CHAPTER 7. Definition and Properties. of Laplace Transforms

Transcription

1 SERIES OF CLSS NOTES FOR 5-6 TO INTRODUCE LINER ND NONLINER PROBLEMS TO ENGINEERS, SCIENTISTS, ND PPLIED MTHEMTICINS DE CLSS NOTES COLLECTION OF HNDOUTS ON SCLR LINER ORDINRY DIFFERENTIL EQUTIONS (ODE") CHPTER 7 Definiion and Properie of Laplace Tranform. Compuaion of he Laplace Tranform Uing he Definiion. Parial Table of (Indefinie) Inegral (niderivaive) 3. Properie of he Laplace Tranform: Lineariy 4. Oher Properie of he Laplace Tranform 5. Parial Table of Laplace Tranform You Mu Memorize 6. Exra Homework Shee on Laplace Tranform Ch. 7 Pg.

2 Laplace Tranform- COMPUTTION OF THE LPLCE Handou # TRNSFORM USING THE DEFINITION Profeor Moeley Read he inroducion and Secion 6. of Chaper 6 of ex (Elem. Diff. Eq. and BVP by Boyce and Diprima, evenh ed.). Pay paricular aenion o Example -6 page REVIEW OF IMPROPER INTEGRLS. The Laplace ranform i defined a an improper inegral. Hence we begin wih a brief review of improper inegral. DEFINITION #. f()d = lim f()d provided he limi exi. c = c EXMPLE #. Compue d Soluion. = = = d lim d lim (ln ) = lim (ln - ln ) = lim (ln ) = (Increae wihou bound and hence he limi doe no exi) EXMPLE #. Compue d where p >. p -p Soluion. d = lim d = lim d = p p = = -p lim ( / (- p)) = lim ( -p -p = = p p ) lim ( -p p p - ) p - Recall ha hee improper inegral, ogeher wih he inegral e, imply he divergence of he harmonic erie and he convergence of he p erie if p >. n n p n DEFINITION # (Laplace Tranform). Le I = [,) and f:i R. Then he Laplace Tranform of f() i he funcion F() {f()} = F() = f() e d () provided he improper inegral exi. (Sufficien condiion for he Laplace Tranform o exi are given below.) Since i arbirary, he Laplace ranform map a given funcion f() in he n Ch. 7 Pg.

3 funcion pace we will denoe by T (ime domain) o he funcion F() in he funcion pace of all Laplace ranform which we denoe by F (complex frequency domain). EXMPLE# Compue {f()} = F() where f() = for all in [,). {f()} = F() = f() e d = e d = lim d = =. lim EXMPLE# Compue {f()} = F() where f() = e a for all in [,). Soluion. We wih o compue {f()} = F() = {e a }. {e a a } = F() = e e d = e a (a-) (a) d = e d = lim e d (a ) = lim e / (a ) = = = lim e (a ) (a) e = if a - < ( > a) a - a - a We begin a able of Laplace Tranform imilar o having a able of aniderivaive. PRTIL TBLE OF LPLCE TRNSFORMS Time Domain (Complex) Frequency Domain f() F() > e a /(-a) > a SUFFICIENT CONDITIONS FOR THE IMPROPER INTEGRL TO EXIST. To inure ha he improper inegral f() e d exi, we mu fir inure ha he proper inegral f() e d exi for all poiive real number. good ar i o recall ha coninuou funcion are inegrable: THEOREM. Le f be a real valued funcion of a real variable whoe domain include he cloed inerval I = [α,β]. Suppoe ha f i coninuou on I. Then he Riemann Inegral Ch. 7 Pg. 3

