Elzaki transform and the decomposition method for nonlinear fractional partial differential equations

Transcription

1 I. J. Ope Proble Cop. Mh. Vol. 9 No. Deceber ISSN 998-; Copyrigh ICSRS Publicio.i-cr.org lzi ror d he decopoiio ehod or olier rciol pril diereil equio Djelloul ZIN Depre o Phyic Uieriy o Hib Bebouli Pole o Ouled Fre Chle lgeri e-il: djeloulz@yhoo.co brc he i o hi pper i o e chge o he pplicio o he odiied doi decopoiio ehod uggeed i [] d e eed o obiig oluio o olier pril diereil equio ih ie-rciol deriie. he rciol deriie i decribed i he Cpuo ee. Soe illurie eple re gie reelig he eeciee d coeiece o he ehod. Keyord: doi decopoiio ehod; lzi ror ehod; Cpuo rciol deriie; olier equio. Iroducio Diereil equio re o ype: lier diereil equio d olier diereil equio. No-lier diereil equio re he o cople i he oluio copred ih lier diereil equio due o he preece o o-lier pr i he. So e id h lo o reercher re orig o deelop e ehod o ole hi id o equio. hee eor reuled i he coolidio o hi reerch ield i y ehod og he e id he doi decopoiio ehod. hi ehod deeloped ro he 9 o he 99 by George doi []-[]. he e opio eerged recely iclude he copoiio o Lplce ror uudu ror Nurl ror or lzi ror ih hi ehod or olig lier d olier diereil equio. og hich re he doi decopoiio ehod coupled ih Lplce ror [][] doi decopoiio uudu ror ehod [8] url decopoiio ehod [9] d lzi ror decopoiio lgorih [][].

2 Djelloul ZIN he oiio o hi pper i o eed he pplicio o lzi ror decopoiio ehod uggeed i [] o ole olier pril diereil equio ih ie-rciol deriie. he rciol deriie i decribed i he Cpuo ee. he pree pper h bee orgized ollo: I Secio oe bic deiiio o Lzi ror ehod re eioed. I ecio e ill propoe lyi o he odiied ehod. I ecio olier ie-rciol pril diereil equio i udied ih he rciol lzi ror decopoiio ehod FDM. Filly he cocluio ollo. Bic deiiio I hi ecio e gie oe bic deiiio d properie o rciol clculu lzi ror d lzi ror o rciol deriie hich re ued urher i hi pper.. Frciol clculu. I here re eerl deiiio o rciol deriie o order α ee []-[]. he o cooly ued deiiio re he Rie--Liouille d Cpuo. We gie oe bic deiiio d properie o he rciol clculu heory hich re ued urher i hi pper. Deiiio. Le Ω=[b] - <<b<+ be iie ierl o he rel i R. he Rie-Liouille rciol iegrl I α ƒ o order α ϵ R α> i deied by I. d Here i he ell-o G ucio. heore. Le α d le =[α]+. I ƒ ϵ C [b] he he Cpuo rciol deriie c D α ƒ ei lo eeryhere o [b]. I N c D α ƒ re repreeed by : c D d here D=d/d d =[α]+. Proo ee []. Rer. I hi pper e coider he ie-rciol deriie i he Cpuo' ee. Whe α ϵ R+ he ie-rciol deriie i deied :

3 lzi ror d he decopoiio ; u d u u u D c here ϵ N *.. Deiiio o lzi ror e iegrl ror clled lzi ror []-[8] deied or ucio o epoeil order i proclied. hey coider ucio i he e deied by :. : j i Me M j Deiiio. I ƒ i ucio deied or ll i lzi ror i he iegrl o ƒ ie / ro = o. I i ucio o d i deied by [ƒ]. d e heore. lzi ror pliie he coeicie o he poer erie ucio : o he e iegrl ror "lzi ror" gie by :.! heore. Le ƒ be i d Le deoe lzi ror o h deriie ƒ o ƒ he or

