Solving Wave and Diffusion Equations on Cantor Sets
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1 Proceedigs o he Pkis Acdemy o Scieces 5 : Copyrigh Pkis Acdemy o Scieces ISSN: pri olie Pkis Acdemy o Scieces Reserch Aricle Solvig Wve d Disio qios o Cor Ses Jmshd Ahmd * d Syed Tsee Mohyd-Di Deprme o Mhemics Fcly o Scieces HITC Uiversiy Til Pkis Asrc: I his pper we pply he proposed locl rciol Adomi s decomposiio mehod o oi he lyic solios o wve d he eqios wihi locl rciol derivive operors. The ierio procedre is sed o locl rciol derivive. The oied resls revel h he mehodology is very eicie d simple ool or solvig rcl pheome risig i mhemicl physics d egieerig. Keywords: ocl rciol clcls; Frciol diereil eqio; locl rciol Adomi s decomposiio mehod. AMS Mhemics Sjec Clssiicio: 74H; 55. INTRODUCTION The locl rciol clcls heory ws pplied o model d process he o-diereile pheome i rcl physicl pheome [ ]. Here re some locl rciol models sch s he locl rciol Fokker-Plck eqio [] he locl rciol sress-sri relios [] he locl rciol he codcio eqio [9] wve eqios o he Cor ses [] locl rciol plce eqio [] Newoi mechics o rcls sse o rel-lie [] d he locl rciol Helmholz eqio [4]. There re eis some lyicl mehods widely pplied o solve o-lier prolems icldes rciol domi decomposiio mehod [5] he homoopy perrio mehod [6] he he-lce iegrl mehod [7] he comple rsorm mehod [8] he homoopy lysis mehod [9] he rciol s-eqio mehod [] d he rciol vriiol ierio mehod [] d more deils see i []. Recely he pplicio o Adomi decomposiio mehod or solvig he lier d olier rciol pril diereil eqios i he ields o he physics d egieerig hd ee eslished i [ 4]. Adomi decomposiio mehod ws pplied o hdle he ime- rciol Nvier-Sokes eqio [5] rciol spce disio eqio [6] rciol KdV-Brgers eqio [7] lier d olier rciol disio d wve eqios [8] rciol Brgers eqio [9]. The Adomi decomposiio mehod s oe o eicie ools or solvig he lier d olier diereil eqios ws eeded o id he solios or locl rciol diereil eqios [-] d o-diereile solios were oied. I his pper or im is o pply he locl rciol Adomi s decomposiio mehod [4 5] or solvig rciol pril diereil eqios i he sese o locl rciol derivive. To illsre he vlidiy d dvges o he mehod we will pply i o he spce-ime rciol wve d he eqios.. PRIMINARY RSUTS AND DFINITIONS I his secio we prese ew mhemicl dmels o locl rciol clcls d irodce he sic oios o locl rciol Received Decemer 4; Acceped Ferry 5 * Correspodig hor: Jmshd Ahmd; mil: jmshdhmdm@gmil.com
2 8 Jmshd Ahmd e l coiiy locl rciol derivive d locl rciol iegrl o o-diereil cios. Deiiio.. I here eiss he relio [7 6] wih or d. Now is clled locl rciol coios lim. The deoed y is clled locl rciol coios o he iervl deoed y C. Deiiio.. A cio is clled odiereile cio o epoe which sisy Hölder cio o epoe he or y X sch h [76] y C y. Deiiio.. A cio is clled o e coios o or shorly coios whe we hve he ollowig relio [76] o. 4 Compred wih 4 q. is sdrd deiiio o locl rciol coiiy. Here is iied locl rciol coiiy. Deiiio.4. Seig C rciol derivive o is deied [76] d d locl order lim o 5 where. For y here eiss D deoed y D. ocl rciol derivive o high order is wrie i he orm k imes k D... D d locl rciol pril derivive o high order k imes k.... k d Deiiio.5 Seig C locl rciol iegrl o o order i he iervl is deied [6] I lim N j j j d 6 where j j j m... d j... N j j N is priio o he iervl. For y here eiss I deoed y I. I D C. Here i ollows h I I or i. I I i.. I we hve For y C we hve locl rciol mliple iegrls I k imes k I.. I For hve. k k I k k C he we
