当前位置：网站首页>Mathematical modeling -- what is mathematical modeling

Mathematical modeling -- what is mathematical modeling

2022-07-07 15:04:00 【WangLanguager】

This paper mainly introduces two examples of data modeling ： Make dumplings 、 Roadblock

Introduce the whole process of data modeling

Introduce the basic methods and steps of mathematical modeling

One 、 introduction

mathematics ： The foundation of each discipline , A tool for social progress

Solve any practical problem with mathematical methods , We must build a bridge between practice and Mathematics .

To solve the process ： Practical problems are transformed into mathematical problems ; Solving mathematical problems ; Mathematical solutions return to practical problems .

This process of solving is called mathematical modeling , That is to establish a mathematical model for practical problems .

Two 、 Examples of mathematical modeling 1： Make dumplings

Usually ,1kg Noodles ,1kg Stuffing , package 100 A dumpling （ Or Tangyuan ）, today 1kg The face does not change , But the stuffing ratio 1kg More , ask ： It should be more packages （ Each is smaller ）, It's better to pack a few less （ Each is bigger ）？

1、 problem ： Round area $S_{all}$ A skin of , Pack into volume $V_{all}$ Dumplings ; It is divided into n Skin , The area of each small circle is ： $s_{small}$ , The wrapped volume is ： $v_{small}$ Dumplings .

$V_{all}$ and $n\cdot v_{small}$ Who is big ？【 qualitative analysis 】

$V_{all}$ Than $n\cdot v_{small}$ How much larger ？【 quantitative analysis 】

2、 hypothesis （1）、 The thickness of dumpling skin is the same （2）、 The shape of dumplings is the same

modeling ： $S_{all}$ = $n\cdot s_{small}$ (1)

$S_{all}=k1\cdot R^{2}_{big}$ , $V_{all}=k2\cdot R^{2}_{big}$ , among $R_{big}$ For the radius of dumplings , be $V_{all}=k\cdot S^{\frac{3}{2}}_{all}$ (2)

$s_{small}=k1\cdot r^{2}_{small}$ , $v_{small}=k2\cdot r^{2}_{small}$ , among $r_{small}$ For the radius of small dumplings , be $v_{small}=k\cdot s^{\frac{3}{2}}_{small}$ (3)

from （1）、（2）、（3） Available ： $V_{all}=n^{\frac{3}{2}}v_{small}$

application ： $V_{all}=\sqrt{n}\times (nv_{small})\geqslant nv_{small}$ , $V_{all}$ yes $nv_{small}$ Of $\sqrt{n}$ times .

if 100 A dumpling can be made 1kg Stuffing , be 50 A dumpling can be made $\sqrt{2}\approx 1.4$ kg Stuffing

3、 The basic key steps of dumpling modeling process

(1) In mathematical language （ Volume and surface area ） Indicates a real phenomenon （ Stuffing and skin ）.

（2） Make simplified and reasonable assumptions （ Same thickness , Same shape ）.

（3） Use the inherent law contained in the problem （ Volume 、 Geometric relationship between surface area and radius ）

The results of this model can be used to explain many phenomena in daily life .

Over time, the unit price of large package goods is cheaper than that of small package goods .

3、 ... and 、 Examples of mathematical modeling 2： Car barricade

1、 background ： campus 、 In the middle of the road in the residential area , Roadblocks are often set up to limit the speed of cars .

2、 problem ： If you want to limit the speed to no more than 40km/h, How far away should a barricade be set ？

3、 analysis ： The speed of the car passing the barrier is close to zero , Accelerate after passing the barrier , The car accelerated to 40km/h when , Slow down because there is the next roadblock ahead , The speed at the barricade is close to zero .

So fast 、 The deceleration cycle alternates to achieve the purpose of speed limit .

4、 hypothesis ： The car performs equal acceleration and equal deceleration between two adjacent roadblocks .

You need to get the acceleration and deceleration of the car .