4 8 Djelloul ZIN 8. o obi lzi ror o pril deriie e ue iegrio by pr d he e he : 9. Properie o lzi ror c be oud i Re.[]] e eio oly he olloig : ;. ;!. ;..!.. lzi ror o rciol deriie o gie he orul o lzi ror o Cpuo rciol deriie e ue he Lplce ror orul or he Cpuo rciol deriie []. ; c F D L here -<α ϵ N. heore. [9] Le deied boe. Wih Lplce ror F he he lzi ror o ƒ i gie by :. F heore. Suppoe i he lzi ror o he ucio ƒ he :. ; c D Proo ee []. Frciol lzi decopoiio ehod FDM I hi ecio e e chge o he ehod propoed i [] d e

5 lzi ror d he decopoiio 9 eed o olig olier pril diereil equio ih ie-rciol deriie. o illure he bic ide o hi ehod e coider geerl olier ohoogeeou rciol pril diereil equio : C R N g D here -<α = d he iiil codiio :... c here D i he Cpuo rciol deriie o he ucio R i he lier diereil operor N repree he geerl olier diereil operor d g i he ource er. pplyig lzi ror deoed i hi pper by o boh ide o q. e ge : c [ D ] [ R ] [ N ] [ g ]. Uig he propery o lzi ror e obi : here -<α =. [ ] R N [ g ] Operig ih he iere lzi ror o boh ide o q. e ge : 8 G [ R N ] here G repree he er riig ro he ource er d he precribed iiil codiio. he ecod ep i lzi decopoiio ehod i h e repree oluio iiie erie gie belo: 9 d he olier er c be decopoed : N

6 Djelloul ZIN here re He' polyoil [] o hich c be clculed by he olloig orul :...!... N p i i i Subiuig 9 d i 8 e he: ] [ N R G O coprig boh ide o q. e ge: R R G I geerl he recurie relio i gie :. ] [ R Filly e pproie he lyicl oluio by ruced erie:. li N N he boe erie oluio geerlly coerge ery rpidly []. pplicio o he FDM I hi ecio e pply he rciol lzi ror decopoiio ehod FDM or he Cpuo rciol deriie o ole olier pril diereil equio ih ie-rciol deriie. ple. Coider he olloig ie-rciol pril diereil equio : < c D ubjec o he iiil codiio. pplyig lzi ror o boh ide o q.. hu e ge:

7 lzi ror d he decopoiio 8. ] [ ] [ ] [ c u D Uig he diereiio propery o lzi ror e he : 9 ]. [ ] [ By pplyig he oreid decopoiio ehod e id : ] [ he ir e copoe o polyoil [] or eple i gie by : O coprig boh ide o q. e ge: Uig He' polyoil d he ierio orul e obi: ] [ he pproie oluio o q. i gie by:

8 Djelloul ZIN he pproie oluio o q. i he pecil ce α= i gie by. h gie : hich i ec oluio o KdV equio preeed i []. Fig. : c oluio b he pproie oluio i he ce α= c he ec oluio d pproie oluio o q. or diere lue o α he =. ple. Coider he olier ie-rciol pril diereil equio o order α <α c D ih he iiil codiio

9 lzi ror d he decopoiio 8. I α= d ih he iiil codiio 8 he ec oluio o he olloig equio : 9 i gie by :. By uig e c obi he ierio orul :. he ir e copoe o polyoil [] or eple i gie by: I iilr y boe e he: 8

10 Djelloul ZIN he pproie oluio i erie or i gie by : α= e obi : 8 hich i ec oluio o he olier pril diereil equio. Rer For grph o he pproie oluio e oo oly our er i he o preiou eple. Fig. : he ec oluio B he pproie oluio he α=.9 C he pproie oluio o q. or diere lue o α he =.

11 lzi ror d he decopoiio ple. Coider he olier ie-rciol pril diereil equio o order α <α D c ih he iiil codiio. I α= d ih he iiil codiio he ec oluio o he olloig equio : 8 i gie by : 9. By uig e c obi he ierio orul : he ir e copoe o polyoil [] or eple i gie by: I iilr y boe e he :

12 Djelloul ZIN here :!!!.! 8!!!! 8!! 9! 8! 9 8 he ir er o he pproie oluio o q. i gie by : α= e obi : h gie : 9 hich i ec oluio o he olier pril diereil equio 8 preeed i [].