3 Solvig Wve d Disio qios o Cor Ses 8 where I k imes d k I.. I k imes k D... D.. Deiiio.6 Mig-eer cio i rcl spce is deied y k. 7 k k The ollowig rles hold k k cos 8 k k. k k si 9 k k.. OCA FRACTIONA DRIVATIVS AND INTGRAS Some sel ormls d resls o locl rciol derivive were smmrized [77]. d d d d k d k k k.. d k k k. d. d cos cos. si 4 k k k k k. d 5 4. ANAYSIS OF OCA FRACTIONA ADOMIAN DCOMPOSITION MTHOD Cosider he geerl locl rciol diereil eqio i locl rciol diereil operor orm R 6 I q. 5 h is locl rciol order diereil operor which y he deiiio reds S d R S d d d S d d S d S S lim h is locl rciol order diereil operor d S is locl rciol coios. Applyig he iverse operor o oh sides o 6 yields R 7 I he iverse diereil operor eiss ccordig o he locl rciol decomposiio mehod meioed ove we hve r 8 R. Filly we c id solio i he orm. 9 Hece we c oi h he ollowig codiio where rciol dimesio y. is eql o or 5. NUMRICA APPICATIONS mple 5.. Cosider he disio eqio ivolvig locl rciol derivive sjec o he iiil codiios. Mkig se o q. 8 he recrrece relio reds s
4 . The compoe o he solio c e deermie rom iiil codiios s Applyig he recrsive relio d we ge he ollowig resls d so o. Ths he pproime solio o i he orm 7 is give y. y The resl is he sme s he oe which is oied i [8]. mple 5. Cosider he ollowig disio eqio o cor se 4 sjec o he iiil codiios 5 Accordig o locl rciol Adomi s decomposiio mehod he recrrece relio reds s. 6 Applyig he recrsive relio 6 d he iiil codiios 5 we ge he ollowig resls d so o. Ths he il series solio is reds s The closed orm solio is. This resl is he sme s oied y Yg [8]. mple 5. Cosider he ollowig wve eqio o cor se wih he rcl vle codiios give y. Accordig o locl rciol Adomi s decomposiio mehod he recrrece relio reds s 84 Jmshd Ahmd e l
5 . 4 Applyig he recrsive relio 4 d he iiil codiios we ge he ollowig resls d so o. Filly we oied. 9 This is he sme s oied y Yg [8]. mple 5.4 Filly cosider he ollowig wve eqio o cor se c 4 wih he rcl vle codiios give y 4 where c is cos. Accordig o locl rciol Adomi s decomposiio mehod he recrrece relio reds s. 4 Applyig he recrsive relio 4 d he iiil codiios we ge he ollowig resls 4 c c c c c c 46 d so o. Filly we oied. sih cosh c c 47 This is he sme s oied y Yg [8]. 6. CONCUSION I his pper he o-diereile solio or he he d wve eqios ivolvig locl rciol derivive operors i mhemicl physics rcl vle codiios re ivesiged y sig he proposed locl rciol Adomi s decomposiio mehod. The oied resls demosre he reliiliy o he mehodology d is wider ppliciliy o locl rciol diereil eqio risig i mhemicl physics egieerig d hece c e eeded o oher prolems o diversiied olier re. 7. RFRNCS. Kolwkr K.M. & A.D. Ggl. ocl rciol Fokker-Plck eqio. Physicl Review eers 8: Crpieri A. B. Chii & P. Corei. Sickiemic dliy d he priciple o virl work i he mechics o rcl medi. Comper Mehods i Applied Mechics d gieerig 9-: 9. Solvig Wve d Disio qios o Cor Ses 85
6 86 Jmshd Ahmd e l. Be Add F. & J. Cresso. Ao odiereile cios. Jorl o Mhemicl Alysis d Applicios 6: Bkhi A. & V. Drdr-Gejji. O clcls o locl rciol derivives. Jorl o Mhemicl Alysis d Applicios 7: Jmrie J. Tle o some sic rciol clcls ormle derived rom modiied Riem- ioville derivive or o-diereile cios. Applied Mhemics eers : Che W. H. S X. Zhg & D. Korosk. Aomlos disio modelig y rcl d rciol derivives. Compers & Mhemics wih Applicios 595: Yg X.J. ocl Frciol Fciol Alysis d Is Applicios. Asi Acdemic Plisher Hog Kog Chi. 8. Yg X.J. Advced ocl Frciol Clcls d Is Applicios. World Sciece Plisher New York NY USA 9. Yg X.J. & D. Ble. Frcl he codcio prolem solved y locl rciol vriio ierio mehod. Therml Sciece 7: Yg X.J. H.M. Srivsv J.H. He & D. Ble. Cor ype cylidricl-coordie mehod or diereil eqios wih locl rciol derivives. Physics eers A 778 : H M.S. R. P. Agrwl & X. J. Yg. ocl