# IX.2 THE FOURIER TRANSFORM

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1 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 7 IX. THE FOURIER TRANSFORM IX.. The Fourier Trnsform Definiion 7 IX.. Properies 73 IX..3 Emples 74 IX..4 Soluion of ODE 76 IX..5 Soluion of PDEs in he Infinie Region The He Equion Guss s Kernel Green s Funcion The Lplce Equion Poisson s Inegrl Formul 73 IX..6 Fourier Inegrls (Fourier Inegrl Represenions 736 IX..7 Soluion of PDE s in he Semi-Infinie Regions 738 IX..8 Review Quesions nd Eercises 75 Appendi 75

2 7 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 IX.. DEFINITION: Inroduce he Fourier inegrl rnsform pir: The comple Fourier rnsform of funcion f ( is defined s i F( f e d nd he inverse Fourier Trnsform is defined s i f( F e d The Fourier rnsform rnsles he funcion f ( from he ime domin o is specrum he funcion F ( in he frequency domin. The inverse Fourier rnsform reconsrucs he funcion f ( from is specrum: ξ ξ i iξ i f( F e d f e d e d Fourier inegrl formul The quesion rises if he funcion reconsruced from is specrum coincides wih he originl funcion. The nswer o his quesion is given for funcions sisfying cerin condiions. f ( iξ i f ( ξ e dξ e d f ( f ( is coninuous poin of disconinuiy f ( + f ( + Theorem 9. (The Fourier Inegrl Theorem Le he funcion f ( sisfy he condiions (Dirichle s condiions: i f ( hs only finie number of finie disconinuiies (jumps in he inervl (, nd hs no infinie disconinuiies. ii f ( hs only he finie number of erem (mim nd minim in he inervl (,. And le he funcion f ( be bsoluely inegrble on (, : Then he Fourier inegrl f d <. f f is con. iξ i f ( ξ e dξ e d + f ( f ( + f is discon., f converges o he funcion f ( he poins ( where is coninuous, nd i converges o ( + ( + of he jump he poins where he funcion f f (verge vlue f is disconinuous. The Theorem 9. provides only he sufficien condiion of he eisence of he Fourier nd he inverse Fourier rnsforms here re mny oher funcions for which he rnsform lso eiss. I lso shows h he funcion reconsruced from is specrum coincides wih he funcion where i is coninuous nd converges o he verge vlue of he jump he poins of disconinuiy.

3 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 73 IX.. PROPERTIES Le f ( sisfy he condiions of he Fourier inegrl heorem on (, Denoe he Fourier rnsform nd he inverse Fourier rnsform by { } ˆ i ˆ F f f f e d { } ˆ i F f f e d Consider some imporn properies of he Fourier rnsform:. Lineriy: F { f ( + bg ( } F { f ( } + bf { g ( } ˆ ˆ F α f + βgˆ αf f + βf { gˆ } i Shifing in : { } { } { } ˆ ( { ˆ i } { } ˆ F e f f F f e F f 3 Shifing in : { } i i F f e F{ f ( } e ˆf 4 Scling: { } 5 Duliy: F f ˆ f { ˆ } ( F f f f g f g d 6 Convoluion: ( F f g F f F g ˆf g { } { } { } ˆ { ˆ ˆ } F f g f g Operionl properies 7 Derivives: Assume lim f nd { } ˆ F f i f { } ˆ F f f ( n n { } ˆ F f i f ( k lim f, k,,... Fourier Trnsform in vrible of he funcion u, Denoe he Fourier rnsform in he vrible of he funcion u(, s Assume h ± i { } ˆ ( F u(, u, u, e d lim u, nd ± k u, lim k, k,,..., hen i 8 Derivive in : F u(, u (, e d u ˆ (, 9 Derivives in : F u(, iu ˆ (, F u(, u ˆ, ( n F u(, i u, n n ˆ (

4 74 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 IX..3 EXAMPLES [Debnh]. (Heviside funcion Uni Pulse Funcion f ( H( + H( ˆf ( The uni pulse funcion cn be defined wih he help of he Heviside uni sep funcion < f ( H( + H( < > > The Fourier rnsform of his funcion cn be deermined s ˆf f i ( e d e i d e i e i e i i i i i e e i sin ( sin Euler s formul. (Two-sided eponenil funcion Consider he even wo-sided eponenil funcion: f ( e > Then he Fourier rnsform of his funcion cn be evlued s f ( e ˆf ( ˆf f i ( e d i e e d ( i ( + i e d + e d ( + i ( + i e e i + i + i + i +. (Gussin funcion Consider he Gussin funcion: f e > The Fourier rnsform of his funcion cn be evlued s ˆf e e i d e i ep 4 (self-reciprocl under Fourier rnsformion. d

