A Comparison of Chebyshev polynomials and Legendre polynomials in order to solving Fredholm integral equations

Transcription

1 tertol Jourl o Scetc Reserch ulctos Volue 6 ssue 3 Mrch 6 35 SSN A Coprso o Cheshev polols Leere polols orer to solv Frehol terl equtos Mlr Astrct- ths reserch we use the uercl soluto etho tht s se o Cheshev polols Leere polols to solve o-sulr terl equto t s kow s Frehol terl equto o the seco k We use these epsos ecuse o ther coverece recurrece propertes Also oth o the c e represete s trooetrc ucto o -] Frst we ep the ukow ucto the terl equto se o the relte oruls the evelop kerel o terl equto o these we shoul tr to ucto whch c e represete s the soluto o ler eretl equto he susttuto to the terl equto we the coecets o the ucto At the e o reserch the etho wll e llustrte the e o eple Mthetcs Suect Clsscto- 5B5 A C5 C e ers- Cheshev pproto Frehol terl equto Leere seres pproto the uercl solutos o terl equtos NRODUCON he e terl equto or equto volv the ukow ucto uer the terl s ws trouce u Bos Reo 888 However the erl hstor o terl equtos oes ck coserle te eore tht to Lplce 78 Lter Ael ws le to terl equto coecto wth echcl prole ote two soluto o t; ter ths Louvlle vestte terl equto whch rose the course o hs reserches o eretl equto scovere portt etho or solv terl equtos soe proles thetcl represetto pper rectl the or o the terl equtos Soe proles hve rect represetto ter o eretl equtos wth ulr cotos lso e reuce to terl equtos Further orto e ou ] ] ] terl equtos re oe o the useul thetcl tools pple lss; here we trouce soe pplctos o the terl equtos such tht the proles o echcl vrto the prole o orecst hu populto eter the eer spectru o eutros utotc cotrol o rott sht torso o wre etc But ost o these equtos re ver cult to solve t s worth ot tht terl Equtos ote o ot hve ltcl soluto ust e solve uercll here re solutos such Lplce trsor Fourer trsor Mell trsor or soe terl equtos ut o terl equtos cot e solve these ethos shoul e solve uercl ethos see ] he ost coo etho o soluto o terl equto s the use o te ereces 3] Fo Goow use the Greor qurture orul or the evluto the terl equtos ths reserch we tr to the uercl soluto o o-sulr ler terl equtos the rect epso o the ukow ucto to seres o Cheshev polols o the rst k to seres o Leere polols s scusse Ellott ] he we use ve terl equto to ot coecet 5] we see the propertes o the Cheshev polols toether prouce pprot polol whch zes error ts pplcto hs s eret ro the lest squres pproto where the su o the squres o the errors s ze; the u error tsel c e qute lre the Cheshev pproto the vere error c e lre ut the u error s ze Cheshev pprotos o ucto re soetes s to e - pprotos o the ucto Cheshev polols or specl clss o polols especll sute or pprot other uctos he re use res o uercl lss t s ssue tht epsos o ve uctos c e ou or uctos whose epsos cot e ou ve ers soe curve tt techque c e use he Leere polols 7] re oe o the portt sequeces o orthool polols whch hs ee etesvel vestte pple terpolto pproto theor uercl terto the soluto o the seco- ourth-orer ellptc equtos coputtol lu cs etc t s ot ol powerul tool or the pproto o uctos tht re cult to copute ut lso essetl reet o uercl terto pprote soluto o eretl terl equtos he Leere spectrl ethos hs ecellet error propertes the pproto o sooth ucto he orthool polol epso occurs we re o prctcl proles pplctos pls portt role els o thetcs phscs wwwsrpor