4 f() d and f() e d (4) boh exi. (Since produc of coninuou funcion are coninuou, if f i coninuou on [α,β], hen o i f() e for all value of.) However, we would like o conider a larger cla of funcion where he Riemann inegral exi. DEFINITION #3. funcion f i aid o be piecewie coninuou on a cloed finie inerval I = [α,β] if he inerval can be pariioned by a finie number of poin α = < < <... < n = β, o ha:. F i coninuou on each of he open ubinerval I n = ( i-, i ) i =,,...,n.. f approache a finie limi a he end poin of each ubinerval are approached from wihin he ubinerval; ha i he limi: lim f(), lim f(), i =,,..., n, all exi. i i EXMPLE. Le f:ir where I=[,5] be defined a follow: ( ) f() \ / \)) ))) / * *)))))))) )))))))))))) CLSS EXERCISE True or Fale. f i coninuou on (,3].. f i coninuou on [,3]. 3. f i coninuou on [3,5]. 4. f i piecewie coninuou on [,5]. 5. f i piecewie coninuou on [,3]. 6. f i piecewie coninuou on [,3]. THEOREM #. Sum and produc of piecewie coninuou funcion are piecewie coninuou. Ch. 7 Pg. 4

5 lo calar muliple of piecewie coninuou funcion are alo piecewie coninuou funcion. THEOREM #3. The e of piecewie coninuou funcion on he cloed inerval I = [α,β], denoed by PC( I ) = PC[α,β], i a ubpace of he vecor pace F( I ) of all real valued funcion on he cloed inerval I. In mahemaical (and phyical) problem he value of a piecewie coninuou funcion a he poin of diconinuiy uually doe no maer and we uually do no wih o diinguih beween wo piecewie coninuou funcion ha differ only a heir poin of diconinuiy. One way o eliminae he diincion beween wo piecewie coninuou funcion ha differ only a heir poin of diconinuiy i o ue he concep of an equivalence cla. (Look up he definiion of an equivalence relaion in a modern algebra ex.) ll piecewie coninuou funcion ha differ only a heir poin of diconinuiy are aid o be in he ame equivalence cla and we perform Laplace ranformaion on equivalence clae of funcion. n eaier way concepually i o imply pecify a unique funcion in each equivalence cla which we place in a new e we call PC a [α,β] (a i for average). Le fpc[α,β]. If f i diconinuou a α, hen redefine f(α) a f(α) = lim f(). Similarly, if f i diconinuou a β, redefine f(β) a f(β) = lim f(). any oher poin of diconinuiy le f(x) = [f(x) + f(x+)]/ where f() = lim f(h) and f(x+) = lim f(). Since he new f only differ from he old f a he poin of h h diconinuiy, hey are in he ame equivalence cla. (There i a mo one coninuo funcion in each equivalence cla.) Le PC a [α,β] be he e of all uch funcion (a and for average value). We ee ha we have exacly one funcion from each equivalence cla. lo, if we add uch funcion ogeher, we ge uch a funcion. Such funcion are alo cloed under calar muliplicaion. Hence hey form a ubpace of PC[α,β], bu each having unique value a he poin of diconinuiy. THEOREM #4. Le f be a real valued funcion of a real variable whoe domain include he cloed inerval [α,β]. Suppoe ha f i piecewie coninuou on I. Then he Riemann Inegral f() d and f() e d boh exi. lo, if fpc a [α,β], hen f() d = for all x[α,β] implie ha f() = for all [α,β]; ha i, he zero funcion i he only funcion in PC a [α,β] wih hi propery. (Such funcion are called null funcion and we wih he zero funcion o be he only null funcion in our funcion pace.) DEFINITION #4. If he domain of a funcion i [,), and f PC[,] >, hen we ay f i piecewie coninuou on [,). If for all >, f PC a [,], hen f PC a [,). THEOREM #5. The e of piecewie coninuou funcion on I = [,), denoed by PC[,) i a ubpace of he e of all real valued funcion on I. PC a [,) i a ubpace of PC[,). x Ch. 7 Pg. 5

