A CHEBYSHEV-LAGUERRE-GAUSS-RADAU COLLOCATION SCHEME FOR SOLVING A TIME FRACTIONAL SUB-DIFFUSION EQUATION ON A SEMI-INFINITE DOMAIN

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1 THE PUBISHIG HOUSE PROCEEIGS OF THE ROAIA ACAEY See A OF THE ROAIA ACAEY Volme 6 mbe 4/5 pp 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

2 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

3 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 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

5 494 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 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 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.

7 496 A.H. Bhy.A. Abdely A.A. Alzhn. Blen E.O. Alzhn 7 Tble mm bole eo o poblem hee γ Tble Abole eo 4 6 γ.5 o he popoed poblem E E E E 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

8 8 A Chebyhev-ee-G-Rd olloon heme o olvn me-onl b-don eqon 497 hndln moe enel lne nd nonlne onl pl deenl eqon. A nmel emple ven o demone he pplbly nd he vldy o he popoed lohm. ACKOWEGEETS Th ppe nded by he enhp o Sen Reeh SR Kn Abdlzz Unvey Jeddh nde n no RG. The ho heeoe nolede h hn SR ehnl nd nnl ppo. REFERECES. C. CAUTO.Y. HUSSAII A. QUARTEROI T.A. ZAG Spel ehod: Fndmenl n Snle omn Spne-Vel e Yo 6.. J.P. BOY Chebyhev nd Foe pel mehod nd ed. e Yo ove.. E.H. OHA A.H. BHRAWY R.. HAFEZ R.V. GORER Job onl-g olloon mehod o ne-emden eqon o ophyl nne onlne Anl. odel. Con. 9 4 pp A.H. BHRAWY e l. e pel ehnqe o yem o onl deenl eqon n onl-ode enelzed ee ohoonl non F. Cl. Appl. Anl. 7 4 pp E.H. OHA A.H. BHRAWY. BAEAU R.. HAFEZ A ne Job onl-g olloon mehod o nmel olon o enelzed pnoph eqon Appl. me. h. 77 pp B.Y. GUO Geenbe ppomon nd pplon o deenl eqon on he hole lne J. h. Anl. Appl. 6 pp A.H. BHRAWY A.A. A-ZAHRAI Y.A. AHAE. BAEAU A ne enelzed ee-g olloon heme o nmel olon o enelzed onl pnoph eqon Rom. J. Phy. 59 pp E.H. OHA A.H. BHRAWY R.. HAFEZ.A. ABEKAWY A Chebyhev-G-Rd heme o nonlne hypebol yem o ode Appl. h. Inom. S. 8 pp E.H. OHA A.H. BHRAWY.A. ABEKAWY R.A. VA GORER Job-G-obo olloon mehod o he nmel olon o + nonlne Shödne eqon J. Comp. Phy. 6 pp J.R. GRAEF. KOG. WAG A Chebyhev pel mehod o olvn Remnn ovlle onl bondy vle poblem Appl. h. Comp. 4 pp R.K. SAEE J.S. HASSA Solvn nl nel eqon by n olloon mehod h. S. e. pp UI RBF-PS mehod nd Foe pedopel mehod o olvn nonlne pl deenl eqon h. S. e. pp A.H. BHRAWY. BAEAU A pel eende-g-obo olloon mehod o pe-onl dveon don eqon h vble oeen Rep. h. Phy. 7 pp A.H. BHRAWY e l. A pel lohm bed on Job opeonl m o nmel olon o me onl don-ve eqon J. Comp. Phy. do.o/.6/.p H. KHAI R.A. KHA The e o Job polynoml n he nmel olon o opled yem o onl deenl eqon Inen. J. Comp. h. do:.8/ E.H. OHA A.H. BHRAWY An een de olve o mldmenonl ellp Robn bondy vle poblem n eende pel-glen mehod Comp. h. Appl. 64 pp H. WAG X. ZHAG A hh-y peevn pel Glen mehod o he hle bondy-vle poblem o vble-oeen onevve onl don eqon J. Comp. Phy. 8 pp A. E-KHATEB.E. A-HOHAY H.S. HUSSIE Spel Glen mehod o opml onol poblem ovened by nel nd neo- deenl eqon h. S. e. pp R. GARRAPPA. POPOIZIO On he e o m non o onl pl deenl eqon h. Comp. Sml. 8 pp J.W. KIRCHER X. FEG C. EA Fl em hemy nd mplon o onnn npo n hmen e 4 pp A. AKIAR A. SECER A. SECER. BAYRA mel olon o onl Benney eqon Appl. h. Ino. S. 8 4 pp H.A.A. E-SAKA The onl-ode SIR nd SIRS epdem model h vble poplon ze h. S. e. pp GIOA H.E. ROA Fonl don eqon o npo phenomen n ndom med Phy A 85 pp J. F. GÓEZ AGUIAR. BAEAU Solon o he eleph eqon n onl ll ppoh Po. Romnn Ad. A 5 pp V.. CHICHAE.B. PACHPATTE e onl neqle nvolvn So onl nel opeo h. S. e. pp A.A.. ARAFA S.Z. RIA. KHAI Solon o onl ode model o hldhood dee h onn vnon ey h. S. e. pp. 7.

