THE PUBISHIG HOUSE PROCEEIGS OF THE ROAIA ACAEY See A OF THE ROAIA ACAEY Volme 6 mbe 4/5 pp. 49 498 A CHEBYSHEV-AGUERRE-GAUSS-RAAU COOCATIO SCHEE FOR SOVIG A TIE FRACTIOA SUB-IFFUSIO EQUATIO O A SEI-IFIITE OAI A.H. BHRAWY.A. ABEKAWY A.A. AZAHRAI. BAEAU 45 E.O. AZAHRAI Kn Abdlzz Unvey Fly o Sene epmen o hem Jeddh Sd Ab Ben-Se Unvey Fly o Sene epmen o hem Ben-Se Eyp Kn Abdlzz Unvey epmen o Cheml nd el Enneen Fly o Enneen Jeddh Sd Ab 4 Cny Unvey Fly o A nd Sene epmen o hem nd Compe Sene An Tey 5 Ine o Spe Sene ele-bhe Romn Coepondn ho:. Blen E-ml: dm@ny.ed. We popoe ne een pel olloon mehod o olvn me onl b-don eqon on em-nne domn. The hed Chebyhev-G-Rd nepolon mehod dped o me dezon lon h he ee-g-rd olloon heme h ed o pe dezon on em-nne domn. The mn dvne o he popoed ppoh h pel mehod mplemened o boh me nd pe dezon hh llo o peen ne een lohm o olvn me onl b-don eqon. Key od: me onl b-don eqon em-nne domn Chebyhev-G-Rd olloon heme ee-g-rd olloon heme Cpo devve.. ITROUCTIO Sevel omponl poblem n dvee eeh e e ondeed on em-nne domn. The ehqe enneen eld nd ndee o poblem n be modeled pl deenl eqon on em-nne domn. Spel mehod povde omponl ppoh h beome popl dn he l dede []. They hve ned ne poply n om ompon o de l o phyl poblem n ld nd he lo. Reenly pel mehod ee ed o nmelly olve poblem on em-nne domn [ 7]; n h e he hoe o he b non o ned ee epnon o he olon depend on ohoonl yem o nnely deenble lobl non dened on he hl lne. In een ye hee h been hh level o nee o employn pel mehod o nmelly olvn mny ype o nel nd deenl eqon de o he ee o pplyn hem o boh ne nd nne domn [ 8 9 ]. Spel mehod no only hve eponenl e o onveene b lo hve hh level o y. Thee e hee mn ype o pel mehod nmely olloon [ ] [4 5] nd Glen [6 7 8] mehod. Fonl deenl eqon FE model mny phenomen n evel eld h ld mehn hemy boloy voely enneen nne nd phy [9 9]. o FE do no hve e nly olon o ppomon ehnqe m be ed. Fne elemen mehod ee peened n [ ] o obn he nmel olon o FE. enhle he nmel emen bed on ne deene mehod o FE popoed n [4 6]. oeove evel pel lohm ee lo dened o FE ee o emple [7]. Tme onl b-don eqon on em-nne domn ded n h ppe n ne olloon mehod. The olloon mehod h de ne o pplon de o ee o e nd dpbly n vo poblem [8 4]. The m o h le o eend he pplon o hed Chebyhev-G-Rd nepolon mehod n ombnon h enelzed ee-g-rd olloon heme o he nmel emen o he me onl b-don eqon on em-nne domn. The hed Chebyhev- G-Rd nepolon mehod dped o me dezon nd he ee-g-rd
A Chebyhev-ee-G-Rd olloon heme o olvn me-onl b-don eqon 49 olloon heme ed o pe dezon on em-nne domn. The ndelned heme povde yem o leb eqon. Fnlly e demone he y o h ne mehod by peenn e emple.the olne o h ppe ollo. In Se. e peen e elevn popee o enelzed ee-g-rd nepolon Chebyhev-G-Rd nepolon nd onl nel. The menoned heme mplemened o he me-onl don model n Se.. A e emple ven n Se. 4. Fnlly ome onldn em e ven n he l eon.. ORTHOGOA POYOIAS A FRACTIOA ITEGRAS In h eon e ell ome elevn popee o he enelzed ee polynoml Chebyhev polynoml nd he onl nel o hee polynoml [4 7 4]. o le Λ nd e be eh non on Λ n he l ene. ene Λ { v v meble on Λ nd v < } eqpped h he ollon nne pod nd nom e le hve Λ v v d v v v. be he enelzed ee polynoml o deee. Aodn o [4] o > e + + [ + + + + ] hee nd +. The e o enelzed ee polynoml Λ ohoonl yem nmely d hδ Γ + + hee δ he Konehe non nd h.! The nlyl om o enelzed ee polynoml o deee on he nevl Λ ven by ee e.. [4] The pel vle q Γ + + Γ + +!!. q q! q! q! Γ + + hee ll be o mpon e le o en he nl ondon o he ven Γ +! FE. e hen my be epeed n em o enelzed ee polynoml Λ
