Corresponding to various models, their original reference objects in the real world are generally called prototypes. This part first discusses the relationship between prototype and model, especially mathematical model, and then introduces the significance of mathematical model.
Prototypes and models? Prototype and model are a pair of duality. Prototype refers to the actual object that people care about, study or engage in production and operation in the real world. In the field of science and technology, words such as system and process are usually used, such as urban transportation system, socio-economic system, ecosystem, electric power system, mechanical system and life system, as well as missile flight process, chemical reaction process, pollution diffusion process, production and sales process, planning and decision-making process and iron and steel smelting process. The objects, research objects and practical problems mentioned in this book refer to the prototype. A model refers to a prototype substitute built by simplifying and refining some information of the prototype for a specific purpose.
The purpose of building the model is particularly emphasized here. The model is not an intact copy, the prototype has the characteristics of various aspects and levels, and the model only needs to reflect the levels of those aspects related to a certain purpose. A prototype can have many different models for different purposes. For example, the model plane placed in the exhibition hall should be realistic in appearance, but it may not fly. The model airplanes participating in the model airplane competition should have good flight performance and do not need to have high requirements on appearance. As for the mathematical model and computer model used in the process of aircraft design and trial-manufacture, only the data are required to truly reflect the flight dynamic characteristics of the aircraft, and the entity of the aircraft is not involved. Therefore, the basic characteristics of the model are determined by the purpose of building the model.
We have seen various forms of models. Models can be classified by replacing prototypes with models. Models can be divided into material models (image models) and ideal models (abstract models). The former includes intuitive model and physical model, while the latter includes thinking model, symbolic model and mathematical model.
Intuitive model? Refers to physical models, toys, photos, etc. For exhibitions. Usually, the size of the prototype is scaled down or enlarged, mainly in pursuit of realistic appearance. The effect of this model is clear at a glance.
? Physical model? Mainly refers to the model built by scientific and technological workers for a certain purpose according to the similarity principle. It can not only display the shape or some characteristics of the prototype, but also conduct simulation experiments to indirectly study some laws of the prototype. For example, the ship model in the wave flume is used to simulate the navigation performance of the ship under the impact of waves, and the aircraft model in the wind tunnel is used to test the aerodynamic characteristics of the aircraft in the airflow. It is very difficult to study some phenomena directly with prototypes, so such models can be used, such as earthquake simulation devices and nuclear explosion response simulation equipment. Attention should be paid to verifying the similarity between the prototype and the model to ensure the reliability of the simulation results. Physical model can often get practical and valuable results, but it also has some shortcomings such as high cost, long time and inflexibility.
Thinking mode? It means that through people's repeated understanding of the prototype, the acquired knowledge is directly stored in the human brain in the form of experience, so that corresponding decisions can be made according to thinking or intuition. For example, the steering wheel is controlled by the driver of the car, and the operation of some technical jobs (such as locksmiths) is generally carried out through this mode. It is often said that some leaders make decisions based on experience. Thinking mode is easy to be accepted, and satisfactory results can be obtained under certain conditions, but it is often vague, one-sided, subjective and accidental, so it is difficult to test its assumptions and facilitate people to communicate with each other.
Symbolic model? Under some conventions or assumptions, with the help of special symbols, lines, etc. The prototype is described in some form. Such as maps, circuit diagrams, chemical structural formulas, etc. , characterized by simplicity, convenience, strong purpose and non-quantification.
The mathematical model specially discussed in the book is a mathematical formula, figure or algorithm composed of numbers, letters or other mathematical symbols, which describes the law of the number of real objects.
Mathematical simulation, closely related to mathematical model, mainly refers to the use of digital computers for computer simulation. According to the characteristics of the actual system or process, it uses computer programming language to simulate the actual operation according to certain mathematical laws, and quantitatively analyzes the system according to a large number of simulation results. For example, the bottleneck in the production process can be identified by simulating the processing of various workpieces on different machines in a certain process sequence; Through the simulation of expressway traffic flow, we can analyze the distribution of vehicles on the road section, especially the congestion situation. Compared with the physical model simulation experiment, computer simulation has obvious advantages: low cost, short time, high repeatability and strong flexibility. Some people regard computer simulation as one of the means to establish a mathematical model, but the mathematical model describes the quantitative relationship of the internal characteristics of the object in a certain sense, and the result is easy to popularize, especially when the analytical form answer is obtained. However, computer simulation completely imitates the actual evolution process of objects, and it is difficult to separate the internal laws of objects from the obtained digital results. Of course, for those practical objects whose internal mechanism is too complicated to establish a mathematical model at present, it is an effective means to obtain some quantitative results by computer simulation.
What is a mathematical model? The mathematical model should be said that everyone is familiar with it. As early as when we were studying elementary algebra, we had already solved practical problems by establishing mathematical models. Of course, many of these questions are artificially set by teachers in order to teach students knowledge. For example, you must have solved the so-called "navigation problem":
The distance between Party A and Party B is 750 kilometers. It takes 30 hours for the ship to sail along the water from Party A to Party B, and 50 hours for the ship to sail against the water from Party B to Party A. Ask the speed and water of the ship.