EL-GENDI NODAL GALERKIN METHOD FOR SOLVING LINEAR AND NONLINEAR PARTIAL FRACTIONAL SPACE EQUATIONS

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1 Inernon Jorn o es Reserch n Scence nd Technoogy Vome Isse 6: Pge o-7 ovemer-ecemer 3 hp://wwwmnornscom/rshm ISS (Onne): E-GEI OA GAERKI ETO FOR SOVIG IEAR A OIEAR PARTIA FRACTIOA SPACE EQATIOS * E-Kdy Sh E-Syed 3 * e E Sem eprmen o hemcs Fcy o Scence ewn nversy Cro Egyp eprmen o Scenc Compng Fcy o Compers nd Inormcs Benh nversy Benh 358 Egyp 3 eprmen o hemcs Fcy o Scence Benh nversy Benh Egyp Asrc- In hs pper n ecen nmerc echnqe s presened o sove he pr rcon spce eqons wh vre coecens on ne domn Ths echnqe sed on nod Gern mehod The rcon dervves re descred n he Cpo sense Aso y dscree scheme s gven or ype o nonner spce-rcon nomos dvecon-dson eqon In hs pper he proems cn e redced o se o ner gerc eqons y sng he Cheyshev nod Gern mehod The esence nd nqeness o he soon or he ner sem dscree we orm soons re proved And he sy nyss or he ner sem nd y dscree schemes re dscssed merc soons oned y hs mehod re n eceen greemen nd ecen o se wh hose oned y prevos wor n he erre Keywords - Shed Cheyshev poynom; od Gern mehod; Frcon dson eqon; Cpo dervve ITROCTIO In recen yers o o enon hs een devoed o he sdy o rcon deren eqons Frcon dervves rse n mny physc nd engneerng proems sch s eecrc rnsmsson rsonc wve propgon n hmn cnceos one modeng o speech sgns modeng he crdc sse eecrode nerce vscoescy wve propgon n vscoesc horns nd d mechncs [3] nd [3] In hs pper we presen drec compon echnqe or he one-dmenson spce rcon dson eqon o he orm: ( ) ( T () wh n nd homogenos ondry condons s oows: ( ) ( ) ( ( T () he nomos em s he h order rcon dervve o wh respec o he spce vre n he Cpo sense whch w e nrodced er on We wys consder: ( ) re consns The rcon order dson eqons re generzons o cssc dson eqons These eqons py mporn roes n modeng nomos dson nd s-dson sysems descrpon o rcon rndom w ncon o dson nd wve propgon phenomen see eg [] nd he reerences heren ny nmerc nvesgons were crred o y mny hors o sove hs proem In [] he cwrd Eer ne derence scheme s pped n order o on nmerc soons or he eqon Esence nd sy o he pprome soons re crred o y sng he rgh shed Grünwd orm or he rcon dervve erm n he sp drecon In [] ppromon echnqes sed on he shed egendre- de re presened o sove css o n-ondry ve proems or he rcon dson eqons The echnqe s derved y epndng he reqred pprome soon s he eemens o shed egendre poynoms In [] egendre psedo-specr mehod wh he ne derence mehod s sed o on he nmerc soon o he rcon dson eqon Aso we mny sdy one nd o ypc nonner spce-rcon pr deren eqons whch s ced rcon nomos dson nd hs he oowng orm: w ( ( w) w) ( ) w ( w) () T (3) wh n nd ondry condons s oows: w( ) ( ) w( w( T (4) *Correspondng hors: 55 S #7 om Cro Egyp E-m: mm_e_dy@yhoocom *E-m: drhe783@gmcom ISS:78-599

2 Inernon Jorn o es Reserch n Scence nd Technoogy s he h order rcon dervve wh respec m d ( s) o he spce vre n he Cpo sense ow he J ( ) m m ( n ) d ( s) rcon nomos dson ecomes ho opc ecse o s wdey ppcons n he evoon o vros m m m dynmc sysems nder he nence o sochsc orces For empe s we-sed oo or he descrpon o nomos rnspor processes n oh sence nd presence o eern veoces or orce eds oreover he rcon ( m) ( s) nomos dson hve nmeros ppcons n ssc ( ) ds m physcs ophyscs chemsry hydrogeoogy nd oogy ( m ) ( s) see or more des [8][6] nd [7] There re some