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# Laplace transform and its applications

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Laplace transform and its applications

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### Laplace transform and its applications

1. 1. Made By:- S.Y. M-2 Shah Nisarg (130410119098) Shah Kushal(130410119094) Shah Maulin(130410119095) Shah Meet(130410119096) Shah Mirang(130410119097) Laplace Transform And Its Applications
2. 2. Topics  Definition of Laplace Transform  Linearity of the Laplace Transform  Laplace Transform of some Elementary Functions  First Shifting Theorem  Inverse Laplace Transform  Laplace Transform of Derivatives & Integral  Differentiation & Integration of Laplace Transform  Evaluation of Integrals By Laplace Transform  Convolution Theorem  Application to Differential Equations  Laplace Transform of Periodic Functions  Unit Step Function  Second Shifting Theorem  Dirac Delta Function
3. 3. Definition of Laplace Transform  Let f(t) be a given function of t defined for all then the Laplace Transform ot f(t) denoted by L{f(t)} or or F(s) or is defined as provided the integral exists,where s is a parameter real or complex. 0t )(sf )(s dttfessFsftfL st )()()()()}({ 0     
4. 4. Linearity of the Laplace Transform  If L{f(t)}= and then for any constants a and b )(sf )()]([ sgtgL  )]([)]([)]()([ tgbLtfaLtbgtafL  )]([)]([)}()({ )()( )]()([)}()({ Definition-By:Proof 00 0 tgbLtfaLtbgtafL dttgebdttfea dttbgtafetbgtafL stst st           
5. 5. Laplace Transform of some Elementary Functions asif a-s 1 )( e.)e( Definition-By:Proof a-s 1 )L(e(2) )0(, s 1 1.)1( Definition-By:Proof s 1 L(1)(1) 0 )( 0 )( 0 atat at 00                              as e dtedteL s s e dteL tas tasst st st
6. 6. |a|s, a-s s at]L[coshly,(5)Similar |a|s, a-s a 11 2 1 )]()([ 2 1 2 e Lat)L(sinh definitionBy 2 e atcoshand 2 e atsinhhave-We:Proof a-s a at]L[sinh(4) -as, 1 ]L[e3)( 22 22 at atat 22 at-                                asas eLeL e ee as atat at atat
7. 7. 0s, as s at]L[cosand as a at]L[sin getweparts,imaginaryandrealEquating as a i as s as ias 1 )L(e1 ]e[]sin[cos sincose Formula]s[Euler'sincosethatknow-We:Proof 0s, as s at]L[cosand as a at]L[sin(6) 2222 222222 at iat iat ix 2222                                as ias LatiatL atiat xix 
8. 8.    n!1n0,1,2...n n! )(or 0,n-1n, 1 )( 1 ust,.)-L(:Proof n! or 1 )()8( 1 0 1 1 0 1)1( 1 0 0 11                                        n n nx n n nu n n u nstn nn n S tL ndxxe S n tL duue S s du s u e puttingdttet SS n tL
9. 9. First Shifting Theorem )(f]f(t)L[e, )(f]f(t)L[e )(f)(f ra-swhere)(e )(e )(ef(t)]L[e DefinitionByProof )(f]f(t)L[ethen,(s)fL[f(t)]If shifting-stheorem,shiftingFirst-Theorem at- at 0 rt- 0 a)t-(s- 0 st-at at asSimilarly as asr dttf dttf dttfe as at              
10. 10. 22)-(s 2-s )4cosL(e 2s s L(cosh2t) )2coshL(e(1) 43)(s 3s )4cosL(e 4s s L(cos4t) )4cosL(e(1) : 22 2t 22 2t 22 3t- 22 3t-           t t t t Eg
11. 11. Inverse Laplace Transform )()}({L bydenotedisand(s)foftransformlaplaceinverse thecalledisf(t)then(s),fL[f(t)]If-Definition 1- tfsf  
12. 12. 2 1 2 1 12 1 )2( 2 1 )1( 1 2 1 C than0s 2 1 B than-2s -1A than-1s 2)1)(sc(s1)(s)B(s2)(s)A(s1 )2()1())(2)(1( 1 2)(s)1)(s(s 1 L)1( 21 1 1                                     tt ee sss L If If If s C s B s A sss L
13. 13. Laplace Transform of Derivatives & Integral   f(u)du(s)f 1 LAlso (s)f 1 f(u)duLthen(s),fL{f(t)}If f(t)ofnintegratiotheoftransformLaplace (0)(0)....ffs-f(0)s-(s)fs(t)}L{f f(0)-(s)fsf(0)-sL{f(t)}(t)}fL{ and0f(t)elimprovidedexists,(t)}fL{then continous,piecewiseis(t)fand0tallforcontinousisf(t)If f(t)ofderivativetheoftransformLaplace t 0 1- t 0 1-n2-n1-nnn st t                      s s
