Application of Homotopy Analysis Method for Solving Linear and Nonlinear Differential Equations with Fractional Orders

Transcription

1 Zarqa Uivrsi Facl o Grada Sdis جبهعة الضسقبء كلية الذساسبت العليب Applicaio o Homoop Aalsis Mhod or Solvig Liar ad Noliar irial Eqaios wih Fracioal Ordrs B Haa Marai Mohammd Sprvisor r. Gharib Mosa Gharib Pro This Thsis was Sbmid i Parial Flillm o h Rqirms or h Masr s gr o Scic i Mahmaics Facl o Grada Sdis Zarqa Uivrsi Zarqa Jorda Novmbr-6

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3 iii االهذاء بسم هللا الرحمن الرح م قل إعملوا فس رى هللا عملكم ورسوله والمؤمنون صدق هللا العظ م إلى من بلغ الرسالة وأدى األمانة.. ونصح األمة.. إلى نب الرحمة ونور العالم ن "س دنا محمد صلى هللا عل ه وسلم" إلى من كلله هللا باله بة والوقار.. إلى من علمن العطاء بدون انتظار.. إلى من أحمل أسمه بكل افتخار والدي العز ز:مرع المه ر القطعان إلى معنى الحب وإلى معنى الحنان والتفان.. إلى بسمة الح اة وسر الوجود.. إلى من كان دعائها سر نجاح وحنانها بلسم جراح أم الحب بة:وداد عبدالمج د العماري إل االنسان الذي علقت عل ه امال ف اجت از هذا الدرب الطو ل..ال زوج الذي جعلت إرضاها هللا غا ته زوج عبدالغن جمعة الحوت إلى من أرى التفاؤل بأع نهم.. والسعادة ف ضحكتهم... إلى شعلة الذكاء والنور.. إلى الوجوه المفعمة بالبراءة بنات غن ه/ وداد

4 iv سبحبى هللا و عوه ببهلل هى هي وهب الي ب هزا العلن و االدساك لكي سعي الي التقذم و االصدهبس. تىجه ببلشكش لكل أعضبء هيئة التذسيس والعبهليي في جبهعة الضسقبء و ختص ببلزكش كلية العلىم قسن الشيبضيبت و كلية الذساسبت العليب هع ركش االستبر الذكتىس: سبهي العلي ولهن هضيذ هي التألق و االبذاع...

5 v شكش وتقذيش ولو أنن أوت ت كل بالغة ****** وأفن ت بحر النطق ف النظم والنثر لما كنت بعد القول إال مقصرا ***** ومعترفا بالعجز عن واجب الشكر من الصعب إن نكتب بما ل ق بك.. مهما كتبت فلن نوف ك حقك.. عود الفضل بعد هللا ف نجاح وتوف ق.. مهما كتبت ومهما قلت فمكانك اكبر من ذالك كل الشكر والعرفان والتقد ر لك بارك هللا ف ك وف ما اعطاك ونفع بعملك الجم ع وجعله خالص لوجه هللا تقبل فائق أحترام وتقد ري األستاذ دكتور:غر ب موس غر ب

6 vi Tabl o Cos COMMITTEE ECISION... ii... iii االهداء... v شكر وتقدير Tabl o Cos... vi Lis O Figrs... viii Lis O Tabls... i Lis O Smbols Ad Abbrviaios... Absrac... i Irodcio... Chapr o: Fracioal Calcls..... Spcial Fcios..... Gamma Fcio Ba Fcio Error Fcio Mig-Llr Fcio Mlli Ross Fcio Rima-Liovill Fracioal Igral Rima- Liovill Fracioal rivaiv....4.capo Fracioal rivaiv... 7 Chapr Two... 8 Mhods Solvig Fracioal irial Eqaios Variaioal Iraio Mhod Adomia composiio Mhod...

7 vii.4.homoop Aalsis Mhod Zroh Ordr ormaio Eqaio High- ordr ormaio Eqaio... 8 Chapr Thr:... 4 Applicaio o Homoop Aalsis Mhod For Solvig Liar ohomogos Fracioal Tlgraph Eqaio No-Liar Tim Fracioal BBM-Brgr Eqaio No Liar Fracioal Forbrg Whiham Eqaio Chapr For... 5 Coclsio ad Fr Rcommdaio... 5 REFERENCES... 5

8 viii Lis O Figrs Figr...Eplici mrical solios wih α = = Figr.. Absol Error E 5 a α = = Figr... Eplici mrical solios wih α = =-... 5

9 i Lis O Tabls Tabl.5. Som Laplac Trasorm... 4 Tabl.5. Ivrs Laplac Trasorm... 7

10 Lis O Smbols Ad Abbrviaios Th Gamma Fcio. s Th Lowr Icompl Gamma Fcio. s Th Uppr Icompl Gamma Fcio. Th Ba Fcio. Er Th Error Fcio. E Th Miag -Llr Fcio. E Th Gralizd Miag -Llr Fcio. E a Th Mlli Ross Fcio. Th Rima-Liovill Fracioal Igral Ordr α. Th Rima-Liovill Fracioal rivaiv Ordr α. * Th Capo Fracioal rivaiv Ordr α. VIM Th Variaioal Iraio Mhod. AM Th Adomia composio. HPM Th Homoop Prrbio Mhod. HAM Th Homoop Aalsis Mhod.

