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# Laplace transform and its application

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its ppt for the laplace transform which part of Advance maths engineering. its contains the main points and one example solved in it and have the application related the chemical engineering

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### Laplace transform and its application

1. 1. Year: 2016-17 Subject: Advanced Engineering Maths(2130002) Topic: Laplace Transform & its Application Name of the Students: Gujarat Technological University L.D. College of Engineering Agnihotri Aparna 160283105001 Agnihotri Shivam 160283105002 Kansara Sagar 160283105004 Makvana Yogesh 160283105005 Padhiyar Shambhu 160283105006 Patil Dipak 160283105008 Patil Mayur 160283105009 Rohit Chetan 160283105010 Sindhav Jaydrath 160283105011 Vasava Yogesh 160283105012
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 › Application in Chemical Engineering
3. 3. Definition of Laplace Transform › Let f(t) be a given function of t defined for all then the Laplace Transform of 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. Application In Chemical Engineering › A fast numerical technique for the solution of P.D.E. describing time-dependent two- or three-dimensional transport phenomena is developed. It is based on transforming the original time-domain equations into the Laplace domain where numerical integration is performed and by subsequent numerical inverse transformation the final solution can be obtained. The computation time is thus reduced by more than one order of magnitude in comparison with the conventional finite-difference techniques.
29. 29. Continue… › Application of Laplace transforms for the solution of transient mass- and heat-transfer problems in flow systems › Application to mass-transfer in single and multi-stream laminar parallel-plate flow systems
30. 30. Thanks…