Mathematical modeling (mathematical modeling) is a method of solving various practical problems by establishing mathematical models. There is no fixed format and standard for mathematical modeling, and there is no clear method. There are usually 6 steps:

1. Clarify the problem
2. Reasonable assumption
3. Build a model
4. Solve the model
5. Analysis test
6. Model interpretation

1. Clear the problem

The problems dealt with by mathematical modeling are usually practical problems in various fields. These problems themselves are often ambiguous, it is difficult to directly find the key points, and it is not clear what method should be used. Therefore, the first task of establishing a model is to identify the problem, analyze the relevant conditions and problems, make the problem as simple as possible at the beginning, and then gradually improve it according to the purpose and requirements.

2. Reasonable assumptions

Making reasonable assumptions is a key step in modeling. It is difficult to directly translate a practical problem into a mathematical problem without simplification and hypothesis, even if it may be too complicated to solve. Therefore, according to the characteristics of the object and the purpose of modeling, the problem needs to be simplified reasonably.

In addition to simplifying the problem, reasonable assumptions also limit the scope of the model.

The basis for making assumptions is usually based on the understanding of the inherent laws of the problem, or from the analysis of data or phenomena, or a combination of the two. When making assumptions, you should not only use professional knowledge related to the problem, such as physics, chemistry, biology, economics, and machinery, but also give full play to your imagination, insight and judgment, identify the priority of the problem, and try to simplify the problem.

In order to ensure the reasonableness of the hypothesis, the hypothesis and the inference of the hypothesis should be tested when there is data, and attention should be paid to the existence of implicit hypothesis.

3. Build a model

To build a model is to establish the relationship between variables based on the basic principles or laws of actual problems.

To describe the change of a variable with the change of another variable, the easiest way is to draw a graph, or draw a table, you can also use mathematical expressions. In modeling, one form is usually transformed into another form. It is easier to convert mathematical expressions into graphs and tables, but the reverse is more difficult.

A combination of some simple typical functions can be used to form various functional forms. Using functions to solve specific practical problems is more than giving the value of each parameter, and seeking realistic explanations for these parameters can often capture some of the essential characteristics of the problem.

4. Solve the model

Solving the model often involves professional knowledge of different disciplines. The development of modern computer science has provided powerful auxiliary tools. There have been many software packages and simulation tools that can carry out engineering numerical calculations and mathematical derivation. Proficiency in mathematical modeling simulation tools can greatly enhance modeling capabilities.

Different mathematical models are difficult to solve. In general, many practical problems cannot be solved analytically. Therefore, it is necessary to use a computer to solve the problem numerically. Before writing the code, the algorithm and calculation steps must be clarified, and the initial value and step length must be clarified. The influence of other factors on the results.

5. Analysis and inspection

After finding the solution of the model, the model and the “solution” must be analyzed. What is the scope of application of the model and the solution, how stable and reliable the model is, whether the purpose of modeling is reached, and whether the problem is solved?

Compared with the objective reality, the mathematical model will inevitably bring certain errors. On the one hand, the allowable range of errors must be determined according to the purpose of modeling. On the other hand, the sources of errors must be analyzed and ways to reduce the errors must be analyzed.

General errors have the following sources, which need to be carefully analyzed and tested:

• Errors in model assumptions: Generally speaking, the model cannot fully reflect the objective reality, so different assumptions need to be made. When analyzing the model, these assumptions need to be carefully tested to analyze and compare the impact of different assumptions on the results.
• The error of the approximate solution method: Generally speaking, it is difficult to obtain the analytical solution of the model. When the numerical method is used to solve the problem, the numerical calculation method itself will have errors. Many of these errors can be controlled.
• Rounding errors of calculation tools: When using calculators or computers for numerical calculations, rounding errors are inevitable due to the limited word length of the machine. If a large number of calculations are performed, the accumulation of these errors cannot be ignored.
• Data measurement error: When using sensors, questionnaires and other methods to obtain data, pay attention to the error of the data itself.

6. Model interpretation

The final stage of mathematical modeling is to translate the model in real-world language, which is very important for people who use the model to understand the results of the model. Whether the model settlement has practical significance and whether it is consistent with actual evidence. This step is a crucial step to make the mathematical model have practical value.