# IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704

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1 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion of he He Equion in Semi-Infinie Region 76 IX..6 Wve Equion 7 IX..7 Soluion of Inegrl Equion 74 IX..8 Review Queion nd Eercie 76 IX..9 plce Trnform wih Mple 78

2 7 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 IX.. DEFINITION: The plce rnform of he funcion f ( ), i defined plcetrnform { f ( ) } ( ) f ( ) φ e d > M Me f ( ) There re everl rdiionl noion ued for he plce rnform: e : ( ) φ( ) ( ) Y ( ) ( ) u( ) f : y : u The funcion i he kernel of he rnform nd i he rnform vrible. The eience condiion for he plce rnform i eblihed for funcion growing no fer hn n eponenil funcion: if here ei conn > nd M > uch h ( ) Me of eponenil order. f for ll hen he funcion ( ) f i clled Theorem 9. (ufficien condiion for eience of he plce rnform) e he funcion f ( ) be piecewie coninuou on [,) nd of eponenil order wih conn > nd M >, hen ) The plce rnform { f ( ) } ( ) f ( ) ) φ ( ) M 3) ( ) φ when 4) ( ) φ i bounded when φ e d ei for ll > Emple:. f ( ) e, M f ( ) e d e >. f ( ) Me f ( ) M > Invere plce rnform We define he invere plce rnform of he funcion φ ( ) n operion which yield funcion f ( ) uch h { f ( ) } φ( ) : { φ } f Here, we conider he plce rnform rericed o rel vlue of. A more generl definiion i bed on he Fourier rnform pplied o funcion which re equl o zero for negive vlue, nd he vrible i n imginry frequency: iω, < ω < f F f f e d f e d iω { } π { } Then he invere plce rnform i defined by φ f α i α + i e d π i, >, α Re( )

3 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 7 IX.. PROPERTIES To clcule he plce rnform nd i invere we will ue moly he ble of he plce rnform nd i properie. e { f ( ) } φ ( ) hen: ) ineriy: Boh nd re liner: { f ( ) + bg( ) } { f ( ) } b{ g( ) } + ) Shifing in : { αφ( ) + βψ ( ) } α { φ( ) } + β { ψ ( ) } { } φ e f { } φ ( + ) e f { φ ( )} e f { φ ( + )} e f 3) Shifing in : e H( ) ( ) H hen 4) Similriy: { f ( ) } be he Heviide uni ep funcion if > if < { } φ f H e > { φ } ( ) ( ) e f H φ d d φ 5) Differeniion: { f ( ) } ( ) 6) Inegrion: f φ d 7) Convoluion: f g f ( ) g( )d Convoluion Theorem { f g} { f} { g} F( ) G( ) 8) Trnform of derivive: { } F G f g { f ( ) } φ ( ) f { f ( ) } φ ( ) f f ( n ) n n n ( n { f ( ) } φ ( ) f f... f )

4 7 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 IX..3 EXAMPES ) Uing he definiion, clcule he plce rnform of f ( ) { } e d d e : d ( e ) inegrion by pr + e e d + + { } e lime ) Derive he propery { f ( ) H( ) } e φ ( ). According o definiion of he plce rnform { ( ) ( )} ( ) ( ) f H f H e d f e d ubiue τ + ( τ) ( τ + ) f e dτ e f e τ d ( τ) e φ ( ) 3) Derive he propery { f ( ) } ( ) f { ( ) } f 4) Evlue { }. f e d e d f τ φ. move o differenil f ( ) e f ( ) d e inegrion by pr f f e d + φ f e f ( ), hen f ( ) nd { f ( ) } φ ( ) f { } { } f { } { } { } 3 In generl, for n, pplicion of propery (7) yield n! { } n n + ume lim f e f. Apply he propery

5 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, ) Inverion of he formul { } { { }} n n! n+ n! n+ n n n+ n! n! : n n + 6) Invere rnform of rionl funcion (pril frcion decompoiion): Evlue 9. + Conver rionl funcion o pril frcion (ee Secion ): + 9 ( + ) + ( ) + A B A 3 B A 3 6 B Therefore + 9 A 3 B e e ) Invere rnform of rionl funcion (convoluion heorem): Evlue. ( ) Noice h { } ( ) nd { e } { { } { e }} e ( ) e d e d e d e e e e e + e e. Then by convoluion

