Generally speaking, the methods of establishing mathematical models can be divided into two categories, one is mechanism analysis and the other is inspection analysis. Mechanism analysis is based on the understanding of the characteristics of real objects, analyzing their causal relationship, and finding out the laws that reflect the internal mechanism. The established model usually has clear physical or practical significance. The following steps are given: first of all, we should understand the actual background of the problem in model preparation. Clarify the purpose of modeling and collect all kinds of information needed for modeling, such as phenomena and data. , and try to clarify the characteristics of the object, so as to initially determine which model to use. In a word, we should make preparations for modeling. This step cannot be ignored. When you encounter problems, please consult comrades who are engaged in practical work humbly and try to master first-hand information. The model assumes that the problem is necessary and reasonably simplified according to the characteristics of the object and the purpose of modeling. Using accurate language to make assumptions can be said to be a key step in modeling. Generally speaking, it is difficult to turn a practical problem into a mathematical problem without simplifying the hypothesis, and even if it is possible, it is difficult to solve it. Different simplified assumptions will lead to different models. Unreasonable or oversimplified assumptions will lead to model failure or partial failure, so the assumptions should be revised and supplemented. If the assumptions are too detailed, trying to take all the factors of complex objects into account may make it difficult or even impossible for you to continue your next work. Usually, the basis of making assumptions is based on the understanding of the inherent law of the problem, the analysis of data or phenomena, or the combination of the two. When making assumptions, we should not only use the knowledge of physics, chemistry, biology and economy related to the problem, but also give full play to our imagination, insight and judgment. We should be good at distinguishing the primary and secondary problems, firmly grasp the main factors, abandon the secondary factors, and linearize and homogenize the problems as much as possible. Experience often plays an important role here. When writing assumptions, the language should be accurate, just like writing known conditions when doing exercises. Model construction is based on the assumptions made, using the internal laws of the object and appropriate mathematical tools to analyze the causal relationship of the object. Construct equation (or inequality) relations or other mathematical structures between various quantities (constants, variables). In addition to the professional knowledge of some related disciplines, we often need more extensive application of mathematical knowledge to broaden our thinking. Of course, we can't ask to master mathematics subjects, but we should know what problems these subjects can solve and how to solve them generally. Similarity analogy method, that is, according to some similarities of different objects, borrows mathematical models in known fields, and is also a method of building models. One principle to follow in modeling is to use simple mathematical tools as much as possible, because the model you build always wants to be understood and used by more people, not just a few experts. The model can be solved by various traditional and modern mathematical methods, such as solving equations, drawing, proving theorems, logical operation, numerical calculation and so on. Especially in computer technology, model analysis is a mathematical analysis of model solution, sometimes it needs to analyze the dependence or stability of variables according to the nature of the problem, sometimes it needs to give a mathematical prediction according to the obtained results, and sometimes it may need to give a mathematical optimal decision or control. In either case, error analysis, model stability or data sensitivity analysis are often needed. Model checking transforms the results of mathematical analysis into practical problems. Compared with the actual phenomenon and data, the rationality and applicability of the model are verified. This step is very important for the success or failure of modeling and should be taken seriously. Of course, some models, such as nuclear war models, cannot be required to be tested in practice. If the results of model verification are not in line with reality or partially, the problem usually lies in model assumptions, and some models need to be modified, supplemented and re-modeled. Keep improving until the test results are satisfactory to some extent. The application mode of the model naturally depends on the nature of the problem and the purpose of modeling, which is beyond the scope of this book. It should be pointed out that not all modeling processes have to go through these steps, and sometimes the boundaries between steps are not so clear. Don't stick to formalization step by step when modeling. The modeling examples in this book adopt flexible expressions.