Chaper 6. Laplace Tranform Kreyzig by YHLee;45; 6- An ODE i reduced o an algebraic problem by operaional calculu. The equaion i olved by algebraic manipulaion. The reul i ranformed back for he oluion of ODE. 6. Laplace Tranform. Invere Tranform. Lineariy. Shifing A funcion f ( ) i defined for. I Laplace ranform i defined a F( ) = i ( f) = f( ) e d The invere ranform i defined a f = i F Noe i i( f) = f, i i ( F) = F Noaion f i a funcion of. F i a funcion of. Example Laplace Tranform Le f( ) = when. Find F( ). F( ) = i ( f) = e d : > Example Laplace ranform a Le f( ) = e for. Find he Laplace ranform. i a a a ( ) ( e ) e e d e When ( a) > i a ( e ) = a = a
Theorem Lineariy of he Laplace Tranform If f( ) = F( ) and g( ) = G( ) i i, hen i af ( ) + bg( ) = af ( ) +bg( ) Kreyzig by YHLee;45; 6- Proof: i af( ) + bg( ) = af( ) + bg( ) e d a f e d b g e d = + i = ai f + b g Example 3 Hyperbolic Funcion Find he ranform of coh a and inha. a a e + e i[ coha] = i = + = a + a a a a e e a i[ inha] = i = = a + a a Example 4 Coine and Sine ω Prove i[ co ω ] =, i [ inω ] = +ω +ω By inegraing by par e ω ω i[ coω ] = e coω d = coω e inω d = [ in ω], i i e ω ω [ inω ] = e inω d = inω + e coω d = i[ coω] ω ω i[ coω ] = i[ co ω], ω ω i[ inω ] = J i[ inω] i i co ω =, + ω ω [ ] [ in ] ω = + ω By complex mehod From he reul of Example iω ω i e = + i ω i + ω +ω From Euler formula i iω e = i [ co ω + i in ω ] ω i[ co ω ] =, i [ inω ] = +ω +ω
Baic Tranform Kreyzig by YHLee;45; 6-3 Proof of (4) We make he inducion hypohei ha (4) hold for. n+ n+ n+ n + n i = e d = e + e d i n+ = Le prove (5) ( n + )! = n+ I aifie he inducion hypohei. So (4) i proved. n i a a a x x dx x a a+ i = e d e e x dx Γ a +, gamma funcion x
Shifing: Replacing by a in he Tranform Kreyzig by YHLee;45; 6-4 Theorem Shifing Proof: i = a e f F a = i ( ) a e f F a a a a F e f d F a e f d e e f d e f ( ) = ( ) = i Example 5 Shifing Prove and of he able 6. a ( a) i[ co ω ] = i e coω = ω + ω + a ω ω i[ in ω ] = i e inω = ω + ω + a Find he invere of a Exience and Uniquene of Laplace Tranform The growh rericion i defined a k f( ) Me f() i piecewie coninuou if i i coninuou and ha finie limi in ubinerval Theorem 3 Exience Theorem for Laplace Tranform If f() i piecewie coninuou aifying he growh rericion for all, hen i Laplace ranform exi for all > k. Proof: Since f() i piecewie coninuou, e f( ) i inegrable. Uniquene If he Laplace ranform of a given funcion exi, i i uniquely deermined.
6. Tranform of Derivaive and Inegral. ODE Kreyzig by YHLee;45; 6-5 Theorem Laplace Tranform of Derivaive i[ f' ] = i[ f] f( ) i f" = i f f f' [ ] [ ] The neceary condiion: f' and f" on he lef ide are piecewie coninuou. f and f ' on he righ ide are coninuou for all aifying he growh rericion. Proof: Uing inegraion by par i[ f '] = i [ f] f( ) Similarly, uing he fir relaion wice, = i [ f ] Uing growh rericion e f e Me k e k M for > k. = = = e f f = Only lef i Theorem If and f, f',..., f ( n ) i coninuou for all aifying he growh rericion, ( n) f i piecewie coninuou, Example Formula 7 and 8 in Table 6. f = ω Le co, ', " co f = f = f = ω ω Uing he ranform of a econd derivaive, = ω + i[ f "] = i[ f] f( ) f' ( ) i [ f] i [ f ] ω i = [ f ] Similarly, g= in ω, g =, g' =ωcoω ω ω = ω ω + i[ g' ] = i [ g] i[ g] i [ co ] ωi coω = [ ]
Laplace Tranform of he Inegral of a Funcion Kreyzig by YHLee;45; 6-6 Theorem 3 Laplace Tranform of Inegra If Laplace ranform of f() i given a F(), for >, >k and > Proof: Le Then i f ( τ) dτ = F( ) and f d F( ) g = f τ dτ τ τ= i Since f() i piecewie coninuou, g() i coninuou and aifie he growh rericion. Example 3 Find he invere of and +ω +ω. Uing (8) of Table 6. inω i = +ω ω Uing Theorem 3 inωτ i d = τ coω +ω ω ω Uing Theorem 3 again inω i = ( co ωτ ) d τ 3 +ω ω ω ω
