Ministry of Education and Science of Ukraine National Technical University Ukraine "Igor Sikorsky Kiev Polytechnic Institute"

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1 Minisry of Educaion and Scinc of Ukrain Naional Tchnical Univrsiy Ukrain "Igor Sikorsky Kiv Polychnic Insiu" OPERATION CALCULATION Didacic marial for a modal rfrnc work on mahmaical analysis for sudns of nd yar nginring faculis Comilrs: Zadry Nadiya, Candida of hysico-mahmaical scincs, associa rofssor Mamsa Karyna, Candida of hysico-mahmaical scincs, associa rofssor Nfodova Galyna, Candida of hysico-mahmaical scincs Prsyuk Mariya, Candida of hysico-mahmaical scincs, Dircor of EQMI NTUU Igor Sikorsky KPI Kiv 6

2 OPERATION CALCULATION Didacic marial for a modal rfrnc work on mahmaical analysis for sudns of nd yar nginring faculis / Comilrs: Zadry Nadiya, Mamsa Karyna, Nfodova Galyna, Prsyuk Mariya К, NTUU Igor Sikorsky КPI, 6 4 Undr h hading of scinis council of PМD NTUU Igor Sikorsky КPI (Physics - Mahmaics Darmn) Educaional ublicaions Undr diion of comilrs Th lcronic vrsion Excuiv dior Y P Busnko, Candida of hysico-mahmaical scincs, associa rofssor Rviwr AM Kulik, Candida of hysico-mahmaical scincs, associa rofssor Kiv 6

3 3 Oraional calculus Conrol work Inroducion Oraional calculus is sudid in h cours of mahmaical analysis in h hird smsr Surgry (symbolic) calculus widly usd in various filds of scinc and chnology A aricularly imoran rol i lays in h sudy of ransins in linar hysical sysms hory of lcrical circuis, auomaion, radio nginring, mchanics Didacic marial conains 3 varians of modular conrol work bing don by h scond yar sudns of chnical scialis in h hird smsr Th work consiss of fiv asks and is dsignd for 9 minus In h firs ask, you find h rsn original imag I hls o larn h dfiniion of h Lalac ransform and is roris Th scond ask of h rsn mus Find imag of h original In h hird ask, you solv h Cauchy roblm for linar diffrnial quaions wih icwis coninuous righ-hand sid Th fourh ask is roosd o solv h Cauchy roblm for linar diffrnial quaions using Duhaml ingral In h fifh ask roosd Volrra ingral quaion of convoluion y Each vrsion of h conrol modul aachd rly

4 4 Th us of of oraional calculaions I Soluion of Cauchy roblm for linar diffrnial quaion wih consan cofficins in finding h righ ar of imag According o h lan: By mans of Lalac ransformaions ugrad linar diffrnial quaion in rlaiv algbraic imag Find ou in his algbraic quaion of h dsird imag of h original (calld oraional soluion) 3 According o rroduc h original imag (answr) Examl y"+y'+y = sin y()=, y'() = - y()y() y'()y() y() = Y() - = Y() y"() Y() y() - y'() = Y()+ sin hav h oraor quaion Y()++Y()+Y() = Y()( ++) = -+ Y() = - + Find h original: ( ) = а) - ( ) ( ( ) - - ) (oraional soluion) alid h horm of diffrniaion of h original F'() -f() W hav F()= - b) rlaiv o h scond rm w aly h scond horm of dcomosiion ( ) ( ) P = -i Pol of ІІ ordr P = -і siml ol P 3 = i siml ol = ( ) ( i)( i) F() = ( ) ( ) = rs F() + rs F() + rs F() = ( )! d d =lim - ( ) +lim i ( ) ( i) 3 +lim -i ( ) = ( i)

5 =lim ( ) - + i + i = ( ) i( i ) i( i ) 4 i i = cos as h soluion of a linar diffrnial quaion is a funcion of: y() = - + Examl 5 cos = ( cos ) y"() + y()= f(), whr f() givn grahically, y()=y'()= i i i + i i( i) = - Th soluion y()y() y"() Y()-y(o)-y'()= Y() f()=ɳ() - ɳ(-)-ɳ(-)+ɳ(-)= =ɳ()-ɳ(-)+ɳ(-) - hav h oraional quaion P Y()+Y()= - Y()= ( ) ( ) ( ) This is h soluion in h oraional form, find h original: A M N а) F() = = = ( ) P ɳ()-cos ɳ() = (-cos)ɳ()=sin ɳ()

