2019美赛数学建模A题论文

1、Winter is approaching, may the dragon s wings grow more abundantSummaryIn the game of thrones, Daenerys Targaryen depicts the image of a dragon. In eastern and western cultures, the phenomenon of dragons is not uncommon. If dragons live in modern society, how can we raise these war monsters? Researc

2、h, and applied the cross disciplines of biology, physics, and chemistry to build a mathematical model and solve it to achieve the maximum growth of the dragon. Of course, dragons do not exist in real life, so we likened pterosaurs, modern Aircraft and chemical burner to derive the specific physiolog

3、ical characteristics of the dragon to ensure the rationality and scientificity of the research.First, we studied the flight and fire-spitting models of dragons. Through analogical reasoning, our hypothetical dragons fire-spitting principle is similar to modern alcohol flamethrowers. For dragon fligh

4、t, we used fluid mechanics to get the dragons flight speed. And glucose energy loss. Combining the two to get the energy loss model of the dragon. Second, we studied the basic physical characteristics of the dragon. For the relationship between the body length and body age of the dragon, we establis

5、hed an elastic model of growth. Because the weight and body length of dragons have upper and lower limits, in order to comply with basic ecology, we have defined the dragons bone saturation value as the cut-off value, and conducted a segmented study. When studying the relationship between weight and

6、 body length, We know that the weight of the dragon is proportional to the cube of the body length. Then, because the dragon needs resources to replenish like other animals, we built a dragons food supply model. Suppose that the three dragons have the same competitiveness and the daily sheep Resourc

7、es are the same. According to ecology, when the number of sheep in a certain area reaches k / 2, we need to migrate the dragon. Finally, the temperature will affect the living environment of the dragon, so the dragon needs to followMigration was selected for changes in temperature, and we selected t

8、hree areas of drought, cold, and warmth to study the dragon, and integrated the model of the regional area of the dragon by the appealing model.In addition, we wrote a letter to the author of the Song of Ice and Fire, giving some suggestions on the actual ecological foundation of the dragon, hoping

9、to be adopted. Although the dragon does not exist in our real life, the dragon can be broken down into Part of our modern society. For the dragons flying spitfire energy loss model, we can further study the aircrafts fluid mechanics and modern flamethrowers. The study of non-existent organisms also

10、prepares us for the arrival of new species .table of ContentsWinter is approaching, may the dragon s wings grow mor.e.a.b.u.n.d.a.n.t 1Summary 1table of Contents21 Introduction 31.1 restatement31.2 Problem Analysis32 Assumptions and reasons 43 Symbol Definition 44 . Mathematical modeling 54.1 About

11、Dragon Flight and Spitfire Consumption 54.2 About the relationship between dragons body length and weight and age74.3 About Dragons FoodSupply 84.4 Regulating the area of dragons by region95 Sensitivity analysis 106 Model evaluation and outlook 116.1 Model evaluation 116.2 Further discussion127 to a

12、 letter from George RR Martin 128 .Appendix: 138.1 References 138.2 Matlab code 137 / 131 Introduction1.1 restatementIn the magical TV series Game of Thrones, Daenerys Targaryen, known as the Mother of Dragons, raised three dragons as an aggressive army. Dragons have always been the most mysterious

13、monsters in Eastern and Western cultures, but if Dragons live in the present era, how should we feed the three dragons in pursuit of maximum growth? In this article, we assume that the growth rules of dragons are in line with basic biology. To study them, we build mathematical models to solve proble

14、m.a. Analyze the change of the dragons weight length with age, and estimate the value of the dragons weight length corresponding to the age group.b. Investigate the loss of self energy during dragon fire, flight, and breathing, so as to estimate the minimum supply value of dragon for external activi

15、tiesc. Dragons need food and survival areas like other animals in the real world. Through certain assumptions and calculations, we can determine the total amount of food that dragons need daily and the size of living areas in three areas.d. Sensitivity analysis: As temperature and climate change, dr

16、agons will also migrate to different regions. Therefore, we need to analyze the differences in the impact of dragons on the survival of arid regions, temperate regions, and cold regions.1.2 Problem AnalysisBecause dragons do not exist in real life, we need to use some things in the real world to com

17、pare dragons in order to achieve the purpose of studying dragons. In analyzing the biological morphological characteristics of dragons, we use the knowledge of ecology and basic elements of biology Lets conceive the basic biological characteristics of the dragon such as weight and body length. For t

18、he energy loss model of the dragon, we have studied three aspects to describe its loss. Here we compare the modern flamethrower and establish related chemical equations to achieve the research of the dragon. Spitfire loss. In addition, in TV series such as Game of Thrones we will find that dragons c

19、an fly in common sense, so we have derived the dragons flight loss. Of course, all aerobic organisms can breathe. Dragons are no exception, so there is a loss of breathing to maintain body temperature. At the same time, in order to make up for the loss of dragons in daily activities, we have establi

20、shed a material reserve model, in which materials are cattle and sheep in real life, etc. Finally, during the cyclical changes in climate and food, the dragons we feed will also migrate to some extent, so we analyzed the impact of different regions on the growth of dragons.Into account various facto

21、rs that we can more scientific training of dragons, have achieved our purpose.2 Assumptions and reasonsAfter a comprehensive analysis of the problem, in order to increase the enforceability, we make the following assumptions to ensure the rationality of our model establishment.2.1 Assumptions: The b

22、asic biological characteristics of dragons are in line with the law of biological growth. In modern life, the growth and development of dragons should also be similar to other animals and conform to basic biology.2.2 Assumption: The dragon will spit fire and fly, and its flight conforms to the physi

23、cal environment of fluid mechanicsReason: In Game of Thrones, the image of the dragon was once able to fly and spit fire.2.3 Assumption: In the single field we are studying, the environment of a certain area will not change abruptly and maintain a dynamic stability.2.4 Hypothesis: Dragons are top pr

24、edators in the food chain, but dragons do not cause devastating harm to the biosphere.2.5 Assumption: The weight distribution of the dragon is uniform, and the body length reaches 30 to 40 cm at the time of birth.Reason 2.6: We refer to ancient biology and some dinosaur fossils.2.7 Hypothesis: Excep

25、t for the skull, heart, liver, lungs, kidneys, bones, etc., the sum of other body masses is proportional to the cube of height.Reason: The hypothesis is obtained by counting the relationship between body length and weight of modern organisms.2.8 Hypothesis: The dragon is a constant temperature anima

26、l whose body temperature is not affected by external factors.Reason: A few pterosaur fossils have traces of hair on the surface, while the dragons in Game of Thrones are similar to pterosaurs.2.9 Hypothesis: The dragon is fully aerobic during the flight to provide energy2.10 Hypothesis: A certain fi

27、xed ratio of the amount of energy that is not assimilated by the growth and metabolism of the dragons breathing and other organisms2.11 Hypothesis: Dragons Flight Similar to Modern Fighter3 Symbol DefinitionsymboldescriptionMqualityLBody lengtha%Unassimilated amount is a fixed ratio of intakeb%Respi

28、ratory and other biological growth metabolic consumption can be a fixed ratio of assimilation amountt、T、itimeAAccelerationWConsume energyNAmount of cattle and sheepd%Assimilation rate of cattle and sheep grazing in the areaKEnvironmental capacitye%Solar utilizationQSolar energy per unit areaSFlying

29、distance of dragonSareaRParameters of environmental carrying capacityVSpeed or rateYage4 . Mathematical modeling4.1 About Dragon Flight and Spitfire Consumption4.1.1 Proposed modelConsidering that dragons fly and spit fire during activities, we have established an energy loss model. Comparing the pr

30、inciple of dragons spitfire with modern flamethrowers, modern flamethrowers consume hydrocarbons or alcohols. It does not cause any impact, so the dragons firebreathing principle is in line with the alcohol flame-thrower principle. Considering that the formaldehyde produced by the metabolism of meth

31、anol in the animal body is harmful to the body, we stipulate that ethanol is the fuel used by the dragons flame. In the process, the relationship between the dragons flight speed and glucose energy consumption is obtained according to fluid mechanics. In this process, we assume that the aerobic resp

32、iration is completely performed, and the energy consumed by the dragon due to flight is obtained according to the glucose consumption. In summary, the dragon energy loss model is obtained.4.1.2 Establishment and Solution of Dragons Spitfire ModelThe thermochemical equation for ethanol combustion is:

33、 C2H50H (l) + 3O2 (g) = 2CO2 (g) + 2H2O (l) H = -12KJ / gSpecify the energy released per unit mass of ethanol combustion x1When the dragon spit fire in unit time t, the unit mass of ethanol consumption is a fixed value The energy consumed by the fire time t1 is w1The mass consumed by the fire time T

34、1 is m4Let the energy emitted by the combustion of unit mass of ethanol be w1 Then W1 = x1 * t m4=W1/WlSolve m4 = x1 * t / W1 4.1.3 Establishment and Solution of Dragon Flight ModelDuring the flight of the dragon, it will be affected by the air resistance. In the ideal situation, the dragons flight

35、can be considered as a uniform acceleration and then a uniform speed, and it will decelerate when it is about to reach its destination.When Long uniform acceleration is specified, the acceleration is aSince the flight of the dragon is similar to that of a fighter, a = 30m / s A 2The speed of the dra

36、gon during uniform motion is v0The total flight length of the dragon during flight is sBecause air resistance is proportional to the speed of movement, that is, F1 = k * v (where k is a constant)Since the dragons flight is similar to an airplane, we can get k = 3.2325Available according to the relev

37、ant kinematic formulaThe flying distance of the dragon during uniform acceleration is s1 = (v0) A 2 / 2aThe flying distance of the dragon during uniform deceleration is s3 = (v0) A 2 / 2aThe flying distance of the dragon during uniform motion is s2 = s-s1-s3Average air resistance during uniform acce

38、leration F1 = k * (0 + v0) / 2The average air resistance during uniform motion is F1 = k * v0Average air resistance during uniform deceleration f1 = k * (v0 + 0) / 2According to the law of conservation of energyThe energy w2 consumed by the dragon during flight is all used for air resistance workW2=

39、F1 *s1+F1 *s2+F1 *s3Solve W2 = 3.2325 * v0 * s-3.2325 * (v0) A 3 / (2 * 30)During the flight of the dragon, the principle of energy provided by aerobic respiration is C6H12O6+6O2=6CO2+6H2OAmong them, the energy produced when 1g of glucose is completely consumed is 16KJThen the weight consumed in thi

40、s process is m6 = W2 / 16v,s=meshgrid(0:0.1:100;0:0.1:100);m=3.2325*v*s-3.2325*vA3/60mesh(v,s,m)4.2 About the relationship between dragons body length andweight and age4.2.1 Proposed ModelFirst, in order to study the relationship between the weight, length, and age of the dragon, that is, morphologi

41、cal characteristics, we established a model of elasticity during growth. The abovementioned change curve is continuous, so we use the weight of the dragon at birth, and consider the weight and length of the dragon. The relationship between age changes can be used to derive the normal weight and body

42、 length of dragons in all ages. When analyzing the weight changes of dragons, biological knowledge shows that the amount of assimilation of the dragon is equal to the intake amount minus the amount of unassimilated amount Considering that the growth rate of the dragon in adulthood is a watershed, we

43、 use the saturation value of the dragons head, heart, and liver as a cutoff value to estimate the relationship between the dragons weight and age, respectively. When studying the body length of the dragon, according to the existing morphological knowledge, the head to hip of the dragon is used as th

44、e length standard. Because the weight of the dragon is proportional to the cube of the dragons length, we get the weight and length Functional relationship. Of course, the daily weight gain of the dragon must be less than the daily energy consumption. In summary, we have a dragon intake model.4.2.2

45、Model establishmentSpecify the weight of the dragon as mDragon was born with a weight of m0 (known m0 = 10kg)Assume that the mass of cattle and sheep fed by a train every day is m2The assimilation amount of the dragon is fixed at a%A certain fixed ratio of the amount of unabsorbed energy due to grow

46、th and metabolism of organisms such as dragons respiration, recorded as b%The weight gain of the dragon is m The sum of the weight of the dragons head, heart, liver, lungs, kidneys, bones, etc. m1 increases with age y until adulthoodDragon is y1 when he is an adultThe growth rate of m1 is v1The mass

47、 of m1 at birth is m0Before the dragon reaches y1m1=m0+v1*yAfter the dragon reaches y1m1 =m0+v1*y14.2.3 Model Solvingm =m2*(1-a%)*(1-b%)-m4-m6So the weight of the dragon m = m + m0Except for the dragon, except for the head, heart, liver, lungs, kidneys, bones, etc., the sum of other body masses is p

48、roportional to the cube of height, and the body length is recorded as lWhen the age of the dragon does not reach y1, l = (m-m1) A (1/3)When the age of the dragon reaches y1, l = (m-m1) a (1/3)M2 =y=0:0.1:20functiony= (m2*(1-a%)*(1-b%)-m4-m6-v1*y)y=20:0.1:100functiony= (m2*(1-a%)*(1-b%)-m4-m6-v1*20)p

49、ower(y,1/3)4.3 About Dragons Food Supply4.3.1 Proposed modelBased on the above analysis, we studied the living area of the three dragons in the region and8 / 13their impact on the ecological community in the region. For the sake of research, we assume that the other creatures in the region are cattl

50、e and sheep, and the competitiveness of the three dragons is comparable, Being a top predator in the food chain.4.3.2 Model establishmentThe local food chain can be approximated as: grassf cow or sheep f dragonAssume that the weight of the grass in the arid region, the warm temperate region, and the

51、 Arctic region is the same as m8.Remember that the mass of each cow and sheep is the same as m7We provide the same initial number of cattle and sheep in all three regionsAssume that the daily growth rate of cattle and sheep is c%The initial number of cattle and sheep is n1And n1 is the number of pop

52、ulations reaching k in the regionDragons live in this area. When the number of cattle and sheep reaches k / 2, in order to ensure the balance of the ecological environment, the dragons need to be moved to other regions.4.3.3 Model SolvingThe initial amount of cattle and sheep on day 1 is: n1The init

53、ial amount of cattle and sheep on the second day is: N2 = (N1-3 * m2 / m7) * (1 + c%)The initial amount of cattle and sheep on the third day is: N3 = (N1-3 * m2 / m7) * (1 + c%)-3 *m2 / m7) * (1 + c%)From this we can get the initial amount of Ni of cattle and sheep on day iI can be solved by the equ

54、ation Ni = K / 2That is, the dragon needs to change a living area after living in the area for i days.4.4 Regulating the area of dragons by region4.4.1 Proposed modelIn order to ensure the normal growth of the dragon, we provide fixed-quality cattle and sheep as the supply of resources for the survi

55、val of the dragon region, and assume that the number of cattle and sheep is proportional to the size of the regional living area. Considering the growth rate of cattle and sheep, we have established a differential The equation draws the relationship between the growth rate of cattle and sheep and th

56、e age of the dragon. However, cattle and sheep will reach a growth saturation value at a certain moment, we will consider it in segments to ensure that the data is more scientific. In order to comply with ecology, cattle The supply of sheep should also have a lower limit. In summary, we have establi

57、shed a dragon-cow-sheep-living area function model.4.4.2 Model establishmentRemember that the assimilation rate of cattle and sheep grazing in this area is d%Because the solar energy received by the surface area of the three areas is different, the total area required for the grass under the same qu

58、ality conditions is different. The utilization rate of the solar energy is required to be e% (0.5 e 1 under the natural conditions of the search data)The solar energy per unit area in the arid area is q1Unit area solar energy in warm zone is q2Solar energy per unit area in the Arctic is q34.4.3 Model SolvingAccording to the utilization of solar energy, we can find:Area required to support the