Mon Apr 2: Laplace transform and initial value problems like we studied in Chapter 5

Transcription

1 Mah Week 2 April 2-6 coninue.-.3; alo cover par of.4-.5, EP 7.6 Mon Apr 2:.-.3 Laplace ranform and iniial value problem like we udied in Chaper 5 Announcemen: Warm-up Exercie:

2 Recall, The Laplace ranform i a linear ranformaion " " ha conver piecewie coninuou funcion f, defined for and wih a mo exponenial growh ( f Ce M for ome value of C and M), ino funcion F defined by he ranformaion formula F = f f e d. Noice ha he inegral formula for F i only defined for ufficienly large, and cerainly for M, becaue a oon a M he inegrand i decaying exponenially, o he improper inegral from = o converge. The convenion i o ue lower cae leer for he inpu funcion and (he ame) capial leer for heir Laplace ranform, a we did for f and F above. Thu if we called he inpu funcion x hen we would denoe he Laplace ranform by X.

3 Exercie ) (o review) Ue he able enrie we compued la Wedneday, o compue a) 4 5 co 3 2e 4 in 2 b) f F f Ce M f e d for M c f c 2 f 2 c F c 2 F 2 e ( ( a co k in k 2 k 2 ( k 2 k 2 ( e a co k a a 2 k 2 ( a e a in k k a 2 k 2 a f F f f 2 F f f Laplace ranform able

4 Exercie 2) (o review) Ue Laplace ranform o olve he IVP for an underdamped, unforced ocillaor DE. Compare o Chaper 5 mehod. x 6 x 34 x = x = 3 x =

5 We'll fill in more able enrie oday. (Compare o fron cover of your ex, which conain hi informaion bu maybe more compacly.) f, wih f Ce M F f e d for M verified c f c 2 f 2 c F c 2 F 2 2 n, n e ( n! n ( co k in k coh k inh k e a co k e a in k f f f n, n f d f 2 f n f, n f co k 2 k in k 2 k 2 ( k 2 k 2 ( 2 k 2 ( k k 2 k 2 ( k a a 2 k 2 ( a k a 2 k 2 a F f 2 F f f n F n f... f n F F F n F n F d 2 k 2 2 k k 2 2

6 2 k 3 in k k co k 2 k 2 2 e a f F a e a n e a, n a 2 n! a n Laplace ranform able work down he able... 3a) coh k = 3b) inh k = 2 k 2 k 2 k 2.

7 Exercie 4) Recall we ued inegraion by par on Wedneday o derive g = g g. Ue ha ideniy o how a) f = 2 F f f, b) f = 3 F 2 f f f, c) f n = n F n f n 2 f... f n, n. d) f d = F. Thee are he ideniie ha make Laplace ranform work o well for iniial value problem uch a we udied in Chaper 5... wih Laplace ranform he "free parameer" when you wrie down he oluion x = x P x H are exacly he iniial value for he differenial equaion, raher han he linear combinaion coefficien in he general homogeneou oluion, o you definiely ave a ep here, in olving IVP.

8 Exercie 5) Find 2 4 a) uing he reul of 4d. b) uing parial fracion. Exercie 6) Show n n! =, n, uing he reul of 4, namely n f n = n F n f n 2 f... f n, n.

9 Mah Tue Apr Laplace ranform, and applicaion o DE IVP, including Chaper 5. Today we'll coninue o fill in he Laplace ranform able, and o ue he able enrie o olve linear differenial equaion. One focu oday will be o review parial fracion, ince he able enrie are e up preciely o how he invere Laplace ranform of he componen of parial fracion decompoiion. Announcemen: Warm-up Exercie:

10 F = f f e d. Exercie ) Check why hi able enry i rue - noice ha i generalize how he Laplace ranform of co k, in k are relaed o hoe of e a co k, e a in k : e a f F a Exercie 2) Ue he able enry n, n n! n and Exercie o ge he able enry n e a n! a n

11 A harder able enry o underand i he following one - go hrough hi compuaion and ee why i eem reaonable, even hough here' one ep ha we don' compleely juify. The able enry i f F We recognize ha i will be helpful for applicaion problem where reonance occur. Here' how we ge i: F = f d d F = d d f e d = f e d d d f e d. I' hi la ep which i rue, bu need more juificaion. We know ha he derivaive of a um i he um of he derivaive, and he inegral i a limi of Riemann um, o hi ep doe a lea eem reaonable. The re i raighforward: d d f e d = f e d = f.