5 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, Derive he Fourier rnsform of he derivive (propery 7: d F d f d f i ( e d d e i d f i i e f ( f ( de inegrion by prs i + i f e d iˆf 5. Derive he Fourier rnsform of he second derivive (propery 7: d F d f d d i f ( e d

6 76 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 Applicion of inegrl rnsform for he soluion of PDE According o properies 7 nd 9, pplicion of he Fourier Trnsform elimines he derivives wih respec o ime or o spce vribles. This fc will be used for he soluion of he differenil equions. Firs, we rnsform differenil equion o elimine he derivive of unknown funcion, hen we solve (lgebric rnsformed equion in he frequency spce, nd finlly, he soluion of he originl problem will be obined by he inverse rnsform: DE soluion of DE F F TE soluion of TE IX..4 SOLUTION OF THE ORDINARY DIFFERENTIAL EQUATIONS Emple 4 (Sedy-Se Conducion Solve he nd order ordinry differenil equion d y y+ f ( (, d wih he help of he Fourier rnsform. Soluion: Apply he Fourier rnsform o he i ŷ y e d given equion, using Propery (7 for he rnsform of he nd lim u, derivive, ssuming ± ˆ ˆ ± k u lim k, k, : ˆy y + f rnsformed equion Solve he rnsformed equion for ŷ ŷ ( ˆf + rnsformed soluion Noe h he funcion ĝ is he Fourier + rnsform of he wo-sided eponenil funcion (Emple : ĝ F e F { g( } + Then he rnsformed soluion cn be wrien s he produc of wo rnsformed funcions: ĝ ˆ f ŷ (

7 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 77 The soluion of he differenil equion cn be found by inverse rnsform wih pplicion of he convoluion heorem: The problem cn be inerpreed s sedy se conducion in he hin rod infinie in boh direcions (lumped cpcince model eposed o convecive enviromen emperure, hp ka (Ch c per, p.4 wih poin he source Emple: { } y( { ˆ F y } F gˆ ˆf g f convoluion g s f s ds s e f ( s ds T, h δ f Emple: Consider he cse when he source funcion f ( is he Dirc del funcion δ f defining he poin impulse. Then inegrion using he propery of he del funcion δ ( u s s ds u yields he soluion of he differenil equion: δ ( s y( δ ( e s ds y e y( e soluion The grph of he soluion is he grph of he double-sided eponenil funcion (Emple cenered.

8 78 IX..5 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 SOLUTION OF PDEs IN THE INFINITE REGION IX..5. He Equion Guss s Kernel - Green s Funcion Emple 5 (-D he conducion wih he generion in he infinie region - Guss s Kernel Consider he iniil-boundry vlue problem for non-homogeneous he equion in he infinie slb (see Chper 4: (, u + S(, (, u >, (, wih he iniil condiion:, u (, u The source funcion defines he he generion in he slb: g (, W S,, where g (,, 3 k m There re no boundry condiions for he infinie region; bu from physicl considerions, we ssume h boh he unknown funcion nd is derivive vnish when pproching ± : u (, u (, ± ± This ssumpion will llow pplicion of propery (7 of he Fourier rnsform which will be used for soluion of he given IBVP. Trnsformed equion Tke he Fourier inegrl rnsform of he equion (pplying properies ( nd (7 for he rnsform of he derivives û (, û, + Ŝ, wih he rnsformed iniil condiion: û (, û where he following noions for he rnsformed funcions re used i [ ] ˆ ( F u(, u, e d u, i [ ] ˆ F u ( u e d u i [ ] ˆ ( F S(, S, e d S, Soluion of rnsformed equion The obined rnsformed equion is n ordinry liner differenil equion in vrible wih consn coefficiens: (, û + û (, Ŝ (, (, û û he generl soluion of which cn be obined by he vriion of prmeer formul (see Tble ODE: u, u ˆ e + e e S, d ( rnsformed soluion ˆ ( ˆ (

