6- Chaper 6. Laplace Tranform 6.4 Shor Impule. Dirac Dela Funcion. Parial Fracion 6.5 Convoluion. Inegral Equaion 6.6 Differeniaion and Inegraion of Tranform 6.7 Syem of ODE
6.4 Shor Impule. Dirac Dela Funcion. Parial Fracion 6- 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 a d g a 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 L δ a = e a
Example Ma-Spring yem under a quare wave Inpu i of he form of a recangular funcion 6-3 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 =. 6-4 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, y y =, ' = 5 6-5 The ubidiary equaion The oluion (6) The parial fracion of he fir erm 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 6-6 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 6-7 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 ω in in * ω in in co ω Uing convoluion heorem : ω in xin y = / [ co( x + y) + co( x y)] Example 3 Unuual Properie of Convoluion f * f in general h = ωτ ω τ dτ ω + ω ω ω ω ( 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 = ( + ) ( ) + ' + : Q( ) = Y a y y Q R Q 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 = ' =, ranfer funcion + a + b The oupu i given by he convoluion of he impule repone q() and he driving force r(). Example 5 Ma-pring yem
Solve y" + 3 y' + y = r, for r = < < = oherwie y y = ' = 6-8 The ranfer funcion I invere Since y y = ' =, he oluion i given by he convoluion of q and r. ( ) ( ) ( 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. τ= ( τ) ( τ) ( ) ( ) y = e e e e + τ= τ= y = e e e e e e τ= τ= τ= ( τ) ( τ) ( ) ( ) ( ) ( ) 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. 6-9 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 L f( ) = F' ( ) and L F' ( ) = f ( ) Example Differeniaion of Tranform The able can be proved uing differeniaion of F(). The econd one d β β L [ inβ ] = d +β +β Inegraion of Tranform If f() ha a ranform and f( ) lim / exi, + f( ) = L F ( ) d and L F d = f Proof: From he definiion f( ) F ( ) d = e f ( ) d d f ( ) e d d L Revere he order of inegraion. = e /
Example Differeniaion and Inegraion of Tranform 6- Find he invere ranform of F() = I derivaive Take he invere ranform L F ' L 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 L [ y] = Y [ y' ] = Y y( ) L. Uing differeniaion of ranform d dy L [ y' ] = Y y( ) Y d d Similarly, uing L [ y" ] = Y y( ) y' L d d dy d [ y" ] = Y y( ) y' Y + y( )
Example 3 Laguerre Equaion Laguerre ODE i y" + y' + ny = n=,,, 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 = L Y n [ ] n=,, Prove Rodrigue formula Uing -hifing Uing he n-h derivaive of f, Afer anoher -hifing
6.7 Syem of ODE 6- 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( ) = vol only for.5 and i =' i = From Kirchhoff volage law in he lower and he upper circui, Rearrange i = i = The ubidiary equaion uing 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 6-3 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