ON A NEW SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATION USING COMPLEX TRANSFORM IN THE UNIT DISK

Transcription

1 Mhemicl Compionl pplicions Vol 9 No pp ON NEW SOLUTION OF FRCTIONL IFFERENTIL EQUTION USING COMPLEX TRNSFORM IN THE UNIT ISK Rbh W Ibrhim Mslin rs Insie of Mhemicl Sciences Universiy Mly 563 Kl Lmpr Mlysi School of Mhemicl Sciences Fcly of science Technology Universiy Kebngsn Mlysi ngi 436 Selngor rl Ehsn Mlysi rbhibrhim@yhoocom mslin@kmmy bsrc-the Möbis rnsform of frcionl differenil eqion Ricci ype is employed o consrc new exc solions for some nonliner frcionl differenil eqions The frcionl operors re ken in sense of he modified Srivsv-Ow frcl in he ni disk Exmples re illsred for problems in biology economic physics Keywords- Frcionl clcls frcionl differenil eqions Srivsv-Ow frcionl operors ni disk INTROUCTION Trnsform mehod is mhemicl echniqe h is pplied in vrios fields sch s qnm mechnics ncler physics omic physics This echniqe generes he solions of pril differenil eqions; reles solions of difficl pril differenil eqions o well known eqions pplies o inegrble eqions For exmple Ricci eqion is employed o consrc generlied solions for ordinry pril differenil eqions Vrios prcicl rnsforms for solving vrios problems were merilied in open lierre sch s he Lplce rnsform he Forier rnsform he rveling wve rnsform he äcklnd rnsformion he inegrl rnsform he frcionl inegrl rnsforms he frcionl complex rnsform Mellin rnsform Frcionl differenil eqions re viewed s opion models o nonliner differenil eqions Vrieies of hem ply imporn roles ools no only in mhemics b lso in physics dynmicl sysems conrol sysems engineering o cree he mhemicl modeling of mny physicl phenomen Frhermore hey re employed in socil science sch s food spplemen clime economics Frcionl differenil eqions concerning he Riemnn-Lioville frcionl operors or Cpo derivive hve been recommended by mny hors see -6] eermining pproxime nmericl exc solions for frcionl differenil eqions plys significn role Nmericl solions or nlyic solions re ypiclly difficl o be comped I is herefore reqired o impose process o solve he problem of nonliner frcionl differenil eqions Recenly one of he mos essenil sefl mehods for frcionl clcls ppered s complex frcionl rnsform inegrl derivive 7-] Frcionl rnsform is devised o renove he frcionl differenil eqions

2 On New Solion of Frcionl ifferenil Eqion 53 ino ordinry differenil eqions yielding he solion procedre remendosly simple In his pper we shll se he Möbis rnsform of frcionl differenil eqion Ricci ype o consrc he exc solions for some nonliner frcionl differenil eqions The frcionl operors re ken in sense of Srivsv-Ow frcl in he ni disk Exmples re illsred o explin he solion procedre inclding problems in complex domin In ] Srivsv Ow provided he definiions for frcionl operors derivive inegrl in he complex -plne C s follows: efiniion The frcionl derivive of order is defined for fncion f by d f f : ; < d d where he fncion f is nlyic in simply-conneced region of he complex -plne C conining he origin he mlipliciy of is removed by reqiring log o be rel when > efiniion The frcionl inegrl of order is defined for fncion f by I f : f d ; > where he fncion f is nlyic in simply-conneced region of he complex -plne C conining he origin he mlipliciy of is removed by reqiring log o be rel when > Remrk From efiniions we hve > ; < I > ; > In or ex we shll se he following operor: efiniion 3 The modified frcionl derivive of order is defined for fncion f by d f f f : ; < d d where he fncion f is nlyic in simply-conneced region of he complex -plne C conining he origin he mlipliciy of is removed by reqiring log o be rel when > Noe h efiniion 3 sisfies he nlyic fncion of he form n f n C n