9 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 79 Inverse rnsform Soluion of he given IBVP now cn be obined by he inverse Fourier inegrl rnsform of he epression ( : { ˆ ( } ˆ u, F u, ( ˆ F u e + F e S (, d ( Funcions e nd e which pper in he inegrnd re he Fourier rnsforms of he Gussin disribuion funcions: Guss s Kernel G(, e G(, e ( ( ( ( Ĝ, e ( Ĝ, e { ˆ ( } ˆ u, F u, F { u G ˆ, } F S ˆ (, G ˆ (, d + { } { ˆ ˆ ( } ˆ ˆ ( F u G, + F S, G, d Then he firs inegrl in he soluion is he inverse Fourier rnsform of he produc of wo rnsformed funcions û Ĝ which ccording o propery (3 is convoluion of hese wo funcions F { ug ˆ ˆ } u G ( u G s, ds ( s e u ( s ds Consider he second erm in he soluion. I leds o he convoluion: { GS} F ˆˆ G S ( G s, S s ds ( ( s ( e S s, dsd Then forml soluion of he given IBVP is: Soluion: (, ( s ( s ( e e u ( s ds + k u S s, dsd

10 73 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 Priculr cses: Consider wo priculr cses: Homogeneous equion (no he generion, S (, Le he iniil emperure disribuion hve sep-wise vriion: ( + ( u H H Then soluion is given by he inegrl over he finie region: u (, e ( s ds Use chnge of vrible: v ( s u (, dv ds ( ( s e ds v e dv ( + ( + ( v e dv e v erf ( ( + erf dv Therefore, he soluion of he He Equion in he infinie region wih no he source is given by u(, erf ( ( + erf The grph shows he emperure profiles for ( + ( u H H

11 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 73 b Non-homogeneous equion - Green s funcion Consider he conducion problem wih zero iniil condiion u ( nd poin he source loced which insnneously relesed energy ime of srengh S (impulse poin source: impulse poin source S(, Sδ ( δ ( Then he firs erm in he soluion disppers becuse of he zero iniil condiion, nd he soluion becomes ( s S ( u (, e δ ( s δ ( dsd k ( Then inegrion of he epression wih he Dirc del funcions yields he soluion Green s Funcion u (, ( ( S e > k ( This soluion of he problem wih he impulse poin source is clled he Green funcion for he infinie region in he Cresin coordine sysem. I is used for soluion of non-homogeneous pril differenil equions. Emple The soluion curves differen momens of ime for, S, k,, re shown in he grph ( δ δ ( δ ( g, g u (,

12 73 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 IX..5. Wve Equion Emple 6 Wve equion for n infinie sring D Almber soluion u (, u(, v >, < < wih iniil condiions: (, u ( u (, u < < iniil displcemen u ( iniil velociy The coefficien in he Wve Equion v is physicl propery of he sring nd represens speed of wve propgion long he sring. I is deermined hrough he equion Tg m v w s where T is ension, g is he ccelerion of grviy, w is weigh of he sring per uni lengh. Trnsformed equion Apply he Fourier rnsform o he wve equion nd iniil condiions: û (, v (, û û (, û û û (, This is he iniil vlue problem for nd order liner ODE in wih consn coefficiens (, û + v û (, The generl soluion is û(, c cosv + c sinv where coefficiens cn be found from iniil condiions. c û û c v hen he soluion of he ODE becomes û u ˆ(, uˆ cosv + sinv v Inverse rnsform The soluion of he IBVP cn now be obined using he inverse Fourier rnsform û i u (, uˆ cosv sinv e d + v This is he generl form of he soluion which includes in he inegrnd he Fourier rnsform of he iniil condiions. For some funcions, i cn be inegred eplicily.

13 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, I cn be shown h D Almber soluion (V.S.Vldimirov Equion of Mhemicl Physics, p.76 wih rbirry funcions ( nd u ( C u C : + v u (, u ( v u ( v u ( s ds v v is lso soluion of he IBVP. Moreover, he soluion of he IVP for he Wve Equion is unique. Priculr cse: Consider he IBVP for n infinie sring wih iniil condiions u ( H ( + H ( u ( Fourier rnsform of iniil condiions yields û û i i e δ e δ sin Then he soluion of he IBVP becomes: u (, H ( v H ( v H ( v H ( v (compre o D Almber soluion Soluion curves for c re shown in he figure (F-.mws ( + ( u H H u (,