Differenial Equaion, Iniial Value Problem Kreyzig by YHLee;45; 6-7 ODE can be olved by Laplace ranform mehod. An iniial value problem y" ay' by r + + = y( ) = K, y' = K r() i inpu and y() i oupu. o Sep Seing up he ubidiary equaion The Laplace ranform of he ODE Y y( ) y' ( ) + a Y y( ) + by= R : i[ y] = Y( ), i [ r] = R( ) ( ) '( ) + a+ b Y = + a y + y + R : ubidiary equaion Sep Solve he ubidiary equaion by algebra ( ) '( ) Y = + a y + y Q+ RQ where Q = ( + a+ b) + a + b a 4 : ranfer funcion Noe, if y( ) = y' ( ) = Y [ oupu] Q = = i R i [ inpu] Sep 3 Inverion of Y for y Example 4 Iniial value problem Solve Find he ubidiary equaion Ue parial fracion expanion
Example 5 Comparion wih he uual mehod Solve Kreyzig by YHLee;45; 6-8 Find he ubidiary equaion The oluion ( ) ( ).6 + +.8 +.8 + + + + + + + 35 4.6 35 35 35 4 4 4 35 4 Ue hifing heorem and formula for coine and ine. / 35.8 35 y = i Y =.6e co + co [ ] ( 4 ) ( 4 ) Advanage of he Laplace Mehod. Nonhomogeneou ODE can be olved wihou olving he homogeneou ODE.. Iniial value are auomaically aken care of. 3. Complicaed inpu r() can be handled very efficienly. 35 4 Example 6 Shifed Daa Problem The iniial condiion do no ar a = y" + y= y π /4 =π/, y' π / 4 = We e = +π/4 = ( +π /4) = ( ) y y y, dy( ) dy( ) dy d = = = y ' d d d d The ODE become y " + y = ( +π /4) y = =π /, y' = = The ubidiary equaion / Y π π ( ) + Y = + Solve for Y Parial fracion expanion / / / / Y π π π π = + + + + + + + + + + + + y = +π/ in y= in( π /4)
Kreyzig by YHLee;45; 6-9 6.3 Uni Sep Funcion. hifing The uni ep funcion or Heaviide funcion i defined a u a = < a if = if > a The Laplace ranform e i u a = e u ad= e d= e i u ( a) = a a = a Le f( ) = for <. Then f ( a) u( a) i f ( ) hifed o he righ by he amoun of a.
Kreyzig by YHLee;45; 6- Time Shifing ( hifing): Replace by a in f() Theorem Shifing F() i he ranform of f() Proof: The hifed funcion i given a f ( a) u( a) = if < a = f a if > a I Laplace ranform i a ( ) ( ) = i f a u a e F Le τ+ a= The lower limi i changed Example Ue of Uni Sep Funcion Wrie f() uing uni ep funcion and find i ranform Uing uni ep funcion Perform he ranform afer wriing erm a f ( a) u( a) a You migh have ued i f ( u ) ( a) = e i f ( + a)
Kreyzig by YHLee;45; 6- Example Applicaion of Boh Shifing Theorem Find he inverion of Wihou he exponenial funcion a he numeraor, F() would be in π, in π, e π π Nex, he exponenial funcion mean hifing, ( 3) f( ) = in π( ) u ( ) + in π( ) u ( ) + ( 3) e u 3 π π Example 3 RC Circui Find i() if a ingle recangular wave wih volage V o i applied. The circui wa quiecen before he wave i applied. The inpu volage i ( ) ( ) Vo u a u b The circui i modeled by he inegro differenial equaion The ubidiary eq. : Solve for I() : Vo / R I( ) = e e + / V o i e R ( a b ) RC /( RC) V o ( a )/( RC ) ( b )/( RC ) i = e u a e u b R
Kreyzig by YHLee;45; 6- Example 4 RLC circui wih a inuoidal inpu The inpu i given by a inuoidal funcion only for a hor ime inerval. The curren and charge are iniially zero. The model for i() The ubidiary equaion I 4 π.i + I + = ( e ) + 4 Solve for I 4 π 4 π I= ( e ) e I I + + + 4 + + + 4 Parial fracion expanion Conan A, B, D, K are found a follow. Afer muliplicaion by he common denominaor, Iner = A=.776 Iner = B=.644 3 Collec erm =A+B+D D=.3368 Collec erm =A+B+D+k K=58.66 Then i can be obained uing hifing ( π) ( π) { } { } i =.776e +.644e.3368co 4( π ) +.6467in 4( π) u π The final oluion i i = i( ) i( )
