Numerical Solution of Second order Integro- Differential Equations(Ides) with Different Four Polynomials Bases Functions

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1 Numercl Soluto of Secod order Itegro- Dfferetl Equtos(Ides) wth Dfferet Four Polyomls Bses Fuctos wo O A R M Deprtmet of Mthemtcs Uersty of Ilor Deprtmet of Mthemtcs d Sttstcs he Poly Id Astrct: - I ths pper method sed o the collocto methods wth some ses fuctos re deeloped to fd the umercl soluto of Fredholm Itegro-Dfferetl Equtos; four dfferet polyoml ses fuctos used were: Legedry Leguerre Hermte d Focc polyoml ses fuctos he dfferetl prt pperg the tegro-dfferetl equto s re-defed d used to geerte ech of the polyoml ses fuctos Some umercl results re ge to demostrte the superor performce of the rous collocto methods prtculrly the tle of error wth the rous lue of N Keywords: Collocto method Itegro-dfferetl equtos (IDEs) Cheyshe Hermte Focc d Leguerre Polyoml INRODUCION Itegro-dfferetl equtos (IDEs) plys mportt role my rches of ler d No-ler fuctol lyss I ddto to ther occurrece the feld of mechcs d mthemtcl physcs Itegrodfferetl equtos he foud wde pplctos the theory of egeerg chemstry stroomy ology ecoomcs potetl theory d electro-sttc hs pper cocers the deelopmets of the rous polyoml ses fucto (see Ortz [5] d Ortz d Smr [7]) wth Legedre Leguerre Focc d Hermte polyomls for the umercl soluto of tegro- dfferetl equtos (IDEs) he polyomls hs foud extese pplcto recet yers preseted seres of ppers for exmple [-8] for the cse of umercl soluto of ordry dfferetl equtos (ODEs) d [49] for the cse of umercl soluto of prtl dfferetl equtos (PDEs) Applcto of the Cheyshe d Legedre polyomls d ther umercl merts solg ODEs d PDEs umerclly he ee dscussed seres of ppers (for exmple [-3]) We re therefore motted to work ths drecto of extedg to rous polyomls proposed the lterture to hdle IDEs umerclly [3-7] Ylcs d Sezer [9] proposed pproxmtg soluto terms of ylor polyomls whch we elee t s prtculr cse of the method preseted ths pper Also ths pper s orgzed s follows I secto the formulto of the rous polyomls ses re deeloped (such s: Focc Hermte Legurre d Legedry); the mtrx represettos of ech prt of IDE d ts supplemetry codtos re oted I secto 3 s effcet u error estmtor s troduced I secto 4 prelmry steps towrds costructo of rous polyoml ses fuctos were cosdered d tkg Cheyshe polyoml s exmple Flly secto 5 some umercl results re proded to demostrte the effcecy of usg rous polyomls ses d compred wth those of [9] CONVERING INEGRO-DIFFERENIAL EQUAION O A SYSEM OF LINEAR ALGEBRAIC EQUAIONS L : g d dx We shll wrte for g g x : g x wwwertorg 99

2 Where s the degree of g d g g g X x x Uless otherwse stted x wll lwys e the depedet rle of the fuctos whch pper throughout ths pper d wll e defed fte terl Let y e the exct soluto of the tegro-dfferetl equto Ly x mx tyt dt f x x 3 Wth m m m c y c y d m m 4 Where f x d x t m re ge cotuous fuctos d c m c m d d some ge costts Mtrx represetto for the dfferet prts d degree Let : x x x V e polyoml ss y V : V X for Also for y mtrx P P VPV where V s o-sgulr lower trgulr mtrx Now we coert the Eqs (3) d (4) to the correspodg ler lgerc equtos three prts; () () d (c) () Mtrx represetto for Dy: Ortz d Smr proposed [7] lterte for the u techque whch they clled the opertol pproch s t reduces dfferetl prolems to ler lgerc prolems he effect of dfferetto shftg d tegrto o the coeffcets ector ~ : ( ~ ~ ~ ) Of polyoml u ~ X s the sme s tht of post-multplcto of ~ y the mtrces d respectely du dx ~ X u ~ X x d u dt ~ X where We recll ow the followg theorem ge y Ortz d Smr [7] heorem For y ler dfferetl opertor L defed y () d y seres wwwertorg 99