3 54 R W Ibrhim M rs This clss of nlyic fncions hs wide pplicions in he geomeric fncion heory nivlen fncion heory when Recenly he hors employed hese operors widely in he geomeric fncion heory by exending some clsses of nlyic fncions ino clsses of nlyic fncions of frcionl power see 3-6] Noe h he rel cse of he Srivsv Ow frcionl operors coincides wih he Riemnn-Lioville frcionl operor which re given by he following definiion efiniion 4 The frcionl rbirry order inegrl of he fncion f of order > is defined by I f f d When we wrie I f I f I ws shown h f f * where * denoed he convolion prodc see ] > s where is he del fncion efiniion 5 The frcionl rbirry order derivive of he fncion f of order < is defined by d d f f d I f d d Remrk From efiniion 4 efiniion 5 we hve > ; < < I > ; > I TRNSFORM METHO The frcionl Ricci eqion in complex domin kes he form: ] where is rel consn : U C; U : { C } I is well known h he solion of kes he form in erms of he generlied hyperbolic rigonomeric fncions see ]

4 On New Solion of Frcionl ifferenil Eqion 55 nh for < coh for < > n for co for > for where is consn We shll ssme he complex frcionl differenil eqion wih independen vrible dependen vrible j where j F j re he modified Srivsv-Ow frcls Or mehod cn be smmried s follows: Sep : Using he complex wve rnsform j j 3 j j where i j re consns Eq3 becomes i 4 d where d Sep : ssming h 4 hs solion of he form n m 5 m m where m m n re consns o be clcled compes from he Möbis rnsform 6 where Sep 3: Sbsiing 5 in 4 seing he coefficiens of he powers of o be ero we impose nonliner lgebric sysem in m Sep 4: Solving he sysem o obin hese vles sbsiing hem ino Eq5 we my receive he exc solions of 3 3 PPLICTIONS In his secion we shll illsre wo exmples o exmine or mehod 3 Exmple Wer s liqid moves hrogh he vdose region in response o grviy

5 56 R W Ibrhim M rs grdiens of pressre Recll h he vdose region hs hole spces filled wih boh ir liqid wer The wer pressre depends on he wer srion reled cpillry forces ecse he soil is only prilly sred he pressre is negive de o cpillriy If he soil is niform in is properies sch s composiion cpillry pressres re mos negive where he soil is dry mos posiive where i is we s n FE i cn be represened s 7 where is he posiion in his model is he so-clled volmeric wer conen I denoes he proporion of he spce filled by wer is he so-clled soil moisre diffsiviy is he srion dependen hydrlic condciviy Eqion 7 describes he infilrion in he vdose region The dvecion is de he grviy he diffsion is de o cpillry wicking Using he complex wve rnsform we receive 8 y pplying he bove mehod yields sch h is he solion of Ricci eqion defined in Now for < we impose he solions While for > implies he solions solions of Eq8 nh nh coh coh n n co co ] ] ] ] ] ] ] ]

6 On New Solion of Frcionl ifferenil Eqion 57 3 Exmple In 973 Fischer lck Myron Scholes 7] sggesed he fmos heoreicl vlion forml for opions The min ficionl ide of lck Scholes excies in he exre of riskless porfolio king posiions in bonds csh opion he nderlying sock Sch n process srenghens he se of he no-rbirge principle s well The lck-scholes model for he vle of n opion cn be described by he frcionl eqion r T 9 where is he Eropen cll opion price sse price posiive rel nmber ime ; r is he risk free ineres re represens he voliliy fncion of nderlying sse y employing he wve rnsform we exrdie r Now in vire of he bove mehod we hve r where For < we impose he solions r r r r nh nh ] ] coh ] r r coh ] While for > implies he solions solions of Eq n r r n ] ] co ] r r co ] 33 Exmple dringe eqion is n eqion chrceriing he relevnce beween deph spcing of prllel sbsrfce drins deph of he wer ble deph hydrlic condciviy of he soils I is employed in dringe design which reds for frcionl ime-spce s follows:

7 58 R W Ibrhim M rs < U T The fom dringe eqion is pern of he flow of liqid hrogh chnnels nodes inersecion of for chnnels beween he bbbles driven by grviy cpillriy 8] Now by sing he complex rnsform where is consn Sbsiing ino Eq we receive he frcionl ordinry differenil eqion: 3 y blncing he highes order derivive erms nonliner erms in Eq 3 we ssme h Eq 3 hve he following forml solion * 4 where sisfies Eq Sbsiing Eq 4 long wih Eq ino Eq 3 hen seing he coefficiens of o ero we cn impose se of lgebric eqions bo Solving he lgebric eqions yielding where is liner fncion in > Sbsiing he bove sserion ino 4 implies new ypes of exc solions of Eq s follows: For < we impose he solions ] nh ] nh ] coh ] coh while for > he solions ] n ] n ] co ] co

8 On New Solion of Frcionl ifferenil Eqion 59 4 CONCLUSION From bove we conclde h he Möbis rnsform of frcionl differenil eqion Ricci ype ffeced on he exc solions of frcionl differenil eqions in complex domin The frcionl operors re ken in senses of he Srivsv-Ow frcionl operors he Riemnn-Lioville frcionl operors We pplied he proposed mehod on differen ypes of frcionl differenil eqions sch s liqid movemen eqion lck-scholes frcionl differenil eqion frcionl dringe eqion in order o cree new exc solions cknowledgemen- The second hor is flly sppored by LRGS/T//UKM/ICT/3/ 5 REFERENCES I Podlbny Frcionl ifferenil Eqions cd Press London 999 J Wes M ologn P Grigolini Physics of Frcl Operors Insie for Nonliner Science Springer New York NY US 3 3 Kilbs H M Srivsv J J Trjillo Theory pplicions of Frcionl ifferenil Eqions vol 4 of Norh-Holl Mhemics Sdies Elsevier Science V mserdm The Neherls 6 4 J Sbier O P grwl J Mchdo dvnce in Frcionl Clcls: Theoreicl evelopmens pplicions in Physics Engineering Springer ordrech The Neherls 7 5 V Lkshmiknhm S Leel J Vsndhr Theory of Frcionl ynmic Sysems Cmbridge cdemic Pblishers Cmbridge 9 6 len Gvenc J Tenreiro New Trends in Nnoechnology Frcionl Clcls pplicions Springer New York NY US 7 Z Li JH He pplicion of he frcionl complex rnsform o frcionl differenil eqions Nonliner Science Leers -6 8 R W Ibrhim Frcionl complex rnsforms for frcionl differenil eqions dvnces in ifference Eqions :9 doi:86/ R W Ibrhim Complex rnsforms for sysems of frcionl differenil eqions bsrc pplied nlysis ricle I pges H M Srivsv M rs R W Ibrhim Clsses of nlyic fncions wih frcionl powers defined by mens of cerin liner operor Inegrl Trnsforms Specil Fncions 7 8 S Sivsbrmnin M rs R W Ibrhim On he srlikeness of cerin clss of nlyic fncions Mhemicl Comper Modelling HM Srivsv S Ow Univlen Fncions Frcionl Clcls Their pplicions Hlsed Press John Wiley Sons New York Chicheser risbn Torono RW Ibrhim M rs ifferenil operor generlied by frcionl derivive Miskolc Mhemicl Noes RW Ibrhim M rs Sbordinion sperordinion for nivlen solions for frcionl differenil eqions Jornl of Mhemicl nlysis

9 6 R W Ibrhim M rs pplicions R W Ibrhim M rs Sbordinion sperordinion for nlyic fncions involving frcionl inegrl operor Complex Vribles Ellipic Eqions R W Ibrhim M rs On nlyic fncions ssocied wih he iok-srivsv liner operor Srivsv-Ow frcionl inegrl operor rbin Jornl for Science Engineering F lck M S Scholes The pricing of opions corpore libiliies Jornl of Poliicl Economy Weire S Hler S Cox M lonso renckhn The Flid ynmics of Foms Jornl of Physics: Condensed Mer