rciol Forier series wih pplicio o wve eqio i rcl virig srig. Asrc d Applied Alysis 5: Aricle ID Yg Y.J. D. Ble & X.J. Yg. A locl rciol vriiol ierio mehod or plce eqio wihi locl rciol operors. Asrc d Applied Alysis : Aricle ID 65.. Golmkheh A.K. V. Fzlollhi & D. Ble. Newoi mechics o rcls sse o rellie. Romi Repors i Physics 65: Ho Y.J. H.M. Srivsv H. Jri & X.J. Yg. Helmholz d disio eqios ssocied wih locl rciol derivive operors ivolvig he Cori d Cor-ype cylidricl coordies. Advces i Mhemicl Physics : Aricle ID JmshdA. & S.T. Mohyd-Di. Adomi s Decomposiio Mehod or Solvig Frciol Gizrg-d qio Arisig i Mhemicl Physics Usig Jmrie s Frciol Derivive. Sciece Ieriol 6: Kh Y. & S.T. Mohyd-Di. Coplig o He s polyomils d plce rsormio or MHD viscos low over srechig shee. Ieriol Jorl o Nolier Scieces d Nmericl Simlio : Hrisov J. He-Blce iegrl o rciol Hl-Time he disio s-model. Therml Sciece 4: He J.H. S.K. lg & Z.B. i. Geomericl eplio o he rciol comple rsorm d derivive chi rle or rciol clcls. Physics eers A 764: Jri H. & S. Seii. Homoopy lysis mehod or solvig lier d olier rciol disio-wve eqio. Commicios i Nolier Sciece d Nmericl Simlio 45: Zhg S. & H.Q. Zhg. Frciol s-eqio mehod d is pplicios o olier rciol PDs. Physics eers A 757: Kh Y. N. Frz A. Yildirim & Q.B. W. Frciol vriiol ierio mehod or rciol iiil-odry vle prolems risig i he pplicio o olier sciece. Compers d Mhemics wih Applicios 65: Ble D. K. Diehelm. Scls & J.J. Trjillo. Frciol Clcls Models d Nmericl Mehods. Series o Compleiy Nolieriy d Chos World Scieiic Boso USA.. i C. & Y. Wg. Nmericl lgorihm sed o Adomi decomposiio or rciol diereil eqios. Compers & Mhemics wih Applicios 57: Jri H. & V. Drdr-Gejji. Solvig sysem o olier rciol diereil eqios sig Adomi decomposiio. Jorl o Compiol d Applied Mhemics 96: Momi S. & Z. Odi. Alyicl solio o ime-rciol Nvier-Sokes eqio y Adomi decomposiio mehod. Applied Mhemics d Compio 77: Ry S.S. & R.K. Ber. Alyicl solio o rciol disio eqio y Adomi decomposiio mehod. Applied Mhemics d Compio 74: Wg Q. Nmericl solios or rciol KdV-Brgers eqio y Adomi decomposiio mehod. Applied Mhemics d Compio 8: Jri H. & V. Drdr-Gejji. Solvig lier d olier rciol disio d wve eqios y Adomi decomposiio. Applied Mhemics d Compio 8: Sri M. D.D. Gji & M. Moslemi. Applicio o He s vriiol ierio mehod d Adomi s decomposiio mehod o he
7 Solvig Wve d Disio qios o Cor Ses 87 rciol KdV Brgers Krmoo eqio. Compers & Mhemics wih Applicios 58: l-shhed M. Adomi decomposiio mehod or solvig Brger s eqio wih rciol derivive. Jorl o Frciol Clcls 4: Yg X.J. D. Ble & W. P. Zhog. Approime solios or disio eqios o cor spce ime. Proceedigs o he Romi Acdemy Series A Mhemics Physics Techicl Scieces Iormio Sciece 4: 7.. Jmshd A. S.T. Mohyd-Di & X.J. Yg. Applicios o locl rciol decomposiio mehod o iegrl eqios. Jorl o Sciece d Ars 6: Yg A.M. Y.Z. Zhg & Y. og. The Yg- Forier rsorms o he-codcio i semiiiie rcl r. Therml Sciece 7: i C.F. S.S. Kog & S.J. Y. Recosrcive schemes or vriiol ierio mehod wihi Yg-plce rsorm wih pplicio o rcl he codcio prolem. Therml Sciece 7: Yg Y.J. D. Ble & W.P. Zhog. Approimio solio o disio eqio o Cor ime-spce. Proceedig o he Romi Acdemy Series A 4: Yg X.J. & Y.D. Zhg. A ew Adomi decomposiio procedre scheme or solvig locl rciol Volerr iegrl eqio. Advces i Iormio Techology d Mgeme 4: Yg X.J. ocl Frciol Iegrl Trsorms. Progress i Nolier Sciece 4: Yg A.M. Z.S. Che H.M. Srivsv & X.J. Yg. Applicio o he ocl Frciol Series psio Mehod d he Vriiol Ierio Mehod o he Helmholz qio Ivolvig ocl Frciol Derivive Operors. Asrc d Applied Alysis : Aricle ID 595.
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