11 i APPLICATION OF HOMOTOPY ANALYSIS METHO FOR SOLVING LINEAR AN NONLINEAR IFFERENTIAL EQUATIONS WITH FRACTIONAL ORERS B Haa Marai Mohammad Sprvisor r. Gharib Mosa Gharib Pro. Absrac I his hsis w prs a modiicaio o a aalic chiq aml h homoop aalsis mhod o obai smbolic approima solios or liar ad oliar dirial qaios o racioal ordr. This mhod was applid o hr ampls: Liar ohomogos racioal lgraph qaio No Liar im- Fracioal BBM-Brgr Eqaio ad No Liar Fracioal Forbrg Whiham Eqaio.

12 Irodcio I rc ars cosidrabl irs i racioal parial dirial qaios FPEs has b simlad d o hir mros applicaios i h aras o phsics ad girig. Som damal wors o varios aspcs o h racioal calcls ar giv b Abbasbad [4]Capo [] bah [] ihlm.al. [6] Jaari ad Sii [8] Kilbas ad Trjillo [5] Oldham ad Spair [7] Kiraova [4] c. Th solio o racioal dirial qaios is mch ivolvd. I gral ; hr iss o mhod ha ilds a ac solio or racioal dirial qaios sch as Laplac rasorm mhod [4647] Forir rasorm mhod [46] Adomia s dcomposiio mhod AM [84] Variaio iraio mhod VIM[6] Homoop prrbaio mhod HPM [4]. Th Homoop aalsis mhod HAM was proposd irsl b Liao [456] or solvig liar ad oliar dirial ad igral qaios. Th HAM coais a crai ailia paramr ailiar cio H ad ailiar liar opraor L ha ca adjs ad corol h covrgc. rgio o sris solio. Thror h HAM ca ovrcom h orgoig rsricios ad limiaios o prrbaio chiqs so ha i provids s wih a possibili o aalz srogl liar ad oliar problms. Mor impora h abov procdr is js a algbraic algorihm ad ca b applid i a smbolic compaio ssm so h wll-ow smbolic sowar Mapl ca b sd. [78] Th hsis is orgaizd as ollows: chapr o coais iv scios. I h irs scio w giv a bri rviw o h Gamma Ba Error Miag-Llr ad Mlli-Ross cios. w prs Rima-Liovill racioal igral drivaiv

13 ad Capo racioal drivaiv. Th las scio coais h Laplac rasorm diiio ad som o is propris [975]. Chapr wo irodcds h basic ida o h mhods o solvig racioal dirial qaios. I chapr wo also w prs Variaioal iraio mhod VIM Adomia dcomposiio mhod AM ad Homoop prrbaio mhod HPM. Scio.4 coais h Homoop aalsis mhod [4846]. Chapr hr Coais h applicaio o h Homoop aalsis mhod.i h irs scio h liar ohomogo racioal Tlgraph qaio. I scio..no liar im- racioal BBM-Brgr Eqaio.I scio.. No Liar Fracioal Forbrg Whiham Eqaio [97544]. Chapr or Coclsio ad Fr Rcommdaio.

14 Chapr o: Fracioal Calcls I his chapr w irodc spcial cios ha ar dd i or wor. Th liar opraor o racioal igraio ad racioal diriaio i h ramwor o h Rima- Liovill ad Capo racioal calcls is irodcd. For h cocp o racioal drivaiv w will adop capo diiio which is a modiicaio o h Rima-Liovill diiio... Spcial Fcios I his scio w will sd h basic diiios ad propris o h ollowig spcial cios: Gamma Ba Error Miag- Llr ad Mlli- Ross cios... Gamma Fcio Th mos basic irpraio o h Gamma cio is simpl h gralizaio o h acorial or all mbrs. iiio.. Th Gamma cio R : is did b: iiio.. d. L R sch ha... h h Gamma cio is did b:! Lim....

15 4 propris...!.4.5 si d d 4 d l.6 iiio.4: Th lowr icompl Gamma cio s is did as: s s Ad h ppr icompl Gamma cio s d is did as:.7 s s d..8 * Th icompl Gamma cio is did as:... Ba Fcio * d.9 iiio.5. Th Ba cio B is did b: B d.

16 5 iiio.6. Th Ba cio ca also b did i rms o h gamma cio : B. Th Ba cio B has h ollowig propris: B B. B B B. B B.4 4 B si cos > >.5 d iiio.7. Th icompl Ba cio B B d is did b:.6... Error Fcio iiio.8. Th Error Fcio is did b:.7 Er d R Th complmar rror cio Erc is a closl rlad cio ha ca b wri i rms o h Error cio as

17 6 Erc Er.8 Er ad Er...4.Mig-Llr Fcio Th Mig- Lllr cio is amd ar h Swdish mahmaicia who did ad sdid i i 9.Th cio is a dirc gralizaio o h poial cio ad i plas a major rol i racioal calcls. iiio.9. Th o- wo paramr rprsaios o h Mig Llr cio ca b did i rms o a powr sris as: E ; i α =..9 E.. As a rsl h ollowig rlaios ar hold : E E. E d E E. d Obsrv ha Eq.. implis ha d d E E E.