2 tertol Jourl o Scetc Reserch ulctos Volue 6 ssue 3 Mrch 6 36 SSN wwwsrpor MEHODS Ler terl equtos c e ve to two tpes epe upo the lts o the terl A portt terl equto o eerl tpe s K F where F s ve cotuous ucto λ s preter re te costts K s clle the kerel s the ukow ucto hs terl equto s kow s Frehol equto o the seco k t ws oserve Volterr tht equto o ths tpe coul e rere s lt ro o sste o ler equtos F equls zero the we hve hooeeous Frehol equto o the seco k Whe the upper lt o the terl s the vrle the equto s kow s Volterr equto o the seco k A Cheshev polols etho Cheshev seres se o the Cheshev polols o the rst k re the ost useul oes hve ster uor coverece For coveece we wrte the Cheshev seres s Where cos cos cos All Cheshev polols sts three ter recurrece relto 3 orer to solve Frehol terl equto we ee the terl o prouct o two uctos Frst we ust the Cheshev epso o Suppose Let h = h we et h We kow tht s coecet o Hece Cotue wth h let the rst seres let the seco seres we et ] h Accor to equto we c wrte ] 5 We wt the epso o where Aother trslterto o the e s cheche

3 tertol Jourl o Scetc Reserch ulctos Volue 6 ssue 3 Mrch 6 37 SSN wwwsrpor t t 6 we set put equto 6 we et ] cos cos cos t t Let us ow copre ths result wth epso o ; we see tht 7 8 For coput set thereore 9 t shoul e ote tht to use Cheshev polols Leere polols we ust che the re o the vrle ro to - So De s orul us equtos 5 9 we B Leere polols etho he Leere epso o ukow ucto the re ee c ths epso kow s the Fourer-Leere seres where c 3 Leere polols sts cert recurrece reltosoe o the ost portt reltos s the relto kow s Boet s recurso orul ee slr er suppose tht c B us ths relto Kroecker elt s result

4 tertol Jourl o Scetc Reserch ulctos Volue 6 ssue 3 Mrch 6 38 SSN C Solv Frehol terl equto o the seco k c c 5 We clss Frehol terl equto ccor to the kerel two tpe: Frehol terl equto wth o-seprle kerel Frehol terl equto wth seprle kerel ths reserch we ocus o the rst tpe ecuse the re ot usull solvle route ethos C Frehol terl equto o the seco k wth o-seprle kerel ost prole where we ee uercl etho the kerel wll e o-seprle Now suppose we hve ths Frehol equto F K where we hve to We tr to pprote the kerel ucto wth oe epeet vrle choose soe vlues or other vrle the use ethos etoe We wrte s eore Cheshev or Leere epso So N Cheshev epso 6 N Leere epso 7 Where N s eerll ukow t e ve the prole or t c e estte ro perhps soe phscl rous ht e we hve N+ ukow coecets orer to etere these costts we wrte ow the epso o terl equto t ech N pots o where N he ep F Cheshev or Leere polols wrte ow or ech N vlues he we ust the kerel epso Suppose tht h K K We ep h ccor to rtrr etho hve ee tol he or ech vlue o we copute the epso or kerel We ot F or N 8 Fll we hve sste o N+ equtos or the N+ ukow coecets whch c e solve C Frehol terl equto o the seco k wth seprle kerel Whe the kerel s seprle we c wrte K h Hece Frehol terl equto o the seco k tkes the or F h slr er s hs ee s the prevous secto rst ep ccor to the orul 6 or 7 se o the esre etho he ep the other ucto the ters o Cheshev polols or Leere olols Fro equto or equto 5 we c copute vlue o ths cse we hve oe epeet vrle so st s ver spler th prevous cse At the e o the soluto we c ot coecets o the ukow ucto equt coecets o polols o the se eree o ech se o Frehol terl equto We wt to ro solv EXAMLE Whch the kerel K s ot seprle We ssue s epeet vlue tr to pprote to the kerel choos soe vlues o oth ethos we pprote to the ucto e o sth eree polol ourth eree polol wwwsrpor

5 tertol Jourl o Scetc Reserch ulctos Volue 6 ssue 3 Mrch 6 39 SSN wwwsrpor For the rst cse we kow s eve ucto hece se o 6 7 we hve our ukow coecets oth ethos; so we choose these our vlues 8 5 A Cheshev polols Approch he kerel stses the equto ve K K K 9 Suppose tht K Fro equto 9 us equto 3 we et Hece ] ] 3 he copute coecets these equtos Cheshev epso o the kerel or ech vlue o re ve tle le : Coecets o the kerel prouce Cheshev epso Evlute or ech vlue o es o equto v 6 7 So we ot

