Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful ool for olving a wide variey of iniial-value problem. The raegy i o ranform difficul ODE or PDE, Iniial-Value Problem, ino imple algebra problem, where oluion can be eaily obained; hen apply he Invere Laplace ranform o rerieve he oluion of he original problem. Thi can be illuraed a follow: Iniial-Value Problem Difficul Soluion o he IVP Laplace Tranform Invere Laplace Tranform Algebra Problem Eay Soluion o he Problem Le f() be a piecewie coninuou on he inerval [, ) and of exponenial order, a. Tha i, here exi real conan K, a, and T, uch ha f() K e a for all > T. (1) Then he Laplace Tranform of F () denoed by L {f() i: L {f() e f()d. () Noe ha he piecewie coninuiy and of exponenial order are ufficien, bu no neceary for L {f() o exi. 1. > a L {f()e a L a {f(). n IN L { n n! n+1 3. n p + 1 L { n 1 p+1 ( n n 3 1 π )
Maoud Malek Laplace Tranform By differeniaing f(), N ime and hen aking he Laplace Tranform, we obain: 4. L {f (N) () N L {f() N 1 f() N f ()... f N 1) () By differeniaing N ime he Laplace Tranform of F () wih repec o he parameer, we obain 5. d N d N (L {f()) L {( ) N f() An immediae conequence of he above formula i: 6. L { N f() ( 1) N dn d N (L {f()) By inegraing he Laplace Tranform of f(), we obain: { f() 7. L L r {f()dr Periodic Funcion A Period funcion f() of period ω, i a funcion, where f( + ω) f(), D f Theorem 1. If f() i piecewie coninuou on [, ) of exponenial order and periodic wih period omega, hen 8. L {f() 1 ω e u f(u)du 1 e ω Proof. Wrie he Laplace ranform of f() a: L {f() ω e f()d + ω ω e f()d + + (n+1)ω nω e f()d + Now le u 1 + ω, u + ω,......, u 1 + nω... hen we have:
Maoud Malek Laplace Tranform 3 L {f() A Special Sep Funcion ω e f()d k (k+1)ω kω e (u k+kω) f(u k + kω) du k (k+1)ω e kω e u k f(u k ) du k k e kω k 1 1 e ω kω ω ωe u f(u) du e (u) f(u) du In engineering one frequenly encouner funcion ha are eiher off or on. I i convenien, hen, o define a pecial funcion ha i he (off) up o a cerain ime a and hen he number 1 (on) afer ha ime. Thi funcion i called he uni ep funcion or he Heaviide funcion. The uni ep funcion, denoed by U(), i defined o be:, if <, U() f(), for., if < a, Hence U( a)f() f(), for a. Thu By inegraing: we obain: f(), if a b [ U( a) U( b)] f(), oherwie; b a e [ U( a)f() ] d 9. L {U( a)f( a) e a L {f() e a [ ] e f() d Gamma Funcion For x >, we define he Gamma funcion a follow: Γ(x) e x 1 d L { x 1. (3) 1. > Γ(x + 1) xγ(x) x+1 L { x 11. n IN Γ(n + 1) nγ(n) n! 1. Γ( 1 ) π
Maoud Malek Laplace Tranform 4 Invere Laplace Tranform The Laplace Tranform i a one o one funcion, herefore i invere funcion i alo a funcion and i called he Invere Tranform denoed by {F (); o f() {L {f(). (4) Condiion for he Exience of a he Invere Tranform of {F () are a follow: Convoluion Theorem If F () L {f() and G() L {g(), hen lim L 1 {F () ; (5) lim L 1 {F () i finie. (6) 13. {F () G() f(u)g( u)du f( u)g(u)du We hould alway chooe he impler inegral. From he fac ha L {U() 1, we obain 14. { F () f(u)du Mo of he ime we can ue parial fracion in order o fine he invere Laplace ranform, bu here are ime ha he convoluion heorem may be eaier or he only way in finding he invere Laplace ranform of he produc of wo Laplace ranform. For example, { 1 ( + k ) can no be evaluaed by he ue of parial fracion, o we mu ue he convoluion heorem a follow: Le F () G() 1 + k o ha f() g() 1 k L 1, { 1 1 in k (7) + k k