9 498 A.H. Bhy.A. Abdely A.A. Alzhn. Blen E.O. Alzhn 9 7. G. W. WAG T. Z. XU The mpoved onl b-eqon mehod nd pplon o nonlneonl pl deenl eqon Rom. Rep. Phy. 66 pp X.J. YAG. BAEAU J.H. HE Tnpo eqon n l poo med hn onl omple nom mehod Po. Romnn Ad. A 4 pp ROSTAY. AIPOUR H. JAFARI. BAEAU Solvn ml-em ode onl deenl eqon by opeonl me o BP h onveene nly Rom. Rep. Phy. 65 pp J. A J. IU Z. ZHOU Conveene nly o movn ne elemen mehod o pe onl deenl eqon J. Comp. Appl. h. 55 pp Y. JIAG J. A Hh-ode ne elemen mehod o me-onl pl deenl eqon J.Comp. Appl. h. 5 pp H. ZHAG F. IU V. AH Glen ne elemen ppomon o ymme pe-onl pl deenl eqon Appl. h. Comp. 7 6 pp I. XU. UO Alenn deon mpl Glen ne elemen mehod o he o-dmenonl onl don-ve eqon J. Comp. Phy. 55 pp EERSCHAERT C. TAJERA Fne deene ppomon o o-ded pe-onl pl deenl eqon Appl. me. h. 56 pp Z. IG A. XIAO. I Wehed ne deene mehod o l o pe onl pl deenl eqon h vble oeen J. Comp. Appl. h. 8 pp H. WAG. U F lenn-deon ne deene mehod o hee-dmenonl pe-onl don eqon J. Comp. Phy. 58 pp A.H. BHRAWY.A. ZAKY A mehod bed on he Job ppomon o olvn ml-em me-pe onl pl deenl eqon J. Comp. Phy. n Pe Z. XIAO-YOG. JUI Conveene nly o Job pedo-pel mehod o he vole dely neodeenl eqon Appl h. Ino. S. 9 pp SARBOA A. AIATAEI An een nmel heme o opled nonlne Be eqon Appl. h. Ino. S. 9 pp E.H. OHA A.H. BHRAWY. BAEAU.A. ABEKAWY mel emen o opled nonlne hypebol Klen-Godon eqon Rom. J. Phy. 59 pp B.-Y. GUO X.-Y. ZHAG A ne enelzed ee pel ppomon nd pplon J. Comp. Appl. h. 8 pp K. IITROV F. ARCEA F.R. RAFAEI onoony o zeo o ee-sobolev-ype ohoonl polynoml J. h. Anl. Appl. 68 pp G. SZEGÖ Ohoonl Polynoml Ame. h. So. Colloq. Pbl. Ame. h. So. Povdene RI 4 h edon 975. Reeved eembe 4