6 To inure ha he Laplace ranform exi, we no only need for he proper inegral on [,] o exi for all, we need for he funcion f() e o grow ufficienly lowly o ha he improper inegral exi. DEFINITION #5. funcion f:[,)r i aid o be of exponenial order if here exi conan K, a, and M uch ha *f()* Ke a M. THEOREM #6. Sum and produc of funcion of exponenial order, are of exponenial order. Noe alo ha calar muliple of funcion of exponenial order are of exponenial order. In fac, THEOREM #7. The e of funcion of exponenial order, denoed by E xp, i a ubpace of he e (I,R) of all real valued funcion on I = [,). THEOREM #8. Suppoe ha ) f i piecewie coninuou on [,); ha i, f i piecewie coninuou on he inerval [,] > and ) f i of exponenial order o ha here exi K, a, and M uch ha *f()* Ke a M, hen he Laplace ranform {f()} = F() = f() e d exi wih domain (a,). i.e. for > a. THEOREM #9. PC[,) E xp i a ubpace of T. So i T pcexp = PC a [,) E xp. EXMPLE #3 Compue {f()} = F() where f() = for. Soluion. We wih o compue {f()} = F() = {}. = {e a lim e / ( ) } = f() e d = lim e d = = = = lim e = if - < ( > ) lim e () e We coninue our able of Laplace Tranform. Ch. 7 Pg. 6

7 PRTIL TBLE OF LPLCE TRNSFORMS Time Domain (Complex) Frequency Domain f() F() > e a /(-a) > a / > Noe ha he new enry i ju a pecial cae of he old enry, e a, wih a =. EXMPLE #4 Compue {f()} = F() where f() = in(a) for. Soluion. We wih o compue {f()} = F() = {in(a)}. {in(a)} = [in (a)] e d = lim [in (a)] e d = We fir compue he aniderivaive (or look i up in a able) Compuaion of he niderivaive Ue inegraion by par. Le e in a e a co a e in a e a co a I = in(a) e d = d = + d u = in(a) dv = e du = a co(a) v = e /() Bu co(a) e d = e co a - e a in a e co a e a in a d = d u = co(a ) dv = e du = -a in(a) v = e /(-) Hence e in a a e in a e co a e a in a I = + co(a) e d = + [ d ] a e in a e co a a e in a e co a a = + [ I ] = I Hence a e in a e co a + a e in a e co a I + I = o ha I =, - e in a - a e co a ( + a ) I = e in(a) a e co(a) and hence I =. a Back o compuing he Laplace Tranform a Ch. 7 Pg. 7

8 F() = {in(a)} = lim - e in a - a e co a a () () - e in a - a e co a - e in a() - a e co a() = lim a a - e co a - a e co a a a = lim = >. a a a We coninue our able of Laplace Tranform. PRTIL TBLE OF LPLCE TRNSFORMS Time Domain (Complex) Frequency Domain f() F() > e a /(-a) > a / > in(a) a a > EXERCISES on Compuaion of he Laplace Tranform Uing he Definiion EXERCISE#. Provide a more rigorou aemen of Theorem # for funcion defined on he cloed inerval I = [α,β]. EXERCISE #. Provide a more rigorou aemen of Theorem #6 by including pecific. Ch. 7 Pg. 8