6 74 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 IX..4 Soluion of IVP for ODE The plce rnform elimine derivive nd i cn be ued for he oluion of differenil equion in emi-infinie domin, <. Bu becue he plce rnform of he derivive include vlue of he funcion nd i derivive zero, he plce rnform i more uible for iniil vlue problem rher hn for boundry vlue problem like he Fourier rnform. Emple (Soluion of IVP by he plce Trnform) Conider he nd order differenil equion y + y y in ( ) π wih wo iniil condiion π y π y π ) Trnle iniil condiion by he chnge of he vrible τ o τ : π y + y y in τ + Then iniil condiion re τ : y y. ) Apply he plce rnform Y ( ) y( τ ) Solve for Y Y + Y Y + Y e ( + )( )( + ) 3) Then by invere plce rnform y τ d τ o equion ( ): τ τ ( τ ) e + e coτ + inτ π Ue bck ubiuion τ o ge he oluion of he originl IVP: y 6 π 5 3 π + + π e e co + in π π y( )

7 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 75 Emple (Soluion of IVP by he plce Trnform) Solve he 3 rd order differenil equion y y y + y e ( ) ubjec o iniil condiion y y y ) Apply he plce rnform Y ( ) y( τ ) e τ d τ o equion ( ): Y y y y Y y y Y y + Y + 3 Y Y Y + Y Y + Solve for Y Y + 3 ( + )( )( )( + ) Conver hi epreion o pril frcion (ee Emple 6): Y ( + ) ( ) ( + ) ( ) ) Then he oluion of he IVP cn be found by invere plce rnform 5 y e e e e y( )

8 76 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 IX..5 SOUTION OF THE HEAT EQUATION IN SEMI-INFINITE REGIONS ) Dirichle Problem Conider he He Equion in he emi-infinie lb for u (, ) boundry condiion u, f u (,) u (,) u (,) iniil condiion u (, ) < u u (, ) > u (,) iniil condiion (,) f ( ) u > Dirichle b.c. lim u, < > bounded oluion α Trnformed equion Apply he plce rnform in he vrible (,) u(,) he He Equion nd o he boundry condiion U e d o u, U, u, U U (,) F ( ) where U F f e d Thi i he nd order liner ODE wih conn coefficien. Generl oluion (,) c e c U + e Soluion of he HE nd correpondingly i plce rnform hould be bounded when, herefore, we hve o ign c : (, ) c e U Applying boundry condiion, one end up wih he oluion for he rnformed funcion U (,) F ( ) e Noe (ee Mple Emple 7) h he funcion rnform of he funcion e i plce 4 G( ) { g(,) } e e 3 π g 4 ( ) e π 3 Then he rnformed oluion i produc of wo plce rnform: rnformed oluion (,) F ( ) G( ) U Apply he invere plce rnform uing he convoluion heorem u (, ) { } ( ) F G f g f τ g τ dτ Finlly, he oluion i given in erm of convoluion inegrl 4( τ ) e d 3 Soluion of IVP: u(,) f ( τ ) π ( τ ) τ

9 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 77 Conn b.c. f f Conider ce when he boundry condiion pecifie conn emperure f f, hen u, Mke he ubiuion: 4 ( τ ) f e π ( τ ) 3 z limi of inegrion: τ dτ ( τ ) ( ) 3 ( τ ) dz dτ z τ z z e dz π f z z u (,) e dz f e dz ferfc π π erf Then he oluion i given by u, f erfc ( ) f Plo he oluion for nd f u (,) 5 The emperure diribuion in he emi-infinie lyer pproche he edy e conn vlue of he boundry condiion f. θ, u, u Generl ce: u (,) u Chnge he vrible: Iniil condiion: θ (,) u (,) u Boundry condiion: θ (,) u (,) u f u Soluion: θ u, u f u erfc, f u erfc +

10 78 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 b) Robin Problem Conider he He Equion in he emi-infinie lb for u (, ) u (,) u u (, ) > α u u (,) u u, iniil condiion u u (, ) < u k + hu hu lim u, > Robin b.c. < > bounded oluion ) Reduce o IBVP wih zero iniil condiion: Chnge of vrible: θ (,) u (,) u u (,) θ (,) + u Iniil condiion: θ (,) u (,) u ( θ + u ) Boundry condiion: ( θ ) imiing condiion: limθ (,) He Equion: k + h + u hu θ + Hθ H u u θ ) Soluion by plce rnform: < θ, h H k Trnformed equion: Boundry condiion: Θ Θ Θ H u + HΘ limθ (,) < ( u ) Soluion: Θ, c e c e Θ + (,) c e Derivive: Subiue ino b.c.: Θ, ce H u c e + He H u c H + c ( u ) H u H + ( u ) c ( u )

11 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 79 Trnformed Soluion: Θ (,) ( u u ) H e H + Ue Mple o find he invere plce rnform: H A, b > invlplce(a*ep(-b*qr())//(qr()+a),,); + b~ + A~ erf e ( b~ A~ + A~ ) + erf b~ b Ae Ab A b b e e erfc A + + erfc ( + A ) Invere plce rnform: H H+ H θ (,) ( u u) e erfc + + ( u u) erfc Soluion: H H+ H u, u + u u erfc e erfc + Emple: u, u 5,, k 5, h u (,) u, 3 u

12 7 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 c) Conc Problem Conider he idel herml conc of wo emi-infinie lb of uniform conduciviie k nd k, nd coefficien of hermodiffuiviy α nd α, correpondingly. Iniilly lb re he emperure T nd T, hen he rnien emperure diribuion i decribed by u (,) u,. u, nd T u k (,) T k u (,) T u u u u (,) > (, ) > α α u, T iniil condiion u, T u, u, > coninuiy k u, k u, > conervion of flu u (,) Symmeric eenion of u (, ) u (,) T u (,) T T Soluion of he Dirichle problem for he emi-infinie lb: + u, T T T erfc + u, T T T erfc Here, he condiion of emperure coninuiy he conc i umed wih conn emperure of he conc T : u, u, T

13 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 7 Apply condiion of conervion of flu he conc: u(,) ( T T ) ep π u(,) ( T T ) ep π k( T T ) ep π k( T T ) ep π k( T T ) π k T T π ( ) k ( T T) k T T Solve for conc emperure: Conc Temperure: kt T k + kt + k Conc emperure i conn for >. Emple: T 3. 3 T 4.3 T. k 4.. k 5..

14 7 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 IX..6 WAVE EQUATION Conider vibrion of he emi-infinie ring (WE) u u + g(,) u (,) : (, ) > g (,) force per uni lengh of he ring (,) f ( ) u > Dirichle boundry condiion (, ) u ( ) u(, ) u ( ) u iniil condiion iniil condiion ) Trnformed equion Apply he plce rnform in he vrible U G F (,) u(,) e (,) g(,) ( ) f ( ) d e d e d o he WE nd boundry condiion U u u U + G (,) F ( ) U u u G U U Q Q noion Generl oluion: e + ce U p U c + where coefficien c, c re, in generl, funcion of. Hence he oluion hould be bounded, c, nd he generl oluion become e U p U c + Ce u u G Conider ce Q : iniilly he ring i re, here i no eernl force. Then he oluion become: U F Noe h ( ) e δ δ e d δ e d e for >

15 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 73 ) Invere rnform: (,) F ( ) e { f ( ) } δ δ f ( ) u Clcule convoluion δ f δ τ f ( τ) dτ δ τ f dτ f H The oluion become: (,) f H u Emple of ime dependen b.c. 3) Conider he ce of periodic b.c.: ( ) in u (,) in H Plo he oluion (T-.mw): f hen u (,) 4 5 Eercie: Conider differen ce: wih grviionl force fied end, iniil hpe nd velociy Coefficien in he Wve Equion: gt w m peed of propgion of ound wve in he medium; g ccelerion of grviy; T enion; w weigh of he ring per uni lengh.

