(1) model preparation
To establish a mathematical model of practical problems, we must first deeply understand the actual background and internal mechanism of the problems to be solved, and make clear what the problems are through appropriate investigation and research. What is the main purpose to be achieved?
In this process, it is necessary to conduct in-depth investigation and study, collect and master information and materials related to the research problem, consult relevant literature, discuss with relevant personnel familiar with the situation, find out the characteristics of actual problems, collect data more reasonably according to the purpose of solving problems, and initially determine the type of modeling.
(2) Model assumption
Generally speaking, practical problems in the real world are often complicated and involve a wide range. If such a problem is not abstracted and simplified, people cannot accurately grasp its essential attributes, and it is difficult to turn it into a mathematical problem. Even if it can be transformed into a mathematical problem, it is difficult to solve it.
Therefore, in order to establish a mathematical model, it is necessary to analyze the problems studied and the relevant information collected, abstract the morphological quantities reflecting the essential attributes of the problems and their relations, and simplify those non-essential factors, so as to get rid of the collective complex forms of practical problems and form useful information resources and preconditions for establishing the model.
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.
However, the abstraction and simplification of practical problems are not unconditional (unreasonable assumptions or too simple assumptions will lead to the failure of the model), and must be carried out in accordance with certain rationality principles. The rationality principle of hypothesis has the following points.
① Purpose principle: according to the characteristics of the research problem, abstract the factors related to the modeling purpose, and simplify those factors that have nothing to do with modeling or have little to do with it.
(2) Simplicity principle: The assumptions given should be simple and accurate, which is conducive to the construction of the model.
(3) The principle of authenticity: assume that the conditions should conform to the situation.
The error caused by simplification should meet the allowable error range of practical problems.
④ Principle of comprehensiveness: While assuming the problem, we should also give the environmental conditions of the actual problem.
In a word, model hypothesis is to abstract and simplify the problem reasonably according to the characteristics of the actual object and the purpose of modeling, and put forward some appropriate assumptions in accurate language.
It should be said that this is a difficult process and a key step in the modeling process. Often it is not done at once, but it needs to be done repeatedly.
(3) Model construction
On the basis of model assumptions, firstly, distinguish which are constants, which are variables, which are known quantities and which are unknown quantities; Then find out the position, function and relationship of various quantities.
Describe the relationship between variables (equal or unequal) with appropriate mathematical tools, and establish corresponding mathematical structures (propositions, tables, graphs, etc.). ), so as to construct the mathematical model of the studied problem.
What mathematical tools to use when building a model depends on the characteristics of the problem, the purpose of modeling and the mathematical expertise of the modeler. It can be said that any branch of mathematics can be used to construct models, and different mathematical models can be constructed by different mathematical methods for the same practical problem.
However, under the premise of achieving the expected goal, simple mathematical tools should be used as much as possible so that the obtained model can be widely used. In addition, what method to use when building the model depends on the nature of the problem and the information provided by the model assumptions.
With the continuous development of modern technology, modeling methods emerge one after another, each with its own advantages and disadvantages. When establishing the model, we can use them at the same time, learn from each other's strong points and finally achieve the goal of modeling.
After the mathematical model is initially established, it is generally necessary to analyze and simplify it to make it easy to solve, and check it according to the purpose and requirements of the research problem, mainly to see whether it can represent the actual problem studied.
(4) model solving
After the mathematical model is established, according to the known conditions and data, the characteristics and structural features of the model are analyzed, and the mathematical methods and algorithms for solving the model are designed or adopted, including various traditional and modern mathematical methods such as solving equations, drawing, logical operation and numerical calculation, especially the application of modern computer technology and mathematical software, which can solve the model quickly and accurately.
(5) Analysis and test of the model
According to the purpose and requirements of modeling, the numerical results of the model solution are analyzed mathematically. The main methods are: correlation analysis between variables, stability analysis, sensitivity analysis of system parameters, error analysis and so on. Through analysis.
If it does not meet the requirements, modify or increase or decrease the model assumptions and re-establish the model until it meets the requirements; If it meets the requirements, the model can also be evaluated, predicted and optimized.
After the model analysis meets the requirements, we should return to the actual problems to test the model, and use actual phenomena and data to test the rationality and applicability of the model, that is, to test the correctness of the model.
If the theoretical value calculated by the model is in good agreement with the actual value, the model is successful; If the difference between the theoretical value and the actual value is too large or partially inconsistent, the model will be invalid.
If we can be sure that the modeling and solving process is accurate, generally speaking, the problem often lies in the model assumptions. At this point, the primary and secondary factors in practical problems should be re-analyzed, and if some factors fail because of being ignored, they should be re-considered when establishing the model.
When modifying, some variables may be deleted or added, and the properties of some variables may also be changed; Either adjust the parameters or change the mathematical method, usually a model needs to be revised repeatedly to succeed. Therefore, the test of the model is very important for the success or failure of the model.
(6) Model application
At present, the application of mathematical models has been very extensive, and it has increasingly penetrated into social science, life science, environmental science and other fields. The application of the model is the purpose of mathematical modeling and the most objective and fair test of the model.
Therefore, a successful mathematical model must analyze, study and solve practical problems according to the purpose of modeling, and give full play to the important role and significance of mathematical model in production and scientific research.