9 Laplace Tranform- PRTIL TBLE OF NTIDERIVTIVES Handou # (INDEFINITE) INTEGRLS Profeor Moeley Ch. 7 Pg. 9

10 Laplace Tranform- PROPERTIES OF LPLCE TRNSFORMS: Handou #3 LINERITY Profeor Moeley Read he inroducion and Secion 6. of Chaper 6 of ex (Elem. Diff. Eq. and BVP by Boyce and Diprima, evenh ed.) again. Pay paricular aenion o Example -6 page Pay pecial aenion o he paragraph afer Example 6, in paricular Equaion (5). The e T = { f:[,)r * f ha a Laplace ranform } i a ubpace of he funcion pace ([,),R) vecor pace and hence i a vecor pace in i own righ. We have een ha he e T pcexp = PC[,) E xp = { f T : f i piecewie coninuou on [,) and of exponenial order } i a ubpace of T and hence a vecor pace in i own righ. THEOREM #. The Laplace ranform {f()} = f() e d = F() () i a linear operaor acing on he vecor pace T Proof: To verify ha i a linear operaor from T o F, we fir ae he definiion of a linear operaor. Definiion. n operaor T:V W (which map he vecor pace V o he vecor pace W) i aid o be linear if, V and calar α,β we have x x y y x y T(α + β ) = α T( ) + β T( ). () pplying () o we ee ha o how ha i i a linear operaor, we wih o verify he ideniy: {c f () + c f ()} = c {f ()} + c {f ()} c, c R, f, f T (3) We ue he andard forma for proving ideniie. Le c,c R and f,f T STTEMENT. {c f ()+c f ()}= [c f (x) c f (x)]e dx RESON. Def n. of Laplace Tran. = lim [c f (x) c f (x)]e dx Def n. of Improper Inegral = lim c [f (x)]e dx c [f (x)]e dx Propery of Riemann Inegral Ch. 7 Pg.

11 = lim c [f (x)]e dx lim c [f (x)]e dx Propery of Limi = c [f (x)]e dx c [f (x)]e dx Def n. of Improper Inegral = c { f () }+ c {f ()} Def n.of Laplace ranform QED. Thu if f() i a linear combinaion of funcion: f() = c f () + c f (), hen i Laplace ranform F() i a linear combinaion of ranform: F() = c F () + c F () where {f()} = F(), {f ()} = F (), and {f ()} = F (). We can ue lineariy o compue he ranform of linear combinaion of funcion in our able. EXMPLE # Compue {f()} = F() where f() = 3 e in() for. Soluion. We wih o compue {f()} = F() = { 3 e in()}. { 3 e in() } = 3 { e 3 }+ 5 {in() } = When dealing wih Laplace ranform, i i uually no neceary or deirable o find a common 3 5 denominaor. However, imple arihmeic i expeced: F() = +. 3 EXERCISES on Properie of Laplace Tranform: Lineariy EXERCISES. Uing he able compued o far, find F() = {f()}() if a) f() = 4 e b) f() = 4 e e c) f() = 8 e e in(3) Ch. 7 Pg.

12 Laplace Tranform- OTHER PROPERTIES OF Handou No. 4 THE LPLCE TRNSFORM Profeor Moeley Read Secion of Chaper 6 of ex (Elem. Diff. Eq. and BVP by Boyce and Diprima, evenh ed.). Pay paricular aenion o Theorem 6.. on page 3, he Corollary on page 3, Theorem 6.3. on page 3, and Theorem 6.3. on page 33. THEOREM # Suppoe f T pcexp ; ha i, f i piecewie coninuou on [,] >, and of exponenial order o ha T,M and σ uch ha *f()*< Me σ >T. Then each of he following properie hold: d ) { f() } = { f() }(), > σ, d ) { e a f(r) } = { f() }(-a), > σ, Shifing propery, r 3) { f(r)dr } = { f() }(), r > σ, If in addiion, f i coninuou and f' piecewie coninuou on [,),hen 4) { f'() } = { f() } - f(), > σ. More generally, if f, f',..., f (n-), are coninuou on [,) and of exponenial order, and f (n) i piecewie coninuou on [,), hen 5) { f (n) () } = n { f() } ) n- f() ) f (n-) () ) f (n-) (), > σ. Specifically 6) { f"() } = { f() } - f() - f'(), > σ. Proof. Thee are all ideniie and can be verified uing he andard forma. We prove only ). ) hrough 6) are lef a exercie. For ) we ar wih he righ hand ide (RHS) STTEMENT. RESON. d d { f() }() = f()e d > σ Definiion of he Laplace Tranform d d d = lim f()e d > σ Definiion of an Improper Inegral d = lim d f()e d > σ Propery of limi and derivaive from d advanced analyi: The limi and derivaive can be wiched under cerain condiion. = lim > σ Propery of derivaive and inegral from advanced f()e d analyi. Derivaive and inegral can be wiched under cerain condiion. Derivaive become parial. Ch. 7 Pg.