18 7 So E E E d d.4 Nw W prov Eq.. B Eq..w g E. E For som spciic vals o α ad β h Miwg Llr cio rdcs o som amiliar. For ampl E!.5 Erc E.6 E!.7 4! Cosh E.8

19 8 5 Sih E!.9 6! Cos E. 7 Si E!. 8 E d E v...5. Mlli Ross Fcio Mlli- Ross cio E a ariss wh idig h racioal igral o a poial a. iiio.. Th icompl Gamma ad Miag Llr cios ar did as: a E a *. W ca also wri a E a a E.4

20 9.. Rima-Liovill Fracioal Igral W sar his scio b irodcig a sccic oaio ha will b rql sd. From ow c will do h racioal igraio o a cio o a arbirar ordr α alog h -ais. I his oaio α is a posiiv ral mbr ad h sbscrips c ad ar h limis o igraio. iiio..r-l Fracioal Igral A ral cio > is said o b i h spac c i hr iss a ral mbr p p sch ha whr c [ ad i is said o b i h spac C i C N {} whr c [ is h s o coios cios o [. iiio.: Th Rima- Liovill igral opraor o ordr α > o a Fcio C is did as: c J d..5 Propris.. I ad ar coios cios c c ad h: J J J J J..6 J c c c J c J.7

21 Thorm.: [Powr Rl] L α > > ad β >- h: J.8 Proo: d J d [ sbsid ] d d B. Corollar.: [Cosa Rl ] L α > > ad is a cosa h : J.9 Proo: Th proo ollows b i w sbsi β = i.8. Thorm.: [Epoial Rl] L α > > ad a> h:

22 J a a a a.4 Whr a is h lowr icompl Gamma cio. Proo: J a a d a a a d [ sbsiig =a-] a a a d a a a a d a a. a Rmar.: Llr cios as: a J ca b prssd also sig h Mlli Ross ad h Miag J a E a E a.4 Thorm.. [Si ad Cosi Rls]L α > > a h : J Sia a E a J Cosa E a.4.4

23 Proo:- J Si a Si a d Sia ad a B sig Eq.. ad Eq.. w g: Proo:- J Cos a a a E a E a Cosad. d B sig Eq.. ad Eq.. w g: E a. Corollar.:Th racioal igral o Sia ad Cosa also ca b prssd i gralizd Si ad Cosi cios as: J Si a Siad S a.44 J Cos a Cosad C a.45

24 ..Rima- Liovill Fracioal rivaiv W irodcd h oaio c or h racioal igraio o a cio. I his chapr w irodc a similar oaio. From ow o c will do h racioal drivaiv o a cio o a arbirar ordr. iiio.. Th Rima-Liovill racioal drivaiv o o Ordr α whr -< α < N is did b [ J ] : d = d d.46 Propris.: I ad ar wo cios did o [b] ad iss almos vr whr C ad C. C C C C.47 J.48 J [ ].49 4 L - < α < m- < β<m ad m N h: m [ ].5

25 4 Rmar.: i r... whr r = mam. Thorm.4 : [Powr Rl] L - < α < N β > L whr ma L ad > h :.5 Proo: ] [ ] [ J. = Corollar.: [Cosa Rl] L - < α < N K is a cosa h: K K.5 Proo: ollows b sig β= i.5. Thorm.5. [ Epoial Rl] L -< α < N ad a h: a E a.5 Proo: sig.4 w hav:

26 5 a E J a a a a J a. Now sig diiio. w g: ] [ a a J ] [ J a a a = a E. Thorm.6: [Si ad Cosi Rl] L -< α < N ad a h: a E a a Si.54 E a a Cos.55

27 6 Proo: Si a [ J Sia] a a [ a E a E E a. a Proo: Cos a [ J Cosa] [ E a E a E a. Corollar.4: [Sish ad Cosh Rl]L -< α < N ad a h: Siha a E a.56 Cosha E a.57

28 7.4.Capo Fracioal rivaiv iiio.4:th Capo racioal drivaiv o ordr α whr - < α < N is did b: * J d.58 Rmar.: Th Capo racioal igral ad Rima-Liovill racioal igral ar qival hc w will s oaio J o do boh o h racioal igral opraors o Capo ad Rima-Liovill. Th oaios * ad ar rspcivl or Capo racioal igral ad Rima-Liovill racioal igral. Thorm.7: L α > - < α < N assms ha is sch ha boh * ad is h: *.59 Propris.4: I C c c * ad * is h: * [ c * * c ] c c ].6 I ad C * is h:

29 8 * J.6 J *!.6 4 * * *.6 whr b a b a C ] [ ad N ad hr iss N wih ad ] [ h: Thorm.8: [Cosa Rl ] L - < α < N h: * K.64 Proo: Sic K N h Kd K * Thorm.9: [Powr Rl]L -< α < N ad > h: N * Proo: Cas- d * d.65

