Introduction to mathematical modeling - from real objects to mathematical modeling [2]

1. Our common models

toy 、 Photo 、 The plane 、 Rocket model ……~ Physical model
Submarine in water tank 、 An airplane in a wind tunnel ……~ The physical model
Map 、 Circuit diagram 、 Molecular structure diagram ……~ Symbolic model

Model For a certain purpose , Abbreviate and abstract a part of objective things , Refined Prototype A substitute for .
Model A concentrated reflection of Prototype The part of features that people need .

• mathematical model Is a kind of abstract 、 Untouchable Symbolic model . Composed of letters and other mathematical symbols , A mathematical formula that describes the quantitative law of real objects 、 Graphics or algorithms .

example ： Navigation problems

So , Mathematical modeling is not new , We have been learning it .

• mathematical modeling Definition ： For one Real objects , For a Specific purpose , According to its Internal laws , Make the necessary Simplify assumptions , Use appropriate Mathematical tools , The one I got Mathematical structure .

For this example ：

Real objects ： Route problem || Specific purpose ： Find the speed of the ship
Internal laws ： Speed times time equals distance
Simplify assumptions ： Both ship and water move at a constant speed
Mathematical tools ： Add, subtract, multiply and divide || Mathematical structure ： Equations

• The most important thing in mathematical modeling is to simplify assumptions ！

（tip： The model cannot be too simple or too complex , Too simple a model is useless , It's too complicated to build a model , therefore We must have a good grasp of the complexity of model assumptions .）

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2. The significance of mathematical modeling

On the macro level

• Mathematical modeling has a long history
• The promotion of scientific and technological progress and social development
• The introduction of mathematical modeling into teaching conforms to the trend of the development of the times

From the specific application

• Analysis and design
• Forecasting and decision making
• Control and optimization
• Planning and management

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3. Life and mathematical modeling

The difference between mathematical modeling problems and mathematical problems is , The conditions of mathematical modeling are often not clear , You should analyze the problem yourself and find the necessary conditions .

3.1 Mathematics in making dumplings

• problem

• analysis

1. Visual analysis —“ For the same skin , There are many fillings in big dumplings ”.
   Need to establish ： When the face does not change , The functional relationship between the number of dumplings and stuffing f(n).
2. qualitative analysis ：n1v1 And n2v2 Which is the big one ？
3. Quantitative calculation ：n1v1 Than n2v2 How much larger ？

• hypothesis

1. The thickness of the skin is the same . namely n1S=n2s
( This is actually unscientific , Because the skin of dumplings should be thicker )
2. The shape of dumplings is the same , namely k1=k2

• modeling

0. Build stuffing 、 The connection between skin and mathematical concepts ：
   Stuffing —— Volume , skin —— Surface area
   
1. R~ Large skin radius
   S=k1R² || V=k2R³ =>V==kS^3/2
2. r~ Small skin radius
   Empathy ：v=ks^3/2
3. elimination S,s,k
   V=(n2/n1)^3/2 v, obtain f(n), That is, get the mathematical model
4. from （3） You know
   n1V=(n2/n1)^3/2 (n2v)

• To sum up, we can see that ,n1V Than n2v Big —— There are many fillings in big dumplings
• Plug in n1V=(n2/n1)^3/2 (n2v), It can be solved ： if 100 A dumpling bag 1Kg Stuffing ,50 A dumpling can make 1.4KG Stuffing .
• Discuss
  if 100 A dumpling bag 1Kg Stuffing ,50 A dumpling can make 1.4KG Stuffing .
  Double the number of dumplings , You can really pack more 40% Is there any stuffing ？

But it is not ,” The thickness of the skin is the same “ The hypothesis of is worth discussing .
It can be done to ” The thickness of the skin increases with the radius “ The quantitative relationship is reasonable 、 Simplified assumptions , Re modeling .

3.2 Barrier spacing design

• problem
  Limit the speed <=40km/h, How far away is a barricade ？
• analysis

1. The speed of the car passing the barricade is close to 0, Accelerate after passing the barrier .
2. The speed is up to 40Km/h Let the driver see the next barricade and slow down , The speed is close to the barricade again 0
3. Cycle like this to achieve speed limit

• hypothesis
  Cars between adjacent roadblocks Wait for accelerated motion and Wait for deceleration .
  ( That is, there is no situation of one foot on the accelerator and one foot on the brake )
• analysis

1. Visual analysis —— Establish barricade spacing , And speed limit 、 The acceleration 、 Function of deceleration

How to know the general acceleration and deceleration of the car ？

• Access to information

1. Search the Internet
2. Experiment and measure by yourself ( More accurate ！)

• modeling
  Accelerate ： distance s1, Time t1, The acceleration a1( Finally achieve 40km/h)
  Slow down ： distance s2, Time t2, deceleration a2( Finally back to 0 km/h)

s1=(1/2)a1t1² || s2=(1/2)a2t2²
Vmax=a1t1|| Vmax=a2t2

Total distance traveled between adjacent roadblocks ：
s=s1+s2=(Vmax² /2)*(1/a1+1/a2)

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Summary of learning skills

Steps to solve practical problems

1. analysis —— Establish a mathematical model about what variables （ Function relation ）
2. hypothesis —— Make assumptions about the uncertainties of the problem
3. modeling —— Mathematicize the relationship between variables , Find all relevant relationships involved in the problem , Reduce it to the target mathematical model .
4. Discuss —— Further analyze the hypothetical conditions
   （ If you come up, consider comprehensively , It will complicate the problem , It can't be solved ）