6 6 b) Availabiliy mulilirs е -a oins o h ossibiliy of alying horm of dlay: е -a F()f(-a) ɳ(-a) bcaus ( ) ( ) 4sin sin ɳ(-) ɳ(-) Answr: y() = sin ɳ() 4sin ɳ(-) + sin ɳ(-) No: funcion y () will saisfy h quaion a all oins whr i is coninuous ІІ Th soluion of h Cauchy roblm wihou finding imag of h righ sid Examl 3 y"() = Th soluion y()=y'()= If F() ( whr F() (som unknown imags) Thn h oraional quaion is: Y()=F() Y()= F() oraional soluion Oraor soluion go as a roduc of wo imags, by Borl horm w hav: G() F()g()*f() W hav G()=, F(), so Y()= F()=G() F()g()*f()= d = g( ) f ( ) d * ( ) d - d - = arcg / / - ln( ) Answr: y()=arcg - ln( ) = arcg ln( ) y( ) o No: Th rquirmn of sing h iniial condiions a h oin = is no ssnial, as h linar chang of variabls y = (-a nw variabl) Cauchy roblm a = is rducd o h Cauchy roblm wih iniial condiions a h oin Similarly, h rlacmn of unknown funcion roblm wih nonzro iniial condiions can b rducd o a roblm wih zro iniial condiions For xaml, if h iniial condiions y()=y y'()=y hn h rlacmn of h funcion y () o z (), whr z ()

7 7 = y() y y w obain:z() = z'() = y'() y / = 3 If h iniial condiions y, y, y, yn- is no considrd a givn, bu arbirary consans, hn y () is no a soluion of h Cauchy roblm i is h gnral soluion of h diffrnial quaion ІІІ Solving sysms of linar diffrnial quaions wih consan cofficins Sysms of linar diffrnial quaions ar solvd similarly, h diffrnc is ha w obain a sysm of oraional quaions Examl 4 x'=x+3y x=x() Iniial condiions y'=x y y=y() x()=, y()= Th soluion x()x() y()y() x'()x()-x()=x()- y'()y()-y()=y() Th sysm of oraional quaions: X()-=X()+3Y() Y()=X()-Y() rwri h sysm: (-)X 3Y= Х= x Х-(+)Y= Y= y - -3 = -( -)+3 = - ( 4) -(+) -3 -(+) = -(+) - = - x X= = ch y Y= = 4 4 Answr: x()=ch+ sh y()= sh sh sh IV Th soluion of ingral quaions Volair of h I and ІІ ordr Ingral quaion is calld h quaion ha conains h rquird funcion undr h ingral sign

8 Considr a siml ingral quaions of Volair wich y is convoluion І kind: k( ) y( ) d ІІ kind: y()+ k( ) y( ) d f ( ) f ( ) 8 whr y () - dsird funcion f () - known funcion k (-) - known funcion, calld h nuclus, and dnd on h diffrnc of argumns If h funcion k (-), f () ar funcions - originals, hn using oraional calculaions can find h soluion of h ingral quaion l y()y() f()f() k(- )K() hn in oraional form of h firs quaion isk() Y() =F() F( ) Y() = y() K( ) Th scond quaion:y()+k() Y() = F() F( ) Y()= y() K( ) In boh cass usd h horm of Borl abou imag convoluion of wo funcions Examl 5 x y(x) = sinx+ ( x ) y( ) d ingral quaion of h II kind Y() = Y( ) Y() = ( ) ( )( ) Answr: y(x)= ( shx sin x), x Examl 6 ( shx sin x), x cos( ) y( ) d sin ingral quaion of h I kind cos*y() = sin Y() = Y()= Answr: y()=, >

9 9 Th srucur of shor-rm modul conrol work SCW 3 Find h Lalac ransform of h funcion - h original Find imag of h original daa by Lalac Solv linar diffrnial quaion wih oraional mhod (45 minus) Srucur of Modul conrol work МCW 3 Find h Lalac ransform of h funcion - h original Find imag of h original daa by Lalac 3 Solv h Cauchy roblm for linar diffrnial quaions by oraional mhod 4 Solv h Cauchy roblm for a linar diffrnial quaion using Duhaml formula 5 To solv h ingral quaion of convoluion oraional mhod (9 minus) For xaml: cos 6 ( ) ( ) 36 5 ( ) 4 ( ) 4 ( ) 4 f()=sin 3 = F() = 3y"+9y=ɳ(-5) y() = y'() = Y() + 9Y()= 5 ( 9) cos 3 sin 9 3( 5) sin ( Y() = ( ) так як y() = ) 4 Find an imag grahically of a givn funcion f () cos sin 3 4 f() = ɳ(-) - ɳ(-3) + (-+4)ɳ(-3) (-+4)ɳ(-4)= = ɳ(-)-(-3)ɳ(-3)-(-4)ɳ(-4) 3 4

10 Examl and soluion of h roblm «oraor calculus» Find h imag funcion 4 f 5 cos 5 5 Find original of h following imag F Solv h Cauchy roblm by oraional calculaions y y y f, y y,, f, 4 Using h formula Duhaml solv h soluion of quaion y y y, y y 5 Solv h ingral quaion y y d cos Exrcis Solving 4 As you know cos, namly cos Bcaus shif horm: 4 cos, bcaus lag horm 4 4 cos Answr: F Exrcis Solving 4 4 ch W know ha, by viru of linar rory, Answr: F Exrcis 3 Solving Th righ sid of quaion analyical xrssion: 3 3ch 4 f is icwis coninuous funcion W wri is

11 By using h linariy rory and alying lag horm f f L y Y Thn y Y, y Y L us wri h oraor quaion, whr Y Y Y Y W found h original of h imag For abl imag hav: By h imag diffrncing horm w hav By h ingraion of original horm: d Thrfor Accoun for lag horm w obain Answr: y Exrcis 4 Solving Find a soluion y subsidiary quaion y y y by iniial condiion y y L y Y Thn y Y, y Y Sinc, w g oraor quaion: Y Y Y, from hr Y Wih h rsuling imag w find h original This can b don in diffrn ways Firs way By xansion horm y rs y rs y lim lim

12 lim y y d d arcg ln y d arcg ln y arcg ln y d y Scond way Tabl imag should: By h horm of original ingraion w g d Third way Dcomos ror raional fracion ino a sum of siml fracions: From abl imags w obain W hav Sinc y By Duhaml formula: y f y d y ln Sinc, y d d arcg ln Ansvr: y arcg Exrcis 5 Solving Considring ha y( ) d is convoluion funcion L y Y Tabl imag should: Y y So ha, oraor quaion is wrin as: and y, cos, by Borl horm 4

13 3, and is soluion Y Y Y can b rrsnd as 4 4 Y From h abl w Find imag of h original y cos sin Answr: y cos sin

14 4 Varian Find imag of h original f sin Find original of h following imag F 3 Solv h Cauchy roblm by oraional calculaions 4,, де y y f y y, f,, 4 Using h ingral Duhaml solv h Cauchy roblm y y h, y y 5 Solv h ingral quaion y sin d cos Varian Find imag of h original 4 5 cos 5 5 f Find original of h following imag F Solv h Cauchy roblm by oraional calculaions,, де f y y y f y y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y y, y y 5 Solv h ingral quaion y y d cos

15 5 Varian 3 Find imag of h original f sin5 Find original of h following imag F Solv h Cauchy roblm by oraional calculaions,, де f y y y f y y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y 5 Solv h ingral quaion y d 3 3 Varian 4 Find imag of h original 4 cos cos5 f Find original of h following imag F Solv h Cauchy roblm by oraional calculaions 4, де f y y f y 3, 3 6, 3 4 Using h ingral Duhaml solv h Cauchy roblm y y y cos, y y

16 6 5 Solv h ingral quaion y d y sin Varian 5 Find imag of h original f cos6 cos Find original of h following imag F 6 3 Solv h Cauchy roblm by oraional calculaions, де f y y f y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y h y y, 5 Solv h ingral quaion y sin d sin Varian 6 Find imag of h original f sin Find original of h following imag F 4 3 Solv h Cauchy roblm by oraional calculaions 3, де f y y f y,, 4 Using h ingral Duhaml solv h Cauchy roblm, y y ch y y

17 7 5 Solv h ingral quaion y cos d Varian 7 Find imag of h original f sin 7sin3 Find original of h following imag F 3 Solv h Cauchy roblm by oraional calculaions, 3де f y y f y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y 5 Solv h ingral quaion y y d Varian 8 Find imag of h original f shd Find original of h following imag F Solv h Cauchy roblm by oraional calculaions, де f y y f y y,,,