30 9 B sbsiig w g d. Cas-: si. Thorm.: [ Epoial Rl] L -< α < N ad a h: * a E a a.66 Proo: b sig horm.59 w hav * a a a a a E a a a

31 a a E a. Thorm.: [Si ad Cosi Rl] L -< α < N ad a h: * ia Sia iia [ E ia E ] * ia Cosa ia [ E ia E ] Proo:-- B sig h ormla i i Si h sig Capo racioal i drivaiv o poials w hav : [ i ia ia * Si a ia ia * * * ] i i ] [ i [ia E ia ia E ] ia ia [ E ia E ] ia. Proo:--B sig h ormla drivaiv o poials w hav : Cos i i h sig Capo racioal ia ia * Coa ia ia * [ ] [ * * ] [ia E ia ia E ] ia ia [ E ia E ] ia

32 Corollar.5: [Sish ad Cosh Rl]ls -< α < N ad a h: * a Siha a [ E a E ] * a Cosha a [ E a E ].69.7 Proo : B sig h ormla Sih w hav: [ a a * Sih a a a * * * ] ] [ [a E a a E ] a a a E ] [ E a Proo : B sig h ormla Cosh w hav: [ a a * Cosh a a a * * * ] ] [ a [a E a a E ] a a E ] [ E a iiio.5. [Capo Tim Ad Spac Fracioal rivaivs] For big h smalls igr ha cds α h Capo im -racioal drivaiv opraor o ordr α> is did b:

33 .7 N d Ad h spac racioal drivaiv o ordr α o is did b: N d.7.5.th Laplac Trasorm W will appl h Laplac rasorm o solv som racioal ordr dirial qaios. To sar ol s bril loo a h diiio o h Laplac rasorm ad pariclarl h Laplac rasorm o h racioal igral ad drivaiv. iiio.6: Th Laplac rasorm o is did b:. d Lim d L s F A s A s.7 W sa ha s L is h iq ivrs Laplac rasorm o s.

34 iiio.7: Th covolio prodc o wo cios ad g is did b: g g d.74 Propris.5 : I L{ } F s ad L{ g } G s N h: L{ g } F s G s.75 L{ g } F s G s.76 { c } cf s L C is cosa.77 4 L { } µ>-.78 S 5 L } s F s s [ ].79 { W sar wih h ollowig abl which shows som ow Laplac rasorm impora Fcios:

35 4 Tabl.5. Som Laplac Trasorm F s L{ } Codiios s> s s> α >- s a s a s a Sia s s> s a Cosa a s> s a Siha a s a s a Cosha s s a s a Lmma--: Laplac rasorm o Rima-Liovill racioal igral o ordr α > is giv b: L { } F s Whr F s L{ }.8 s Proo: L{ } L{ d L{ } L{ } L{ } B sig Laplac rasorm o Propris Eq..76 F s s F s s

36 5 Lmma.: Laplac rasorm o Rima-Liovill racioal drivaiv o ordr α > is giv b: ] [ } { s s F s L.8 Proo: B sig h ollowig orm o.79 } { s s F s L w hav ]} [ { } { L L ] [ } { s L s B sig Lmma Eq.. w g ] [ s s s F s ] [ s s F s. Lmma.: Laplac rasorm o Capo racioal drivaiv o ordr α > is giv b: } { * s s F s L.8

37 6 Proo: B sig Lmma Eq.. h sig Eq..79 w g L{ * } L{ [ ]} L { } s s F s s s s F s s. Rcall i h prss L{}=Fs h cio is calld h ivrs Laplac rasorm o Fs ad is dod b =L - {Fs}. Th ollowig abl givs h ivrs Laplac rasorm o som impora cios dpd o h abl.

38 7 Tabl.5. Ivrs Laplac Trasorm Fs =L - {Fs} Codiios s a s a E a s a s E a s s a a E a s s a E a s s a s E a s a s F ; ; a s a a a b b s a s b a b

39 8 Chapr Two Mhods Solvig Fracioal irial Eqaios I his chapr w sd mhods o solvig racioal dirial qaio which ar Variaioal Iraio Mhod Adomia composiio Mhod Homoop Prrbrbaio Mhod ad Homoop Aalsis Mhod...Variaioal Iraio Mhod Cosidr h dirial qaio: L N g. whr L ad N ar liar ad oliar opraors rspcivl ad g is h sorc ihomogos rm. Th variaioal iraio mhod admis h s o a corrcio cioal or qaio. i h orm L N~ g d.. whr is a Lagrag s mliplir which ca b idiid opimall b variaioal iraio mhod.th sbscrip dos h h approimaio. ~ is cosidrd as a rdisricd variaio i. ~. Th sccssiv approimaios o h solio. Th zro-h approimaios ca b a slcio cio.howvr gsss ' '' h iiial vals ad ar prrabl or h slciv zroh approimaio as will b s lar. Cosql h solio is giv b :

40 9 Lim.. Eampls..: Cosidr h ollowig qaio Th corrcio cioal is giv b: d ] [ Th scssi iraios ar obaid 6 Th sris solio is giv b...!!...!!