49 A.H. Bhy.A. Abdely A.A. Alzhn. Blen E.O. Alzhn d. h In pl pplon he enelzed ee polynoml p o deee + e ondeed. Then e hve. 5 The ell-non Chebyhev polynoml e dened on he nevl [ ] by T o o. 6 The Chebyhev polynoml y he ollon elon T ± ± T T. 7 e hen e dene he ehed pe h epe o he eh non e dened ollo:. The nne pod nd he nom o v v d. 8 4 The e o Chebyhev polynoml om omplee -ohoonl yem nd ς π T h ς ς. 9 Fo Chebyhev-G-Rd qde oml [] π o + ϖ π + π + ; hee nd ϖ e he node nd he oepondn Choel nmbe o he Chebyhev-G-Rd qde oml on he nevl [ ] epevely. o e node he ollon dee nne pod nd nom v v ϖ. In ode o e Chebyhev polynoml T n on he nevl e dene he o-lled hed Chebyhev polynoml by nodn he hne o vble. e he hed Chebyhev polynoml T n be denoed by T n. The nly om o he hed Chebyhev polynoml T n o deee n ven by
4 A Chebyhev-ee-G-Rd olloon heme o olvn me-onl b-don eqon 49 T n n n n +! n n!! hee T n n nd T n. The ohoonly ondon T T m n d δ h mn n hee n nd hn π h. Any non qe neble n my be epeed n em o hed Chebyhev polynoml hee he oeen e ven by T 4 T d. 5 h Thee e evel denon o he onl neon o ode ν > nd no neely eqvlen o eh ohe. Remnn-ovlle nd Cpo onl denon e he o mo ed om ll he ohe denon o onl ll h hve been noded eenly. enon.. The onl nel o ode ν odn o Remnn-ovlle ven by ν ν J ζ ζd ζ ν > > Γ ν 6 J ν hee Γ ν e d he mm non. enon.. The Cpo onl devve o ode ν dened m ν mν ζ d ζd ζ m< ν m > m Γ m ν 7 d hee m he eln non o ν.. FUY COOCATIO ETHO The mn obeve o h eon o develop he olloon mehod o olve nmelly he me onl b-don eqon on em-nne domn n he ollon om be o he nl ondon ν + H [ [ ] [ [ ] hee onn hle H nd e ven non. Hee e e he e o enelzed ee nd hed Chebyhev G-Rd pon o he pe nd me 8 9
494 A.H. Bhy.A. Abdely A.A. Alzhn. Blen E.O. Alzhn 5 ppomon epevely. The m o h o o onde he dvne o he olloon pon dbon n peed domn. o e olne he mn ep o he olloon mehod o olvn he pevo me onl b-don eqon on em-nne domn. Ame e ppome he olon ne doble epnon o he om T hee T. Then he pl pl devve nd my be en T T hee T nd T. Fhemoe he ppomon o he me onl devve ν n be omped hee ν ν T ν T ν +! Γ + ν ν T T.!! Γ + ν o dopn enble one o e 8 9 n he om: + H [ [ ]. 4 Fom he nl ondon mmedely e e. 5
6 A Chebyhev-ee-G-Rd olloon heme o olvn me-onl b-don eqon 495 The non nd n be eplly obned by n he nomon nlded n Seon. o Eq. 4 yeld leb eqon n + + nnon epnon oeen ; H F 6 hee F nd he nl ondon ve 7 8 9 nd h n n yeld + + leb eqon nmely. ; H F The eln yem o leb eqon hen olved by ny ndd olve. 4. UERICA SIUATIO In h eon e peen nmel emple o ho he y nd pplbly o he popoed mehod. Conde he ollon onl b-don eqon ] [ [ 4 6 T e Γ + γ γ γ h he nl ondon ] [ [ T nd he e olon ven by ]. [ [ T e Fo he nmel mplemenon e onde he domn [] []. Fom Tble e ee he hhly e el bed on mmm bole eo n h mehod. enhle e l bole eo o evel pon n Tble. In F. e ee he behvo o bole eo o nmel olon.5 h 4 6 nd.5 γ.
496 A.H. Bhy.A. Abdely A.A. Alzhn. Blen E.O. Alzhn 7 Tble mm bole eo o poblem hee γ.5 66 6 6 46 5.8 5.8 4.95 4.96.49 5. 5..8 4 4.6.4.8.8.8.8 4.6 4 4 5 Tble Abole eo 4 6 γ.5 o he popoed poblem E E E. 9.76.5 9 8.6 9 5.4649 4 9 5.86684 4 9.56594 4 9 4.644 6 8.766 6 8.97 6 8 6.988 8 9.697 8.588 8 8.5964 9.947 8.4967 8.675 9.8 8 4.657 7.6885 4 8.748 4 7.6998 4 6.55 6 8 8.67 6 6.567 6 5.7 8 7.989 8 6. 8 5 59 7.74 6.9654 5.799 6. 6 5. 6 4. 6 E. 6. 6. 6 5 5 F. -deon ve o bole eo o poblem. 5. COCUSIOS In h ppe e hve developed nd mplemened ne nmel lohm o olve me onl b-don eqon on em-nne domn. The nmel el ven n h o demone he ood y o h lohm. oeove he lohm noded n h ppe n be ell ed o
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