18 8 4 Using h ingral Duhaml solv h Cauchy roblm y y y, y y 5 Solv h ingral quaion y y sin d Varian 9 Find imag of h original f sh cos3 Find original of h following imag F Solv h Cauchy roblm by oraional calculaions,, де f y y f y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, y 3 5 Solv h ingral quaion cos y y d Varian 3 Find imag of h original f cos Find original of h following imag F 4

19 9 3 Solv h Cauchy roblm by oraional calculaions,, де f y y f y y,,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y ch cos 5 Solv h ingral quaion y y d Varian Find imag of h original f 4 ( ) Find original of h following imag F 3 Solv h Cauchy roblm by oraional calculaions,, де f y y f y y,,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y 5 Solv h ingral quaion y d Varian Find imag of h original f d sin

20 Find original of h following imag F 3 3 Solv h Cauchy roblm by oraional calculaions,, де f y y f y y,, 4 Using h ingral Duhaml solv h Cauchy roblm y 4 y, 3 ch y y 5 Solv h ingral quaion y y d Varian 3 Find imag of h original f sh sin5 Find original of h following imag F ( 4) 3 Solv h Cauchy roblm by oraional calculaions 4,, де f y y f y y,, 4 Using h ingral Duhaml solv h Cauchy roblm, y y y y ch ( sin ) 5 Solv h ingral quaion y y d Varian 4 Find imag of h original sin cos f

21 Find original of h following imag F Solv h Cauchy roblm by oraional calculaions,, де f y y f y y,,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y cos 5 Solv h ingral quaion y sin y d Varian Find imag of h original f Find original of h following imag F Solv h Cauchy roblm by oraional calculaions 4,, де f y y f y y,,, 4 Using h ingral Duhaml solv h Cauchy roblm y y y, y y 5 Solv h ingral quaion y sin y d Varian 6

22 Find imag of h original 4 5 cos 5 5 f Find original of h following imag F Solv h Cauchy roblm by oraional calculaions,, де f y y f y y,,, 4 Using h ingral Duhaml solv h Cauchy roblm, y y y y ch 3 cos 5 Solv h ingral quaion y y d Varian 7 Find imag of h original f ch sin3 Find original of h following imag F ( 9) 3 Solv h Cauchy roblm by oraional calculaions 3,, де f y y f y y, 4, 4 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y Solv h ingral quaion y y d

23 Varian 8 3 Find imag of h original f sin 4 Find original of h following imag F 3 Solv h Cauchy roblm by oraional calculaions,, де f y y f y y, 4, 4 Using h ingral Duhaml solv h Cauchy roblm y y y, y y 5 Solv h ingral quaion y sh y d Varian 9 Find imag of h original f sin 5 4 Find original of h following imag F 4 3 Solv h Cauchy roblm by oraional calculaions 4,, де f y y f y y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y 5 Solv h ingral quaion y sh ch y d

24 4 Varian Find imag of h original f sh cos3 Find original of h following imag F 3 ( 4 3) 3 Solv h Cauchy roblm by oraional calculaions 9,, де f y y f y y, 3, 3 4 Using h ingral Duhaml solv h Cauchy roblm y sh y, y y ch 5 Solv h ingral quaion y sin y d Varian Find imag of h original f sin d Find original of h following imag F 3 ( ) 3 Solv h Cauchy roblm by oraional calculaions 4,, де f y y f y y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y ( ), y y 5 Solv h ingral quaion y y d

25 5 Varian Find imag of h original f cos3 ( ) Find original of h following imag F 3 Solv h Cauchy roblm by oraional calculaions, де f y y f y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y y, y y 5 Solv h ingral quaion ch y d Varian 3 Find imag of h original f ch ( ) Find original of h following imag F 3 Solv h Cauchy roblm by oraional calculaions 3,, де f y y f y y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y y, y y ch 5 Solv h ingral quaion y sh y d

26 6 Varian 4 Find imag of h original f sin Find original of h following imag F 3 ( ) 3 3 Solv h Cauchy roblm by oraional calculaions 3,, де f y y f y y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y y, y y ch 5 Solv h ingral quaion cos y d Varian 5 3 Find imag of h original f Find original of h following imag F 7 3 Solv h Cauchy roblm by oraional calculaions 3, де f y y f y,,, 4 Using h ingral Duhaml solv h Cauchy roblm y y y, y y 5 Solv h ingral quaion y sh y d