13 lzi ror d he decopoiio Fig. : he ec oluio B he pproie oluio he C he pproie oluio o q. or diere lue o he d e oo oly hree er. Cocluio I hi pper he rciol lzi ror decopoiio ehod FDM h bee pplied or idig he ec or he pproie oluio o he olier rciol pril diereil equio. he FDM c eily be pplied o y proble d i cpble o reducig he ize o copuiol or. he reul ho h he rciol lzi ror decopoiio ehod FDM i pproprie ehod or olig olier pril diereil equio o rciol order. Ope Proble I hi or he rciol lzi ror decopoiio ehod FDM o be eecie or olig olier pril diereil equio ih ierciol deriie. Oe c pply he rciol url ror

14 8 Djelloul ZIN decopoiio ehod FNDM...o he e proble olier rciol pril diereil equio. I i poible o ole olier pril iegrodiereil equio o rciol order by hi ehod FDM? Reerece [] G. doi Nolier Sochic Sye heory d pplicio o Phyic Kluer cdeic Publiher Neherld 989. [] G. doi R. Rch quliy o pril oluio i he decopoiio ehod or lier or olier pril diereil equio Copu. Mh. ppl [] G. doi Solig Froier Proble o Phyic: he Decopoiio Mehod Kluer cdeic Publiher Boo 99. [] G. doi Soluio o phyicl proble by decopoiio Copu. Mh. ppl [] G. doi Soluio o olier P.D.. ppl. Mh. Le [] S.. Khuri Lplce decopoiio lgorih pplied o cl o olier diereil equio J. Mh. l. ppl. -. [] S.. Khuri e pproch o Bru proble ppl. Mh. Copu. -. [8] D. Kur J. Sigh S. Rhore Suudu Decopoiio Mehod or Nolier quio I. Mh. For. -. [9] M.S. Rhdeh S. Mi Solig Coupled Sye o Nolier PD Uig he Nurel Decopoiio Mehod I. J. o Pure d ppl. Mh [] M. Khlid M. Sul F. Zidi U. rhd lzi ror Decopoiio lgorih pplied o Cl o No-Lier Diereil quio J. N. Sci. Re. 8-. [] D. Zie M. Hdi Cheri Reoluio o Nolier Pril Diereil quio by lzi ror Decopoiio Mehod J. ppro. heo. ppl. Mh. -. [].M. lzi S.. lheeb Modiicio o Suudu ror "lzi ror" d doi Decopoiio Mehod ppl. Mh. Sci. 9. [] I. Podluby Frciol Diereil quio cdeic Pre S Diego C 999.

15 lzi ror d he decopoiio 9 [].. Kilb H. M. Sri J. J. rujillo heory d pplicio o Frciol Diereil quio leier erd. [] K. Diehel he lyi Frciol Diereil quio Spriger- Verlg Berli Heidelberg. [].M. lzi S.M. lzi.. lour O he Ne Iegrl ror "lzi ror Fudel Properie Ieigio d pplicio" Glo. J. Mh. Sci. -. [].M. lzi.m.. Hill Hooopy Perurbio d lzi ror or Solig Nolier Pril Diereil quio Mh. heor. Mod. -. [8].M. bd lohoud.m. lzi lzi ror o Deriie preed by Heiide Fucio W. ppl. Sci. J [9].M. lzi S.M. zi; O he Coecio Beee Lplce d Lzi ror d. heo. ppl. Mh. -. []. Ney B. gheli R. Drzi; Ne Iegrl ror or Solig Nolier Pril Diereil quio o rciol order heo. ppr. ppl [] Y. Zhu Q. Chg S. Wu e lgorih or clculig doi polyoil ppl. Mh. Cop [] M.M. Hoeii H. Nbzdeh O he coergece o doi decopoiio ehod ppl. Mh. Copu [].M. Wzz he riiol ierio ehod or riol oluio or KdV K Burger d cubic Bouieq equio J. Cop. ppl. Mh. 8-.