16 74 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 IX..7 INTEGRA EQUATIONS Recll he Volerr inegrl equion of he nd kind λ + u K, y u y dy f If he kernel of he inegrl equion K (,y ) i funcion of y K (,y) g( y) hen he inegrl equion i id o be convoluion inegrl equion: convoluion inegrl equion u ( ) λ g ( y) u ( y) dy + f ( ) We will conider pplicion of he plce rnform o he oluion of he convoluion inegrl equion. The mehod i bed on he convoluion heorem for he plce rnform: { f g} f g Apply he plce rnform o he convoluion inegrl equion λ ( ) + u g y u y dy f λ { } + λ + u g u f u g u f Solve for he rnformed funcion u u ( ) f λ g Then he forml oluion of he inegrl equion i given by he invere plce rnform f u( ) λg Emple (non-homogeneou Volerr inegrl equion of he nd kind) Solve he inegrl equion λ u + u y dy Apply he plce rnform (ue ble propery 49): u u ( y) dy u u + λ

17 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 75 Solve for u u λ Then he oluion i given by he invere plce rnform u e λ λ Emple (Abel inegrl equion) Abel inegrl equion Tble T: { } Γ Solve he inegrl equion: u ( y) ( y) f dy Rewrie (AE) in he form ( ) f u y y dy nd pply he plce rnform < < (AE) f ( ) u ( y)( y) dy { } Γ { } { } u Γ u u Solve for he rnformed unknown funcion { } u y u( ) f ( ) u ( ) ( ) Γ ( ) Γ + Γ + f u Γ ( ) Γ u { f ( ) } { } + Γ Then pplicion of he invere plce rnform nd convoluion heorem, yield forml oluion for he Abel inegrl equion u ( ) u + ( y) f ( y) dy Γ ( ) Γ

18 76 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 IX..8 REVIEW QUESTIONS ) How i he plce rnform defined? ) Wh condiion gurnee he eience of he plce rnform? 3) How i he invere plce rnform defined? 4) Wh re he min properie of he plce rnform nd he invere plce rnform? 5) How cn he convoluion heorem be pplied for evluion of he invere plce rnform? 6) How cn he invere plce rnform of he rionl funcion be found? 7) Wh propery llow pplicion of he plce rnform for oluion of differenil equion? 8) Wh re he min ep in he procedure of pplicion of he plce rnform for oluion of differenil equion? EXERCISES. ) Derive he imilriy propery { f ( ) } φ. f φ f f. b) Uing inegrion by pr derive { }. Evlue uing he definiion of he plce rnform: 3 ) { in( )} b) { e } 3. Evlue uing he Tble nd he properie of he plce rnform: e b) ( + 5) 3 ) { } c) { } 3 { } d) { e co } e

19 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, Evlue he following invere plce rnform uing pril frcion: ) 3 + ( 9)( + ) uing he convoluion heorem: c) + 9 b) d) ( 3)( ) ( + 9) 5. Solve he following Iniil Vlue Problem: ) y y 6y 3 y + y + y y b) y y y y in3 y 6. Solve he Neumnn Iniil-Boundry Vlue Problem for u (,) : u u (, ) > T u, k u, q lim u, > < bounded oluion Skech he oluion curve. 7. Solve he Iniil-Boundry Vlue Problem for (,) dependen periodic boundry condiion: u wih ime- u u (,) > T u, k u, + hu, Tm in u (,) ( ω ) > Chooe ome vlue for he conn nd kech he oluion curve.

20 78 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 IX..9 APACE TRANSFORM WITH MAPE ) Evluion of he plce Trnform uing he definiion (for > ): Emple : > f():;ume(>); f( ) : > phi():in(f()*ep(-*),..infiniy); φ( ~ ) : ~ ) Evluion of he plce Trnform nd he invere plce rnform wih he help of commnd in he pckge inrn: > wih (inrn): Emple : > lplce(ep(),,); Emple 3: Emple 4: Emple 5: Emple 6: > lplce(^*in(),,); ( + 3 ) ( + ) 3 > invlplce(/(^+9),,); co( 3 ) > invlplce(ep(-*)/,,); Heviide ( ) > Y:(+3)/(+)/(-)/(-)/(+); > y():invlplce(y,,); + 3 Y : ( + ) ( ) ( ) ( + ) y( ) : + 5 ) + e( 3 e e ) 3 e( Emple 7: > ume(>):ume(>): > G():ep(-**qr()); > invlplce(g(),,); G( ) : e ( ~ ~ ) ~ ~ e ~ ~ 4 π ( 3/ )

21 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 79

22 7 Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 Rue plce

### IX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704

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