A NEW SPECTRAL ALGORITHM FOR TIME-SPACE FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS WITH SUBDIFFUSION AND SUPERDIFFUSION

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1 HE UBISHIG HOUSE ROCEEDIGS OF HE ROAIA ACADEY Seres A OF HE ROAIA ACADEY Vole 7 ber /6 pp A EW SECRA AGORIH FOR IE-SACE FRACIOA ARIA DIFFEREIA EQUAIOS WIH SUBDIFFUSIO AD SUERDIFFUSIO A.H. BHRAWY Kng Abdlzz Unversy Fcly o Scence Depren o hecs Jeddh Sd Arb Ben-Se Unversy Fcly o Scence Depren o hecs Ben-Se Egyp E-l: lbhry@yhoo.co.k Absrc. hs pper repors ne specrl collocon lgorh or solvng e-spce rconl prl derenl eons h sbdson nd sperdson. In hs schee e eploy he shed egendre Gss-obo collocon schee nd he shed Chebyshev Gss-Rd collocon pproons or spl nd eporl dscrezons respecvely. We ocs on pleenng he ne lgorh or o physcl probles nely e rconl oded nolos sbdson nd rconl nonlner sperdson eons. he nercl resls obned by sng hs lgorh hve been copred h noher nercl schee n order o deonsre he hgh ccrcy nd ecency o he proposed ehod. Key ords: e nd spce rconl dson eon sbdson nd sperdson legendre Gss-obo nd Chebyshev Gss-Rd drres.. IRODUCIO In recen yers here hs been hgh level o neres o eployng specrl ehods or nerclly solvng ny ypes o negrl nd derenl eons de o her ese n pleenon over ne nd nne dons [ 6]. he speed o convergence s one o he gre dvnges o specrl ehods. Besdes specrl ehods hve eponenl res o convergence; hey lso hve hgh level o ccrcy. Specrl ehods hve been clssed o hree ypes nely collocon [7 8] [9 ] nd Glerkn [ ] ehods. Frconl derenl eons FDEs [3 8] odel ny phenoen n severl elds sch s ld echncs chesry bology vscoelscy engneerng nnce nd physcs [9 3]. os FDEs do no hve ec solons so pprove ehods nd nercl echnes hve been proposed nd developed o nd he solons o sch eons. Fne eleen ehods hve been presened n [4 6] o obn he nercl solons o rconl derenl eons. ercl reens bsed on ne derence ehods ere developed or solvng FDEs [7 9]. Recenly severl specrl lgorhs ere desgned nd developed or nercl solons o FDEs see or eple [3 34]. he collocon ehod hs de rnge o pplcons de o s ese o se nd dpbly n vros probles. We ocs bsclly on proposng ne specrl collocon ehod or o ell-knon FDEs ro hecl physcs besdes deonsrng he ccrcy o hs proposed collocon ehod. he n obecve o hs rcle s o propose ne collocon ehod or he nercl solons o o ypes o rconl prl derenl eons FDEs nely he oded nolos rconl sbdson nd he rconl nonlner sperdson eons. he proposed ehod s bsed pon he shed egendre Gss-obo collocon schee or spl dscrezon n conncon h Chebyshev Gss-Rd collocon schee or eporl dscrezon. hereore e presen lly collocon schee or solvng sch probles. he proble s hen redced o syse o lgebrc eons. Fnlly he ccrcy o he proposed ehod s deonsred by es proble hch corresponds o physclly enngl cse. o he bes o or knoledge here re no resls on he collocon ehod or solvng nonlner rconl sbdson or sperdson eons. hs pper s orgnzed s ollos. We presen e relevn properes o rconl dervves nd shed Jcob polynols n Sec.. he proposed schee s

