Content summary:

Mathematics, as a tool and means of modern science, should understand what mathematical models and mathematical modeling are, and understand the general methods and steps of mathematical modeling.

Key words:

Mathematical model, mathematical modeling, practical problems

With the rapid development of social science and technology, mathematics has penetrated into various fields, and mathematical modeling is particularly important. Mathematical modeling plays an important role in people's lives, and with the development of computer technology, mathematical modeling plays an important role in human activities and serves human beings better.

I. Mathematical model

Mathematical model is a mathematical structure formed by making some necessary assumptions and using appropriate mathematical tools according to the unique internal laws for specific objects and specific purposes in the real world.

Simply put, it is a mathematical expression of the essence of a certain feature of a system (or a description of a part of the real world in mathematical terms), that is, mathematical formulas (such as functions, graphs, algebraic equations, differential equations, integral equations, difference equations, etc.). ) are used to describe (express and simulate) the existence law of an objective object or system studied in a certain aspect.

With the development of society, biology, medicine, society and economy, a large number of practical problems have appeared in various disciplines and industries, which urgently need people to study and solve. However, the social demand for mathematics is not only mathematicians and talents who specialize in mathematical research, but also requires people in various departments to be good at using mathematical knowledge and mathematical thinking methods to solve a large number of practical problems they face every day. Realize economic and social benefits. They don't apply mathematics knowledge to find practical problems (just like doing math application problems at school), but need to use mathematics to solve practical problems. Moreover, they need not only mathematics, but also knowledge, work experience and common sense in other disciplines and fields. Especially in modern society, it is almost inseparable from computers to really solve a practical problem. There are few problems that can be solved with ready-made mathematical knowledge. What you can meet is a mixture of mathematics and other things, not "clean" mathematics, but "dirty" mathematics. The mystery of mathematics is not waiting for you to solve, but hidden in the depths. In other words, you should analyze and find out the complicated practical problems.

The mathematical model has the following characteristics: an important feature of the mathematical model is its high abstraction. Through the mathematical model, the thinking in images can be transformed into abstract thinking, thus breaking through the constraints of the actual system and making use of the existing mathematical research results to conduct in-depth research on the research object. Another characteristic of mathematical model is economy. Using mathematical model to study can save a lot of equipment operation and maintenance costs. The use of mathematical models can greatly speed up the progress of research work and shorten the research cycle, especially today when electronic computers are widely used. However, the mathematical model is limited, which will inevitably cause some distortions in the process of simplification and abstraction. The so-called "model is model" (not prototype) refers to this attribute.

Second, mathematical modeling.

Mathematical modeling is a practice of solving practical problems by mathematical methods. That is, after abstracting, simplifying, assuming and introducing variables, the actual problem is expressed by mathematical methods, and then a mathematical model is established, and then it is solved by advanced mathematical methods and computer technology. In short, the process of establishing mathematical model is called mathematical modeling.

The model is a simulation of the related properties of objective entities. The model plane displayed in the window should look like a real plane. It doesn't matter whether it can really fly. But the airplane models participating in the model airplane competition are completely different. If the flight performance is poor and looks like an airplane, it is not a good model. A model is not necessarily an imitation of an entity, but also an abstraction of some basic attributes of the entity. For example, a geological map can reflect the geological structure of the area with abstract symbols, characters and numbers without physical simulation. Mathematical model is also a kind of simulation, using mathematical symbols and mathematics. The abstract and concise description of the essential attributes of practical topics by programs and graphs may explain some objective phenomena, or predict the future development law, or provide the best or better strategy in a sense for controlling the development of a phenomenon. Generally, mathematical model is not a direct copy of real problems, and its establishment often requires people to observe and analyze real problems deeply and carefully. It also requires people to use all kinds of mathematical knowledge flexibly and skillfully. This kind of applied knowledge is abstracted from practical problems, and the process of extracting mathematical models is called mathematical modeling. There are many factors in practical problems, so you can't and don't need to consider them all. You can only consider the most important factors and discard the secondary factors. When the mathematical model is established, the actual problem becomes a mathematical problem. You can use mathematical tools and methods to solve this practical problem. If only there were ready-made mathematical tools. If there are no ready-made mathematical tools, it will prompt mathematicians to find and develop new mathematical tools to solve them, which in turn will promote the development of mathematics itself. For example, Kepler summed up Kepler's three laws from the observation data of planetary motion. Newton tried to explain them with his own mechanical laws, but the existing mathematical tools were not enough at that time. This promoted the invention of calculus. To solve mathematical models, besides mathematical reasoning, we usually have to deal with a lot of data and do a lot of calculations, which was difficult to achieve before the invention of electronic computers. Therefore, although many mathematical models have been solved theoretically, they can't get useful results because of too much calculation, so they can only be shelved. The appearance and rapid development of electronic computers have opened up a broad road for solving practical problems with mathematical models. Now, it is almost impossible to really solve a practical problem without a computer. If a mathematical model is established and solved by mathematical or numerical methods, will everything be fine? No, because the mathematical model can only approximately reflect the relations and laws in practical problems, it needs to be tested whether it reflects well or not. The mathematical model is not well established, the given practical problem is not described correctly, and the correct mathematical solution is useless. Therefore, after the mathematical solution is obtained, the conclusion should be tested to see if it is reasonable and feasible. If it is not practical, we should try to find out the reason, modify the original model, re-solve and test until it is more reasonable and feasible, and then we can get the answer and put it into practice first. However, there is no perfect answer. Or come to an end for the time being, and then make improvements after there are new situations and requirements.

Applying mathematical knowledge to study and solve practical problems, the first task we encounter is to establish a suitable mathematical model. In this sense, it can be said that mathematical modeling is the basis of all scientific research. Without a better mathematical model, it is impossible to get better research results. Therefore, establishing a better mathematical model is one of the keys to solving practical problems. Mathematical modeling comprehensively uses all kinds of knowledge to solve practical problems, which is to cultivate and improve students' ability to apply what they have learned to analyze and solve problems.

Third, the general method of mathematical modeling

There is no certain model for establishing mathematical model, but an ideal model should reflect all the important characteristics of the system: the reliability and availability of the model.

General methods of modeling:

Mechanism analysis of 1.

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.

(1) Proportional analysis-the most basic and commonly used method to establish the functional relationship between variables.

(2) Algebraic method-the main method to solve discrete problems (discrete data, symbols, graphics).

(3) Logical method is an important method to study mathematical theory, which is practical for sociology and economics.

Problems are widely used in decision-making and countermeasures.

(4) Ordinary differential equation-The key to solving the change law between two variables is to establish the "instantaneous change rate".

The expression.

(5) Partial differential equation-solving the variation law between the dependent variable and more than two independent variables.

2. Test analysis method

The test and analysis method regards the research object as a "black box" system, and cannot directly seek the internal mechanism. By measuring the input and output data of the system, and on this basis, using the statistical analysis method, the model with the best data fitting is selected from a certain type of model according to the predetermined standard.

(1) regression analysis-used to determine the expression (xi, fi) i = 1, 2, ..., n of the function f(x) from a set of observations. Because it deals with static independent data, it is called mathematical statistics method.

(2) Time series analysis-dealing with dynamic related data, also known as process statistics.

(3) Regression analysis is used to determine the expression of function f(x) from a set of observed values (xi, fi)i= 1, 2, …, n, which is called mathematical statistics method because it deals with static independent data.

(4) Time series analysis-dealing with dynamic related data, also known as process statistics.

It is also a common modeling method to combine these two methods, that is, to establish the structure of the model through mechanism analysis and to determine the parameters of the model through system testing. In the actual process, which method to use for modeling mainly depends on our understanding of the research object and modeling purpose. The specific steps of modeling by mechanism analysis are shown in the left figure.

3. Simulation and other methods

(1) Simulation-essentially a statistical estimation method, equivalent to sampling inspection.

(1) discrete system simulation-there is a set of state variables.

(2) continuous system simulation-there are analytical expressions or system structure diagrams.

(2) Factor testing method-local testing of the system, and then continuous analysis and modification according to the test results to obtain the required model structure.

(3) artificial reality method-based on the understanding of the past behavior and future goals of the system, and taking into account the possible changes in related factors of the system, artificially form a system. (See: Qi Huan's Mathematical Model Method, Huazhong University of Science and Technology Press, 1996)

Fourthly, the classification of mathematical models.

Mathematical models can be classified in different ways. Here are some commonly used models.

1. According to the application fields (or disciplines) of the model, it can be divided into: population model, traffic model, environment model, ecological model, urban planning model, water resource model, renewable resource utilization model, pollution model, etc. A larger category has formed many marginal disciplines such as biological mathematics, medical mathematics, geological mathematics, quantitative economics, mathematical sociology and so on.