14 734 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 IX..5.3 Lplce s Equion Emple 7 (Lplce s Equion in semi-infinie plne Dirichle problem Poisson inegrl formul Consider -dimensionl Lplce s equion in he semi-infinie plne y, : >, bu wih u(,y u,y + y (,, y (, wih he boundry condiion y : (, u ( u Trnsformed equion vrible Dirichle i [ ] ˆ ( F u(,y u,y e d u,y We will pply he Fourier rnsform in he i ˆ F f ( u e d u Trnsformed equion û û + y hs he generl soluion: û c y y e + ce The soluion is bounded c for > Then wo erms of he soluion cn be combined û ce y Applicion of he boundry condiion yields û û e y Inverse rnsform Funcion y Poisson s kernel + y y e c for < is Fourier rnsform of hen he soluion of he rnsformed equion cn be wrien s y û ûf + y which is produc of wo Fourier rnsforms. Then he inverse rnsform of û cn be wrien s convoluion of hese wo funcions: Poisson s inegrl formul u ( s ( s y ds u(, y + y ( + ( u H H u(,y I is soluion of he Dirichle problem for Lplce s equion in he semi-plne which is clled Poisson s inegrl formul for he upper hlf-plne. For he iniil condiion ( H ( + H ( u he soluion is given by: u(, y + rcn rcn y y

15 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, F- Poisson inegrl.mws Soluion of Lplce's Equion in he upper hlf-plne - Poisson's inegrl formul > resr; > u:heviside(s+-heviside(s-; Poisson's Inegrl Formul: u : Heviside ( s + Heviside ( s > u(,y:simplify(in(u/((-s^+y^,s-infiniy..infiniy*y/pi; u (, y > plo3d(u(,y,-4..4,y..,grid[5,5]; + rcn + : y rcn + y u(,y > wih(plos: > densiyplo(u(,y,-4..4,y..,grid[5,5],sylepchnogrid,esboed; > conourplo(u(,y,-4..4,y..,esboed,coloring[blue,yellow],filledrue;

16 736 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 IX..6 FOURIER INTEGRALS Suppose h he funcion f: sisfies he Dirichle condiion in every finie inervl of (see p.696 nd suppose h here eiss n improper inegrl f ( f ( d < Comple Fourier inegrl represenion i f F e d where he coefficien funcion is given by i F( f ( e d Suppose h he funcion f :(, sisfies he Dirichle condiion in every finie inervl of nd f d < b Sndrd Fourier inegrl represenion + f A cos B sind where he rel coefficien funcions re given by A f ( cosd B f ( sind c Fourier cosine inegrl represenion f A cosd where he rel coefficien funcions re given by A f ( cosd d Fourier sine inegrl represenion f B sind where he rel coefficien funcions re given by B f ( sind

17 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, Convergence All give Fourier inegrl represenions converge o: f f ( f + f + if f is coninuous if f is disconinuous FI-.mws Fourier Inegrl Represenion W he upper limi in he Fourier Inegrl (defines he ccurcy of pproimion > W:; W : > f(:heviside(-heviside(-; f( : Heviside( Heviside ( Sndrd Fourier Inegrl: > A(w:in(f(*cos(w*,-infiniy...infiniy/Pi; sin( w A( w : w > B(w:in(f(*sin(w*,-infiniy...infiniy/Pi; B( w : + cos( w w > u(:in(a(w*cos(w*+b(w*sin(w*,w..w; u( : Si ( Si( > plo({f(,u(},-..,color[red,blck]; Fourier Cosine Inegrl: > A(w:*in(f(*cos(w*,-infiniy...infiniy/Pi; sin( w A( w : w > u(:in(a(w*cos(w*,w..w; Si ( + Si ( u( : > plo({f(,u(},-..,color[red,blck]; Fourier Sine Inegrl: > B(w:*in(f(*sin(w*,-infiniy...infiniy/Pi; B( w : ( + cos( w w > u(:in(b(w*sin(w*,w..w; u( : Si( Si ( + Si ( > plo({f(,u(},-..,color[red,blck];