Maoud Malek Laplace Tranform 5 Then (7) give { 1 ( + k ) 1k in kτ in k( τ)d τ 1 [co k(τ ) co k ] d τ k 1 [ 1 k k ] in k(τ ) τ co k in k k co k k 3 Inead of uing parial fracion, o evaluae { k ( + k ), we may ue he convoluion heorem. From 14, we have: Inegral Equaion { k ( + k ) in ku d u [ 1 ] k co ku 1 co k k An equaion in which an unknown funcion appear under an inegral ign i called an inegral equaion. There i a cloe connecion beween differenial and inegral equaion, and ome problem may be formulaed eiher way. The convoluion heorem i ueful o olve inegral equaion. Example. Solve he following inegral equaion: f() 3 e e τ f(τ) d τ. Le F () L {f() and g() e, wih G() L {g() L {e 1 1. Noe ha Take he Laplace ranform of boh ide. e τ f(τ) d τ {G() F (). L {f() f() 3 3 1 + 1 f() 1 1. f() The invere ranform, hen give 6( 1) 4 1 ( ) 6 3 6 4 + 1 + 1. { 6 f() 6 3 + 1 4 3 3 + 1 e. + 1
Maoud Malek Laplace Tranform 6 Any equaion in he form f() g() + g(τ)g( τ) d τ i called a Volera inegral equaion for f(). Trigonomeric and Hyperbolic Funcion Here are ome ueful formula for he invere ranform. { k + k { k k {arcan( k ) { in(k) + k { inh(k) in(k) { ln co(k) coh(k) k ( ) + k k inh(k) Soluion of Linear Differenial Equaion wih Conan Coefficien We now conider how he Laplace Tranform may be applied o olve he iniial-value problem coniing of he nh-order linear differenial equaion wih conan coefficien. Conider he iniial-value problem: a y (n) + a 1 y (n 1) + + a n 1 y + a n y b y() c, y () c 1,..., y (n 1) () c n 1. By aking fir, he Laplace Tranform of boh ide of he equaion and hen he Invere Tranform of he reuling equaion, we may find a oluion o he problem. We illurae our mehod wih he following example. We hall denoe L {x by X() and L {y by Y (). Solve y 6y + 9y e 3 ubjec o: y(), y () 6. Sep 1. L {y 6L {y + 9L {y L { e 3 Sep. [ Y () 6] 6 [Y () ] + 9Y () Sep 3. [ 6 + 9] Y () ( 3) + Sep 4. Y () 3 + ( 3) 5 ( 3) 3 ( 3) 3 Soluion: y() {Y () e 3 + 1 1 4 e 3
Maoud Malek Laplace Tranform 7 Finally, we find he oluion of a yem of linear differenial equaion wih iniial condiion. Solve x x y e, y + x y e, ubjec o : x(), y(). L {x L {x L {y L {e Sep 1. L {y + L {x L {y L {e ( 1)X() Y () Sep. 1 X() + ( 1)Y () 1 1 [ ] [ ] 1 X() 1 1 Sep 3. 1 Y () 1 1 1 1 X() ( 1)( Sep 4. + 1) 1 [ 1 1 + 1 + 3 ] [ + 1 3 + 1 Y () ( 1)( + 1) 1 + 1 + 1 ] + 1 x() {X() 1 [ e co + 3 in ] Soluion: y() {Y () [e co + in ] Dirac Dela Funcion A uni pule funcion, called Dirac Dela Funcion i a generalized funcion (no a real funcion, a we hink of hem), depending on a real parameer uch ha i i zero for all value of he parameer excep when he parameer i zero. Some imporan properie of he Dirac Dela funcion are a follow: 1. δ( a), for a.. δ(a) 1 a δ(). 3. δ( a ) 1 [δ( + a) + δ( a)]. a 4. f()δ( a) f(a). 5. For ɛ >, a+ɛ a ɛ f()δ( a) f(a). 6. L {δ( a) e a, provided a >.
Maoud Malek Laplace Tranform 8 Click here o ee an animaion of he Dirac Dela Funcion Solve y + y 15y 6 δ( 9), y() 5, y () 7. Sep 1. Take he Laplace Tranform of he equaion Y () y() y () + ( Y () 15Y () 6 e 9 ( + 15) Y () + 5 + 3 6 e 9 Sep. Solve for Y () 6 e 9 Y () ( 3)( + 5) 5 + 3 ( 3)( + 5) 6 e 9 F () G() Sep 3. Ue parial fracion F () 1 [ 1 8 3 1 ] ( + 5) G() 1 4 [ 9 3 11 ] ( + 5) Sep 4. Find he invere ranform f() 1 [ e 3 e 5] g() 1 [ 9 e 3 11 e 5] 8 4 Soluion: y() 1 [ e 3( 9) e 5( 9)] U( 9) 1 [ 9 e 3 11 e 5] 8 4