2. According to the mathematical method (or branch of mathematics) of establishing the model, it can be divided into initial mathematical model, geometric model, differential equation model, graph theory model, Markov chain model, planning theory model and so on.

In the textbook of mathematical models classified by the first method, we focus on establishing models in a specific field with different methods, while in the book classified by the second method, we use ready-made mathematical models belonging to different fields to explain the application of some mathematical skills. In this book, we focus on how to apply the basic mathematical knowledge that readers have mastered to build models in different fields.

3. According to the performance characteristics of the model, there are several points:

Deterministic model and stochastic model depend on whether the influence of random factors is considered. In recent years, with the development of mathematics, the so-called catastrophe model and fuzzy model have appeared.

Static model and dynamic model depend on whether the changes caused by time factors are considered.

Linear model and nonlinear model depend on the basic relationship of the model, such as whether the differential equation is linear or not.

Discrete model and continuous model refer to whether the variables (mainly time variables) in the model are discrete or continuous.

Although most practical problems are random, dynamic and nonlinear in nature, deterministic, static and linear models are easy to handle and can often be used as a preliminary approximation to solve problems, so deterministic, static and linear models are often considered first in modeling. Continuous model is easy to be solved by calculus for theoretical analysis, while discrete model is easy to be calculated by computer, so which model to use depends on specific problems. In the concrete modeling process, the continuous model is decentralized.

4. According to the purpose of modeling, there are description model, analysis model, prediction model, optimization model, decision model and control model.

5. According to the understanding of the model structure, there are so-called white box model, gray box model and black box model. This is to compare the research object to an organ in a box and reveal its mystery through modeling. White box mainly includes phenomena described by some disciplines with quite clear mechanisms, such as mechanics, heat and electricity, and corresponding engineering and technical problems. Most of the models in this area have been basically determined. The main problem that needs further study is the optimal design and control. Grey box mainly refers to the phenomenon whose mechanism is not very clear in the fields of ecology, meteorology, economy and transportation. And there is still a lot of work to be done to establish and improve the model in different degrees. As for the black box, it mainly refers to the phenomenon that the mechanism (quantitative relationship) in the fields of life science and social science is not very clear. Although some engineering and technical problems are mainly based on physical and chemical principles, they are often regarded as gray box or black box models due to many factors, complicated relationships and difficult observation. Of course, there is no obvious boundary between white, gray and black, and with the development of science and technology, the "color" of the box is bound to gradually change from dark to bright.

Five, the general steps of mathematical modeling

The steps of modeling are generally divided into the following steps:

1. Model preparation. First of all, we should understand the actual background of the problem, clarify the requirements of the topic and collect all kinds of necessary information.

2. Model assumptions. On the basis of clarifying the modeling purpose and mastering the necessary data, the main factors are found out through the analysis and calculation of the data, and some assumptions that are in line with the objective reality are put forward after necessary refining and simplification, so as to highlight the main characteristics of the problem and ignore the secondary aspects of the problem. 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 get different results. If your 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.

3. Model composition. According to the hypothesis and the relationship between things, the relationship between variables is described with appropriate mathematical tools, and the corresponding mathematical structure is established-that is, the mathematical model is established to turn the problem into a mathematical problem. We should pay attention to using simple mathematical tools as much as possible, because simple mathematical models can often better reflect the essence of things and are easy to be mastered and used by more people.

4. Model solving. Using known mathematical methods to solve the mathematical problems obtained in the previous step often requires further simplification or hypothesis. When it is difficult to get an analytical solution, we should also get a numerical solution with the help of a computer.

5. Model analysis. The solution of the model is analyzed mathematically. Sometimes it is necessary to analyze the dependence or stability of variables according to the nature of the problem. Sometimes it is necessary to give a mathematical prediction according to the obtained results, and sometimes it is possible to give a mathematical optimal decision or control. In either case, it is usually necessary to analyze the error, the stability or sensitivity of the model to the data, etc.

6. Model test. Analyze the practical significance of the results and compare them with the actual situation to see if they are in line with reality. If the result is not satisfactory, the hypothesis should be modified, supplemented or re-modeled. Some models need to be repeated several times and constantly improved.

7. Model application. The established model must be applied in practice to produce benefits, and it is constantly improved and perfected in application. The application mode naturally depends on the nature of the problem and the purpose of modeling.

References:

(1) Qi Huan's Mathematical Model Method, Huazhong University of Science and Technology Press, 1996.

(2) Practice and Understanding of Mathematics (quarterly), edited and published by chinese mathematical society.