2 4 A.H. BHRAWY nvesged nd pleened or he e rconl dson odel n Sec. 3. Secon 4 s devoed o solve e-spce rconl sperdson eon. A nercl slon s presened n Sec. 5. Fnlly soe concldng rerks re gven n he ls secon.. FRACIOA CACUUS AD JACOBI OYOIAS here re severl denons o rconl negron o order > nd no necessrly evlen o ech oher [35]. Renn-ovlle nd Cpo rconl denons re he o os sed ro ll he oher denons o rconl clcls. Denon.. he negrl o order rconl ccordng o Renn-ovlle s gven by Γ J d > > J he operor J sses he ollong properes + β Γ β + β + J J J J J J J J. Γ β + + Denon.. he Renn-ovlle rconl dervves o order s dened s d D d < > Γ d 3 here s he celng ncon o. Denon.3. he Cpo rconl dervves o order s dened s c d D d < >. Γ 4 d Here e rend he reder o soe sel properes o shed Jcob polynols h re os relevn o specrl pproons [36 37]. I ncldes egendre nd Chebyshev polynols s o specl cses. he θ ϑ h dervve o Jcob polynol cn be obned ro D Γ + θ + ϑ + + Γ + θ + ϑ + θ ϑ θ + ϑ +. 5 θ ϑ he se o k consss coplee θ ϑ -orhogonl syse h θ + ϑ + θ ϑ Γ k + θ + Γ k + ϑ + k θ ϑ hk. 6 k + θ + ϑ + Γ k + Γ k + θ + ϑ + θ ϑ θ ϑ We denoe by k k > he shed Jcob polynol o degree k dened on he nervl []. In vre o 5 e dedce h D θ ϑ k k Γ k + ϑ + k + θ + ϑ + Γ k + Γ + ϑ + 7 D θ ϑ k Γ k + θ + k + θ + ϑ + Γ k + Γ + θ + 8

3 3 A ne specrl lgorh or e-spce rconl prl derenl eons h sbdson nd sperdson 4 D θ ϑ Γ + k + θ + ϑ + θ + ϑ k k. Γ k + θ + ϑ e le θ ϑ θ ϑ hen e dene he eghed spce θ ϑ [ ] ollong nner prodc nd nor n he sl y h he θ ϑ v θ ϑ v d v θ ϑ v v θ ϑ. he se o shed Jcob polynols ors coplee [ ] -orhogonl syse. De o e obn θ ϑ We denoe by θ ϑ k θ ϑ θ ϑ θ + ϑ + h θ ϑ k h θ ϑ k he nodes o he sndrd Jcob-Gss nerpolon on he nervl θ ϑ [ ]. her correspondng Chrsoel nbers re ϖ. he nodes o he shed Jcob- θ ϑ θϑ Gss nerpolon on he nervl [] re he zeros o + hch e denoe by. θ ϑ θ ϑ Clerly + nd her correspondng Chrsoel nbers re θϑ θ + ϑ + ϖ θϑ ϖ. e S be he se o polynols o degree os. hnks o he propery o he sndrd Jcob-Gss drre ollos h or ny φ S +. θ+ ϑ+ θ ϑ θ ϑ φ d + φ + d θ+ ϑ+ θϑ θϑ θϑ θϑ ϖ φ + ϖ φ. he bove negrl s ec or φ S nd θ ϑ θ ϑ φ S n cses o selecng ϖ he zeros nd eghs o Jcob Gss-Rd nd Jcob Gss-obo drres respecvely. 3. IE FRACIOA ODIFIED AOAOUS SUBDIFFUSIO EQUAIO We propose egendre-chebyshev collocon ehod nd descrbe s pleenon or he nercl solon o e-rconl oded nolos sbdson eon. We pproe he solon o sch eon or spl nd eporl dscrezons by dpng he egendre Gss-obo [38] ehod n conncon h Chebyshev Gss-Rd collocon ehod [39]. Recenly soe odels hve been sed or descrbng processes h becoe less nolos s e progresses by he nclson o secondry rconl e dervve cng on dson operor h nonlner sorce er [4 4 4]: h he nl condon + + R [ ] [ ] 3 g [ ] 4

4 4 A.H. BHRAWY 4 nd he bondry condons ] [ g g 5 here < re rel consns he nonlner sorce er ] [ C R nd g g g re gven ncons. he rconl e dervve operors nd re dened n ers o he Renn-ovlle rconl dervve. o e olne he n seps o pplyng he shed egendre Gss-obo nerpolon pons s collocon nodes or he spl pproon enhle he shed Chebyshev Gss-Rd nerpolon pons s nvesged s collocon nodes or he eporl pproon. o hs end le s epnd he pproe solon by ens o egendre nd Chebyshev seres n he or 6 here. Frherore he pproon o he spl prl dervve cn be coped s 7 here. he pproon o he eporl prl dervve cn be coped s 8 nd 3 9 h. 3 Bsed on he bove reen o eporl nd spl prl dervves he rs coponen o he rgh hnd sde o 3 cn be obned eplcly by here. 4 + he bove relon s epressed eplcly by dpng he Renn-ovlle rconl dervve o he poer seres o he shed Chebyshev polynol. hereore dopng 6 enbles one o epress 3 n he or:

