overview

数学模型：沟通现实世界与数学世界的桥梁。
数学建模:：建立数学模型的全过程。
算法：解决问题的手段和策略。

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overview
foreword
1. Elementary model
Summarize

foreword

With the continuous development of science and technology, education, and culture, our living standards continue to improve, and we pay more attention to high-quality life. In this way, many high-quality problems will be faced. The technology of mathematical modeling is becoming more and more important, he can help us

Solving practical problems: Mathematical modeling can transform practical problems into mathematical problems, so as to use mathematical methods and tools to solve and analyze problems, and provide ideas and methods for solving problems.

Optimal design: Mathematical modeling can play a role in the design of products, systems and processes, through mathematical model analysis and optimization of design parameters, to improve the performance and efficiency of products and systems.

Prediction and prevention: Mathematical modeling can help us predict and prevent some major events, such as natural disasters, epidemics, etc., so as to take effective measures to deal with them.

Promote development: Mathematical modeling can help us explore new fields and solve new problems, and promote the development and innovation of science and technology. Many people have started to learn mathematical modeling,

This article introduces the basics of mathematical modeling.

1. Elementary model

1.1 Examples of mathematical modeling

From ancient times to the present. We are in contact with mathematical models in our life and learning process, and some of these models have been established by predecessors.
For example:

Euclidean geometry constructed the first mathematical model for the spatial forms of the real world.
Newton used calculus and Kepler's third law to establish the law of gravity in a simulated reality.
In the 19th century, mathematics was applied to electromagnetism, and the electronic equations established by Maxwell-a mathematical model composed of four calculus.
Mark used mathematical modeling to implement a mathematical model of how a CT scanner works.
The prediction of weather forecast can establish a corresponding mathematical prediction model to predict the weather.
There are many successful cases of mathematical modeling in aerospace, medical care, economy, manufacturing, agriculture, etc., to improve our quality of life.

1.2 Mathematical modeling methods and steps

1. Model

模型原型It is formed by reducing and refining part of the information for a specific purpose 原型替代物.

2. Image model

形象模型A model made according to a certain proportion based on the actual size of an object is called an image model, and an image model is also called an image model 直观模型. 比如:car, house, etc.
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3. Symbolic models

符号模型It is to use some vivid and vivid features 符号to describe the characteristics of something. 比如:Aerial view, circuit diagram, cell molecular diagram
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The ones in the water tank 舰艇模型and the ones in flight 飞机模型are all physical models. The physical model is mainly a model constructed by scientific researchers according to the principle of similarity for a certain purpose. They can not only show the shape or similar features of the prototype, but also can be used 模拟实验，间接的研究模型的某些规律. For example 水箱中的建筑模型用来模拟波浪冲击下舰艇的航行性能. The flying airplane model is used for experimentation 飞机在气流中的空气动力学特性.
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In later chapters, we will introduce some of 量纲分析方法the designs suitable 物理模型for inference .原型物理指标

4. So how to define the mathematical model?

特定目的A mathematical model is a kind of model, which is necessary to make a real world for one , according to its internal laws 简化假设. Take advantage of the appropriate 数学工具obtained one 数学表示.

A mathematical model is an abstract and succinct description of the essential attributes of an actual subject. It may explain some objective phenomena, or predict the law of future development, or provide the ultimate solution in a certain sense for controlling the development of a certain line. Better strategy , get better strategy.

5. Mathematical modeling process:

The whole process of establishing a mathematical model is called mathematical modeling. The whole process of mathematical modeling includes:

translate practical problems into mathematical problems;
Mathematical problem solving;
Mathematical solution checks, followed by answers to practical questions;

图示：
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The above picture is to translate a realistic information representation into a mathematical model. Then obtain the solution of the mathematical model in the mathematical world. Then use the math section to answer the actual problem, and verify whether it can reasonably explain the actual problem. The left half represents the new world, and the right half represents the mathematical world. The process of mathematical modeling is a two-way translation process. The first time translates the practical problem into a mathematical problem, and the second time translates the mathematical solution back to the practical problem. The process of mathematical modeling realizes the cycle from practice to theory to practice

6. Modeling method:

It is roughly divided into 机理分析方法two 测试分析方法types.
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机理分析方法It is the understanding of the characteristics of objective things, and finding out the establishment laws that reflect the internal mechanism.
测试分析方法It is to regard the object as a black box system, that is, its internal mechanism is not clear.
We need to conduct statistical analysis through measurement data. For many practical problems, we often combine the two methods for modeling, that is, use mechanism analysis to establish a model structure. After the structure of the model is established, there is a test analysis to determine the detailed and accurate parameters of each model. This ensures the robustness of the model.