6.4 Shor Impule. Dirac Dela Funcion. Parial Fracion Kreyzig by YHLee;45; 6-3 Dirac dela funcion or uni impule funcion i defined a if δ a = = a a+ε a ε = oherwie ( a) d δ = The dela funcion can be obained by aking he limi of f k δ( a) = lim f k ( a) k Sifing propery of dela funcion a+ε a ε δ( ) = g ad ga The Laplace ranform of dela funcion. Sar from f k fk ( a) = u( a) u ( a k) k + { } Take he limi k and apply l Hopial rule o he quoien. k k a e a e lim e e lim k k k i δ a = e a
Example Ma Spring yem under a quare wave Inpu i of he form of a recangular funcion Kreyzig by YHLee;45; 6-4 The ubidiary equaion Ue he parial fracion expanion : The invere ranform : Uing hifing ( ) ( ) ( ) ( ) y = e + e u ( ) e e u + Example Hammer blow repone of a ma pring yem The inpu i given by a dela funcion Solving algebraically The oluion ( ) ( ) y = e u e u
Example 3 Four Terminal RLC Nework 4 Find he oupu volage if R= Ω, L= H, C = F. The inpu i a dela funcion and curren and charge are zero a =. Kreyzig by YHLee;45; 6-5 The volage drop on R, L, C hould be equal o he inpu. Uing i= q' The ubidiary equaion Uing hifing and 99 99.5 The oluion More on Parial Fracion F( ) The oluion of a ubidiary equaion i of he form Y = G( ) Parial fracion repreenaion may be needed. () Unrepeaed facor ( a) in G() Parial fracion hould be () Repeaed facor Repeaed facor 3 a in G() Parial fracion a in G() Parial fracion α +β (3) Unrepeaed complex facor Parial fracion A + A ( a) B ( a) ( a) A B C + + 3 ( a) ( a) ( a) A + B α +β
Example 4 Unrepeaed Complex Facor. A damped ma pring yem under a inuoidal force. r = in for < <π y" + y' + y = r( ), = for >π The ubidiary equaion, y y Kreyzig by YHLee;45; 6-6 =, ' = 5 The oluion The parial fracion of he fir erm (6) Muliplying he common denominaor Term of like power of hould be equal on he righ and lef ide A=, B=, M=, N=6 Therefore he fir erm become The invere ranform (8) The invere of he econd erm of (6) i obained from (8) uing hifing { } u π () Rewrie he hird erm of (6) 3 ( + ) 4 + + + + The invere uing hifing e co 4in (7) The final oluion For < <π y()= Eq. (8) + Eq. (7) For >π y()= Eq. (8) + Eq. (7) + Eq. ()
6.5 Convoluion. Inegral Equaion Kreyzig by YHLee;45; 6-7 The convoluion of wo funcion f and g i defined a ( * ) f g f τ g τ dτ : Noe he inegraion inerval Theorem Convoluion heorem If F and G are Laplace ranform of f and g, repecively, he muliplicaion FG i he Laplace ranform of he convoluion (f*g) Proof: Se p= τ, hen Calculae he muliplicaion G can be inide of F For fixed τ, inegrae from τ o. becaue and τ are independen. ( The inegraion over blue region ) The inegraion can be changed a e f g d ( * ) Some properie of convoluion
Example Convoluion Le H( ) =. Find h(). a Kreyzig by YHLee;45; 6-8 Rearrange : H( ) = a F() G() a Invere ranform : f( ) e g( ) =, = aτ a Uing convoluion heorem : h = f( ) * g e dτ ( e ) a Example Convoluion Le H( ) = +ω. Find h(). Rearrange : H( ) Invere of ( +ω ) : inω ω = + ω +ω ( +ω ) ( )( ) in in in h ω * ω in in d co ω = ωτ ω τ τ ω + ω ω ω ω ω Uing convoluion heorem : Example 3 Unuual Properie of Convoluion f * f in general ( f* f) may no hold Applicaion o Nonhomogeneou Linear ODE Nonhomogeneou linear ODE in andard form y" + ay' + by= r( ) : a and b, conan The oluion Y( ) = ( + a) y( ) + y' ( ) Q( ) + R( ) Q( ) : Q =, ranfer funcion + a+ b The invere of he fir righ erm can be eaily obained. The invere of he econd erm, auming y y y = q τ rτ dτ = ' = The oupu i given by he convoluion of he impule repone q() and he driving force r().
Kreyzig by YHLee;45; 6-9 Example 5 Ma pring yem Solve y" + 3 y' + y = r, The ranfer funcion for r = < < = oherwie y y = ' = I invere Since y y = ' =, he oluion i given by he convoluion of q and r. i i ( ) ( τ) ( τ) ( τ) ( τ) y = q τ rτ dτ q τ uτ uτ dτ e e dτ e e For < : y() = r()= only for << For << : The upper limi i, For > : The upper limi i, Noe he change in he lower limi. hould be le han. y = e e e + e τ= τ= τ= ( τ) ( τ) ( ) ( ) τ= τ= ( τ) ( τ) ( ) ( ) ( ) ( ) τ= y = e e e e e e Inegral Equaion Convoluion can be ued o olve cerain inegral equaion Example 6 A volerra Inegral Equaion of he Second Kind Solve Convoluion of y and in( ) Uing he convoluion heorem we obain he ubidiary equaion + Y( ) Y( ) = Y ( ) Y ( ) = + 4 4 + + The anwer i
6.6 Differeniaion and Inegraion of Tranform. Kreyzig by YHLee;45; 6- Differeniaion of Tranform If F() i he ranform of f(), hen i derivaive i df ( ) F( ) = f( ) e d F' ( ) = = f( ) e d d Conequenly i f( ) = F' ( ) and i F' ( ) = f ( ) Example Differeniaion of Tranform The able can be proved uing differeniaion of F(). The econd one d β β i [ inβ ] = d +β +β ( ) Inegraion of Tranform If f() ha a ranform and lim f( ) / exi, + f( ) = i F( ) d and f i F d = Proof: From he definiion f ( ) F ( ) d = e f ( ) d d f ( ) e d d i Revere he order of inegraion. = e /
Example Differeniaion and Inegraion of Tranform Kreyzig by YHLee;45; 6- Find he invere ranform of F() = I derivaive Take he invere ranform i F' i coω = f +ω f( ) coω = Uing inegraion of ranform Le Then = ' F F F d G d Take he invere ranform of boh ide g coω f( ) = f( ) = Special Linear ODE wih Variable Coefficien Ue differeniaion of ranform o olve ODE. Le i [ y] = Y [ y' ] = Y y( ) i. Uing differeniaion of ranform d dy i [ y' ] = Y y( ) Y d d Similarly, uing i [ y" ] = Y y( ) y' ( ) i d d dy d [ y" ] = Y y( ) y' ( ) Y + y( )
Example 3 Laguerre Equaion Laguerre ODE i y" + y' + ny = n=,,, Kreyzig by YHLee;45; 6- The ubidiary equaion Separaing variable, uing parial fracion dy n + n n + = d d Y ( ) lny = nln n+ ln ln n + n Y = ( ) n n + The invere ranform i given by Rodrigue formula l = i Y n [ ] n=,, Prove Rodrigue formula Uing hifing Uing he n h derivaive of f, Afer anoher hifing
6.7 Syem of ODE Kreyzig by YHLee;45; 6-3 The Laplace ranform can be ued o olve yem of ODE. Conider a fir order linear yem wih conan coefficien The ubidiary equaion Rearrange a Y + a Y = y G ( ) a Y + ( a ) Y = y ( ) G ( ) Solve hi yem algebraically for Y ( ) and Y ( ) and ake he invere ranform for y ( ) and y ( ) Example Elecrical Nework i and i. Find he curren v = volonly for.5 and i =' i = From Kirchhoff volage law in he lower and he upper circui, Rearrange The ubidiary equaion uing i = i = Solve algebraically for I and I ( + ) 5 5 5 65 I = e e 7 7 ( / ) ( / ) + + 7 3 + + 5 5 5 5 I = e + e 7 7 ( / ) ( / ) + + 7 3 + +
The invere ranform of he quare bracke erm uing hifing. 5 5 / 65 7 / e e 7 3 5 5 / 5 7 / e + e 7 3 Kreyzig by YHLee;45; 6-4 Uing hifing 5 5 / 65 7 / 5 5 (.5)/ 65 7(.5)/ i ( ) = e e e e u(.5) 7 3 7 3 5 5 / 5 7 / 5 5 (.5)/ 5 7(.5)/ i ( ) = e + e e + e u(.5) 7 3 7 3 Noe ha he oluion for i differen from ha for due o he uni ep funcion. Example 3 Two mae on Spring Ignoring he ma of he pring and he damping Newon econd law(ma X acceleraion) Hooke law (reoring force) Iniial condiion y = y = y ' = 3 k, y ' = 3k The ubidiary equaion The algebraic oluion uing Cramer rule The final oluion