Mathematical modeling is an innovative activity, and the subject it faces is a problem that people must solve to further develop their understanding and practice in production and scientific research. "What is the problem? The problem is the contradiction of things. Where there are unresolved contradictions, there are problems. " Therefore, the process of finding a topic is the process of analyzing contradictions. The fundamental contradiction between production and science and technology is the contradiction between understanding and practice. We analyze these contradictions and find the unresolved contradictions, that is, the practical problems that need to be solved. If these practical problems need to be quantitatively analyzed and answered, then we can establish these practical problems as mathematical modeling topics. The preparation of modeling is to understand the actual background of the problem, clarify the purpose of modeling, grasp all kinds of information of the object, understand the characteristics of the actual object, and understand the situation clearly and correctly.
(2) Modeling assumptions
As a discipline, prototype is complex and concrete, and it is the unity of quality and quantity, phenomenon and essence, contingency and inevitability. If such a prototype is not abstracted and simplified, it will be difficult for people to understand it, and it will be impossible to accurately grasp its essential attributes. Modeling hypothesis is to abstract and simplify the prototype according to the characteristics of the actual object and the purpose of modeling, and abstract the form, quantity and their relationship that reflect the essential attributes of the problem. Simplifying those non-essential factors, making them get rid of the specific complex form of the prototype, forming useful information resources and preconditions for modeling, and making assumptions in accurate language is a key step in the modeling process. The abstraction and simplification of the prototype is not unconditional. We must be good at distinguishing the main and secondary aspects of the problem, firmly grasp the main factors, discard the secondary factors, and try to homogenize and linearize the problem as much as possible, in accordance with the principle of hypothetical rationality, as follows.
① Purpose principle: abstract the factors related to the modeling purpose from the prototype, and simplify the factors that have nothing to do with the modeling purpose or have little to do with it.
Principle of Simplicity: The assumptions given should be simple and accurate, which is conducive to the construction of the model.
(3) Principle of authenticity: Assuming the conditions are reasonable, the error caused by simplification should meet the allowable error range of practical problems.
Principle of comprehensiveness: while making assumptions about the prototype itself, we should also give the environmental conditions in which the prototype is located.
(3) Model construction
On the basis of modeling hypothesis, the conditions of modeling hypothesis are further analyzed. First, distinguish what is a constant, what is a variable, what is a known quantity and what is an unknown quantity. Then find out the position, function and relationship of various quantities, establish the equality or inequality relationship between various quantities, list, draw pictures or determine other mathematical structures, choose appropriate mathematical tools and methods of constructing models to express them, and construct a mathematical model depicting practical problems.
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 when building a model, and different mathematical models can be built for the same practical problem. Generally speaking, the simpler the mathematical tools used, the better.
How to build the model depends on the nature of the actual problem and the modeling information given by the modeling hypothesis. As far as mechanism analysis and system identification proposed in system theory are concerned, they are two basic methods to construct mathematical models. System identification method is to construct a model by using the input and output information of the system or the actual test data of the system given by the modeling hypothesis without knowing the internal mechanism of the system. With the development of computer science, computer simulation has strongly promoted the development of mathematical modeling and become the basic method of modeling. These modeling methods have their own advantages and disadvantages, and can be used at the same time to learn from each other's strengths and achieve the goal of modeling.
(4) model solving
After the mathematical model is established, according to the known conditions and the characteristics and structural features of the data analysis model, the mathematical methods and algorithms for solving the model are designed or selected, including solving equations, drawing, proving theorems, logical operation and stability discussion, especially writing computer programs or using software packages suitable for algorithms to solve the model with the help of computers.
(5) Model analysis
According to the purpose and requirements of modeling, numerical results of model solutions, or correlation analysis between variables, or stability analysis, or sensitivity analysis of system parameters, or error analysis, etc. After analysis, if it does not meet the requirements, modify or add modeling assumptions and re-model until it meets the requirements; If the analysis meets the requirements, the model can also be evaluated, predicted and optimized.
(6) Model checking
After the model analysis meets the requirements, we should return to the objective reality to test the model, and use actual phenomena and data to test the rationality and applicability of the model to see if it conforms to the objective reality. If not, we will modify or increase or decrease the hypothetical conditions, re-model, cycle, and constantly improve until we get satisfactory results. At present, computer technology has provided us with advanced means of model analysis and model testing, which can save a lot of time, manpower and material resources.
(7) Model application
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 special role of mathematical model in production and scientific research.
The basic steps of mathematical modeling introduced above should be mastered flexibly according to specific problems, or alternately, or in parallel, and mathematical modeling should be done in an eclectic way, which is conducive to the modeler to play his own intelligence.
About the software, there are matlab lindo and so on.