The history of mathematical models can be traced back to the time when humans began to use numbers. With the application of numbers, various mathematical models are constantly established to solve various practical problems. A mathematical model can be established to establish an optimal scheme for daily activities such as college students' comprehensive quality evaluation, teachers' work performance evaluation, and visiting friends and purchasing. The establishment of mathematical model is a necessary bridge to communicate the relationship between practical problems and mathematical tools. 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, mathematical models can be roughly divided into two categories: (1) deterministic models that describe the inevitable phenomena of objects, and their mathematical tools are generally algebraic equations, differential equations, integral equations and difference equations; (2) mathematical models of excellent athletes are 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 the definition of the mathematical model in this paragraph. At present, there is no unified and accurate definition of mathematical model, because there can be different definitions from different angles. 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 mathematical symbols such as letters and numbers for a certain purpose, 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.
A good model should be self-consistent, which is verified by our research group. Mutual agreement, confirmed by international peers. Pucha has no objection after repeated scrutiny by laymen with professional basic knowledge.
Model for solving problems
I don't know. The mathematics we established simply explains the material level. The spiritual level can't touch the door at all.