27 7 Varian 6 Find imag of h original cos f Find original of h following imag F Solv h Cauchy roblm by oraional calculaions 4,, де f y y f y y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, 3 y y cos 5 Solv h ingral quaion y y d Varian 7 Find imag of h original f sin d Find original of h following imag F 3 Solv h Cauchy roblm by oraional calculaions, 3 де f y y f y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y sin 5 Solv h ingral quaion y y d

28 8 Varian 8 Find imag of h original f Find original of h following imag F Solv h Cauchy roblm by oraional calculaions, де f y y f y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y 5 Solv h ingral quaion y y d Varian 9 Find imag of h original f ch Find original of h following imag F 4 3 Solv h Cauchy roblm by oraional calculaions 4, де f y y f y, 3, 3 4 Using h ingral Duhaml solv h Cauchy roblm y y, y y 5 Solv h ingral quaion y sin y d

29 9 Varian 3 Find imag of h original f sind Find original of h following imag F 3 3 Solv h Cauchy roblm by oraional calculaions, де f y y f y,, 4 Using h ingral Duhaml solv h Cauchy roblm y y h, y y 5 Solv h ingral quaion y y d Answr: Varian F ln 4 f 5 3 y sin sin sin 4 4 y sh ch arcg h 4 5 y Varian F f ch 3 y

30 3 ln y cos sin 4 y arcg 5 Varian 3 F 5 y cos sin cos sin f 3 4 y ln ln 5 y Varian 4 F f sin sin y sin 5 y y 3 cos Varian 5 F 4 ln 36 f cos 4 3 y

31 3 3 cos 4 4 y sh arcg ch 5 y Varian 6 F ln y 3 3 f 3 4 y sh ch lnch 5 y 3 3 Varian 7 F 6 ln 4 f 3 y 3 4 y ln 3 3 y cos 3 sin 3 5 Varian 8 F f cos 3 sin 3 4 3

32 3 3 y cos cos cos 4 y ( ln ) y Varian 9 F 3 f y y 3 ln y 3 cos sin Varian F 3 4cos cos f 3 y sin sin sin ch y sh ch ln 4 y cos 3 sin Varian F 5 ln

33 f sin 3 33 y y sh ln 4 5 y Varian F arccg 3 f y cos sin ) sin 4 y sh ch cos 3 3 sin 3 y Varian 3 F f cos cos y sin sin cos 4 y sh arcg ch 5 y sin 3 Varian 4

34 F ! 5! f 3 y 4 y 5 y ln 34 Varian 5 F f 8! sin 3 6 y ln y 5 y 4 cos 3 Varian 6 F f y cos cos cos 4 y sh ch 5 ych Varian 7

35 F f 35 cos cos3 8 y cos 3 cos ln y ch cos 4 y 5 Varian 8 F 8 6 f 3 y cos 3 cos ( ln ) 4 y 5 y 3 6 Varian 9 F y sh sh 3 y cos cos y ln y sh 5 5

36 36 Varian F 3 f y 3 cos3 3 3 y h ln ch 4 y sh sin 5 Varian F 4 y y cos 3 ch 4 y ln 5 y Varian F 9 y cos y 4 y arcg y 5 ( ln )

37 37 Varian 3 F sin cos 3 ( y 3 ) 9 f 3 4 ln 5 y y ch 3 6 Varian 4 F 4 4 f y 3 ( 3 ) ln 5 y y ch 3 3 Varian 5 F 3 3 f 3 3 sin 3 3 y ( ) ln y y ch sh 5

38 38 Varian 6 F 4 4 f sin y 4 cos 4 3 y ln y Varian 7 F 3 f 3 y y sh ln 4 5 y 3 6 Varian 8 ( ) F 3 f 3 5 sin cos 3 y 4 y ln ln 5 y sin Varian 9

39 F 3 f ( ch ) y ( 3) y ln 4 y 5 Varian 3 F f 3 y y sh ch arcg h 4 5 y Rfrncs MLKrasnov, AIKislv, GIMakarnko Funcions of a comlx variabl Oraional calculus Sabiliy hory Moscow, "Scinc", 98-3 Problms in Mahmaics for Tchnical Schools Scial scions of analiza-v4ch Par xbook (Edid AVEfimova, BPDmidovicha) Moscow, "Scinc" 98-36

40 4 3 VFChudsnko Collcion of asks for scial courss of highr mahmaics (modl calculaions): Txbook for vuzov- M : Highr School, ІVAlєksєєva, VOGaydy, OODihovichny, LBFdorova Coora for Torіya funksіy komlksnoї zmіnnoї Orasіyn numbrs Worksho Kiїv- 3, 6