5 5 A ne specrl lgorh or e-spce rconl prl derenl eons h sbdson nd sperdson 43 + R [ ] []. 4 I rens o desgn n pproon or he nl nd bondry condons. We y pproe hese condons by ens o egendre nd Chebyshev polynols s g g g o E. s colloced o collocon nodes. oreover he nl-bondry condons n s lso colloced egendre nd Chebyshev collocon nodes. Frs e obn lgebrc eons or he nknon coecens ro F r s. R ζ r η r ; s here Fr s ζ r η s 4 ζ r η s hle ζ r nd η s re he shed egendre Gss- obo nd he shed Chebyshev Gss-Rd drres nodes respecvely. Second reng he nl condon he egendre Gss-obo nodes leds o lgebrc eons r r s 3 ζ g ζ r. 4 Fnlly he pproon o he bondry condons he Chebyshev Gss-Rd nodes leds o + lgebrc eons η s η s g η s g η s hs n rn yelds syse o + + lgebrc eons F r s R ζ r η η g s s g η g ζ η s. ζ r η s ζ r η s r ; s r s s r s s. he bove syse o nonlner lgebrc eons n he epnson coecens y be solved nerclly n sep-by-sep nner by sng eon s erve ehod. 5 6

6 44 A.H. BHRAWY 6 4. IE-SACE FRACIOA OIEAR SUERDIFFUSIO EQUAIO In hs secon e propose n ecen solon or he e-spce rconl dson eon reled o Renn-ovlle nd Cpo rconl dervves h nonlner er [43]: sbec o c R [ ] [ ] 7 g [ ] 8 g g [ ] 9 here nd re consns hle R g g nd g re gven ncons. he pproe solon hs seres o he or he Cpo rconl dervve o he pproe solon s gven by c 3 c 5 3 c nd y be epressed n n eplc or by dpng he Cpo rconl dervve or he poer seres o he shed Chebyshev polynol. oreover he Renn-ovlle rconl prl dervve or spce vrble s obned by here 6 nd y be obned n n eplc or by pplyng he Renn-ovlle rconl dervve o he poer seres o he shed egendre polynol. hereore he lly collocon schee o 7 9 er eployng 3 3 leds o syse o + + lgebrc eons + 5 ζ r η s ζ rη s ζ rη s 6 r s r s ζ r ζ r η s s η s s + ζ η + R ζ η r ; s g r g η s g η s. Fnlly he bove syse o nonlner lgebrc eons n he epnson coecens y be solved by sng eon s erve ehod. 33

7 7 A ne specrl lgorh or e-spce rconl prl derenl eons h sbdson nd sperdson ble bsole errors sng he presen ehod nd he CD schee [44] Or schee CD schee [44] UERICA RESUS In hs secon e shll hghlgh he ccrcy nd he ecency o or ehod. Consder he rconl sbdson eon h enn condons [44]: Γ e Γ + 34 he ec solon s gven by [] []. 35 e + [] []. In ble e sho nd copre he bsole errors sng he egendre Gss-obo ehod n conncon h Chebyshev Gss-Rd collocon ehod or schee nd hose h hve been presened by Ren e l. [44] by pleenng he copc derence CD schee [44]. Fro hs ble e conclde h he presen ehod s ore ccre hn he CD schee [44]. I s lso observed h he bsole error s very sll despe he relvely sll nber o grd pons sed. hs nercl eperen deonsres he ly o he ehod COCUSIOS In hs pper e hve proposed n ecen nd ccre lgorh bsed on he egendre Gss- obo ehod n cobnon h Chebyshev Gss-Rd collocon ehod o obn he nercl solons o e-spce rconl prl derenl eons FDEs h sbdson nd sperdson. he ehod s bsed pon redcng he enoned proble no syse o lgebrc eons n he epnson coecen o he solon. We hve olned he pplcon o egendre nd Chebyshev collocon ehods or solvng FDEs. In prncple hs lgorh y be eended o reled probles sch s o copled nonlner FDEs. One gh lso consder e-spce cople rconl Schrödnger eons nd o-sded spce FDEs. We shold noe h s nercl ehod e re resrced o solvng probles over ne don. Hence hs ehod s prclrly ell sed or bondry vle probles h ne spl dons. We hope o eend he proposed ehod sng generlzed gerre polynols or spl dscrezon or probles on hl-lne [3]. oreover hs ehod y be eended o he o-densonl cse or slr probles.

