MEDZINÁRODNÝ VEDECKÝ ČASOPIS MLADÁ VEDA / YOUNG SCIENCE

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2 MEDZINÁRODNÝ VEDECKÝ ČASOPIS MLADÁ VEDA / YOUNG SCIENCE Číslo 8, ročník 5., vydané v decembri 217 ISSN Konak: info@mladaveda.sk, el.: , Foografia na obálke: Plník v Pieninách. Branislav A. Švorc, foo.branisko.a REDAKČNÁ RADA doc. Ing. Peer Adamišin, PhD. (Kaedra environmenálneho manažmenu, Prešovská univerzia, Prešov) doc. Dr. Pavel Chromý, PhD. (Kaedra sociální geografie a regionálního rozvoje, Univerzia Karlova, Praha) prof. Dr. Paul Rober Magocsi (Chair of Ukrainian Sudies, Universiy of Torono; Royal Sociey of Canada) Ing. Lucia Mikušová, PhD. (Úsav biochémie, výživy a ochrany zdravia, Slovenská echnická univerzia, Braislava) doc. Ing. Peer Skok, CSc. (Ekomos s. r. o., Prešov) prof. Ing. Róber Šefko, Ph.D. (Kaedra markeingu a medzinárodného obchodu, Prešovská univerzia, Prešov) prof. PhDr. Peer Švorc, CSc.,predseda (Inšiú hisórie, Prešovská univerzia, Prešov) doc. Ing. Per Tománek, CSc. (Kaedra veřejné ekonomiky, Vysoká škola báňská - Technická univerzia, Osrava) REDAKCIA PhDr. Magdaléna Kereszesová, PhD. (Fakula sredoeurópskych šúdií UKF, Nira) Mgr. Marin Hajduk (Inšiú hisórie, Prešovská univerzia, Prešov) RNDr. Richard Nikischer, Ph.D. (Minisersvo pro mísní rozvoj ČR, Praha) Mgr. Branislav A. Švorc, PhD., šéfredakor (Vydavaeľsvo UNIVERSUM, Prešov) PhDr. Veronika Trsianska, PhD. (Úsav sredoeurópskych jazykov a kulúr FSŠ UKF, Nira) Mgr. Veronika Zuskáčová (Geografický úsav, Masarykova univerzia, Brno) VYDAVATEĽ Vydavaeľsvo UNIVERSUM, spol. s r. o. Javorinská 26, 8 1 Prešov Slovenská republika Mladá veda / Young Science. Akékoľvek šírenie a rozmnožovanie eu, foografií, údajov a iných informácií je možné len s písomným povolením redakcie.

3 EXAMPLE OF MACROSCOPIC MODELING IN HIGH SCHOOLS PŘÍKLAD MAKROSKOPICKÉHO MODELOVÁNÍ NA STŘEDNÍCH A VYSOKÝCH ŠKOLÁCH Jana Vysoká 1 Auorka působí jako asisen na Kaedře informaiky a přírodních věd při Vysoké škole echnické a ekonomické v Českých Budějovicích. Ve svém výzkumu se věnuje émaice maemaického modelování a je auorkou diserační práce s názvem Maemaické modelování ve výuce na sředních školách. The auhor works as a lecurer a he Deparmen of Informaics and Naural Sciences a he Insiue of Technology and Business in České Budějovice. In her research she deals wih he opic of mahemaical modeling and is he auhor of a disseraion eniled Mahemaical Modeling in Teaching in High Schools. Absrac This paper offers a demonsraion of how some advanced opics discussed a universiies can be presened in an accessible way o sudens in secondary schools or in high schools where mahemaical modeling is no par of he lesson. The main seleced ask is focused on simple eample of mahemaical modeling in solving simple ranspor asks. In his work, we will pay aenion o he issue of raffic flow modeling from a macroscopic perspecive. Firs, we derive a mahemaical model in he form of parial differenial equaions and hen we will focus on solving his equaion using he mehod of characerisics. The inerpreaion is presened in a way ha should be easily grasped by sudens. Key words: mahemaical modeling, parial differenial equaion, mehod of characerisics Absrak Teno článek nabízí ukázku oho, jak lze někerá émaa probíraná na vysokých školách zpřísupni sudenům na sředních školách nebo vysokých školách, kde není maemaické modelování součásí výuky. Hlavní cíl je zaměřen na jednoduchý příklad maemaického modelování při řešení snadné dopravní úlohy. V éo práci budeme věnova pozornos problemaice modelování dopravního oku z makroskopického hlediska. Nejprve odvodíme maemaický model ve formě parciální diferenciální rovnice a pak se zaměříme na řešení éo 1 Affilaion: RNDr. Jana Vysoká, The Insiue of Technology and Business in České Budějovice, Okružní 1, 371 České Budějovice, The Czech republic vysoka@mail.vsecb.cz 234 hp://

4 rovnice meodou charakerisik. Výklad je prezenován způsobem, s jehož pomocí by sudeni mohli problemaiku pochopi. Klíčová slova: Maemaické modelování, parciální diferenciální rovnice, meoda charakerisik Inroducion Mahemaical modeling is a consanly evolving modern discipline, which is used in many areas of human aciviy. Modeling can be used in various disciplines e.g. echnical, biological, economical or social. We mee mahemaical models every day ogeher wih weaher forecasing, he modeling of ranspor neworks, populaion, invenory managemen, groundwaer flow, he deerminaion of flood areas during floods and civil engineering works ec. Models have become an inegral par of a ool for forecasing he developmen of various processes. Using he appropriae mahemaical model has many advanages and helps us o undersand easier comple phenomena and processes and cones, allows he simulaion of differen possible oucomes. Therefore, modeling simple phenomena using procedures and mehods deserves o be included ino he curricula of seleced schools. The effor of some schools is o enhance he eaching he subjecs using of commercial programs for eample Malab or Saisics o demonsrae problems in various subjecs such as mahemaics, physics, biology or chemisry. This maerial could serve as an epansion of preparaion of learning maerials designed for universiy sudens who have chosen advanced mahemaics as an elecive, or for high school sudens who are engaged in mahemaics beyond compulsory curriculum. From here i is only a small sep o he numerical modeling. The graphs in he paper were consruced using a graphical sofware Derive 6 (Kuzler, Kokol-Voljc, 23). The densiy of vehicles and he flow of vehicles The movemen of vehicles on he highway can be described analogously wih he movemen of fluid a fluid paricle moion. The flow of vehicles on he highway can be (from a cerain angle and under given condiions) undersood as he flow of fluid paricles (e.g. when viewed on he road from he aircraf). Therefore i will be necessary o disinguish he concep of a macroscopic speed as he speed of he flow of vehicles and a microscopic speed as he speed of individual vehicles. The moion of fluid paricles is influenced by he movemen of surrounding paricles bu he driving of he driver depends on oher facors. The driver adjuss he speed accordingly o raional consideraions of he siuaion in fron of him or behind him and perceives oher facors which may have impac on he speed. The paricles of he fluid may collide in is movemen wih oher paricles; he driver of he vehicle ries o avoid he collision wih anoher vehicle, of course. Because vehicles represen unis in our model, we will assume ha hey have he idenical lengh. Furher we will assume for simpliciy ha he vehicle raffic will follow only in one direcion. In order o deermine he number of vehicles on he secion of he highway being measured per a given uni of ime, we have o ake ino accoun, firsly, he overall densiy of vehicles and, secondly, heir speed. The densiy of vehicles on he moniored secion is a concep ha would deserve a more deailed analysis. The densiy of vehicles could be defined as he raio of he number of vehicles o he uni lengh of he road and could be will idenified by he symbol : 235 hp://

5 The densiy of vehicles = he number of vehicles / he uni lengh of he road. For eample, if he observed road secion is 1 m long and if 35 vehicles are locaed in his secion, and hen he densiy of vehicles is he value. In he case, ha measured road segmen is blank, hen. If 1.35, hen we are alking abou a raffic jam. We will deermine he densiy of vehicles in a given place and in a given ime. We will denoe i,, where he variable depends as a funcion of wo real variables in he form on he variable. Anoher imporan concep of deermining he characer of road raffic is he flow of vehicles, which can be defined by his way: The flow of vehicles = he number of vehicles / he ime uni. The flow of vehicles can be defined in many ways, he simples one is o define he flow of vehicles as he produc of he densiy and he vehicle speed a which he vehicles are moving. We assume always ha he speed of he observed group of vehicles is consan. If we inroduce a uniform label for he speed of vehicles as v in he e and we use he symbol for he flow of vehicles symbol, hen yields v. (1) In he even, ha he vehicle speed, will be changed as a resul of oher condiions such as a raffic acciden or a speed limiaion he speed or changes in he movemen of vehicles joined wih ligh signals he flow funcion need no be in he form of he linear funcion in he form (1) and may depend on oher facors. Characerisic of he road raffic Wha means a coninuous raffic, a limied raffic or a raffic jam in a pracice? We will assume ha each vehicle has a lengh of 4.5 meers and drivers ry o keep he safe disance (a ime lag of 2 seconds). The rule of he disance of 2 seconds means he following siuaion: Once while driving he vehicle, we will focus on some fied poin in he roadway e.g. a ree or a bollard and from he momen his poin misses vehicle in fron of you, we deduc 2 seconds. If we miss he same poin in less han ha, hen a conrolled vehicle is no in a safe disance from he vehicle ahead. If he vehicle is raveling in he speed of 9 km/h, han i is a disance of approimaely 5 meers. In he secion of 1 m long here are a maimum of 18 vehicles in any given ime. We can hen disinguish he following siuaions: 1. If he disance beween vehicles is 5 m, hen on a srech of road 1 m long i can move freely from o 18 vehicles. The size of he flow of vehicles will be maimized if he vehicles are no limiing each oher during driving. Therefore, he maimum flow of vehicles will be 1,62 vehicles per 1 hour. Whereby, in his case he densiy is greaer he flow is greaer. In pracice, we are alking abou he coninuous raffic. 236 hp://

6 2. If a greaer number of vehicles i.e. in he range 19 o 1 vehicles will move on he same road segmen, hen each vehicle is locaed in he free secion of a leas 1 m long. In his case, he driver has o keep an aenion o he siuaion before and afer he conrolled vehicle and adequaely respond o any changes. The speed of vehicles will change in cerain secions of road and he raffic can be alernaely compressed or dilued. The more is raffic dense, he greaer he flow of vehicles is less. The raffic is hen characerized as limied. 3. If he moniored secion of road conains more han 1 vehicles, han he driver is forced o sop. This hen leads o a local raffic jam. If he road conains approimaely 2 vehicles, han here is a global jam on he road and all vehicles sand. The flow of vehicles is zero. We could give illusraive eamples of he pracice and herefore i will be appropriae o reasonably define he specific form of he funcion of he flow of vehicles. Suppose ha he maimum speed is he same for all vehicles, le i be denoed by he symbol. We will omi some reflecions on he psychological aspecs of driving in a row, for eample, if he driver has he vehicle ahead far, hen he will accelerae he ride. Based on empirical eperience in pracice i can be assumed ha, if he roadway is empy, herefore he vehicle can move in he maimum speed. Conversely, if he road is full, here is a raffic jam (and he vehicles sop). These properies has for eample a funcion, (Hokr, 25) in he form v ma v v ma 1. (2) ma Then we can inroduce he flow funcion in he form where yields v 1 ma, (3) ma v 2 ma v. ma Macroscopic model of he vehicle flow Le us monior he secion of he road on which he vehicle moves in he consan speed in he ime inerval. The densiy of vehicles in he given segmen and ime is specified. The basic consideraion will be based on he fac ha he change of he number of vehicles in he secion being measured by he uni of ime corresponds o he difference beween he number of cars ha ener and leave he given segmen. We deermine he number of vehicles in he ime and in he ime, heir difference will represen a change of he densiy of vehicles deeced in he ime and we calculae he number of he vehicles in he ime which leave he measured secion in he same ime a change of he flow funcion measured in he ma 237 hp://

7 given place. If we denoe he change of densiy in he considered ime by he symbol he change of he flow funcion in he given secion, where locaion of he vehicle on he highway, hen under he previous consideraions is. and is a variable describing he This relaionship is possible o rewrie under cerain assumpions in he form of parial differenial equaions of he firs order in he following form v. (4) This equaion could be simply verbally eplained as follows: The change of he densiy of vehicles wih respec o a cerain ime inerval is idenical o he change of he flow funcion measured in he given secion. If he flow funcion is no given in a linear form, han he equaion (4) is changing ino he form. (5) Equaion (5) can be rewrien under cerain assumpions as ]. [ (6) We will be able o deermine he specific soluion of he given ask afer we will know he disribuion of he densiy of vehicles in a cerain place and ime. The mos common and mos naural choice of condiions is he iniial disribuion of he densiy of vehicles, herefore he descripion of he siuaion in he beginning; le us denoe his funcion by he,. The problem in he form [ ] ogeher wih he iniial symbol condiion, is called he iniial problem (Cauchy problem). The mehod of characerisics One of he ways how o describe he soluion of he menioned problem is he way of using he graphical presenaion wih help of so-called characerisics. The equaion (4) wih he, has a simple form, and herefore is soluion can be easily iniial condiion guessed. A search funcion of variables and, v, > has he form This fac can be easily verified by a es. If we are differeniaing according o variable or, hen we are aking he remaining variable as consan and differeniaion rules remain unchanged. The funcion is consan on he all lines wih slopes v in he form v c, where c is a real consan. In oher words, he densiy of he given iniial condiions is ransmied in a ime on a sraigh line wih he slope v or he derivaive of he funcion in 238 hp://

8 he direcion v is equal o zero. The inersecion of his line and he ais is denoed by he symbol. These lines, on which densiy is consan, are called characerisic lines or characerisic corresponding o he given problem. A drawing characerisics is quie simple and we can ge a good idea abou solving a given ask in a predeermined ime and place by he using he graphical represenaion. The imporance of characerisics remains unchanged even when we are solving a nonlinear equaion (5). Characerisics are defined as a sraigh lines wih he slope. The characerisic line wih he slope equal o he value inersecs he ais in he poin and has an equaion in he form. I is easy o verify by a simply subsiuion ha he given funcion of he form, saisfies he equaion (5). The value of he soluion of he iniial condiions, in he poins of his line. There are some eamples of simple siuaions: is equal o he corresponding value 1. Vehicles are going equally in he consan speed, he raffic densiy is no changed, see fig Vehicles are no moving and saying in he row, his is an eample illusraing he raffic jam, see fig Vehicles are gradually slowing, heir densiy is increasing, and slopes of characerisics are increasing unil here is a raffic jam, see fig Vehicles are saring o creep, heir densiy is decreasing, slopes of characerisics are decreasing, see fig. 4. Fig. 1 Fig hp://

9 Fig. 3 Fig. 4 Combining all hese eamples i is possible o describe he complicaed siuaion as are he raffic direced by raffic lighs or he raffic on secions wih reduced speeds. To be able o solve complicaed cases of iniial condiions, i would be advisable o sop a he cases menioned in eamples 3 and 4. Le us consider he ask in he form wih he iniial condiion in he form of he jump p, where p and l [ v] (7) for and l are given nonnegaive real numbers. This problem is called he Riemann problem. Then we could disinguish wo differen siuaions: for 1. l < p Characerisics for he value p l are illusraed in he red color and characerisics for he value in he blue color, see fig. 5. The figure 5 shows ha in he period beween red and blue areas (areas defined by inequaliy ) characerisics are "missing". If we complee an empy par by lines wih slopes of values beween l p l and p in he green color, han we ge coninuous informaion on he ransmied value of he densiy in he jump, see fig l > p Fig. 5 Fig. 6 The figure 7 again shows ha he characerisics are crossing; i means ha we canno deermine a unique value of a densiy a any ime. The direcion, in which he informaion on 24 hp://

10 jump will propagae from he firs inersecions of he characerisics (so called shock wave), mus be only one because he same informaion is moving in boh direcions. The slope of his single characerisic, which represens he propagaion of shock wave, see fig. 8, is given by so called Rankine-Hugonio condiion in he form l p s. (8) l p Fig. 7 Fig. 8 The derivaion of his condiion eends beyond he scope of his e bu if a suden is ineresed in his opic, we can recommend sudying relevan pages in (Drábek, Holubová, 27) or (Le Veque, 21, 22). Eercises: Draw he following siuaions by using characerisics. Moniored secion is 1 km long, in he firs measuring secion here is deeced he value of he densiy of vehicles 1 and in he second secion, which is measured from he poin B, he value of he densiy of vehicles 2. Skech a soluion and he poin B se in a disance of 1 m in he posiive par of he ais. 1. 1=.21, 2 =.2, v ma = 14 m/s 2. 1=.2, 2 =.46, v ma = 8.3 m/s Soluion: 1. The densiy 1 =.21 represens 21 vehicles per 1 km and vehicles move in he given speed v ma = 14 m/s, which is 5 km/h. The densiy 2 =.2 represens 2 vehicles per 1 km or he raffic jam, vehicles are moving in he zero velociy. Therefore, i is he firs described case, where <. I is necessary o calculae he value of he slopes of characerisics l p corresponding o he given densiies 1 and 2 which means o deermine he value according o (8). Subsiuing given values ino a following relaionship we will ge his resul: 2v v ma,.2 ma ma 241 hp://

11 Then we can calculae he direcion of propagaion of he shock wave as follows.26 s The resuling graph of characerisics is shown in fig. 9. Fig Anoher eample represens a case, where vehicles are in he raffic jam and saring o move aking off a a maimum speed equal o 8.3 m/s = 3 km/h, which is he second described alernaive wih he condiion. The siuaion can be illusraed using l > previous calculaions already easily, see fig. 1. p Fig. 1 The iniial condiion can be defined in a more comple form, where here may be a combinaion of all he evens described above. Moniored secion of road wih a lengh of 1 m is divided ino hree pars: m - 5 m wih a maimum speed of 5 km/ h, vehicles are complying wih he safey disance of 2 5 m - 6 m wih a maimum densiy of vehicles simulaing a row before raffic lighs (maimum speed km/ h) and he secion 6 m - 1 m wih a maimum speed of 3 km/ h. The iniial condiion is specified in he firs segmen by value 1 =.21 vehicles / m (he red color), in he second secion of he road 2 =.2 vehicles/m (he blue color) and in he las par is he iniial densiy value deermined 242 hp://

12 by he value 3 =.46 vehicles/m (he black color), see fig. 11. In he ime s he raffic ligh urns o be green and he vehicles sar o move. To formulae his problem we have deliberaely used he daa of he previous eamples (Jelínek, Vysoká, 213, 214). Fig. 11 Fig. 12 The complee graph can deermine he approimae ime and place where here are changes in he flow of he raffic flow. A he poin where he red characerisic inersecs he green characerisic ransferring he value of densiy approimaely equal o.2 vehicle/m ge in he sum, is he slope of ransiion line equals zero (his condiion occurs in ime approimaely 18 s a he poin approimaely 16 m from he sar). In he ime abou 36 s vehicles are already limied and can freely pass around a raffic ligh. The map of all characerisics given by he graph in fig. 12 was necessary o be downloaded in oher widh, because i would no be appropriaely scaled in he e and i was necessary o draw he inverse characerisics o he coordinae sysem, (previous graphs were consruced ino he coordinae sysem, ) in order o ge well arranged resuls. Therefore, slopes of characerisics quie precisely do no correspond o he realiy, bu he essence of he problem is capured. Conclusion The purpose of his paper was o sugges an epansion of preparaion of learning maerials designed for sudens of seleced schools. In his work, we focused on solving easy raffic problem by mahemaical modeling from a macroscopic perspecive. Inerpreaion was 243 hp://

13 presened in a way ha should be easily grasped by sudens - universiy sudens who have chosen advanced mahemaics as an elecive, or high school sudens who are engaged in mahemaics beyond compulsory curriculum Firs, we derived a mahemaical model in he form of parial differenial equaions. In order o apply he equaion for an illusraive eample of he pracice i was be necessary o inroduce and undersand some imporan conceps such as he raffic densiy, he raffic jam, he coninuous and reduced raffic or he flow funcion. Then we focused on solving his equaion using he mehod of characerisics. The heoreical par is complemened by several pracical eamples describing siuaions encounered in everyday raffic. Teno článok odporúčal na publikovanie vo vedeckom časopise Mladá veda: doc. Vladimíra Perášková, Ph.D. References 1. DRÁBEK, Pavel, HOLUBOVÁ, Gabriela, 27, Elemens of Parial Differenial Equaions, Berlin : Waler de Gruyer. ISBN HOKR, Milan, 25. Transporní procesy. Učební e, Technická univerzia v Liberci. 3. JELÍNEK, Jiří, VYSOKÁ, J., 213. Modelování dynamiky dopravního proudu, Silnice Železnice, Osrava - Víkovice: Konsrukce Media. ISSN X. 4. JELÍNEK, Jiří, VYSOKÁ, Jana, 213. Porovnání modelů pro výpoče husoy dopravního oku, Silnice Železnice, Osrava - Víkovice: Konsrukce Media ISSN X. 5. JELÍNEK, Jiří, VYSOKÁ, Jana, 214. O podchodach v modelirovanii ranspornoj dinamiki, Vesnik Asrachanskogo gosudarsvennogo echničeskogo universiea, Asrachan : Asrachanskij gosudarsvennyj echničeskij universie. ISSN KUTZLER, B., KOKOL-VOLJC, V., 23. Úvod do Derive 6, OEG, Ausria. 7. Le VEQUE R., 22. Finie Volume Mehods for Hyperbolic Problems, Cambridge Universiy. ISBN Le VEQUE R., 21. Some Traffic Flow Models Illusraing Ineresing Hyperbolic Behavior. 244 hp://