18 738 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 IX..7 INTEGRAL FOURIER TRANSFORM IN THE SEMI-INFINITE REGIONS Consider semi-infinie region <. Define he inegrl rnsform pir of he funcion ( F{ u } û( u( K(, F { u} u( û( K(, ˆ d d u s. Fourier Inegrl Trnsform kernel We re looking for specific form of he kernel K (, which cn be pplied for priculr form of he boundry condiion. In he following Tble, we specify he kernel K (, for hree ypes of clssicl homogeneous boundry condiions: Boundry condiion Kernel K (, Inegrl rnsform pir I [ ] u sin F { u } F I û u sind I { uˆ } I u uˆ I sind du II d cos F { u } F II û u cosd II { uˆ } II u uˆ II cosd du III + Hu d cos + H sin + H cos + H sin û u d + H FIII { u } III F III { uˆ } cos + H sin u( uˆ III d + H The kernel corresponding o he Dirichle boundry condiion is bsed on he Fourier sine inegrl rnsform pir (noe, h he coefficien is spli ino : Fourier sine rnsform Fourier cosine rnsform û ( u( sind u( û sind The kernel corresponding o he Neumnn boundry condiion is bsed on he Fourier cosine inegrl rnsform pir: û ( u( cosd u( û cosd The inegrl rnsform pir wih he kernel corresponding o Robin boundry condiion: û cos + H sin ( u( d u( û + H cos + H sin d + H

19 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, The kernel of inegrl rnsform for he cse of Robin boundry condiion is obined from he soluion of he uiliry boundry vlue problem for he, : semi-infinie region wih prmeer [ X ( + X ( [, X + HX I cn be verified h he funcion X cos + H sin is soluion of he uiliry BVP. Convoluion Formuls: [See lso Lokenh Debnh, ITTA, p.5.] û u sind Le FI { u } I F { v } II II ˆv v cosd { ˆ ˆ } I II F u v I I II uˆ vˆ sind (I-I-II u ( s sinsds vˆ II sind u ( s vˆ II sins sind ds u ( s vˆ II cos( s cos( s d ds + u ( s vˆii cos( s d vˆii cos( s d ds + u ( s vˆii cos( s d vˆii cos( s d ds + u( s v( s v( s+ ds { ˆ ˆ } ( ( + F u v I I II u s v s v s ds (I-I-II { ˆ ˆ } ( + + ( F u v II II II u s v s v s ds (II-II-II { ˆ ˆ } ( + + ( F u v II I I { ˆ ˆ } I I I u s v s v s ds (II-I-I F u v (I-I-I { ˆ ˆ } F u v (I-II-II I II II

20 74 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 { ˆ ˆ } II II F u v I II II uˆ vˆ sind (I-II-II u ( s cossds vˆ II sind u ( s vˆ II sin cossd ds u ( s vˆ II sin( s sin( s d ds + + u ( s v( p cospdp sin( s sin( s d ds + + u ( s v ( p cosp sin ( s sin ( s ddp ds + + u ( s v ( p cosp sin ( s sin ( s ddp ds + + undefined { ˆ ˆ } ( + + ( F u v II II II u s v s v s ds (II-II-II

21 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 74 { ˆ ˆ } ( + + ( F u v II I I u s v s v s ds (II-I-I { ˆ ˆ } F u v (I-I-I I I I

22 74 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 OPERATIONAL PROPERTIES: We consider he funcion u ( s funcion describing some physicl quniy in he semi-infinie region [,. A, i is specified by one of he boundry condiions. When pproches infiniy, we u nd is derivive re equl o zero: ssume h boh he funcion u ( ( u u Inegrl rnsform of Apply he inegrl rnsform o he second pril derivive of u u K (, d : Dirichle boundry condiion, [ u], K(, sin u FI u K (, d u ( sind u sind u sin ( ( u( ( u cos cosd [ u( ] d d [ sin] [ cosu( ] + u( d[ cos ] cosu cos u + u d cos u u ( sin d [ ] û (for homogeneous boundry condiion, u If he boundry condiion is non-homogeneous, [ u] f ( u ( K (, d f ( uˆ in he cse of non-homogeneous condiion [ u] f (, hen The inegrl rnsform elimines he derivive. Similr resuls cn be obined for he remining wo boundry condiions:

23 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, du Neumnn b.c., d F II u u K (, d : K(, cos u cosd u cos d u( u cos d cos [ ] u u u cos cos d cos u( u + sind u + sin d u ( [ ] u + sinu sinu u d sin [ ] u u cos d u û û (for homogeneous boundry condiion, If condiion is non-homogeneous, f ( u ( K du d, hen u (, d f ( uˆ du 3 Robin b.c., + Hu d Boundry Condiion I Dirichle [ u] f ( : K(, Kernel K (, sin cos + H sin + H u K, û + f d du d II Neumnn f ( cos û f du cos + H sin d + H III Robin + Hu f ( û +? see p.75