12 For reonance and oher applicaion... Exercie 4) Ue f = F 2 k 2 a) co k = 2 k 2 2 b) 2 k in k = 2 k 2 2 c) Then ue a and he ideniy 2 k 2 2 = 2 k 2 2 k 2 2 k 2 2 k k 2 2 o verify he able enry 2 k 2 2 = 2 k 2 in k co k. k

13 Noice how he Laplace ranform able i e up o ue parial fracion decompoiion. And be amazed a how i le you quickly deduce he oluion o imporan DE IVP, like hi reonance problem: Exercie 5a) Ue Laplace ranform o wrie down he oluion o 2 x x = F in x = x x = v. wha phenomenon do he oluion o hi IVP exhibi? (Compare, in your homework you will re-olve he IVP when he forcing i F co. We worked prey hard in Chaper 5, o derive hi general oluion formula.) 5b) Ue Laplace ranform o olve he general undamped forced ocillaion problem, when x 2 x = F in x = x : x = v wha phenomenon o he oluion o hi IVP exhibi when you'll do hi wih forcing F in. Thi will mirror work we did in Chaper 5.) (bu )? (In your homework,

14 f, wih f Ce M F f e d for M verified c f c 2 f 2 c F c 2 F 2 2 n, n n! n ( e ( a co k in k coh k inh k e a co k e a in k 2 k 2 ( k 2 k 2 ( 2 k 2 ( k k 2 k 2 ( k a a 2 k 2 ( a k a 2 k 2 a e a f F a f f f n, n f d f 2 f n f, n f co k F f 2 F f f n F n f... f n F F F n F n F d 2 k 2 2 k 2 2

15 2 k in k 2 k 3 in k k co k e a n e a, n 2 k k 2 2 a 2 n! a n Laplace ranform able

16 The pendulum applicaion ha we didn' cover carefully in Chaper 5...we'll ue hi for a equence of example uing Laplace ranform over he nex everal lecure.

17 Mah Wed Apr 4.3 parial fracion;.5 piecewie forcing. Announcemen: Warm-up Exercie:

18 parial fracion occur naurally when olving iniial value problem wih Laplace ranform, a we've already een. Here' a moderaely-involved example: Exercie ) Solve he following IVP. Ue hi example o recall he general parial fracion algorihm. x 4 x = 8 e 2 x = x =

19 Wolfram alpha can check mo of your ep, once you've e up he problem. Or, if i' a ridiculou problem don' ry o even work i by hand: Exercie 2a) Wha i he form of he parial fracion decompoiion for X = b) Check exac number wih Wolfram alpha 2c) Wha i x = X? 2d) Have Wolfram alpha compue he invere Laplace ranform direcly. Noice ha being fluid wih Euler' formula i ueful.

20 .5 Piecewie coninuou forcing funcion...e.g. urning he forcing on and off. The following Laplace ranform maerial i ueful in yem where we urn forcing funcion on and off, and when we have righ hand ide "forcing funcion" ha are more complicaed han wha undeermined coefficien can handle. We will coninue hi dicuion on Friday, wih a few more able enrie including "he dela (impule) funcion". f wih f Ce M F f e d for M commen u a uni ep funcion e a for urning componen on and off a = a. f a u a e a F more complicaed on/off f f d F G "convoluion" for invering produc of Laplace ranform The uni ep funcion wih jump a = i defined o be, u =,. IThi funcion i alo called he "Heaviide" funcion, e.g. in Maple and Wolfram alpha. In Wolfram alpha i' alo called he "hea" funcion. Oliver Heaviide wa a an accomplihed phyici in he 8'. The name i no becaue he graph i heavy on one ide. :-) hp://en.wikipedia.org/wiki/oliver_heaviide > wih plo : plo Heaviide, = 3..3, color = green, ile = `graph of uni ep funcion` ; graph of uni ep funcion Noice ha echnically he verical line hould no be here - a more precie picure would have a olid poin a, and a hollow circle a,, for he graph of u. In erm of Laplace ranform inegral definiion i doen' acually maer wha we define u o be.

21 Then, a ; i.e. a u a =, a ; i.e. a and ha graph ha i a horizonal ranlaion by a o he righ, of he original graph, e.g. for a = 2: Exercie 3) Verify he able enrie u a uni ep funcion e a for urning componen on and off a = a. f a u a e a F more complicaed on/off

22 Exercie 4) Conider he funcion f which i zero for 4 and wih he following graph. Ue lineariy and he uni ep funcion enry o compue he Laplace ranform F. Thi hould remind you of a homework problem from he aignmen due omorrow - alhough you're aked o find he Laplace ranform of ha ep funcion direcly from he definiion. In your nex week' homework aignmen you will re-do ha problem uing uni ep funcion. (Of coure, you could alo check your anwer in hi week' homework wih hi mehod.)

23 Exercie 5a) Explain why he decripion above lead o he differenial equaion iniial value problem for x x x =.2 co u x = x = 5b) Find x. Show ha afer he paren op puhing, he child i ocillaing wih an ampliude of exacly meer (in our linearized model).

