Local Fractional Variational Iteration Method for Solving Nonlinear Partial Differential Equations within Local Fractional Operators

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1 Available a hp://pvamed/aam Appl Appl Mah SSN: Vol sse December 5 pp Applicaios ad Applied Mahemaics: A eraioal Joral AAM Local Fracioal Variaioal eraio Mehod for Solvig Noliear Parial Differeial qaios wihi Local Fracioal Operaors Absrac Hossei Jafari Deparme of Mahemaics Uiversi of Mazadara Babolsar ra Jafari@mzacir Hassa Kamil Jassim Deparme of Mahemaics Uiversi of Thi-Qar Nasiriah raq hassaamil@ahoocom eceived: November 8 4; Acceped: November 5 5 his aricle he local fracioal variaioal ieraio mehod is proposed o solve oliear parial differeial eqaios wihi local fracioal derivaive operaors To illsrae he abili ad reliabili of he mehod some eamples are illsraed A compariso bewee local fracioal variaioal ieraio mehod wih he oher merical mehods is give revealig ha he proposed mehod is capable of solvig effecivel a large mber of oliear differeial eqaios wih high accrac addiio we show ha local fracioal variaioal ieraio mehod is able o solve a large class of oliear problems ivolvig local fracioal operaors effecivel more easil ad accrael ad hs i has bee widel applicable i egieerig ad phsics Kewords: Noliear parial differeia eqaio; local fracioal variaioal ieraio mehod; local fracioal derivaive operaors MSC No: 6A33; 34A; 34A34; 35 rodcio The local fracioal calcls has araced a lo of ieres for scieiss ad egieers Several secios of local fracioal derivaive had bee irodced ie he local fracioal derivaive srcred b Kolwaar ad Gagal 997 Kolwaar ad Gagal 998 ad Yag 55

2 56 ar faja a d arir fa H he modified iema- Lioville derivaive give b Yag ad Jmarie 6 he fracal derivaive sggesed b Yag Jmarie ad Parvae 5 he fracal derivaive cosidered b Yag Che 6 ad Che e al he geeralized fracal derivaive proposed Che e al he local fracioal derivaive preseed b Yag Adda ad Cresso he local fracioal derivaive srcred b He Fa ad He ad Yag As a resl he local fracioal calcls heor becomes impora for modelig problems for fracal mahemaics ad egieerig o Caor ses ad i plas a e role i ma applicaios i several fields sch as heorical phsics i Kolwaar 998 ad Yag he hea codcio heor i Yag He Balea ad Yag 3 he fracre ad elasici mechaics i Yag he flid mechaics i Yag ad Balai ad lizarraraz Several aalical ad merical echiqes were sccessfll applied o deal wih differeial eqaios fracioal differeial eqaios ad local fracioal differeial eqaios Wazwaz Scheider ad Wss 989 Zhao ad Li Momai ad Odiba 8 Lasi Zho ad Jiao Momai ad Odiba 6 Tarasov 8 Golmahaeh ad Balea ad Li e al Hrisov The echiqes iclde he hea-balace iegral Hrisov he fracioal Laplace rasform Balea e al he harmoic wavele Caai 5 8 local fracioal variaioal ieraio Yag ad Balea 3 ad S e al 3 he local fracioal decomposiio Yag e al 3 Jafari ad Jassim 4 he local fracioal series epasio Jafari ad Jassim 4 ad he geeralized local fracioal Forier rasform Yag e al 3 mehods his paper we ivesigae he applicaio of local fracioal variaioal ieraio mehod for solvig oliear local fracioal parial differeial eqaios The srcre of he paper is as follows Secio we give he cocep of local fracioal calcls Secio 3 we give aalsis of he mehod sed Secio 4 we cosider some illsraive eamples Fiall i Secio 5 we prese or coclsios Basic Defiiios of Local Fracioal Calcls his secio we give some basic defiiios ad properies of fracioal calcls heor which will be sed i his paper Defiiio We sa ha a fcio f is local fracioal coios a Yag if f f

3 AAM: er J Vol sse December 5 57 wih for ad For a b i is so called local fracioal coios o a b deoed b f C a b Defiiio Seig f C a b local fracioal derivaive of f a is defied as Yag Wag e al 4 ad Ya e al 4 D f L f f lim f f where f f f f Noe ha local fracioal derivaive of high order is wrie i he form D imes f f D D D f 3 ad local fracioal parial derivaive of high order imes f f 4 Defiiio 3 Le s deoe a pariio of he ierval [ a b] as j N ad N b wih j j j j j ad ma{ } Local fracioal iegral of f i he ierval [ a b] is give b Yag Wag e al 4 ad Ya e al 4 a b f b a f d lim N j f j j 5 Defiiio 4 fracal space he Miage Leffler fcio e fcio ad coe fcio are defied as Yag Wag e al 4 ad Ya e al 4 6