6 tertol Jourl o Scetc Reserch ulctos Volue 6 ssue 3 Mrch 6 SSN B coput susttut ote vlues to equto 8 we hve Solv ths sste o equtos ves For pprot to the ucto e o polol o eree we choose 5 B coputto we ot B Leere polols Approch Now we use Leere polols We c wrte ro N N N K K c K Susttuto to equto 9 ves c 3 c Us the orul 3 us Boet s recurso orul hece c he results re ve tle le : Coecets o the kerel prouce Leere epso Fro 5 we hve 6 So we c or ech vlue o us the coecets tle Coput the requre vlues the susttuto to equto 8 ves wwwsrpor

7 tertol Jourl o Scetc Reserch ulctos Volue 6 ssue 3 Mrch 6 SSN Fll solv these equtos us equto 6 we ot A ow choos 5 us tle we C Copr the ccurc o two pproches A coprso o the results wth Fo Goow s ve tle 3 Fo Goow le 3: Coprso o the ccurc o ethos Cheshev th eree Cheshev 6 th eree Leere th eree Leere 6 th eree Fo Goow presete ther result to D wth estte u error o coserle proveet ccurc ote wth etr coputto ue to rou-o error We see V CONCLUSON We c solve Frehol terl equto o the seco k wth the uercl soluto us Cheshev polols hs etho s useul ecuse o Cheshev polols propertes We c lso use the uercl soluto se o Leere polols But oth ethos we hve to orlze the re o the epeet vrle such tht t to t Whe we ppl these ethos o eple we : he recurrece relto etwee coecets o seres re ore coplcte or Leere polols th Cheshev polols Coput wth Leere etho s ver spler th Cheshev etho ths s the crtero or selecto o solv etho t s recoee tht Leere s etter choce he coput te sve us Leere epso ste o Cheshev epso wll e ore th the coput te sve the epso o kerel Due to the lre uer o clcultos the ccurc o the kerel coecets re ver sestve to the rou-o error the Cheshev etho he etho o clcult the coecets o the kerel Cheshev polols shoul cotue utl we rech zero coecets ccor to the esre ccurc ut the etho o Leere polols uer o requre coecets o the kerel s equl to the eree o pproto oth ve etho whe the eree o pproto s ukow we c strt wth low N crese t utl the esre ccurc s reche ACKNOWLEDGMEN woul lke to thk th techer Al Mlr hk ou; ou h stll hve ever oe o our stuet s respect REFERENCES ] A Jerr troucto to terl Equtos wth Applctos Joh Wle & Sos 999 ] Howr L Johso Nuercl retet o Nosulr Frehol terl Equtos o the Seco K S Jose Stte Collee 965 3] Fo L Goow E he Nuercl Soluto o No-sulr Ler terl Equtos hl rs o the Rol Soc pp5-53 ] Ellott D he epso o uctos ultr-sphercl polols Jourl o the Austrl Mthetcl Socet ] Apro Gl Jver Seur Nco ee Nuercl Methos or Specl Fuctos SAM ] JC Mso Dv C Hsco Cheshev olols CRC ress 7] Nsh Lu heor Applctos Leere olols Wvelets he Uverst o oleo 8 wwwsrpor

8 tertol Jourl o Scetc Reserch ulctos Volue 6 ssue 3 Mrch 6 SSN ] Des G Zll Wrre S Wrht Avce Eeer Mthetcs Joes & Brtlett pulshers ] JC Mso Dv C Hsco Cheshev olols CRC ress ] Frcesco Gcoo rco terl Equtos Courer Dover ulctos 985 ] Erw Kresz Avce Eeer Mthetcs Joh Wle & Sos ] Mtur Rh terl equtos ther pplctos W 7 AUHOR Author Mlr MS Flht Mechcs Cotrol Nhv r lr@hooco wwwsrpor