8 46 A.H. BHRAWY 8 ACKOWEDGES hs pper s nded by he Denshp o Scenc Reserch DSR Kng Abdlzz Unversy Jeddh nder grn no. 9/838/434. he hor hereore cknoledges h hnks DSR echncl nd nncl sppor. REFERECES. C. CAUO e l. Specrl ehods: Fndenls n Sngle Dons Sprnger-Verlg e York 6.. A.H. BHRAWY.A. ZAKY A ehod bsed on he Jcob pproon or solvng l-er e-spce rconl prl derenl eons J. Cop. hys. 8 pp A.H. BHRAWY An ecen Jcob psedospecrl pproon or nonlner cople generlzed Zkhrov syse Appl. h. Cop. 47 pp E.H. DOHA e l. An ccre egendre collocon schee or copled hyperbolc eons h vrble coecens. Ro. J. hys. 59 pp H. SCHAE K. ESÄSSER he pplcon o he specrl ehod o nonlner ve propgon J. Cop. hys. pp E.H. DOHA e l. ercl reen o copled nonlner hyperbolc Klen-Gordon eons Ro. J. hys. 59 pp J. A B-W. I J.R. HOWE herl rdon he rnser n one- nd o-densonl enclosres sng he specrl collocon ehod h ll specr k-dsrbon odel In. J. He ss rnser 7 pp X. A C. HUAG Specrl collocon ehod or lner rconl negro-derenl eons Appl. h. odell. 38 pp E.H. DOHA A.H. BHRAWY R.. HAFEZ On shed Jcob specrl ehod or hgh-order l-pon bondry vle probles Con. onlner Sc. er. Sl. 7 pp S.R. AU H. RICE Sprse specrl- ehod or he hree-densonl helclly redced ve eon on o-cener dons J. Cop. hys. 3 pp A. E-KHAEB.E. A-HOHAY H.S. HUSSIE Specrl Glerkn ehod or opl conrol probles governed by negrl nd negro- derenl eons hecl Scences eers pp Y.YAG Jcob specrl Glerkn ehods or rconl negro-derenl eons Clcolo DOI.7/s ZAYEROURI. AISWORH G.E. KARIADAKIS A ned erov-glerkn specrl ehod or rconl DEs Coper ehods n Appled echncs nd Engneerng do:.6/.c G.W. WAG.Z. XU he proved rconl sb-eon ehod nd s pplcons o nonlner rconl prl derenl eons Ro. Rep. hys. 66 pp A. AKIAR e l. ercl Solon o Frconl Benney Eon Appl h. Inor. Sc. 8 pp S.J. SADAI e l. Soe rconl coprson resls nd sbly heore or rconl e dely syses Ro. Rep. hys. 65 pp A..O. AWAR e l. Frconl Cpo he eon hn he doble plce rnsor Ro. J. hys. 58 pp A. BORHAIFAR KH. SADRI A ne operonl pproch or nercl solon o generlzed nconl negroderenl eons J. Cop. Appl. h. 79 pp H. HEYDARI e l. Wveles ehod or he e rconl dson-ve eon hys. e. A 379 pp H.A.A. E-SAKA he Frconl-order SIR nd SIRS Epdec odels h Vrble oplon Sze h. Sc. e. pp G.A. EES D.V. AGHE Frconl eclson sscs n non-hoogeneos nercng prcle syses Ro. Rep. hys. 66 pp A.A.. ARAFA S.Z. RIDA. KHAI Solons o Frconl Order odel o Chldhood Dseses h Consn Vccnon Sregy hecl Scences eers pp D. CAFAGA Frconl clcls: hecl ool ro he ps or he presen engneer IEEE rnscons on Indsrl Elecroncs pp J. A J. IU Z. ZHOU Convergence nlyss o ovng ne eleen ehods or spce rconl derenl eons J. Cop. Appl. h. 55 pp B. JI R. AZAROV Y. IU Z. ZHOU he Glerkn ne eleen ehod or l-er e-rconl dson eon J. Cop. hys. 8 pp I D. XU. UO Alernng drecon plc Glerkn ne eleen ehod or he o-densonl rconl dson-ve eon J. Cop. hys. 55 pp EERSCHAER C. ADJERA Fne derence pproons or o-sded spce-rconl prl derenl eons Appl. er. h. 56 pp Z. DIG A. XIAO. I Weghed ne derence ehods or clss o spce rconl prl derenl eons h vrble coecens J. Cop. Appl. h. 33 pp

9 9 A ne specrl lgorh or e-spce rconl prl derenl eons h sbdson nd sperdson H. WAG. DU Fs lernng-drecon ne derence ehods or hree-densonl spce-rconl dson eons J. Cop. hys. 58 pp A.H. BHRAWY A.A. A-ZAHRAI Y.A. AHAED D. BAEAU A ne generlzed gerre-gss collocon schee or nercl solon o generlzed rconl pnogrph eons Ro. J. hys. 59 pp OKHARY Reconsrcon o eponenlly re o convergence o egendre collocon solon o clss o rconl negro-derenl eons J. Cop. Appl. h. 79 pp E.H. DOHA D. BAEAU A.H. BHRAWY R.. HAFEZ A Jcob collocon ehod or roesch s proble n pls physcs roc. Ronn Acd. Seres A 5 pp A. KAYEDI-BARDEH. ESAHCHI. DEHGHA A ehod or obnng he operonl r o rconl Jcob ncons nd pplcons Jornl o Vbron nd Conrol pp S. IRADOUS-AKCHI e l. ercl solon or clss o rconl convecon dson eons sng he lle oble lveles Jornl o Vbron nd Conrol pp K. IER B. ROSS An Inrodcon o he Frconl Clcls nd Frconl Derenl Eons John Wley & Sons Inc. e York G. SZEGÖ Orhogonl olynols Collo blcons XXIII Aercn hecl Socey Y. UKE he Specl Fncons nd her Approons Vol. Acdec ress e York A.H. BHRAWY.A. ZAKY D. BAEAU e nercl pproons or spce-e rconl Brgers eons v egendre specrl-collocon ehod Ro. Rep. hys E.H. DOHA A.H. BHRAWY R.. HAFEZ.A. ABDEKAWY A Chebyshev-Gss-Rd schee or nonlner hyperbolc syse o rs order Appl. h. Inor. Sc. 8 pp F. IU C. YAG K. BURRAGE ercl ehod nd nlycl echne o he oded nolos sbdson eon h nonlner sorce er J. Cop. Appl. h. 3 pp Q. IU F. IU I. URER V. AH Fne eleen pproon or oded nolos sbdson eon Appl. h. odel. 35 pp A. OHEBBI. ABBASZADEH. DEHGHA A hgh-order nd ncondonlly sble schee or he oded nolos rconl sb-dson eon h nonlner sorce er J. Cop. hys. 4 pp C. I Z. ZHAO Y.Q. CHE ercl pproon o nonlner rconl derenl eons h sbdson nd sperdson Cop. h. Appl. 6 pp J. RE Z-Z. SU X. ZHAO Copc derence schee or he rconl sb-dson eon h enn bondry condons J. Cop. hys. 3 pp Receved Deceber 4

Supporting information How to concatenate the local attractors of subnetworks in the HPFP

Supporting information How to concatenate the local attractors of subnetworks in the HPFP n Effcen lgorh for Idenfyng Prry Phenoype rcors of Lrge-Scle Boolen Newor Sng-Mo Choo nd Kwng-Hyun Cho Depren of Mhecs Unversy of Ulsn Ulsn 446 Republc of Kore Depren of Bo nd Brn Engneerng Kore dvnced