x y exp λ'. x exp λ 2. x exp 1.

Transcription

1 egecosmcd Egevalue-egevector of the secod dervatve operator d /d hs leads to Fourer seres (se, cose, Legedre, Bessel, Chebyshev, etc hs s a eample of a systematc way of geeratg a set of mutually orthogoal bass vectors va the egevalues-egevectors to a operator We are ofte less terested the soluto of the ODE or the secod dervatve operator tself Ma obectve: we are ofte more terested the resultg set of egevectors that possess "ce" propertes (orthogoalty, recursve relatoshps, dervatve & tegral relatoshps, etc Istructor: am Su Wag Problem Statemet Fd the egevalues ad egevectors (egefuctos for the secod dervatve operator L defed =[- ] We wll use the terms egevectors ad egefuctos terchageably because fuctos are a type of vectors L y D y d y λ' y y( y( or ay symmetrc boudary codto d By "symmmetrc BC" above, we mea t as a part of a symmetrc lear trasformato L that does ot chage scalar product upo swappg a put vector to the operator Do ot cofuse the word "symmetrc" wth a "eve" fucto symmetrcal wrt to = ( L y, z ( y, L z defto of a symmetrc lear trasformato (we swap ay y ad z the LVS he fucto that satsfes the dfferetal porto s: y ep λ' d Chec: ep λ' λ' y d If we loosely cosder ust the ODE, there are a fty umber of egevalues ad fte umber of egevectors -- bascally ay epoetal fucto ep(λ'* wth ay costat λ' wll do Specfcally, the dervatve operator comes wth tal codtos or boudary codtos If we apply the gve symmetrcal boudary codto ad cofe ourselves strctly to oe-dmesoal varat subspace, there s o egevector that satsfes the boudary codto If we rela the oe-dmesoal varat subspace restrcto ad allow two-dmesoal varat subspace to be cosdered, we have the followg egevalues ad egevectors (depedg o the sg of λ' for λ' > y ep λ' ad y ep λ' whch s equvalet to a combato of y sh λ' ad y cosh λ' for λ' y ad y for λ' < y ep λ' ep λ ep λ y ep λ' ep λ ep λ y ep( λ ad y ep( λ where λ λ' he square root of a egatve umber s a magary umber he epoetal fuctos wth magary argumets, ep(*λ* ad ep(-*λ*, are equvalet to a combato of for λ' < y s( λ & y cos( λ λ because e cos( λ s( λ

2 egecosmcd I geeral, for each egevalue of the secod dervatve operator, there are two learly depedet, but ot ecessarly orthogoal egevectors he two egevectors s the result of beg a two-dmesoal varat subspace (Remember, two-dmesoal space meas, by defto, that there are two learly depedet vectors For each egevalue, the egevectors are ot uque We ca have dfferet combatos of two egevectors, ad the resultg learly depedet vectors are also egevectors Of these egevectors, hyperbolc se, ad are odd fuctos (atsymmetrc wth respect to =, f(-=-f( that do ot satsfy the symmetrc boudary codto of the dfferetal equato Hyperbolc cose fucto s eve (symmetrc wth respect to =, f(-=f(, but t has a mmum value of at = ad does ot satsfy the BC Oly the cose ad se fuctos are vald Of these, oly the cose fuctos satsfy y(-=y(=, ad oly the se fuctos satsfy y(-=y(= Wthout loss of geeralty, we ca wrte the egevalue-egevector equato terms of -λ, rather tha smply λ' Dog so saves us from havg to wrte the square root sg hus, strctly speag, the set {λ'} are the egevalues for the secod-dervatve operator However, we commoly refer to the set {λ} as the egevalues; t s ust a more geeral use of the term o satsfy the symmetrc BC, we must choose λ to be teger multples of π λ y λ π, y cos λ After elmatg sh ad cosh fuctos, there stll rema a fte umber of egevalues ad a equally fte umber of the correspodg egevectors Although we frst dscuss cose egevectors, se egevectors are smlar Cose egevectors are all mutually orthogoal to oe aother; furthermore, they are already ormalzed (ecept for cos(*= he followg are a few eamples Defe scalar product prod( g I geeral, for f( g( d cos( π cos( π d = cos( π cos( π d = cos( π cos( 3 π d = cos( π cos( π d = cos( π cos( 3 π d = cos( 3 π cos( 3 π d = d = ormalze y for cos( π cos( π d y ( y ( d for, cos( π cos( π d y ( y ( d