hors sdyng he spec nomos dson eqon n heorec nyss nd nmerc smons see [6] [4] m ( ) () nd [3] ( ) J ( ) In hs pper we sed E-gend nod Gern mehod whch s eser echnqe hn he s Gern mehod In Gern mehod ech ss poynom chosen ms ssy he ondry condons ndvdy whch cses he Gern ormon o ecome compced prcry when he ondry condons re me-dependen [] Frhermore he presence o nonner erm compces he compon o he sness mr [9] owever he Gern mehod s sed on vron ormon whch preserves essen properes o he connos proem sch s coercvey conny nd symmery o he ner orm nd sy eds o opm error esmes [] On he oher hnd he mn dvnge o he nod Gern mehod s s smpcy nd ey n mpemenon In ddon hs mehod des wh nonner erms more esy hn Gern mehods oreover he proems wh vre coecens nd gener ondry condons re reed s he sme wy s proems wh consn coecens nd smpe ondry condons In c In E-gend Cheyshev nod Gern mehod we sr rom we orm o he eqons we repce hrd o eve negrs y Egend qdrre The orm o E-gend qdrre s ssyng symmerc propery ence we cn redce he nmer o operons o 5% whch mpes o decrese he rondng error Aso E-gend qdrre s n ernng seres whch converges s ( s he nmer o grd pons) The remnder o hs pper s orgnzed s oows: In secon we presen he procedre o soon or he pr rcon spce eqon n ner nd nonner cse In secon 3 we presen he error nyss In secon 4 we gve nmerc epermens o cry he mehod Frcon ervve Spce In hs secon we w gve he rcon dervve spce Frsy we w gve he oowng denons: enon The rcon dervve n he Remnn- ove verson o ncon () s dened s oows [9] ISS: ds An ernve denon nown s he Cpo rcon dervve s: (5) The wo denons re no n gener eqven hey re reed y he oowng reon: ( ) Genery when we consder he rcon deren eqons he Cpo denon s oen preerred snce s esy or mposng n nd ondry condons on cssc dervves B or he Remnn-ove denon hese condons ms e mposed on rcon dervves nd hs s oen no ve So h we w se he Cpo denon n hs pper enon [9] For he rcon dervve spce I ( s dened s oows: I ( { ( : endowed wh he sem-norm: I ( ( nd he norm I ( / I ( ( m m} ( nd e I ( denoes he cosre o C ( ) wh respec o he ove norm nd semnorm enon 3 [5] The rcon spce E ( dened eow E ( { ( : ( ( m m} endowed wh he semnorm / ( nd he norm E ( / E ( ( nd e E ( denoes he cosre o C ( ) wh respec o he ove norm nd semnorm enon 4 [7] For dene he semnorm F( ) ( ( nd he norm ( / ( (

3 Inernon Jorn o es Reserch n Scence nd Technoogy F( ) s he Forer rnsorm o he ncon nd T nd s he me sep whch cn dene noher rcon dervve spce Then eqon (7) s ppromed s oows: Fnd ( e ( e he cosre o C ( wh ( ) / ( ) sch h respec o he ove norm nd semnorm v ( Theorem [7] The spces I ( E ) ( ( ) ( nd ( ) / v v v ( ) (8) ( re eq n he sense h her sem norms s we ( ) e s norms re eqven v ( ) ( ( ) emm [6 (Frcon Poncré Fredrchs)] nd For ( we hve F( v ( g hen he sem-dscree proem (8) cn e wren n smpe C orm e h: ( ( ( ) / F( v ( ) (9) nd or m / m Z C ( ( emm [7] For I ( ) hen ( ) ( ) 3 ERICA TREATEETS FOR TE PARTIAFRACTIOA SPACE In hs secon we presen he nmerc soon or me spce rcon ner nd nonner eqons respecvey he spce rcon dervve s he Cpo dervve 3 E-GEI OA GAERKI ETO FOR IEAR CASE Ths mehod srs wh he we orm nd he r spce concdes wh he es ncon spce The we orm o proem () nd () n cse s gven s oows: ( ) / Fnd ( ) sch h v ( ) ( ( ) ( v ( ) / v ( ) (7) he nner prodc vs dened s v ( ) v( ) d e we w prove he esence nd nqeness o he we orm (7) So we gve he properes o he rcon dson operor