7. Small Demo

Mathematical modeling should be no stranger, because it is ubiquitous in the learning process in life. 例如：Application questions in middle school, for example 航行问题. 微积分中导数定积分概念的引入Mathematical modeling is present in universities . Let's look at a small case of modeling.

The first is a small case of calculus.
- Case description:
  The Hong Kong-Zhuhai-Macao Bridge has many advantages. It is the longest cross-sea bridge in the world with the most difficult construction and the highest technical content. Engineers need to build bridge piers, and bridge sections need to be drained. How to pump water? Estimating how much it will cost to pump water is a practical issue for engineers to consider. So what quantities should be used to describe this practical problem?
- Problem refinement:
  It can be quantified by the work done, that is, how much work needs to be done to pump clean water to zero? How do we calculate it?
- Questions to ponder:
  Is work equal to the weight of the water times the height of the pool? During the process of drawing blood, the water level gradually drops. If the water is layered from top to bottom, the distance moved by each layer of water cups is different when it is pulled out, so it cannot be calculated by multiplying the gravity of the water by the height of the pool.
  
  最终确定To solve the problem, use the mechanism analysis method to model in the integral course, that is, use it 微元分析方法. Can estimate work done.
- Modeling steps:
  
  The above case数学建模的步骤为
  1. The first step is to build a model that needs to be generated 假设. We assume it is a cylinder with a diameter of 10m and a water depth of 20m.
  2. The diameter of the second step 变量说明is 2r=10. Establish the coordinate system x-axis direction downward x is the integral variable, and its integral interval is 0~20. We divide water into the small multiplication of microelements x to x+dx.
  3. part three建立模型
  4. Finally, the fourth part 数学求解needs to verify the results.
A small case of life.
- Case description:
  Can the chair be stabilized on uneven ground? This problem stems from a common reality in daily life. When a chair with four legs is placed on an uneven ground, usually only three legs are on the ground. We only need to move a few times to make all four feet fall to the ground at the same time. Then we will use mathematical language to describe this real problem, and use mathematical tools to prove it.
- Problem refinement:
  We only need to move a few times to make the four feet fall to the ground at the same time.
- Questions to think about:
  数学建模步骤:
  1. First, make assumptions about the model.
    Assumption 1: The four legs of the chair are of the same length, and the feet are in point contact with the ground. Then the line connecting the four corners forms a square.
    Assumption 2: The height of the ground changes continuously, that is to say, our ground has no steps.
    Assumption 3: The ground is relatively flat, that is to say, there will be no deep grooves or protrusions on the ground. Make sure that in any position of the chair, at least three feet are sweeping the floor.
  2. The second step is variable specification.
    If the chair moves, then we need to find one 变量to describe it 椅子的位置. We just assumed that the line connecting the known angles is a square, which according to the symmetry center of the square is the symmetrical point. Rotating the square around the center means that the position has changed. So 变量旋转角度θuse it for drawing 椅子的位置.
    
    The conclusion we want to prove is that the feet of the chair touch the ground, how to describe it with quantity ?
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脚着地呢就表示脚和地面的距离为零，那θ发生改变后，距离也会发生改变，所以呢距离是它的函数。
已知呢有四个角，那有四个距离，由于正方形的对称性呢，我们只要设两个距离就可以了，可以减少变量。
对角线A，C两角的距离之和，我们把它记作==f(θ)。Bd两角的距离之和呢，我们把它记作记==g(θ)。
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  3. The third step is to build a model.
    
    Just assumed that the ground is a continuous surface.
    Translated into a mathematical language, it means that the functions f(θ) and g(θ) are continuous functions, and it is known that at least three feet are on the ground at any position.
    Translated into a mathematical language, it is equal to 0 for any f(θ) multiplied by g(θ).
    The chair is rotated 90 degrees, whichever is based on the symmetry line of the square and which diagonals must be interchanged.
    We assume that at the initial position, that is, when θ is equal to 0, the two points b and d are on the ground. So g(0)=0, f(0) is greater than 0,
    and after rotating 90 degrees, BD becomes AC. So f(π/2) is equal to g(0)=0.
    In this way, we translate a practical problem into a mathematical problem.
    Shown in the box above
  4. The fourth step is mathematical solution.
    For our calculus tool, use the borrowing theorem of calculus that you have learned. We do a simple proof, because we want to prove it f(θ)×g(θ)=0. It is equivalent to f(θ)-g(θ) having a zero point. Let's construct the function h(θ)=f(θ)-g(θ). According to known conditions, we can get h0>0. The score of H2 is less than 0.

8. Summary of modeling steps

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9. Modeling classification summary