41 ..Adomia composiio Mhod Th Adomia dcomposiio mhod was irodcd b Adomia i arl 99. Th mhod has b provd o b rliabl ad ici chiq or a wid class o dirial ad igral qaios. Th mhod ssiall is a powr sris mhod similar o h prrbaio chiq. Cosidr a racioal dirial qaio i h orm L N.4 c = -. whr L d d is h racioal drivaiv o α ordr h h corrspodig L opraor. Th oliar rm N is prssd b a iii sris o h Adomia polomials. N A.5 d A.6... N[ ]! d Usig h Maclai Sris o racioal ordr ad applig h opraor L o boh sids o Eq..4 w hav L N L < α.7

42 Th iras ar drmid b h ollowig wa N L ad.8 L A Fiall approima h solio b h rcad sris Lim.9 N N N Eampls..: Cosidr h ollowig qaio: Th ac solio lcos Usig h Malar sris o racioal ordr Th gralizd iraio procdrs A d d

43 whr A is a Adomia polomials b applicaio h mhod wih N w hav Thror Homoop Prrbaio Mhod Th Homoop Prrbaio Mhod w cosidr a gral qaio o h p L. Whr L is a igral or dirial opraor w di a cov homoop H p b: H p p F pl. Whr F is a cioal opraor wih ow solios v which ca b obaid asil. I is clar ha or H p. w hav H F H L.

44 This shows ha H p coiosl racs a implicil did crv orms a sarig poi H v o a solio cio H.Th mbddig paramr moooicall icrass orm zro o i as h rivial problm F coiosl dorms h origial problm L. Th mbddig paramr p ] ca b cosidrd as a padig paramr. Th homoop prrbaio mhod ss h homoop prrbaio p as a padig paramr o obai i i p i p p....4 I p h Eq..4 corrspods o Eq.. ad bcoms h approima solio o h orm i Lim p i.5 I is wll ow ha sris Eq..4 is covrg or mos o h cass ad also h ra o covrgc is dpd o L. For mor dails abo h covrgc o h homoop prrbaio mhod ad h rrcs hri [4]. Th comparisos o qal powrs o p giv solios o varios ordrs. H s homoop prrbaio mhod cosidrs h solio o h homoop qaio i a sris o p as i p i p p i....6

45 4 ad h mhod cosidrs h oliar rm N as... i i i H p ph H H p N.7 whr H ar h H s polomials which ca b calclad b sig h ormla i p i i i p N p H... =.8 Eampls..: Cosidr h Hlmholz qaio 8 Wih h iiial codiios si Applig h cov homoop mhod ]... [8... p p p p p p p Ad comparig h coicis o qal powrs o P si : p si : p

46 5 p : 4 si Givs h sris solio as si si 4 si si 4 si cos..4.homoop Aalsis Mhod I his hsis w appl h homoop aalsis mhod [45] o solv h liar ad oliar dirial qaios wih racioal ordrs. This mhod was proposd b Liao [67].W d Liao s basic idas o h racioal parial dirial qaios. Cosidr a dirial qaio NF =.9 whr NF is a oliar racioal parial dirial opraor ad do idpd Variabls ad is a ow cio. W cosrc sch a Homoop: H v ; q q q Lv ; q qnfv ; q.

47 whr 6 q is calld homoop - paramr dos a iiial whr q is calld homoop - paramr dos a iiial propr L wh ad v q is a cio o ad q. wh q H[ v ; q; q] q L[ v ; ] A q H[ v ; q] q NF[ v ;] rspcivl v ;. Solio o h qaio H [ ; q; q] q ad v ;. H [ v ; q; q] q As h mbddig paramr q icrass rom o. Th solio v ; q o h qaio. H [ v ; q; q].

48 7.4..Zroh Ordr ormaio Eqaio I opolog his id o variaio is calld ormaio ad qaios. ad. cosrc h homoop v ; q. For brvi qaios. ad. ar calld h Zroh - ordr dormaio qaios. To ovrcom his rsricio o h arl HAM.Liao irodcd a ailiar paramr qaio. H o cosrc sch a id o zroh ordr dormaio H [ v ; q q] q L[ v ; q ] q H N [ v ; q] q L[ v ; q ] q NF [ v ; q].4 For q ad bsids is mh- ordr drivaiv wih rspc o h mbddig paramr q i.. m [ m] v ; q m q.5 q whr m= or brvi m is calld h mh-ordr dormaio drivaiv. i: [ m] m v ; q m m q.6 m! m! q

49 8 B Talor s Thorm ; w pad v ; q i a powr sris o h Embddig paramr q as ollows: v ; q m m v ; q v ; m q q.7 m m! q Form qaio. ad.6 h abov powr sris : m m m v ; q q.8 A q h sris.8 Bcoms m m v ;.9 Thror sig qaio. w hav m m. which ms b o o h solios o h origial oliar qaio as prov b Liao..4..High- ordr ormaio Eqaio i h vcors: [... ]. iria h Zroh- ordr dormaio qaio.4 m-ims wih rspc o q ad h dividig hm b m! ad iall sig q. W g h ollowig m-ordr dormaio qaio:

50 9 L[ ] NFR. m m m m whr NFR m! NF[ v ; q] q m m m q. ; m m.4 ; m Th so calld m-h ordr dormaio qaio.4 is liar which ca b asil solvd sig Mahmaics pacag. For m r m mm L [ FR m ].5 m NF[ v ; q] m mm L [ ]. m q m! q whr L is h ivrs o h liar opraor[i. ivrs o diriaio is igraio]. Th m h ordr approimaio o is giv b m.6 m For h covrgc o h abov mhod w rr h radr o Liao[4567]. I qaio. admis iq solio h his mhod will prodc h iq solio.

51 4 Chapr Thr: Applicaio o Homoop Aalsis Mhod For Solvig Fracioal irial Eqaio Applicaio o Homoop Aalsis Mhod For Solvig Fracioal irial Eqaio. I his chapr w ar solvig Liar ohomogos racioal Tlgraph qaio o-liar im racioal BBM-Brgr qaio ad Forbrg Whiham Eqaio...Liar ohomogos Fracioal Tlgraph Eqaio Th lgraphr's qaios or js lgraph qaios ar a pair o copld liar dirial qaios ha dscrib h volag ad crr o a lcrical rasmissio li wih disac ad im. Th qaios com rom Olivr Havisid who i h 88s dvlopd h rasmissio li modl. Th modl dmosras ha h lcromagic wavs ca b rlcd o h wir ad ha wav pars ca appar alog h li. Th hor applis o rasmissio lis o all rqcis icldig high-rqc rasmissio lis sch as lgraph wirs ad radio rqc codcors adio rqc sch as lpho lis low rqc sch as powr lis ad dirc crr. c a b g. Whr a bad c ar cosa ad g h sorc rm is a cio o ad.w assm ha h iiial ad bodar codiios ar as ollows:

52 4 h. as >. B maiplaig h abov procdr w di h liar opraors as ollows: [ v ; q] v ; q.. NF[ v ; q] v ; q c v ; q av ; q bv ; q g..4 Usig h abov diiio w gai h m h-ordr liar racioal opraors as ollows: NFR m m c m a m b m m g..5 Now or m w hav m m m m NFR L..6 From.5 ad.6 w ow sccssivl obai.7 c b g..8 i c b g.9 i c i bi.

53 4 Th. Ad so o.ths or h solio is as ollows:.. W s h iiial codiio h sorc g sih ad ailiar opraor L. Noic ha i c a b. Ths w hav g sih.. Applig h abov procdr ilds { cosh d}. whr or α = w hav sih..4

54 4 Figr...Eplici mrical solios wih α = =-..No-Liar Tim Fracioal BBM-Brgr Eqaio Th Bjami Boa Maho qaio or BBM qaio also ow as h rglarizd log-wav qaio RLWE. This qaio was sdid i Bjami Boa ad Maho 97 as a improvm o h Korwg d Vris qaio KdV qaio or modlig log srac gravi wavs o small amplid propagaig i-dircioall i + dimsios. Th show h sabili ad iqss o solios o h BBM qaio. This corass wih h KdV qaio which is sabl i is high wav mbr compos. Frhr whil h KdV qaio has a iii mbr o igrals o moio h BBM qaio ol has hr..5 sc h 4 To solv h abov problm wih h HAM mhod w choos h liar oigr ordr opraor

55 44 ; ] ; [ q v q v L.6 Eq..5 approachs s o di h oliar racioal parial dirial opraor.7 Usig.7 w hav h Zroh-ordr dormaio qaio ; ] ; [ q NFv q q L v q.8 Also o ha wh q= ad q= rspcivl w g ; ; q v q v.9 Eq...4ild h m h-ordr dormaio qaio ] [ NFR L m m m m. whr. NFR m i i m i m m m m. Ths h solio o Eq.. or m bcoms. NFR L m m m m. Emploig Eqs..5.9ad.w ow sccssivl obai 4 sc h. ; ; ; ; ] ; [ q v q v q v q v q v NF

56 a 4 sc cosh ] [ 4 h sih 4 7sih 4 5 sih 4 sc 4 4 a 4 sc cosh 8 4 sc cos 6 cosh cosh 4 ] [ h h h.5 Procdig i his mar h rs o h compos or ca b compll obaid ad h sris solios ar hs irl drmid. Fiall w hav 5. m m I h abov rms w sbsi h domia rms will rmai ad h rs rms vaish bcas h icld acor o N m m ad or α = w hav 4 4 sc h.6

57 46 Figr.. Absol Error E 5 a α = = -.-No Liar Fracioal Forbrg Whiham Eqaio Th Whiham qaio was proposd as a alra modl qaio or h simpliid dscripio o i-dircioal wav moio a h srac o a iviscid lid. As h Whiham qaio icorporas h ll liar disprsio rlaio o h war wav problm i is hogh o provid a mor aihl dscripio o shorr wavs o small amplid ha radiioal log wav modls sch as h KdV qaio. W idi a scalig rgim i which h Whiham qaio ca b drivd rom h Hamiloia hor o srac war wavs. Th Whiham qaio is igrad mricall ad i is show ha h qaio givs a clos approimaio o iviscid r srac damics as dscribd b h Elr qaios. Th prormac o h Whiham qaio as a modl or r srac damics is also compard o wo sadard r srac modls: h KdV ad h BBM qaio. I is od ha i a wid paramr rag o amplids ad wavlghs h Whiham qaio prorms o par wih or br ha boh h KdV ad BBM qaios.

58 To solv h oliar qaio wih h HAM mhod w choos h liar oigr ordr opraor L[ v : q] v ; q..8 opraor Eq..6 dircs s o di h oliar racioal parial dirial NF[ v ; q] v ; q v ; q v v ; q v v. ; q v v ; q.9 Th zroh-ordr dormaio qaio ca b cosrcd [**]as q L[ v ; q ] qnfv ; q. Obviosl q= ad q= rspcivl ild v ; v ;. Cosidrig Eqs..4 w gai h mh ordr dormaio qaio L[ m mm ] NFR m. m whr NFR m m m i mi i m.. i i mi m m

59 48 Now h solio o Eq. or m bcoms. NFR L m m m m.4 Usig Eqs..7. ad. w ow sccssivl obai 8.5 ] [ ] [ ] 9 [ 4.6 [ ].7 [

60 49.8 I h abov rms w sbsi ol will rmai domia rm ad h rs rms vaish bcas h icld acor o m N m ad or ach α= w hav Ths h solio o h giv problm is as ollows: ] [.4

61 5 Figr... Eplici mrical solios wih α = =-

62 5 Chapr For Coclsio ad Fr Rcommdaio I his chapr w irodc som op problms ha appard i or sd o h racioal opraors. W hav rrrd o som o hs problms i propr placs i his hsis wh w discssd similar idas. Morovr som idas also ar giv i his chapr ha w ca appl hm o h sdid racioal diriaios ad igraios. Ths problms ca b smmarizd as ollow: Problm 4.. How o d ad sd h all rsls o Rimma-Liovill ad Capo racioal opraors o ral ordrs o h cas o compl mbr? Problm 4.. How o id w applicaios o racioal calcls ha comig rom ohr scics? Problm 4.. How o d h s o o Tlgraph Eqaio prs h solios o h gral cas o liar racioal dirial qaios wih dir racioal ordrs?

63 5 REFERENCES. Abdl Majid wazwaz 4 Th Variaioal Iraio Mhod For Solvig liar ad Noliar OE s ad Sciiic Modls Wih Variabl Coicis.. A.Chigal ad M. Rghioa Homoop Prrbaio Mhod For solvig Som Iiial Bodar Val Problms Wih Nolocal Codiios.. Ahmad El-Ajo Zaid Odiba Shahr Momai Ahmad Alamh Cosrcio O Aalsis Solios o Fracioal irial Eqaios Usig Homoop Aalsis Mhod IAENG Iraioal Joral O Applid Mahmaics. 4. A.Roozi E.Alibii SS.Hossi S.M.Sha- o H.Ebrahimi Homoop Prbio Mhod For Spcial Noliar Parial irial Eqaios joral o.sad.u A.A.Kilbas ad J.J.Trjillo irial Eqaios O Fracioal Ord:r Mhods Rsls Problms Appl Aal Bal.R ad Wog.R Spcial Fcios: AGrada T Cambridg Uivrsi Prss Cambridg UK. 7. Bahma Ghazaari F.visi Homoop Aalsis Mhod For Th Fracioal Noliar Eqaios. 8. Bahma Ghazaari Amah Spahvadzadh4 Adomia composiio Mhod For Solvig Fracioal Bra-p Eqaios: Joral o Mahmaics ad Compr Scic Crompo B A Irodcio o Fracioal Calcls ad h Fracioal isio Wav Eqaio.Upblishd Masr Thsis Uivrsi o Massachss Lowll.. as S Fcioal Fracioal Calcls Hidlbrg : Sprigr.. i hlm K 4 Th Aalsis O Fracioal irial Eqaios Hidlbrg :Sprigr.. aardar - Gjji Varsha 4 Fracioal Calcls Thor ad Applicaios Idia: Norosa pblishig Hos pv.ld.

64 5. Ishva M 5 Propris ad Applicaios O Th Capo Fracioal Opraor ;Upblishd Masr Thsis Karirl TH Us vri Soia.Blgaria. 4. Josph M. Kim 9 Fracioal Parial irial Eqaios Usig Th Homoop Aalsis Mhod. 5. Kilbas. R Srivasava.H ad Trjillo.J 6 Thor ad Applicaios O Fracioal irial Eqaios Elsvir. 6. K.ihlm N.J.Ford ad A..Frd A Prdicor Corrcor Approach For h Nmrical Solio o Fracioal irial Eqaio Noliar K.B. Oldham ad J.Spair 974Th Fracioal Calcls Acadmic Prss Nw Yor. 8. H. Jaari ad S.Sii 9 Homoop Aalsis Mhod or Solvig Liar ad Noliar Fracioal isio-wav Eqaio Comm Noliar Sci Nmr Siml Hasa Sabri i Rza bzhabadi Solvig Fracioal Forbrg- Whiharm Eqaio B Homoomp Aalsis Mhod: Joral o Applid Mahmaics Saisics ad Iormaics JAMSI 7 NO.. O. Abdlaziz I. Hashim ad A. Sai 8 Sris Solios o Tim- Fracioal PEs b Homoop Aalsis Mhod: Hidawi Pblishig Corporaio irial Eqaios ad Noliar Mchaics Volm 8 Aricl I pags.. Lszczshi Jac A Irodcio O Fracioal Mchaics Czsochowa : Th Pblishig Oic O Czsochowa Uivrsi O Tcholog.. L.bah ad.bhaa7ihgral Trasorms ad Thir Applicaiosscod diiochapma ad Hall/CRCBoca Rao.. L.bah Rcs Applicaios O Fracioal Calcls o Scic ad Egirig I J Mah. Mah Sci Mahai A ad Habold H 8 Spcial Fcios For Applid Sciiss Nw Yor : Sprigr Scic +bsiss Mdia LLC. 5. Millr. K ad Ross. B 99 A Irodcio o h Fracioal Calcls ad Fracioal irial Eqaios Now Yor : Joh Wil.

65 54 6. M.O. Olaiwola Th Variaioal Iraio Mhod or Aalic Tram o Homogos ad Ihomogos Parial irial Eqaios. 7. Mhdi Gajiai Solio O Noliar irial Eqaios Usig Homoop Aalsis Mhod. 8. Mhdi Taari Mhdi hgha_ Mohs Razzaghib 6 Applicaio o h Adomia composiio Mhod or h For Plac qaio: Mahmaical ad Compr Modllig Marcl B. Fia Laplac Trasorms ;Thor Problms ad Solios.. Mhdi hgha.jalil Maaia ad Abbas saadamadi Th Solio OF Th Liar Fracioal Parial irial Eqaios Usig h Homoop Aalsis Mhod.. Mhdi hgha.jalil Maaia ad Abbas saadamadi 9 Solvig Noliar Fracioal Parial irial Eqaios Usig h Homoop Aalsis Mhod.. M.Capo 967Liar Modls o issipaio Whos Q is Almos Frqc Idpd J.R. Asroomic Pascal Sbah Xavi Gordo Irodcio o Gamma Fcio Nmbr. Compaio. Fr. Fr/Cosa / cosas.hml. 4. S.J Liao Bod Prbaio:Irodcio o h Homoop Aalsis Mhod Chapma ad Hall/CRC prss Boca RATON 5. S.J.Liao ad A.Campo A alsis Solios o h Tmprrar isrbio i Blasis Viscos Flow Problms Jlid mch S.J.Liao 9Nos o h Homoop Aalsis Mhod Som iios ad Thorms Comm i Noliar Sci.ad Nmr. Simla S.J.Liao 99 Th Proposd Homoop Aalsis Tchiq or Solio o Noliar Problms phd hsis.shaghai Jiao og ivrsi. 8. S.J Liao 4 Bod Prbaio:Irodcio o h Homoop Aalsis Mhod Chapma ad Hall/CRC w Yor. 9. S.J.Liao98 Topolog ad Gomr or Phsicss Acadmic prss.florida prss.

66 55 4. Sd Tas Mohd-i ad Mhammad Aslam Noor 8 Homoop Prrbaio Mhod For Solvig Parial irial Eqaios. 4. S.Abbasbad7 A Approimaio Solio o a Noliar Eqaio Wih Rima- Liovill s Fracioal rivaivs b H s Variaioal Iraio Mhod J Compa Appl Mah S.V.Kiraova Mlipl mliid Miag Llr Fracios ad rlaios o Gralizd Fracioal Calcls J Comp Appl Mah S.Momai ad N.T.Shawagh 6composiio Mhod or Solvig Fracioal Riccai irial Eqaios Appl Mah Comp Sil Kmar vdra Kmar Fracioal Modllig or BBM-Brgr qaio b Usig Nw Homoop Aalsis Trasorm Mhod: Joral o h Associaio o Arab Uivrsiis or Basic ad Applid Scics T.Haa S. Nad ad S.Asghar 4 Priodic Uidircioal Flow o A Viscolasic Flid Wih h Fracioa Mawll ModlAppl Mah Comp W.sh 9 Lcr Nos For Laplac Trasorm. 47. Zriga M Solvig Fracioal Oscillaors Usig Laplac Homoop Aalsis Mhod Mahmaics ad Compr Scic Sris 8 4.

67 56 تطب ق طر قة الهموتوب عل المعادالت التفاضل ة ذوات الرتب الكسر ة إعداد هناء مرع محمد المشرف األستاذ الدكتور غر ب موس غر ب ملخص الخط ة و غ ر الخط ة ف هذه الرسالة قام الباحث بتقد م طر قة تحل ل الهموتوب للحصول عل حلول تقر ب ة للمعادالت التفاضل ة الخط ة وغ ر الخط ة ذوات الرتب الكسر ة طبقت هذه الطر قة عل ثالثة أمثلة لحل معادلة من رتبة كسر ة معادلة تلغراف الخط ة غ ر متجانس ة من رتبة كسر ة معادلة بن ام ن بونا غ ر خط ة من رتبة زمن كسر ة و معادلة Forbrg- Whihamغ ر خط ة من رتبة كسر ة.