3 , 99 3 egecosmcd Cose Egefuctos cos( π cos( π cos( π cos( 3 π 5 5 ow, wth these egevectors, we ca epress ay twce-dfferetable eve fucto =[- ] as a lear combato of these egevectors hs s a huge, otrval statemet, because t meas the seres has to coverge for the statemet to hold, ad we ca state t wth cofdece eve wthout ay d of covergece test or formal proof (Actually, t turs out, wth rgorous proo the sort that mathematcas le, that we ca rela the LVS to clude all cotuous fuctos that has at most a fte umber of dscotutes =[- ] Sce there are a fte umber of egevectors, we eed a fty umber of terms theory (pure mathematcs However, we have to trucate to a fte umber of terms practce (egeerg mathematcs f = a y a = a cos λ I practce f a = a cos λ Sce the egevectors are all mutually orthogoal, the followg proecto formula apply f( cos λ d y a y, y cos λ cos λ d he cose egevectors are also ormalzed; thus, we ca elmate the deomator a y f( cos λ d We apply the secod dervatve operator ad estmate the secod dervatve of ay twce-dfferetable fucto =[- ] that satsfes f(-=f(= he secod dervatve of f s, d d f d d = a y = a d d y = a λ y = a λ y

4 4 egecosmcd Eample Epress a costat fucto f(= as a lear combato of the egevectors a a cos λ where a y, y cos λ d = Let us appromate f(= wth lmted umber of terms f( λ a f( d λ π a f( cos λ d f appro ( a = a cos λ a = 44 "average" of f( f appro ( 9 Actually, sce f(==y, whch s tur orthogoal to all the cose egevectors, all the coeffcets other tha a are hus, ths case s rather trval he pot of ths eercse s to demostrate how we ca represet ay twce-dfferetable fucto as a a lear combato of the egevectors It s always a good practce to start wth somethg smple to mae sure the method wors We try aother fucto below

5 5 egecosmcd Eample Epress the followg fucto as a lear combato of the egevectors α 5 f( rewrte as f( a α = a cos λ 5 Chage the umber of terms ad see how the appromato chages λ a f( d λ π a f( cos λ d f appro ( a = a cos λ a = 388 f appro ( f( Fourer Seres Appromato Gve Fucto he frst dervatve of the gve fucto s, Aalytcal soluto: f'( Lear combato of egevectors: f' appro ( he secod dervatve of the gve fucto s, α α = Aalytcal soluto: f''( α 3 α α 3 Lear combato of egevectors: f'' appro ( = a λ s λ a λ cos λ

6 6 egecosmcd 5 Frst Dervatve 5 Secod Dervatve f' appro ( f'' appro ( f'( f''( 5 Fourer Seres Appromato Gve Fucto 5 Fourer Seres Appromato Gve Fucto Eample Epress the followg Bessel's fucto as a lear combato of the egevectors f( z J( z for z=[α, β] α β rewrte as f( a ( z ( z α β α β α z( ( α = a cos λ Chage the umber of terms ad see how the appromato chages λ a f( z( d λ π a f( z( cos λ d b f( z( s λ d f appro ( a = a cos λ = b s λ zz α, α β f appro ( 5 f appro ( ( zz 5 f( z( f( zz 5 Fourer Seres Appromato Gve Fucto 5 zz Fourer Seres Appromato Gve Fucto