24 744 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 IX..7. He equion in he semi-infinie region Consider he homogeneous one-dimensionl he equion (, u(, u [, > Iniil condiion: (, u ( u Boundry condiion: (, f ( u > Dirichle Cse f ( Trnsformed equion. Apply he inegrl rnsform corresponding o he Dirichle problem (Fourier sine rnsform o he differenil equion: û (, û, Applicion of he convoluion formul (I-I-II: This is n ODE for he rnsformed funcion û (, s funcion of wih he vrible reed s prmeer: û(, + û(, wih n iniil condiion which cn be obined by inegrl rnsform of he originl iniil condiion: û (, û u ( K(, d Soluion of his liner s order homogeneus ODE wih consn coefficiens is given by ( ˆ ˆ u, u e Inverse rnsform soluion of IBVP. u (, u (, û e K, d F { uˆ } I ˆ I F u e I I { ˆ } F F u e G(, FI FI { uˆ } FII e + û e sin d This soluion cn be reduced o he rdiionl form [Ozisik]: u (, û u ( s G ( s, G( s, ds u ( ( s ( + s u ( s e e ds u ( ( s ( + s u ( s e e ds e sind ( u sin d e sind e sin sind d ( ( + e e d 8 ( ( + u e e d

25 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, FS--h.mws HE in he semi-infinie spce - inegrl rnsform soluion (Fourier sine rnsform B.C.: f u ( H ( û cos > resr; > :; : Kernel of inegrl rnsform > K(omeg,:sqr(/Pi*sin(omeg*; Iniil condiion: K (, : > u(:-heviside(-; sin( u( : Heviside ( > u(omeg:fcor(in(u(*k(omeg,,..infiniy; u( : ( cos( Inverse rnsform - soluion of IBVP: he upper limi defines he ccurcy of pproimion (similr o he number of erms in he runced Fourier series > u(,:in(u(omeg*ep(-omeg^/^**k(omeg,, omeg..4; u (, : 4 ( cos ( / 4 e( sin( d > u:subs(.,u(,:u:subs(.,u(,: u:subs(.5,u(,:u3:subs(.,u(,: > plo({u(,u,u,u,u3},..4,colorblck; u (, iniil condiion ( u H. boundry condiion f.5.

26 746 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 Cse f ( Non-homogeneous ime-dependen boundry condiion. { } u (, F u ˆ (, I The rnsform of he derivive: u( u( FI sind f ( uˆ Then he rnsformed equion is ( û(, f (, ( τ ˆ FI u e + FI e f ( τ dτ (, û û + û(, f ( The iniil condiion is he sme: û (, û Then he rnsformed soluion is obined by vriion of prmeer τ u ˆ(, uˆ e + e e f ( τ dτ The soluion of he IBVP cn be found by he inverse rnsform: ( s ( + s ( τ u ( s e e ds + FI e f ( τ dτ ( s ( + s ( τ u ( s e e ds + F I e f( τ dτ ( s ( + s u ( s e e ds + ( τ 3 e f ( τ dτ 4 inegrion wih Mple 3 τ ( s ( + s u s e e ds u (, Cse of f cons : + e ( τ ( τ 3 f ( τ dτ u (, ( s ( + s u s e e ds + f erf inegrion wih Mple u (, ( s ( + s u s e e ds + f erfc Aemp of pplicion of convoluion formul (filed so fr: G(, τ ( s ( + s u ( s e e ds ( τ + FI FII FII e f ( τ d τ

27 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, Lplce s Equion in he semi-infinie srip Consider -D Lplce s Equion in he semi-infinie srip: u (, y u(, y, y + (,, y (,M u wih boundry condiions: u (, f ( (,M f ( u u y > Neumnn (insulion > Dirichle > Dirichle M We will pply he Fourier cosine rnsform in he vrible which corresponds o he Neumnn problem in he semiinfinie region: û(, y u(, y K(, d u(, y cosd Trnsformed equion: û û + ( nd order ODE y The inegrl rnsform of he boundry condiions: ( f ( cosd fˆ fˆ M ( f ( M cosd Then he nd order ODE hs o be solved wih boundry condiions. The generl soluion of he ODE is c cosh y + c sinh û y Find coefficiens c nd c from boundry condiions, hen he soluion of he IBVP will be given by he inverse inegrl rnsform: u (, y û(, y K(, d û(, y cosd Emple Le f ( fˆ ( H ( H ( fˆ f M Apply firs boundry condiion y c y M c sinhm fˆ M Then he rnsformed soluion is ˆfM û sinh y sinhm Soluion of IBVP (inverse rnsform: û cosd u (, y [ ] M c fˆ M sinhm