13 = lim (-)f()e d > σ Propery of parial derivaive = lim f()e d > σ Properie of inegral and limi = f()e d > σ Definiion of improper inegral = { f() } Definiion of Laplace Tranform Q.E.D. EXMPLE # Compue {f()} = F() where f() = for. Soluion. We wih o compue {f()} = F() = { }. STTEMENT. RESON. { } = { () } lgebra d = ( { } ) d Par ) of heorem above d = ( ) ) d From he TBLE { } = d = - d lgebra = ( ) - Calculu = lgebra Similarly we can compue > 3 (3)() 3! 3 = > 4 4 nd by mahemaical inducion we can prove ha n! n > n+ We coninue he developmen of our able of Laplace Tranform. Ch. 7 Pg. 3

14 PRTIL TBLE OF LPLCE TRNSFORMS Time Domain (Complex) Frequency Domain f() F() > e a /(-a) > a / > in(a) a a > > > 3 (3)() 3! 3 = > 4 4 n! n > n+ Now by uing he lineariy of he Laplace Tranform, we can compue he ranform for any polynomial. EXMPLE # Compue {f()} = F() where f() = for. Soluion. We wih o compue {f()} = F() = { }. STTEMENT. RESON. { } = 3 { } + 5 { } + 7 { } Lineariy of Laplace Tranform = Table of known Laplace Tranform = lgebra We can ue Theorem # o obain oher ranform pair: EXMPLE #3 Compue {f()} = F() where f() = in(a) for. Soluion. We wih o compue {f()} = F() = { in(a) }. Ch. 7 Pg. 4

15 STTEMENT. RESON. d { in(a) } = ( { in(a) } ) d Par ) of heorem above d a = From he TBLE { in(a) } = d a d = lgebra d a a = a ( ) ( + a ) - () Calculu = a a lgebra EXMPLE #4 Compue {f()} = F() where f() = e a in(ω) for. Soluion. We wih o compue {f()} = F() = { e a in(ω) }. a a STTEMENT. RESON. { e a in(ω) } = { in(ω) }( - a ) Par ) of heorem above = (- a) From he TBLE {in(ω) } = EXMPLE #5 Compue {f()} = F() where f() = co(ω) for. Soluion. We wih o compue {f()} = F() = { co(ω) }. We will ue propery in( ) d in( ) 4): { f'() } = { f() } - f() wih f() = o ha f() = = co(ω). d in( ) in( ( ) ) in( ) Hence { co(ω) } = { } = { }= = We coninue he developmen of our able of Laplace Tranform. Ch. 7 Pg. 5

16 PRTIL TBLE OF LPLCE TRNSFORMS Time Domain (Complex) Frequency Domain f() F() > e a /(-a) > a / > in(a) a a > > > 3 3! 3 > 4 n! n > n+ a in(a) > a e a in(ω) co(ω) (- a) > a Ch. 7 Pg. 6

17 Laplace ranform TBLE OF LPLCE TRNSFORMS Handou No. 5 TO BE MEMORIZED Profeor Moeley lhough i i no reaonable o memorize all Laplace Tranform pair, i i reaonable o memorize he mo common one. Below i a li of he Laplace Tranform pair you mu memorize. f() F(S) = {f()}() Domain of F() )) > o )) > o )) 3 > o n n n! )) n+ > o e a )) a > a in(ω) co(ω) ω )))) + ω )))) + ω > o > o e a in(ω) e a co(ω) ω )))) ( a) + ω > a -a ))))) ( a) + ω > a Ch. 7 Pg. 7

18 EXERCISES on Oher Properie of he Laplace Tranform EXERCISE # Uing a Table and Your Knowledge find F() = {f()}() if ) f() = ) f() = (+) 3) f() = in }Hin: Ue rig ideniie 4) f() = in (3) 5) f() = e 3 6) f() = e - 7) f() = e - co 3 Ch. 7 Pg. 8