7 7 egecosmcd Actually the egevalues λ do ot eed to be teger multples of π he egevectors oly eeds to satsfy the followg geeral symmetrc BC the Sturm-Louvlle probem Ad there are may ways to satsfy t p y' m y y' y m a p y' m y y' y m b p a b Oe way s: y' c y at =a=- ad y' c y at =b=, where c=ay costat y' m c y m y' c y p c y m y c y y m p c y a m y c y y m b Aother way s: y' c y at =a=- ad y' d y at =b= p c y m y c y y m p d y a m y d y y m ths wll volve both se ad cose b Eample c cos(λ aloe (wthout s(λ or s(λ aloe wll do ( λ s( λ c cos( λ λ s( λ c cos( λ λ s( λ c cos( λ ( λ s( λ c cos( λ λ s( λ c cos( λ λ s( λ c cos( λ lambda( λ root( λ s( λ c cos( λ, λ λ lambda( λ = 77 λ lambda λ π λ = Chec orthogoalty coscos, cos λ cos λ d 388 Appromate f( coscos = 6 4 f( cos λ d 3 a f appro ( a cos λ cos λ cos λ d = Cose Egefuctos cos λ f appro ( cos λ 9 Because c ca be ay costat, the umber of possble egevalue-egevector sets s fte cos λ cos λ 3 5 5

8 8 egecosmcd Fourer Seres I the above developmet, we appled the the dea of egevalue-egevector to represet ay gve vector from the LVS For a geeral fucto (ot ecessarly odd or eve, we eep both the cose ad se egevectors I addto, for smplcty, we tae the egevector for λ= to be (whch s ot ormalzed, ad shft the ormalzato factor of to the a term I mathematcal lterature, ths s called the Fourer seres Whe the fte seres s trucated to a fte umber of terms, t s called the trucated Fourer seres a f( a cos λ b s λ where = λ π, a f( d a y = f( cos λ d b z Cose Seres For eve fuctos, eep oly the cose terms f( a = f( s λ a cos λ a f( cos λ d λ π, Se Seres For odd fuctos, eep oly the se terms f( a = b s λ b z f( s λ d λ π, ote that a s the average of the gve fucto f( wth the terval =[-, ] For problems where the gve terval s ot wth =[-, ], we ca always trasform the gve terval from =[α, β] to =[-, ] he eample above demostrates ths Fourer Seres Comple Form cos ad s ca be epressed terms of ep, ad vce versa d e θ cos( θ s( θ Euler's formula cos( θ e θ e θ s( θ e θ e θ where = =- s the magary umber, ot to be cofused wth a de Substtutg the above cos ad s epressos to the Fourer seres formula, we obta, f( a f( A = ep λ a ep λ = A ep λ = = B ep λ A a A a b B a ep λ b ep λ b

9 A f( d A f( ep λ d B z 9 egecosmcd f( ep λ ote that A s the coeffcet for the ep(*λ * term, but A s the scalar product of f( ad ep(-*λ *, ot ep(*λ * Lewse for B Why do we terchage the terms the tegral that defes the scalar product? O the surface, t seems ths s ot followg the proecto formula However, we are dealg wth comple umbers here, ad a scalar product defto that s vald for real Eucldea space s vald for comple space We eed to modfy the defto slghtly for comple space so that t obeys the geeral rules of scalar products ( g or ( g β α β α f( g( d where f( s the comple cougate of f( f Re( f Im( f f( g( d Detour For colums of comple umbers, ether oe of the followg deftos may be sutable (but the defto s ot uque (, y y or (, y y where s the comple cougate of Eample Defto # (, y y = ( = o Defto # (, y y ( = ( = Defto #3 (bad (, y ( y = ( = o d bad eample he 3rd defto s vald, because the legth of a ozero vector should be postve for the defto to be vald Eample he above two deftos gve two dfferet scalar products that are comple cougate of each other However, the legths from both deftos are detcal 3 y 3 Defto # (, y y y = 9 3 = 5 y y = Defto # (, y y y = = 5 y y = Comple cougate Idetcal