4 ar faja a d arir fa H 58 ] [ 7 ] [ 8 The followig resls are valid: d d 9 d d d d a b d b a b a d b a 3 3 Aalsis of he Local Fracioal Variaioal eraio Mehod We cosider a geeral oliear local fracioal parial differeial eqaio: f N L 3 where L deoes liear local fracioal derivaive operaor of order deoes liear local fracioal derivaive operaor of order less ha L N deoes oliear local fracioal operaor ad f is he odiffereiable sorce erm Accordig o he rle of local fracioal variaioal ieraio mehod he correcio local fracioal fcioal for 3 is cosrced as Yag ad Balea 3: ~ ~ f N L 3 where is a fracal Lagrage mliplier Maig he local fracioal variaio of 3 we have ~ ~ f N L 33

5 59 AAM: er J Vol sse December 5 The eremm codiio of is give b Yag 34 view of 34 we have he followig saioar codiios: 35 So from 35 we ge 36 The iiial vale is give b 37 view of 36 we have f N L 38 Fiall from 38 we obai he solio of 3 as follows: lim 39 4 Some llsraive amples his secio we gave some illsraive eamples for solvig he oliear parial differeial eqaios ivolvig local fracioal derivaive operaors b g local fracioal variaioal ieraio mehod ample Le s cosider he oliear local fracioal parial differeial eqaio 4 ad sbjec o he fracal vale codiios 4 From 4 we ae he iiial vale which reads

6 ar faja a d arir fa H 6 43 B g 38 we srcre a local fracioal ieraio procedre as 44 Hece we ca derive he firs approimaio erm 45 The secod approimaio ca be calclaed i he similar wa which is 46 Proceedig i his maer we ge he hird approimaio as 3 47 ad so o Ths we have he local fracioal series solio 48 As a resl he fial solio reads lim 49 ample Cosider he oliear local fracioal parial differeial eqaio 4

7 6 AAM: er J Vol sse December 5 ad sbjec o he fracal vale codiios 4 From 4 we ae he iiial vale which reads 4 B g 38 we srcre a local fracioal ieraio procedre as 43 Hece we ca derive he firs approimaio erm 44 The secod approimaio ca be calclaed i he similar wa which is 45 Proceedig i his maer we ge he hird approimaio as 3 46 ad so o Ths we have he local fracioal series solio 47 As a resl he fial solio reads lim 48

8 ar faja a d arir fa H 6 ample 3 Cosider he oliear local fracioal parial differeial eqaio 49 ad sbjec o he fracal vale codiios 4 From 49 we ae he iiial vale which reads 4 B g 38 we srcre a local fracioal ieraio procedre as 4 Hece we ca derive he firs approimaio erm co 43 The secod approimaio ca be calclaed i he similar wa which is 44 Proceedig i his maer we ge he hird approimaio as 3 45 ad so o Ths we have he local fracioal series solio

9 AAM: er J Vol sse December As a resl he fial solio reads lim 47 5 Coclsios his paper we have sdied oliear parial differeial eqaios ivolvig local fracioal operaor wih he local fracioal variaioal ieraio mehod LFVM The eac solio of he oliear parial differeial eqaios is obaied b he local fracioal variaioal ieraio mehod The resls showed ha he variaioal ieraio mehod is remarabl effecive Compariso wih he local fracioal decomposiio mehod LFDM shows ha he local fracioal variaioal ieraio mehod is a powerfl mehod for oliear eqaios The advaage of he LFVM over he LFDM is ha here is o eed for he evalaios of he Adomia polomials Acowledgeme The ahors are ver graefl o he referees for heir valable sggesios ad opiios FNCS Adda F B Cresso J Abo o-differeiable fcios J Mah Aal Appl 63: Balea D Diehelm K Scalas ad Trjillo J J Fracioal Calcls Models ad Nmerical Mehods World Scieific Boso Mass USA Che W 6 Time-space fabric derlig aomalos diffsio Chaos Solios ad Fracals 8:93-99 Che W Zhag X D Korosa D vesigaio o fracioal ad fracal derivaive oscillaio models J Noliear Sci Nmer Simlaio : 3-9 Che W S H G Zhag X D Korosa D Aomalos diffsio modelig b fracal ad fracioal derivaives Comp Mah Appl 59: Caai C 8 Harmoic wavele solio of Poisso s problem Bala Joral of Geomer ad s Applicaios 3: 7 37 Caai C 5 Harmoic waveles owards he solio of oliear PD Compers & Mahemaics wih Applicaios 5:9-