14. 14. 22 2 22 3 22 2n s a at)L(sin at)L(sins s a- a-at)L(sinssinat}L{-a thisfroma(0)f0,f(0)Also sinat-a(t)fandatcosa(t)fsinat thenf(t)Let:Sol atsinoftransformlaplaceDeriveExample a a a         )1( 1 )( 1 cos cosf(u)-Here:Sol cos 2 0 0                   ss sf s uduL u uduLEg t t
15. 15. Differentiation & Integration of Laplace Transform            0 n n nn ds(s)f t f(t) Lthen ,transformLaplacehas t f(t) and(s)fL{f(t)}If TransformsLaplaceofnIntegratio 1,2,3,...nwhere,(s)]f[ ds d (-1)f(t)]L[tthen(s)fL{f(t)}If TranformLaplaceofationDifferenti
16. 16. 3 2 2 2 2at2 at2 )( 2 )( 1 1 )1()e(-:Sol )e(: as asds d asds d tL tLExample                    
17. 17.                                            ss s s s ds t t LExample s 11 11 1 s 22 cottan 2 tantan tan 1 .t)L(sin-:Sol sin
18. 18. Evaluation of Integrals By Laplace Transform                       1 )1()cos( 1 )(cos cos)cos( cos)(3 )()}({ cos-:Example 2 2 0 0 0 3 s s ds d ttL s s tL tdttettL tttfs dttfetfL tdtte st st t 25 2 100 8 )19( 19 cos cos )1( 1 )cos( )1( 2)1( 1 2 0 3 0 22 2 22 22                        tdtte tdtte s s ttL s ss t st
19. 19. Convolution Theorem g(t)*f(t) g*fu)-g(tf(u)(s)}g(s)f{L theng(t)(s)}g{Landf(t)(s)}f{LIf t 0 1- -1-1    
20. 20.   )1(e e .e . )1( 1 )1( 1 . 1 )1( 1 n theoremconvolutioby )( 1 1 (s)gand)( 1 (s)fhavewe: )1( 1 : t 0 t 0 t 0 2 1 1 2 1 2 2 1                                             t eue dueu dueu ss L ss L ss L eL s tL s HereSol ss LExample tuu t u t ut t
21. 21. Application to Differential Equations 04L(y))yL( sidebothontranformLaplaceTaking . . (0)y-(0)ys-y(0)s-Y(s)s(t))yL( (0)y-sy(0)-Y(s)s(t))yL( y(0)-sY(s)(t))yL( Y(s)L(y(t)) 6(0)y1y(0)04yy: 23 2      eg
22. 22. tt s s 2sin 2 3 2cos 4s 6 4s Y(s) transformlaplaceinverseTaking 4s 6 Y(s) 06-s-4)Y(s)(s 04(Y(s))(0)y-sy(0)-Y(s)s 22 2 2 2          
23. 23. Laplace Transform of Periodic Functions      p 0 st 0)(sf(t)dte e-1 1 L{f(t)} ispperiodwith f(t)functionperiodiccontinouspiecewiseaoftransformlaplaceThe 0tallforf(t)p)f(t if0)p( periodithfunction wperiodicbetosaidisf(t)Afunction-Definition ps-
24. 24.                                                               2w sπ hcot ws w e e . e1 e1 . ws w e1 ws w . e1 1 L[F(t)] e1 ws w wcoswt)ssinwt( ws e sinwtdteNow tallforf(t) w π tfand w π t0forsinwtf(t) 0t|sinwt|f(t) ofionrectificatwave-fulltheoftransformlaplacetheFind 22 2w sπ 2w sπ w sπ w sπ 22 w sπ 22 w sπ w sπ 22 2 w π 0 w π 0 22 st st
25. 25. Unit Step Function s 1 L{u(t)} 0aif e s 1 s e (1)dte(0)dte a)dt-u(tea)}-L{u(t at1, at0,a)-u(t as- a st- a st- a 0 st- 0 st-                   
26. 26. Second Shifting Theorem a))L(f(tea))-u(tL(f(t)-Corr. L(f(t))e (s)fea))-u(ta)-L(f(t then(s)fL(f(t))If as- as as-     
27. 27.   )(cos)2( )2(cos)2()2(L )()()(L theroemshiftingsecondBy (ii)L 33 1 }{. }{)]2(L[e 2,ef(t) )]2((i)L[e 22 1 22 2 1- 1- 22 2 1- )3(2 )62( 362 )2(323t- 3t- -3t ttu ttu s s Ltu s se atuatfsfe s se s e s e eLee eLetu a tuExample s as s s s ts ts                                            
28. 28. Dirac Delta function 1))(( ))(( 0 1 0lim 0        tL eatL tε, a εat, a ε at, -a)δ(t as ε  
29. 29.  sin3tcos3t2ex(t) sin3t2ecos3t2ex inversionon 92)(s 6 92)(s 2)2(s 134ss 102s x 2(1)x13x(0)]-x4[s(0)]x-sx(0)x[s haveweTransform,LaplaceTaking 0(0)xand2x(0)0,t(t),213xx4x 0(0)xand2x(0)0,at therew (t)213xx4xequationthe-Solve:Example 2t 2t-2t- 222 2                   