10 he egefuctos ep(*λ * ad ep(-*λ * are mutually orthogoal egecosmcd ep λ, ep λ ep λ ep λ d ep λ λ d ep λ, ep λ ep λ ep λ d ep( d d ep λ, ep λ ep λ ep λ d ep λ λ d ep λ, ep λ ep λ ep λ d ep λ d We ca further compact the lear combato epresso to a smpler form f( A ep λ A f( ep λ d = Or, equvaletly, f( B ep λ B f( ep λ d = I the Fourer seres approach, we brea up a gve fucto to varous frequecy compoets he coeffcet A tells us how much of the gve fucto f( ca be epressed as a susodal oscllato of frequecy λ hus, Fourer seres allows us to etract the varous frequecy compoets mbedded the gve fucto he set of coeffcets {A } are the ampltudes at the correspodg frequeces, ad the whole set s commoly referred to as the spectrum of f( Fourer seres s very mportat may aspects of egeerg aalyss: harmoc oscllato, spectral aalyss, dgtal sgal processg, ose flterg, A closely related dea s Fourer trasform, whch, too, breas a gve cotuous fucto to cotuous frequecy compoets, rather tha ust at frequeces correspodg to egevalues of a harmoc oscllator We eted the tegrato lmt betwee - ad + ad replace the summato sg the last epresso wth a tegral a cotuous doma F( f( F( ω F( F( ω f( ep( π ω f( d ep( π ω F( ω dω forward Fourer trasform (aalogous to the coeffcets {A } Iverse Fourer trasform (aalogous to the lear combato equato

11 egecosmcd May other versos of Fourer trasform ad verse trasform pars est F( f( F( ω F( F( ω f( ep( π ω f( d ep( π ω F( ω dω I the followg par, we factor out the *π term ad dstrbute t equally betwee the forward ad verse trasform pars F( f( F( ω π F( F( ω f( π ep( ω f( d ep( ω F( ω dω I the followg par, we shft the *π term to the verse trasform We ca also shft the *π term to the forward trasform F( f( F( ω F( F( ω f( π ep( ω f( d ep( ω F( ω dω Our Fourer seres here most closely resembles the followg trasform pars F( f( F( ω F( F( ω f( ep( π ω f( d Dscretzed verso of Fourer trasform: F = ep( π ω F( ω dω f ep π Forward Fourer trasform f F ep π Iverse Fourer trasform = Eamples of Fourer trasforms: Our ears ad the assocated erve cells covert mechacal soud waves the tme-doma to frequecy (ptch soud ad the assocated ampltude (loudess Lewse, our eyes trasform electromagetc waves the vsble rego to frequecy (e, colors ad the assocated ampltude (brghtess I terms of commo household tems, a rado etracts a specfc frequecy compoet, ad the televso also wors a smlar fasho

12 egecosmcd Fourer seres s a method of epressg a gve fucto wth learly depedet fuctos tae from the same LVS; t s ot a lear trasform I other words, Fourer seres, eteded to fte umber of terms, s ust a represetato of the same fucto; t does ot chage oe fucto to aother It s epressg a vector wth a set of learly depedet bass vectors Fourer trasform, o the other had, coverts oe fucto to aother fucto I other words, t s a fucto of fuctos (vectors; thus, Fourer trasform s a lear trasform Fourer seres s aalogous to aylor's seres epaso of a gve fucto aylor's seres s based o the varous dervatves evaluated at oe sgle pot Whereas, the Fourer seres s based o varous tegrals over a terval aylor's seres s eact at oe sgle pot aroud whch epaso s made, ad the appromato degrades as we move farther away from that pot of epaso Fourer seres gves a good appromato over the etre terval