whch s gven n [4] s oows: - ( ) ( ) ) ( ) / ( / on ) ( ) - ) v ( ) ) ( v v ( ) / ( ) / coercvy ( conny on ) / ( / ( ) ) ( ) re consns Appyng he mpc Eer ppromon o pprome he me dervve we dene Theorem (Esence nd nqeness) For ( ) nd or sceny sm sep sze here ess nqe soon ssyng (9) Proo Frsy we w prove he coercvy o he ner orm y sng he properes o he rcon dson operor nd Frcon Poncré Fredrchs neqy ) ( ) ( ( ( ) )) C ( ) ( ) / ( ) ( ) / ( ) ( / hen ner orm ) s coercve over ) ( ) e we w prove he conny o he ner orm ( ) over ) / ( / ( ) ) ( ) s oows: ( ( v ( ) ( ) C ( ( ) ) ( ) / ( / ( ) ) ( ) ( ) / ( ) v v ( ) / ( ) oreover we cn so prove he conny o F() ( ) / ( ) s oows: F ( ( g g v ( ) ( ) C g v ( ) / ( ) ( ) / ( ) over Thereore he hypoheses o -grm heorem re ssed [4] nd hen here es nqe soon or he sem-dscree we orm (9) Theorem 3 (Sy o he sem-dscree proem) For ( ) nd or sceny sm sep sze he proem (9) s se nd hods ISS:78-599

4 Inernon Jorn o es Reserch n Scence nd Technoogy o ensre he ppromons ssy he ondry condons ( ) / ( ) ( ) ( ) we se Aso snce he es ncon v() s Proo For nd v hen proem (8) w e ncon o h order poynoms so we cn wre hese poynoms n he eqven crdn orm ( ) ( ( ) ) v( ) v V ( ) () The rgh hnd sde o () w e he nod ves V re rrry ecep h ( ) ( ( ) ) V V o ensre h v sses he ondry C ( ) / () condons ow he dscree we orm s gven s oows: ( ) nd P The e hnd sde o () v ( ) ( ) v ( ) ( ) (From emm ) ( ) ( ) ( ) ( ) ( ) C () ( ) / ( ) ( ) ( ) From () nd () we hve C ( ) / (3) ( ) C ( ) ( ) For so we hve ( ) / C ( ) 3 (4) ( ) ( ) From (3) (4) we on ( ) / C ( ) 4 ( ) ( ) ow E-gend nod Gern mehod dscrezon proceeds y ppromng he soon he poynoms o hgh degree So we nrodce ne dmenson spce ( ) / P P ( ) P s he spce o poynoms n whch he poynom degree s ess hn or eq o nd he spce s gven s oows P spn ( ) ( ) } { ( ) re gven y: ( ) T ((/ ) or ecep / nd ) T ((/ ) ) (5) ( ) The grd pons re he erem pons o he shed Cheyshev poynom T (( / ) ) e he pprome soon s gven s oows ( ( ( ( ) (6) v P (7) he nner prodc g h s eved s oows g( ) h( ) g h nd ( y ) y cos The qnes re gven y: [9] / 4 s cos 4 (8) Snce hen he mpped weghs w e gven rom he oowng reon Then he rs dscree nner prodc ecomes v nn ( ) Vm m( ) n m snce ( ) hen he sm redces o v V (9) d d For evng he second erm n (7) e ( ) ( ) hen he rcon dervve o he crdn ncon cn e wren ( ) T (( / ) ) T ((/ ) ) s: ISS:

5 ( y ) y cos nd he Cpo rcon dervve o he Shed Cheyshev poynom s: T ((/ ) ) T (( / ) ) d ( ) ( ) or he dervves o Cheyshev poynom T ssy T T T T T T T 4 ( ) ( ) so we cn dedce h he recrrence reons T T ( ) T T 3 T even () T ( T T 3 5T ) odd () Then rom eq (-) we cn dedce he orgn mod derenon mr n he specr spce s sprse pper rngr mr wh neres d ( ) even oherwse Then he Cpo rcon dervve o he Shed Cheyshev poynom s [4]nd gven s oows: ( ) ( / ) ( ) (3) ( ) ( )( ) ( ) 3 4( / ) 4(/ ) ( ) (4) (3 )( )( ) ( )( ) ( ) nd or n 3 4 we hve he oowng recrrence reon ( ( )) ((/ ) )( ( )) ( ) ( ( )) (5) ( ) hence y ssng (3) (4) nd (5) n () hen we hve T ((/ ) ) dn n( ) (6) ( ) n Conseqeny he rcon dervve o he crdn ncon s gven n he oowng orm: ( ) T ((/ ) ) dn n( ) ( ) n The second erm n () cn e eved s oows: ( )( ) V ( [ ( )( ( ) ( ) Inernon Jorn o es Reserch n Scence nd Technoogy he rs dervve o he crdn ncons () ( ) ( ))]) (7) he pons s derved n [] Smry he sorce erm s gven s oows: v V (8) From eqons (9) (7) nd (8) we hve V ( [ ( )[ ( ) ( ) ( ) ( )]] Snce V s re nery ndependen he coecen o ech V ms e zero so nd ( B [ C ( )( ( ) ( ) ( ) ( ))] ) (9) B ( ) ( ) ( ) C ( ) ( ) e A B C ) hen (9) cn e wren s ( A wh he ondry condons Then he y dscree proem s gven n he oowng orm A wh he ondry condons (3) 3 STABIITY FOR F ISCRETE PROBE In hs secon we se rec ery-schder ed pon heorem [8] o prove he esence o he soon or eq (3) n emm 3 For gven open nd onded domn R n connng he orgn nd e : R e connos ncon I ( ) or nd hen hs ed pon n whch s he cosre o We nrodce dscree norm whch ndced rom he dscree nner prodc ISS:

6 Inernon Jorn o es Reserch n Scence nd Technoogy / y he dscree Gronw neqy we on g g g ( ) ( ) g g CT ep A Theorem 4 The eqon (3) hs soon n Proo e B ( ) R e cenered he orgn wh rds nd consder e he ondry o e ( ) e vecor on he ondry sch h or some E ( ) (3) E ( ) A y ng dscree nner prodc o (3) wh hen we hve ( E ( ) ) ( ) ( ) ( A ) ( ) ( ) rom [] nd snce ( s connos on [ ] [ T ] hen y Gronw emm we hve: nd y ppyng Cshy-Schwrz neqy we hve A (3) A s onded y posve nmer vde oh sdes o (3) y A hen we hve hen or rge nd or very sm hen whch mpes o conrdcon So E ( ) ence here s soon sch h E ) ( Theorem 5 For ny ed he dscree scheme (3) s se Proo From eqon (3) nd y ng dscree nner prodc wh nd snce Then rom Cshy-Schwrz neqy we hve A (33) hen y smmng (33) rom o A we on ISS: OIEAR CASE In hs secon we w sre how we cn se he nod Cheyshev Gern mehod o sove he nonner dson eqon So we w gve he we orm o proem (3) nd (4) n cse s gven s oows: Fnd ( ) / ( ) sch h: w v ( ( w) w) v w ( ( / ( w) v v ) ( ) The esence nd nqeness o he we orm s proved n [3] e he pprome soon s gven s oows: w( w ( W ( ( ) so he ove we orm cn e wren s w v ( ( w ) w ) v w ( ( w ) v v P Aer some mnpons we hve W B W C W B W ( w C ( ) ( ) ( ) ( ) ( w ) ) ( ) ( ) nd hence (35) s gven s W B W A W ) (34) (35) ( w (36) A C To pprome he me dervve we w se he cwrd Eer ne derence or he ner prs whe he orwrd Eer ne derence or he nonner pr e s he ppromon o ) Then (36) s ppromed y ( W A W W [ ( W ) ( ( W ) wh he ondry condons )]

7 Inernon Jorn o es Reserch n Scence nd Technoogy W W (37) 4 ERICA EXPERIAETS In hs secon we w gve nmerc empes nd we w se ATAB 8 sowre o on he nmerc ress Empe : Consder he oowng spce rcon order deren eqon: ( () 8 ) 8 ( 3 ( (6 3 ) e wh he n nd ondry condons: 3 ( ) ( ) ( ( T 3 The ec soon s ( ( ) e The nmerc ress re shown n e nd gres In e we gve he soe errors eween he ec soon ( nd he pprome soon ( ) nd we me comprson wh ress oned y mehod n [] he neror pons n me T wh me sep 5 TABE : The soe error eween he ec nd pprome soons n he neror pons T X od ehod [] ehod 533 e-6 4 e-5 86 e e e e e-6 37 e e-6 36 e e-6 94 e e-6 95 e e-6 49 e e-7 83 e-5 ppro 5 I s noed rom Te nd Fg h we cn cheve good ppromon or he ec soon y sng E-gend Gern mehod nd so or ress re n good greemen wh he mehod nrodced n [] Empe : Consder he oowng nonner spce rcon order deren eqon [3]: w ( w d ( e 5 w) d( w w ( w 5 5 ( ) (5) (5) 5 5 ( e (5) (5) wh he n nd ondry condons: w( ) ( ) w ( w( In hs cse he ec soon s ( ) e ( ) The nmerc ress re shown n e nd gre In e we gve he mmm error eween he ec soon w ( nd he pprome soon w ( n he neror pons wh deren me seps oe h he mmm error s dened s oows: w( w ( w w m TABE : The mmm error or T T 3 T 5 T 56 e-3 8 e-3 39 e-4 6 e-6 6 e e-4 79 e-5 69 e e e-4 58 e e e e-4 5 e e-7 Ec () ( Fg: () po o he pprome soon ( po o he ec soon or 3 nd TABE 3: The comprson eween E-gend nod Gern nd psedo-specr mehods or deren nd me T 5 3 od ehod Psedo ehod od ehod Psedo ehod 394 e e e e-4 75 e-5 37 e-4 7 e-5 4 e e-5 e-4 55 e-5 6 e e-5 99 e e-5 3 e-4 ISS:

8 W( W( () Fg () Psedo ehod T 5 Inernon Jorn o es Reserch n Scence nd Technoogy REFERECES Ec Psedo Ec od ( nd 3 ( E-gend od mehod T 5 nd 3 I s cer rom e when he me sep e smer; we on good ccrcy hogh or ong me On he oher hnd n e 3 we me comprson eween he nod mehod nd he psedo-specr mehod or consn n me nd or deren nmer o grd pons deren me seps We noe h he me sep 4 nd or he mmm error o he nod mehod s ( 5 e-5) oreover when he nmer o grd pons ncresed ( 3 ) he mmm error decrese o rech (337e-5) owever psedo-specr mehod he sme me sep nd when he nmer o grd pons ncresed he mmm error ncresed rom (99 e-5) o (3 e-4) Aso we cn oserve h rom Fgre whch ensres or nmerc ress So or mehod s convergen nd se n he nmerc sense 5 COCSIO In hs rce we propose new echnqe or sovng ner nd nonner rcon dvecon-dson eqon nmercy The mehod sed on he Cheyshev poynom nd he rcon dervves re descred n he Cpo sense The soon oned sng he proposed mehod shows h hs pproch cn sove he proem eecvey Comprsons re mde eween he pprome nd ec soons sre he vdy nd he gre poen o he proposed echnqe ACKOWEGET We e o epress sncere pprecon nd deep grde o prcpns n hs wor Boyd JP Cheyshev nd Forer specr mehods over neo Cho W Chng S K ee Y J merc soons or spcercon dsperson eqons wh nonner sorce erms B Koren h Soc r Bshor Appcons o rcon ccs App h Sc ehem K The Anyss o Frcon eren Eqons: An Appcon-Orened Eposon sng eren Operors o Cpo Type Sprnger 4 5 Erry E E-Syed gher order psedospecr derenon mrces Apped merc hemcs Ervn V J eer Roop J P merc ppromon o me dependen nonner spcercon dson eqon SIA Jorn on merc Anyss 45 () Ervn VJ Roop JP Vron ormon or he sonry rcon dvecon dsperson eqon mer eh Pr E (3) enry B I ngnds T A Werne S Anomos dson wh ner recon dynmcs: rom connos me rndom ws o rcon recon-dson eqons Physc Revew E rce (3) 6 9 eshven JS Goe S Goe Specr ehods or Tme-ependen Proems The Cmrdge monogrphs on pped nd compon mhemcs 7 Khder Swem hdy AS An ecen nmerc mehod or sovng he rcon dson eqon Jorn o Apped hemcs & Bonormcs - Kher A Temsh RS ssn A Cheyshev specr coocon mehod or sovng Brgers -ype eqons J o compon nd pped mhemcs Khder On he nmerc soons or he rcon dson eqon Commn onner Sc mer Sm Ks AA Srvsv Tro JJ Theory nd Appcons o Frcon eren Eqons Esever Sn ego 6 4 C P Zho Z G Chen Y Q merc ppromon o nonner rcon deren eqons wh sdson nd sperdson Compers & hemcs wh Appcons XJ X CJ A spce-me specr mehod or he me rcon deren eqon SIA J mer An 47(3) gn R Adh O Ben Zho X J Anomos dson epressed hrogh rcon order deren operors n he Boch-Torrey eqon Jorn o gnec Resonnce 9 () eerscher Benson A Bemer B Operor evy moon nd mscng nomos dson Physc Revew E rce 63 (I) 8 Oreg J Rhenod W C Ierve Soon o onner Eqons n Sever Vres Acdemc Press ew Yor- ondon 97 9 Podny I Frcon eren Eqons vo 98 Acdemc Press Sn ego C SA 999 Ry SS Anyc soon or he spce rcon dson eqon y wo-sep Adomn decomposon mehod Commn onner Sc mer Sm Sdmnd A ehghn A pproch or soon o he spce rcon dson eqon Compers nd hemcs wh Appcons Shen J Tng T Specr nd gh-order ehods wh Appcons Scence Press o Chn 6 3 Zheng Y Zho Z A y dscree Gern mehod or nonner Spce-rcon dson Eqon hemc Proems n Engneerng Arce 76 pges ISS: