Elementary Differential Equations and Boundary Value Problems

Transcription

1 Elemenar Differenial Equaions and Boundar Value Problems Boce. & DiPrima 9 h Ediion Chaper 1: Inroducion คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 ผศ.ดร.อร ญญา ผศ.ดร.สมศ กด วล ยร ชต 1

2 Topics Some Basic Mahemaical Models; Direcion Fields Soluions of Some Differenial Equaions Classificaion of Differenial Equaions 2

3 1.1 Basic Mahemaical Models; Direcion Fields Elemenar Differenial Equaions and Boundar Value Problems 9 h ediion b William E. Boce and Richard C. DiPrima 2009 b John Wile & Sons Inc. Differenial equaions are equaions conaining derivaives. The following are examples of phsical phenomena involving raes of change: Moion of fluids Moion of mechanical ssems Flow of curren in elecrical circuis Dissipaion of hea in solid objecs Seismic waves Populaion dnamics A differenial equaion ha describes a phsical process is ofen called a mahemaical model. 3

4 Some Applicaions of Differenial Equaion 4

5 Example 1: Free Fall Formulae a differenial equaion describing moion of an objec falling in he amosphere near sea level. Variables: ime veloci v Newon s 2 nd Law: F ma m(dv/d) ne force Force of gravi: F mg downward force Force of air resisance: F γ v upward force Then m dv d mg γ v Taking g 9.8 m/sec 2 m 10 kg γ 2 kg/sec we obain dv d v 5

6 v v Example 2: Skeching Direcion Field (1 of 3) Using differenial equaion and able plo slopes (esimaes) on axes below. The resuling graph is called a direcion field. (Noe ha values of v do no depend on.) v v'

7 Example 2: Direcion Field Using Maple (2 of 3) v v Sample Maple commands for graphing a direcion field: wih(deools): DEplo(diff(v())9.8-v()/5v() 0..10v0..80sepsize.1colorblue); When graphing direcion fields be sure o use an appropriae window in order o displa all equilibrium soluions and relevan soluion behavior. 7

8 v v Example 2: Direcion Field & Equilibrium Soluion (3 of 3) Arrows give angen lines o soluion curves and indicae where soln is increasing & decreasing (and b how much). Horizonal soluion curves are called equilibrium soluions. Use he graph below o solve for equilibrium soluion and hen deermine analicall b seing v' 0. Se v 0 : v 9.8 v 0.2 v

9 Equilibrium Soluions In general for a differenial equaion of he form a b find equilibrium soluions b seing ' 0 and solving for : ( ) b a Example: Find he equilibrium soluions of he following ( + 2) 9

10 Example 3: Mice and Owls (1 of 2) Consider a mouse populaion ha reproduces a a rae proporional o he curren populaion wih a rae consan equal o 0.5 mice/monh (assuming no owls presen). When owls are presen he ea he mice. Suppose ha he owls ea 15 per da (average). Wrie a differenial equaion describing mouse populaion in he presence of owls. (Assume ha here are 30 das in a monh.) Soluion: dp d 0.5 p

11 Example 5: Direcion Field (2 of 2) Discuss soluion curve behavior and find equilibrium soln. p 0.5 p

12 Seps in Consrucing Mahemaical Models Using Differenial Equaions 1. Idenif independen and dependen variables and assign leers o represen hem. 2. Choose he unis of measure for each variable. 3. Ariculae he basic principle ha underlies or governs he problem ou are invesigaing. This requires our being familiar wih he field in which he problem originaes. 4. Express he principle or law in he previous sep in erms of he variables idenified a he sar. This ma involve he use of inermediae variables relaed o he primar variables. 5. Make sure each erm of our equaion has he same phsical unis. 6. The resul ma involve one or more differenial equaions. 12

13 1.2 Soluions of Some Differenial Equaions Elemenar Differenial Equaions and Boundar Value Problems 9 h ediion b William E. Boce and Richard C. DiPrima 2009 b John Wile & Sons Inc. Recall he free fall and owl/mice differenial equaions: v v p 0.5 p 450 These equaions have he general form ' a - b We can use mehods of calculus o solve differenial equaions of his form. 13

14 Example 1: Mice and Owls (1 of 3) To solve he differenial equaion p 0.5 p 450 we use mehods of calculus as follows. dp d 0.5 ( p 900) dp / d p p dp d ln p C p 900 e 0.5+ C p 900 ± e 0.5 e C p ke 0.5 k ± e C Thus he soluion is p ke 0.5 where k is a consan. 14

15 p ke

16 Example 1: Inegral Curves (2 of 3) Thus we have infiniel man soluions o our equaion p 0.5 p 450 p since k is an arbirar consan. 0.5 ke Graphs of soluions (inegral curves) for several values of k and direcion field for differenial equaion are given below. Choosing k 0 we obain he equilibrium soluion while for k 0 he soluions diverge from equilibrium soluion. 16

17 Example 1: Iniial Condiions (3 of 3) A differenial equaion ofen has infiniel man soluions. If a poin on he soluion curve is known such as an iniial condiion hen his deermines a unique soluion. In he mice/owl differenial equaion suppose we know ha he mice populaion sars ou a 850. Then p(0) 850 and p( ) ke 0.5 p(0) ke 0 50 k Soluion : p( ) e

18 Soluion o General Equaion To solve he general equaion a b we use mehods of calculus as follows. d d a ln b a b / a b / a ± e a + C Thus he general soluion is a e b a + ke a where k is a consan. C d / d b / a a b / a b / a + ke a d b / a e a+ C k ± e C a d 18

19 Iniial Value Problem Nex we solve he iniial value problem a b (0) From previous slide he soluion o differenial equaion is b + a a ke Using he iniial condiion o solve for k we obain b 0 b (0) 0 + ke k 0 a a and hence he soluion o he iniial value problem is 0 b a + 0 b e a a 19

20 Equilibrium Soluion To find he equilibrium soluion se ' 0 & solve for : se b a b 0 ( ) a From he previous slide our soluion o he iniial value problem is: b + a Noe he following soluion behavior: 0 If 0 b/a hen is consan wih () b/a If 0 > b/a and a > 0 hen increases exponeniall wihou bound If 0 > b/a and a < 0 hen decas exponeniall o b/a If 0 < b/a and a > 0 hen decreases exponeniall wihou bound If 0 < b/a and a < 0 hen increases asmpoicall o b/a b a a e 20

21 Example 2: Free Fall Equaion (1 of 3) อ างจาก slide5 Recall equaion modeling free fall descen of 10 kg objec assuming an air resisance coefficien γ 2 kg/sec: dv / d v Suppose objec is dropped from 300 m. above ground. (a) Find veloci a an ime. (b) How long unil i his ground and how fas will i be moving hen? For par (a) we need o solve he iniial value problem v v v(0) 0 Using resul from previous slide we have b a + b a.2 ( e ) a e v + 0 e v

22 Example 2: Graphs for Par (a) (2 of 3) The graph of he soluion found in par (a) along wih he direcion field for he differenial equaion is given below. v v v 49 1 (.2 e ) v(0) 0 22

23 Example 2 Par (b): Time and Speed of Impac (3 of 3) Nex given ha he objec is dropped from 300 m. above ground how long will i ake o hi ground and how fas will i be moving a impac? Soluion: Le s() disance objec has fallen a ime. I follows from our soluion v() ha s ( ) s(0) v( ) e C s( ) s( ) Le T be he ime of impac. Then s( T ) 49T + 245e.2T Using a solver T sec hence 0.2(10.51) v(10.51) 49 1 e ( ) f/sec + 245e + 245e C

24 1.3: Classificaion of Differenial Equaions Elemenar Differenial Equaions and Boundar Value Problems 9 h ediion b William E. Boce and Richard C. DiPrima 2009 b John Wile & Sons Inc. The main purpose of his course is o discuss properies of soluions of differenial equaions and o presen mehods of finding soluions or approximaing hem. To provide a framework for his discussion in his secion we give several was of classifing differenial equaions. 24

25 Ordinar Differenial Equaions When he unknown funcion depends on a single independen variable onl ordinar derivaives appear in he equaion. In his case he equaion is said o be an ordinar differenial equaions (ODE). The equaions discussed in he preceding wo secions are ordinar differenial equaions. For example dv d dp v 0.5 p 450 d 25

26 Parial Differenial Equaions When he unknown funcion depends on several independen variables parial derivaives appear in he equaion. In his case he equaion is said o be a parial differenial equaion (PDE). Examples: (wave equaion) ) ( ) ( (hea equaion) ) ( ) ( x u x x u a x u x x u α 26

27 Ssems of Differenial Equaions Anoher classificaion of differenial equaions depends on he number of unknown funcions ha are involved. If here is a single unknown funcion o be found hen one equaion is sufficien. If here are wo or more unknown funcions hen a ssem of equaions is required. For example predaor-pre equaions have he form du / d a u α uv dv / d cv + γ uv where u() and v() are he respecive populaions of pre and predaor species. The consans a c α γ depend on he paricular species being sudied. Ssems of equaions are discussed in Chaper 7. 27

28 Order of Differenial Equaions The order of a differenial equaion is he order of he highes derivaive ha appears in he equaion. Examples: d d + 1 e 4 2 d d u + u sin xx 0 We will be suding differenial equaions for which he highes derivaive can be isolaed: ( n) ( ) f 2 ( ( n 1) ) 28

29 Linear & Nonlinear Differenial Equaions An ordinar differenial equaion ( n) F is linear if F is linear in he variables ( n) Thus he general linear ODE has he form Example: Deermine wheher he equaions below are linear or nonlinear. (1) d d (4) 4 2 d d ( ) 0 ( n) ( n 1) a0( ) + a1( ) + + a ( ) g( ) (2) (5) + 3e u xx + uu 2 n sin 0 (3) (6) u xx 2 + sin( u) u 0 cos 29

30 Soluions o Differenial Equaions A soluion φ() o an ordinar differenial equaion saisfies he equaion: Example: Verif he following soluions of he ODE sin 2 ) ( cos ) ( sin ) ( ; ( ) 1) ( ) ( ) ( n n f ( ) 1) ( ) ( ) ( n n f φ φ φ φ φ 30

31 Soluions o Differenial Equaions Three imporan quesions in he sud of differenial equaions: Is here a soluion? (Exisence) If here is a soluion is i unique? (Uniqueness) If here is a soluion how do we find i? (Analical Soluion Numerical Approximaion ec) 31