Method 1 ： Access to information , Method 2 ： To test

（1） Test data of accelerating car

Speed （km/h）	0	10	20	30	40
Time (s)	0	1.6	3.0	4.2	5

（2） Test data of vehicle deceleration

Speed （km/h）	40	30	20	10	0
Time (s)	0	2.2	4.0	5.5	6.8

5、 modeling ： The distance the car accelerates S1, Time t1, The acceleration a1

The distance the car slows down S2, Time t2, deceleration a2, The speed limit is ： $V_{max}$

be ： $S1=\frac{1}{2}\cdot a_{1}\cdot t_{1}^{2}$ , $S2=\frac{1}{2}\cdot a_{2}\cdot t_{2}^{2}$ , $V_{max}=a_{1}\cdot t_{1}$ , $V_{max}=a_{2}\cdot t_{2}$

Total distance traveled between adjacent roadblocks ： $S=S1+S2=\frac{V_{max}^{2}}{2}\cdot (\frac{1}{a_{1}} + \frac{1}{a_{2}})$

Given $V_{max}$ , From the test data a1 and a2, You can figure it out S.

The total distance traveled by vehicles between two adjacent roadblocks is designed as the distance between roadblocks .

Parameter estimation ： The relationship between speed and time during design driving is ： $t=C_{1}\cdot v+C_{2}$ , Test data , Then use the least square method to calculate ：

$a_{1}=\frac{1}{c_{1}}=2.2046m/s^{2}$

$a_{2}=1.6437m/s^{2}$

$S=\frac{V_{max}^{2}}{2}\cdot (\frac{1}{a_{1}} + \frac{1}{a_{2}})$

$V_{max}=40km/h$

be ： S=65.5556m

The distance between roadblocks is 65 rice

6、 The basic key steps of barrier spacing modeling

（1） Make simplified and reasonable assumptions （ Wait for acceleration, wait for deceleration ）.

（2） Use the inherent law contained in the problem （ Time 、 distance 、 Speed 、 The physical relationship between accelerations ）

（3） Estimate the parameters of the model according to the test data （ Acceleration and deceleration ）

Mathematical modeling used in barricade design can also be used to solve other problems , for example ： Design the height of the barricade 、 The shape of the barricade .

Four 、 What is a mathematical model （Mathematical Model） And mathematical modeling (Mathematical Modeling)？

1、 mathematical model ： For a real object , For a specific purpose , According to its inherent law , Make the necessary simplifying assumptions , Use appropriate mathematical tools , Get a mathematical structure .

2、 mathematical modeling ： The whole process of establishing mathematical model

3、 The whole process of mathematical modeling ：

Information of real objects ——>【 describe 】——> mathematical model ——>【 solve 】——> Solutions to mathematical models ——>【 explain 】——> Solutions to real objects

two “ translation process ”： Real phenomena are translated into Mathematics Model , The solution of mathematical model is translated into the solution of real phenomenon .

practice ——> theory ——> practice

5、 ... and 、 The basic analysis method of mathematical modeling

（1） Mechanism analysis ： Understanding of the characteristics of objective things , The quantitative law of internal mechanism .【 White box model 】

（2） Test and analysis ： Statistical analysis of measured data , The model that best fits the data .【 Black box model 】

（3） Mechanism analysis 、 Test and analyze the combination of the two ： Mechanism analysis and establishment of model mechanism , Test and analyze to determine the model parameters .【 Grey box model 】

Mechanism analysis is mainly learned from case studies , Modeling mainly refers to mechanism analysis .

6、 ... and 、 Basic steps of mathematical modeling

Model preparation ——> The model assumes ——> Model composition ——> Model solving ——> model analysis ——> Model test ——> Model application

If the model test finds that the model is not suitable , The model assumptions need to be corrected again

7、 ... and 、 Summary of content

Have a preliminary understanding of what mathematical modeling is

（1） Mathematical modeling refers to transforming practical problems into mathematical problems , Then solve the mathematical problem , The whole process of returning to practical problems after solving mathematical problems and constantly revising mathematical models .

（2） Mathematical modeling widely exists in all fields of social life .

原网站

版权声明
本文为[WangLanguager]所创，转载请带上原文链接，感谢
https://yzsam.com/2022/188/202207071250132786.html

当前位置：网站首页>Mathematical modeling -- what is mathematical modeling

Mathematical modeling -- what is mathematical modeling

边栏推荐

猜你喜欢

随机推荐