Approximate solutions for the time-space fractional nonlinear of partial differential equations using reduced differential transform method

Transcription

1 Global Joral o Pr ad Applid Mahmaics ISSN Volm Nmbr 6 7 pp 5-6 sarch Idia Pblicaios hp://wwwripblicaiocom Approima solios or h im-spac racioal oliar o parial dirial qaios sig rdcd dirial rasorm mhod Mohamd S Mohamd ad Tria T Al- Qarshi Mahmaics parm Facl o Scic Al-Azhar ivrsi Cairo Egp Mahmaics parm Facl o Scic Tai ivrsi Tai Sadi Arabia Absrac I his papr w sablish a modiid rdcd dirial rasorm mhod which ar sccssll applid o obai h aalical solios o im-spac racioal oliar Nwll-Whihad qaio FNWEs Th racioal drivaiv is a i Capo ss Th obaid rsls show ha h proposd chiqs ar simpl ici ad as o implm or racioal dirial qaios W ma h Figrs o compar bw h approima solios W compar bw h approima solios ad h ac solios or h parial racioal dirial qaios wh α β Kwords: Nwll-Whihad qaio; dcd dirial rasorm mhod; Fracioal calcls Capo ss INTOCTION Fracioal dirial qaios hav b h ocs o ma sdis d o hir rq apparac i varios applicaios i lid mchaics viscolasici biolog phsics ad girig Cosql cosidrabl aio has b giv o solv his id o qaios oral mos o hm do o hav ac solios cl svral mrical mhods hav b irodcd or his prpos o o hm ar h amos h rdcd dirial rasorm mhod TM was irs proposd b Ksi ad Orac - Th TM was also applid b ma rsarchrs o hadl oliar qaios arisig i scic ad girig -4

2 54 Mohamd S Mohamd ad Tria T Al- Qarshi I his sd w ar cocrd wih h solio o h oliar Nwll-Whihad- Sgl qaio NWSE was drivd b Nwll ad Whihad 5 ad i has b sd i modlig varios orms o problms ha aris rom lid mchaics I has applicaios i Chmical Bio-Egirig ad Mchaical Egirig c Svral ahors hav proposd dir mhods o solvig his qaio i h pas ars 6 NWSE ar solvd b sig h Adomia dcomposiio mhods 7 ad Homoop aalsis mhod 8 Th mai aim o his aricl is o prs approima aalical solios o im racioal Nwll-Whihad qaio b sig h rdcd dirial rasorm mhod MTM Th Nwll-Whihad qaio wih im-spac racioal drivaivs is wri i opraor orm as 7-8: a b q Whr ad ar paramr dscribig h ordr o h im-spac racioal drivaivs a b ad q is a posiiv igr PELIMINAIES AN NOTATIONS I his scio w giv som basic diiios ad propris o racioal calcls hor which shall b sd i his papr: iiio Th racioal drivaiv o i h Capo s ss is did as 9: d or iiio For h Capo racioal drivaiv o ordr o h whol spac dod b c c is did b : d Propr Som sl ormla ad impora propris or h modiid ima- Liovill drivaiv as ollows : r r r r r

3 Approima solios or h im-spac racioal oliar o parial dirial 55 g g g g g g g g g g g 4 ECE IFFEENTIAL TANSFOM METHO TM I his scio w irodc h basic diiios o h rdcd dirial rasormaios Cosidr a cio o hr variabls w ad assm ha i ca b rprsd as a prodc - w F G Basd o h propris o o dimsioal dirial rasorm h cio w ca b rprsd as i i i i w F i i G W i i i i i i whr W i i F i i G is calld h spcrm o w L dos h rdcd dirial rasorm opraor ad h ivrs rdcd dirial rasorm opraor Th basic diiio ad opraio o h TM mhod is dscribd blow iiio I w is aalic ad coiosl diriabl wih rspc o spac variabls cio ad im variabl i h domai o irs h h spcrm 5 6 w W w 7 is h rdcd rasormd cio o w Th dirial ivrs rdcd rasorm o is did as: W W w W Combiig Eqs 7 ad 8 w g 8 w w 9 Wh Eq 9 rdcs o

4 56 Mohamd S Mohamd ad Tria T Al- Qarshi w w From h Eq 8 i ca b s ha h cocp o h rdcd dirial rasorm is drivd rom h powr sris pasio o h cio iiio I V v ad h covolio dos h rdcd dirial rasorm vrsio o h mliplicaio h h damal opraios o h rdcd dirial rasorm ar show i h Tabl : Tabl - Fdamal opraios o h rdcd dirial rasorm mhod Origial cio rdcd dirial rasorm cio v r r r V V v V N N N N s m s m!! s s m m! 4 ILLSTATIVE EXAMPLES To illsra h capabili ad rliabili o his mhod wo dir cass o imspac racioal oliar Nwll-Whihad qaio ar prsd Firs w will cosidr a im-racioal Nwll-Whihad qaio whil h scod dals wih h sam qaio o boh spac ad im racioal drivaiv Eampl: I qaio i b a ad q h im racioal Nwll- Whihad qaio is wri as:

5 Approima solios or h im-spac racioal oliar o parial dirial 57 sbc o h iiial codiio Sih cosh Applig h TM o Eqs w obai h ollowig rcrrc rlaios i i i i sig h FTM o h iiial codiios w g Sih cosh Applig h iiial codiios 4 io Eqs w hav Sih cosh 5 6Csch Sih Sih 7 5 Cosh 8 4 Cosh 8 Sch 6 Tah 4 5 Th gral solios o Eqs ar : a 6 So w hav: Sih cosh 7 5 Cosh Sih cosh 5 6Csch Sih Sih 4 Cosh 8 So h solio o qaio a α is giv as: Sch 8 6 Tah 7 Sih Sih cosh cosh 8Sih 8 Cash 5 8

6 58 Mohamd S Mohamd ad Tria T Al- Qarshi Ad also w ow ha Talor s sris pasio o So as ds ds o which is h ac solio Figr: Th ac solios giv i a ad h mrical solios o : b b 8 I is as o s ha h wo solios loo almos idical Figr c ad d dpic h volio solio o h cass o α = 5 ad α = 5 rspcivl I is o b od ha as h im-racioal drivaiv paramr α dcrass h volio solio bircas or small vals o

7 Approima solios or h im-spac racioal oliar o parial dirial 59 Figr : Th mrical solios o : c α = 5 ad d α = 5 Figr : Compariso bw approima solios 8 a α ac solio

8 6 Mohamd S Mohamd ad Tria T Al- Qarshi Figrs - shows h volio rsl or h Nwll-Whihad qaio wh α corrspods o h solio obaid b FTM 8 ad corrspods o h ac solio giv i I is as o s ha h or solios loo almos idical Eampl I qaio b a ad q h im-spac racioal Nwll- Whihad qaio is wri as: Sbc o h iiial codiio Applig h FTM o Eqs w obai h ollowig rcrrc rlaios i i i i sig h TM o h iiial codiios 4 w g 4 Applig h iiial codiios 4 io Eqs w hav 5 Th gral solios o Eqs ar : a 6 So w hav: 7

9 Approima solios or h im-spac racioal oliar o parial dirial 6 So h solio o qaio 7 a α is giv as: 6 Figr4 shows h volio rsl or h Nwll-Whihad qaio wih imspac racioal drivaivs: α = β = ad α = 75 β = 75 8 Figr 4: Th mrical solios o : α = ad β = α = 75 ad β = 75 5 CONCLSION I his papr h modiid rdcd dirial rasorm mhod was applid or solvig im- spac racioal Nwll-Whihad qaio wih iiial codiios Th racioal drivaiv was did i h Capo ss Th proposd approimad solios o Nwll-Whihad qaio wih a appropria iiial codiio ar obaid i rms o a powr sris wiho sig a id o discrizaio prrbaio or rsriciv codiios c wo ampls ar illsrad o sd h civss ad accrass o FTM I is od ha FTM solios ar i cll agrm wih hos obaid sig AM 7 HAM 8 Howvr compaios show ha h FTM is vr as o implm ad ds small siz o compaio corar o AM ad HAM This shows ha FTM is vr civ ad ici powrl mahmaical ool which is asil applicabl i idig o h approima aalic solios o a wid rag o ral world problms arisig i girig ad allid scics Mahmaica has b sd or compaios i his papr

10 6 Mohamd S Mohamd ad Tria T Al- Qarshi EFEENCES Y Ksi G Orac Th rdcd dirial rasorm mhod or parial dirial qaios I J Noliar Sci Nmr Siml Y Ksi G Orac Th rdcd dirial rasorm mhod or solvig liar ad oliar wav qaios Ira J Sci Tchol 4 - M S Mohamd K A Gprl dcd dirial rasorm mhod or oliar igral mmbr o Kadomsv- Pviashvili Hirarch dirial qaios Joral o h Egpia Mahmaical Soci M S Mohamd Approima solios o ssm o oliar parial dirial qaios sig dcd dirial rasorm mhod Joral o Iormaio ad Compig Scic L A Sgl isa Sid-walls cas slow amplid modlaio o clllar covcio J Flid Mcha A Saravaa ad N Magsh A compariso bw h rdcd dirial rasorm mhod ad h adomia dcomposiio mhod or h Nwll- Whihad-Sgl qaio Joral o h Egpia Mah Soci Z Hammoch ad T Maoi Approima aalical ad mrical solios o racioal KPP-li qaios G Mah Nos M Saad F AL-Mali ad Talib A Aalic algorihm or h im-spacracioal Nwll-Whihad qaio Iraioal viw Phsics IEPHY K A Gprl M S Mohamd aalical approima solio or oliar im-spac racioal Kli Gordo qaio Chi Phs B -6 M A E Hrzallah K A Gprl Approima solio o h im-spac racioal cbic oliar Schrodigr qaio Appl Mah Modlig G Jmari Nw sochasic racioal modls or Malhsia growh h Poissoia birh procss ad opimal maagm o poplaios Mah Comp Modllig