10 64 ar faja a d arir fa H Fa J He J H Fracal derivaive model for air permeabili i hierarchic poros media Absrac ad Applied Aalsis D 3547 : -7 Golmahaeh A ad Balea D O oliear fracioal Klei Gordo eqaio Sigal Procesg 9: H M Balea D Yag X J 3 Oe-phase problems for discoios hea rasfer i fracal media Mahemaical Problems i gieer D : -3 He J H A ew fracal derivaio Therm Sci He J H Asmpoic mehods for soliar solios ad compacios Absrac ad Applied Aalsis D : -3 Hrisov J Hea-balace iegral o fracioal half-ime hea diffsio sb-model Thermal Sciece 4: 9-36 Jmarie G The Miowsi s space-ime is cosise wih differeial geomer of fracioal order Phs Le A 363: 5- Jmarie G 6 Modified iema-lioville derivaive ad fracioal Talor series of o-differeiable fcios frher resls Comp Mah Appl 5: Lasi N Fracioal Schrodiger eqaio Phsical eview 66: -7 Kolwaar K M Gagal A D 997 Holder epoes of irreglar sigals ad local fracioal derivaives Pramaa J Phs 48: Kolwaar K M Gagal A D 998 Local fracioal Foer-Plac eqaio Phs ev Le 8: 4-7 Li Z B Zh W H ad Hag L L Applicaio of fracioal variaioal ieraio mehod o ime-fracioal Fisher eqaio Advaced Sciece Leers : 6-64 Momai S Odiba Z ad Alaweh A 8 Variaioal ieraio mehod for solvig he space-ime fracioal KdV eqaio Nmerical Mehods for Parial Differeial qaios 4: 6-7 Momai S ad Odiba Z 6 Aalical solio of a ime fracioal Navier-Soes eqaio b Adomia decomposiio mehod Applied Mahemaics ad Compaio 77: Parvae A Gagal AD 9 Calcls o fracal sbses of real lie Formlaio Chaos Solios ad Fracals 7: 53-8 Parvae A Gagal A D 5 Fracal differeial eqaios ad fracal-ime damical ssems Pramaa J Phs 64: Scheider W ad Wss W 989 Fracioal diffsio ad wave eqaios Joral of Mahemaical Phsics 3:34 44 S W H Balea D ad Jafari H 3 Damped wave eqaio ad dissipaive wave eqaio i fracal srigs wihi he local fracioal variaioal ieraio mehod Fied Poi Theor ad Applicaios Aricle 89 3: -6 Tarasov V 8 Fracioal Heiseberg eqaio Phsics Leers A 37: Wazwaz A M Parial Differeial qaios: Mehods ad Applicaios lsevier Balema The Neherlads Wag S Q Yag Y J ad Jassim H K 4 Local Fracioal Fcio Decomposiio Mehod for Solvig homogeeos Wave qaios wih Local Fracioal Derivaive Absrac ad Applied Aalsis Aricle D : -7 Yag X J 9 esearch o fracal mahemaics ad some applicaios i mechaics MS hesis Chia Uiversi of Miig ad Techolog Chiese Yag X J ad Balea D 3 Fracal hea codcio problem solved b local fracioal variaio ieraio mehod Thermal Sciece 7: Yag X J Balea D ad Zhog W P 3 Approimaio solios for diffsio eqaio o Caor ime-space Proceedig of he omaia Academ A 4: 7-33

11 AAM: er J Vol sse December 5 65 Jafari H ad Jassim H K 4 Local Fracioal Adomia Decomposiio Mehod for Solvig Two Dimesioal Hea codcio qaios wihi Local Fracioal Operaors Joral of Advace i Mahemaics 9: Yag X J Balea D ad Machado J A 3 Mahemaical aspecs of Heiseberg cerai priciple wihi local fracioal Forier aalsis Bodar Vale Problems : 3-46 Ya S P Jafari H ad Jassim H K 4 Local Fracioal Adomia Decomposiio ad Fcio Decomposiio Mehods for Solvig Laplace qaio wihi Local Fracioal Operaors Advaces i Mahemaical Phsics Aricle D 658 4: -7 Yag X J Advaced Local Fracioal Calcls ad s Applicaios World Sciece Pblisher New Yor Yag X J Local Fracioal Fcioal Aalsis ad s Applicaios Asia Academic Pblisher Limied Hog Kog Jafari H ad Jassim H K 4 Local Fracioal Series pasio Mehod for Solvig Laplace ad Schrodiger qaios o Caor Ses wihi Local Fracioal Operaors eraioal Joral of Mahemaics ad Comper esearch : Yag X J Local fracioal iegral rasforms Prog Noliear Sci 4: -5 Zho Y ad Jiao F Nolocal Cach problem for fracioal evolio eqaios Noliear Aalsis: eal World Applicaios : Zhao Z G ad Li C P Fracioal differece/fiie eleme approimaios for he ime- space fracioal elegraph eqaio Applied Mahemaics ad Compaio 9: