It is a new discipline developed in recent years, and it is a science that combines mathematical theory with practical problems. It reduces practical problems to corresponding mathematical problems, and on this basis, it uses mathematical concepts, methods and theories to conduct in-depth analysis and research, thus depicting practical problems from a qualitative or quantitative perspective and providing accurate data or reliable guidance for solving practical problems.
[Edit this paragraph] 1. Requirements for establishing mathematical model:
1, true and complete.
1) is true, systematic and complete, and the image reflects the objective phenomenon;
2) It must be representative;
3) Extrapolation, that is, the information of the prototype object can be obtained, and the reasons about the prototype object can be obtained in the process of model research and experiment;
4) It must reflect the various achievements made in completing the basic tasks and should be consistent with the actual situation.
2. Concise and practical. In the process of modeling, we should reflect the essential things and their relationships, and eliminate the non-essential things that have little influence on reflecting the objective truth, so that the model can be as simple and operable as possible and the data can be easily collected while ensuring a certain accuracy.
3. adapt to change. With the change of related conditions and the development of people's understanding, we can adapt to the new situation well by adjusting related variables and parameters.
According to the research purpose, a structure (called real prototype or prototype) that generalizes and approximately expresses the main characteristics and relations of the processes and phenomena studied by formal mathematical language. The so-called "mathematization" refers to the construction of mathematical models. The method of understanding things by studying their mathematical models is called mathematical model method, or MM method for short.
Mathematical model is the product of mathematical abstraction, and its prototype can be concrete objects and their properties and relationships, or mathematical objects and their properties and relationships. Mathematical models can be divided into broad sense and narrow sense. In a broad sense, mathematical concepts, numbers, sets, vectors and equations can all be called mathematical models. In a narrow sense, only the mathematical relationship structure models that reflect specific problems and specific things systems can be roughly divided into two categories: (1) deterministic models that describe the inevitable phenomena of objects, and their mathematical tools are generally substitution equations, differential equations, integral equations and difference equations. The mathematical model of excellent athletes is often mentioned in sports practice. According to investigation and statistics, the model of modern world-class sprinters is about 1.80m in height and 70kg in weight, 100 Mi Yue 100 seconds or better.
An equation or inequality composed of letters, numbers and other mathematical symbols, or a model that describes the characteristics of the system and its internal connection or connection with the outside world with charts, images, block diagrams and mathematical logic. It is an abstraction of a real system. Mathematical model is a powerful tool to study and master the law of system motion, and it is the basis for analyzing, designing, predicting or controlling the actual system. There are many mathematical models and different classification methods.
Static model and dynamic model Static model means that the relationship between variables of the system to be described does not change with time, and is generally expressed by algebraic equations. Dynamic model refers to a mathematical expression that describes the laws of system variables changing with time, and is generally expressed by differential equations or difference equations. The transfer function of the system commonly used in classical control theory is also a dynamic model, because it is transformed from the differential equation describing the system (see Laplace transform).
Distributed parameter model and lumped parameter model describe the dynamic characteristics of the system with various partial differential equations, while lumped parameter model describes the dynamic characteristics of the system with linear or nonlinear ordinary differential equations. In many cases, the distributed parameter model can be simplified to a lumped parameter model with low complexity through spatial discretization.
Continuous-time models and discrete-time models with time variables varying in a certain interval are called continuous-time models, and the above models described by differential equations are all continuous-time models. When dealing with lumped parameter model, time variables can also be discretized, and the obtained model is called discrete time model. The discrete-time model is described by the difference equation.
The relationship between variables in stochastic model and deterministic model is given in the form of statistical value or probability distribution, while the relationship between variables in deterministic model is deterministic.
Parametric and nonparametric models The models described by algebraic equations, differential equations, differential equations and transfer functions are all parametric models. The establishment of parametric model is to determine the parameters in the known model structure. Parametric models are always obtained through theoretical analysis. The nonparametric model is the response obtained directly or indirectly from the experimental analysis of the actual system. For example, the impulse response or step response of the system recorded by experiments is a nonparametric model. Using various system identification methods, parametric models can be obtained from nonparametric models. If the structure of the system can be determined before the experiment, the parameter model can be obtained directly through experimental identification.
The relationship between variables in linear and nonlinear models is linear, and the superposition principle can be applied, that is, several different inputs act on the response of the system at the same time, which is equal to the sum of the responses of several inputs acting alone. The linear model is simple and widely used. The relationship between quantities in the nonlinear model is not linear and does not satisfy the superposition principle. When allowed, nonlinear models can usually be linearized into linear models. The method is to expand the nonlinear model into Taylor series near the working point, keep the first-order term and omit the higher-order term, and then the approximate linear model can be obtained.
[Edit this paragraph] 2. Definition of mathematical model
At present, there is no uniform and accurate definition of mathematical model, because different angles can have different definitions. But we can give the following definition. "A mathematical model is an abstract and simplified structure about a part of the real world, used for special purposes." Specifically, a mathematical model is an equation or inequality established by letters, mathematics and other mathematical symbols, and it is a mathematical structural expression that describes the characteristics of objective things and their internal relations, such as charts, images, block diagrams, etc.
2. Methods and steps of establishing mathematical model
First, model preparation.
First of all, we should understand the actual background of the problem, clarify the modeling purpose, collect all kinds of necessary information, and try our best to understand the characteristics of the object. Second, the model hypothesis.
According to the characteristics of the object and the purpose of modeling, it is a crucial step to simplify the problem reasonably and make assumptions with accurate language. If all the factors of the problem are taken into account, it is undoubtedly a courageous act and the method is very poor. Therefore, a superb modeler can give full play to his imagination, insight and judgment, be good at distinguishing priorities, and linearize and homogenize problems as much as possible in order to simplify the handling methods.
Third, the model composition.
According to the assumptions made, the causal relationship of the object is analyzed, and the equation relationship between various quantities or other mathematical structures are constructed by using the internal laws of the object and appropriate mathematical tools. At this time, we will enter a vast world of applied mathematics, where there are many lovely children at the knees of the elderly with high numbers and probabilities. They are graph theory, queuing theory, linear programming, game theory and many other theories. They are really a great country with unique views. But we should remember that the mathematical model is established for more people to understand and apply, so the simpler the tool, the more valuable it is.
Fourth, the model is solved.
We can use all kinds of traditional and modern mathematical methods, especially computer technology, such as solving equations, drawing pictures, proving theorems, logical operations, numerical operations and so on. Solving a practical problem often requires complicated calculation, and in many cases, the system operation has to be simulated by computer, so programming ability and familiarity with mathematical software packages are very important.
Fifth, model analysis.
Mathematically analyze the model solution. Looking horizontally, the side of the ridge has become a peak with different heights. Whether you can analyze the model results carefully and accurately determines whether your model can reach a higher level. Also remember that in either case, error analysis and data stability analysis are needed.
The sixth mathematical model classification:
According to the application field of the model:
Biological mathematical model
Medical mathematical model
Geological mathematical model
Quantitative economic model
Mathematical sociological model
Mathematical and physical model
Classification according to whether random factors are considered or not:
deterministic model
Stochastic model
Depending on whether to consider the change of the model:
static model
Dynamic model
Classification according to the discrete method or continuous method applied:
discrete model
Continuous model
Classification according to mathematical methods of modeling:
Geometric model
Differential equation model
Graph theory model
Planning theoretical model
Markov chain model
According to people's understanding of the development process of things:
White box model:
Refers to those models with clear internal laws. Such as mechanics, heat, electricity and related engineering and technical problems.
Grey box model:
It refers to those problems whose internal laws are not very clear, and there is still a lot of work to be done to establish and improve the model in different degrees. Such as meteorology and ecological economics.
Black box model:
Refers to some phenomena whose internal laws are still unknown. Such as life science and social science. However, due to many influencing factors and complicated relationships, it can also be simplified as a grey box model to study.