be simple and easy to analyze, but the conclusion drawn by the model may be quite different from the actual situation, thus reducing its practicality. If the "hypothesis" is weak, the consistency between the model and the actual situation will be greatly improved, but the complex model will set a great obstacle to theoretical analysis. Therefore, it is very important and an art for mathematical modeling to give appropriate assumptions according to practical problems and research purposes.

Third, the principle of balance.

In the process of analyzing the main factors and interactive incentives, it is a key issue to explore the balance relationship in practical problems.

"Balance" is a phenomenon that can be seen everywhere in real life, such as the law of conservation of energy in physics and the balance of force in statics, all of which describe some balance relations in physical phenomena. Another example is the change of matter over a period of time or within a certain range. We will find that the change of matter in this period is also in a state of balance with its increase or decrease. We call this law of balance material balance.

The so-called balance principle means that any phenomenon in nature is bound to be dominated by a certain balance relationship in its changing process. The relationship between the factors described in this model is essentially a balanced relationship in practical problems. Therefore, the application of the equilibrium principle is one of the key steps to establish the mechanism model.

Fourth, explore a balanced relationship.

The balance relationship in some problems is obvious and clearly expressed in the problems, but in most problems, this relationship is often hidden behind the problems and needs to be simplified before it can be gradually clarified.

The excavation of equilibrium relationship is often staggered from the proposition of hypothesis. When exploring the balance relationship, it is very important to break the whole into parts and replace the curve with the straight curve.

For example, when studying the dynamic behavior of a system, if the whole system is considered, it is often impossible to start. We can first consider the dynamics in a small time interval, in which the change rate of correlation quantity can be approximately constant, thus simplifying the problem.

List the models obtained by analyzing the equilibrium relationship, which will depend on the modeler's in-depth understanding of the background knowledge of the problem and his ability to synthesize, summarize and analyze the actual problems. The mathematical description of the balance relationship in the problem sometimes directly constitutes the mathematical model of the problem, such as the equation obtained when solving the application problem with the column equation. But sometimes the proposed equilibrium relationship needs mathematical treatment to get an ideal mathematical model.

Mechanism model is essentially a mathematical description of the equilibrium relationship between the main factors in practical problems under certain assumptions. For example, when middle school algebra learns to solve application problems, the equations listed for application problems are a simple incentive model. When listing equations, it is often necessary to find the equivalent relationship in the problem. This relationship is actually the balance relationship of the problem.