24 Picure for he wing: > plo plo. in, =.. Pi, color = black : plo2 plo Pi in, = Pi..2 Pi, color = black : plo3 plo Pi, = Pi..2 Pi, color = black, lineyle = 2 : plo4 plo Pi, = Pi..2 Pi, color = black, lineyle = 2 : plo5 plo., =.. Pi, color = black, lineyle = 2 : plo6 plo., =.. Pi, color = black, lineyle = 2 : diplay plo, plo2, plo3, plo4, plo5, plo6, ile = `advenure a he winge` ; 3 3 advenure a he winge Alernae approach via Chaper 5: ep ) olve for. ep 2) Then olve and e x = y for. x x =.2 co x = x = y y = y = x y = x

25 f, wih f Ce M F f e d for M verified c f c 2 f 2 c F c 2 F 2 2 n, n n! n ( e ( a co k in k coh k inh k e a co k e a in k e a f u a f a u a a f f f n, n 2 k 2 ( k 2 k 2 ( 2 k 2 ( k k 2 k 2 ( k a a 2 k 2 ( a k a 2 k 2 a F a e a e a F e a F f 2 F f f n F n f... f n

26 f d F f 2 f n f, n f co k 2 k in k 2 k 3 in k k co k e a n e a, n F F n F n F d 2 k 2 2 k k k 2 2 a 2 n! a n f g d F G f wih period p e p p f e d Laplace ranform able

27 Mah Fri Apr 6.5, EP7.6 piecewie and implule forcing. Announcemen: Warm-up Exercie:

28 Laplace able enrie for oday. f wih f Ce M F f e d for M commen u a uni ep funcion e a for urning componen on and off a = a. f a u a e a F more complicaed on/off a e a uni impule/dela "funcion" EP 7.6 impule funcion and he operaor. Conider a force f acing on an objec for only on a very hor ime inerval a a, for example a when a ba hi a ball. Thi impule p of he force i defined o be he inegral p a a f d and i meaure he ne change in momenum of he objec ince by Newon' econd law m v = f a a m v d = m v a a a = a f d = p Since he impule p only depend on he inegral of f, and ince he exac form of f i unlikely o be known in any cae, he eaie model i o replace f wih a conan force having he ame oal impule, i.e. o e f = p d a, where d a, i he uni impule funcion given by = p., a, a a d a, =, a. Noice ha a d a, d = a d =. a a Here' a graph of d 2,., for example:

29 2 3 4

30 Since he uni impule funcion i a linear combinaion of uni ep funcion, we could olve differenial equaion wih impule funcion o-conruced. A far a Laplace ranform goe, i' even eaier o ake he limi a for he Laplace ranform d a,, and hi effecively model impule on very hor ime cale. d a, = u a u a d a, = e a e a = e a e. In Laplace land we can ue L'Hopial' rule (in he variable ) o ake he limi a : lim e a e = e a e lim = e a. The reul in ime pace i no really a funcion bu we call i he "dela funcion" a anyway, and viualize i a a funcion ha i zero everywhere excep a = a, and ha i i infinie a = a in uch a way ha i inegral over any open inerval conaining a equal one. A explained in EP7.6, he dela "funcion" can be hough of in a rigorou way a a linear ranformaion, no a a funcion. I can alo be hough of a he derivaive of he uni ep funcion u a, and hi i conien wih he Laplace able enrie for derivaive of funcion. In any cae, hi lead o he very ueful Laplace ranform able enry a uni impule funcion e a for impule forcing

31 Exercie ) Revii he wing from Wedneday' noe and olve he IVP below for x. In hi cae he paren i providing an impule each ime he child pae hrough equilibrium poiion afer compleing a cycle. x x = x = x =.

32 > > wih plo : plo plo. in, =.. Pi, color = black : plo2 plo Pi in, = Pi..2 Pi, color = black : plo3 plo Pi, = Pi..2 Pi, color = black, lineyle = 2 : plo4 plo Pi, = Pi..2 Pi, color = black, lineyle = 2 : plo5 plo., =.. Pi, color = black, lineyle = 2 : plo6 plo., =.. Pi, color = black, lineyle = 2 : diplay plo, plo2, plo3, plo4, plo5, plo6, ile = `Wedneday advenure a he winge` ; 3 3 Wedneday advenure a he winge > impule oluion: five equal impule o ge ame final ampliude of meer - Exercie : > f.2 Pi um Heaviide k 2 Pi in k 2 Pi, k =..4 : > plo f, =..2 Pi, color = black, ile = `lazy paren on Friday` ; > Or, an impule a = and anoher one a =. > g.2 Pi 2 in 3 Heaviide Pi in Pi : > plo g, =..2 Pi, color = black, ile = `very lazy paren` ; > lazy paren on Friday very lazy paren