28 748 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 Lplce s Equion in he s qudrn Consider -D Lplce s Equion in semi-infinie srip: u (, y u(, y, y + [,, y [,] u wih boundry condiions: y u > Dirichle (, f ( u > Dirichle Soluion: We will pply he Fourier sine rnsform in he vrible corresponding o he Dirichle problem in he semiinfinie region: û (, y u(, y K(, d u(, y The rnsformed equion: û + y û sind The inegrl rnsform of he second boundry condiion: fˆ ( f ( sind The generl soluion of he ODE is û c y y e + ce The soluion is bounded c Apply he firs boundry condiion y c fˆ The soluion of he rnsformed equion: ˆ y û f e The soluion of he BVP for Lplce s Equion u (, y fˆ y e sind

29 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, FS--h.mws LE in he s qudrn - Dirichle problem - Fourier sine rnsform > resr; > f(:heviside(--heviside(-; > plo(f(,..5; f( : Heviside ( Heviside ( f ( H( H( > f:simplify(in(f(*sin(omeg*,..infiniy; f : cos( + cos( Trnsformed soluion: > u:f*ep(-omeg*y; u : ( cos( + cos( e ( y Inverse rnsform -soluion of BVP: he upper limi of inegrl defines he ccurcy of pproimion > u(,y:in(u*sin(omeg*,omeg..3*/pi: > plo3d(u(,y,..4,y..4,esboed; u(,y

30 75 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 IX..8 REVIEW QUESTIONS How is he Fourier rnsform defined? Wh condiions gurnee he eisence of he Fourier rnsform? 3 How is he inverse Fourier rnsform defined? 4 Wh re he min properies of he Fourier rnsform nd he inverse Fourier rnsform? 5 How cn he convoluion heorem be pplied for evluion of he inverse Fourier rnsform? 6 Wh propery llows pplicion of he Fourier rnsform for soluion of differenil equions? 7 Wh re he min seps in he procedure of pplicion of he Fourier rnsform for soluion of he differenil equions? EXERCISES. Derive he scling propery δ ( F { f ( } ˆ f b Using inegrion by prs derive he Fourier rnsform of he second order derivive (propery 7b: d ˆ F f f d. Using he definiion of he Fourier rnsform derive he rnsform of he one-sided eponenil funcion: f nd skech he grph of he rel pr Im ˆf of he rnsformed funcion for. e < Re ˆf nd he imginry pr 3. Using he resul of he previous problem nd he convoluion heorem, evlue he inverse Fourier rnsform: F ( + i nd skech he grph of he soluion for. y in he elecricl circui is modeled wih he help of he s 4. The curren order differenil equion dy y g ( d + > (, where f ( represens he pplied elecromgneic force. Use he Fourier rnsform for soluion. Wrie he soluion in he form of convoluion. Then evlue he soluion for he cse of insnneous force g δ. Skech he grph of he defined by he Dirc del funcion soluion curve. y 5. Use he Poisson inegrl formul (p. o derive he soluion of he Lplce equion in he semi-infinie region ( y > wih he boundry condiion defined by he Dirc del funcion: δ u, u nd skech he grph of he soluion.

31 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, Consider IX..7., p.74 Find he soluion of he IBVP for : ( ( u H H 4 f ( [ H ( H ( ] 7. Verify resul of ppendi operionl propery of Inegrl Fourier rnsform in cse of non-homogeneous Robin boundry condiion. 8. Following Secion IX..6, find he sndrd Fourier inegrl represenion, he Fourier cosine inegrl represenion, nd he Fourier sine inegrl represenion of he funcion < f ( cos < < >

32 75 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 du d Appendi 3 Robin b.c., + Hu f ( : K cos + H sin + H (, K (, + H sin + H cos K (, + H H ( K, + H cos H sin K, K, + H ( ( F u II u K (, d u K (, d u u K (, d K (, u u K (, K (, d u K (, K (, du u K (, u K (, + ud K (, u K (, + u K (, + u K (, d u K (, u K (, uk (, d + u H + + H + H u uk (, d u + H + f u + H Hu uk, d ˆ (

33 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 753

34 754 Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7

### 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

Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 699 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

### 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

Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 6, 8 699 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

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