3 y : V ( ) we he : Ly x) ~ ( X V 5 where d g () VV () Mtrx represetto for the tegrl term: Let us ssume tht m( x y( x) m( x y( dt m ( ) x d m V he we c wrte dt MV 6 where M m m o m m wth o for l (c) Mtrx represetto for the supplemetry codtos: Replcg y the left hd sde of (4) t c e wrtte s wwwertorg 99

4 () ( m) () ( m) () m () ( m) c m y ( ) c m y ( ) c m ( ) c m ( ) m m B 7 where for = B c () m ( ) c m m c c () m m () m () m ( m) ( ) ( ) c () m ( ) c () m ( m) ( m ( ) ) ( ) 8 We refer to B s the mtrx represetto of the supplemetry codtos d for computg the elemets of the mtrx B c e deduced from (7): () ( m) () ( m) c ( ) c ( ) for d m m m () ( m) () ( m) c ( ) c ( ) for m km m d ( d d We troduce d codtos tke the form B d It follows from (5) d (6) tht B s ts th colum he followg reltos ) the ector tht cots rght hd sdes of codtos he the supplemetry 9 Ly m( x y( d ( M ) V Let M M : d M stds for ts th colum d let f f f V wth f f f ) he the coeffcet of exct soluto y V ( fte lgerc system: of prolem (3) d (4) stsfes the followg M M B settg f ; ; d ; 3 G ( B B M M ) d g ( d d f f) wwwertorg 993

5 we c wrte sted of (3) G g 4 Defto he polyoml y V wll e clled pproxmte soluto of (3) d (4) f the ector ( ) s the soluto of the ler lgerc equtos G g 5 Where G s the mtrx defed y restrcto of G to ts frst ( + ) rows d colums Remrk 3 For d G t s trsformed to Dfferetl Equto 3 ERROR ESIMAION e Eq (3) s trsformed to Fredholm tegrl equto of secod kd d for I ths secto error estmtor for the pproxmte soluto of (3) d (4) s oted Let us cll x) y( x) y ( ) s the error fucto of the pproxmte soluto y to y where y s the exct ( x soluto of (3) d (4) Hece y stsfes the followg prolem: m( x y dt f H ( x x Ly ) wth 3 () ( m) () ( m) c m y ( ) c m y ( ) d m 3 he perturto term H c e oted y susttutg the computed soluto y to the equto H Ly m( x y dt f 33 We proceed to fd pproxmto ( ) to the error fucto e the wy s we dd efore for the soluto e x N of prolem (3) (4) Sutrctg (3) d (3) from (3) d (4) respectely the error fucto e stsfes the prolem m( x e dt H ( x x Le ) 34 wwwertorg 994

6 wth the homogeeous codtos () ( m) () ( m) me ( ) c m e ( ) m c 35 It should e oted tht order to costruct the pproxmt ( ) to e oly the rght hd sde of system (5) eeds to e recomputed; the structure of the coeffcet mtrx e x N G rems the sme 4 CONSRUCION OF VARIOUS POLYNOMIAL BASES FUNCION V ws cosdered s polyoml ss ge y V V X I secto ( ) : x : where V s osgulr lower trgulr mtrx d degree ( ) It ws used to coert (3) d (4) to system of ler equtos he rous polyomls re ery terestg polyoml ss wth mtrx V of the sme structure We rgue the pplcto of the u method for the cse of Cheyshe polyomls though the cse of Legedre Leguerre Hermte d Focc my lso e smlr But dfferet recurse formule wll e employed For stce the shfted cheyshe polyomls re defed s x ( ) d for x x ( ) ( ) x x I ths cse the fuctos m(x f d the pproxmte soluto re wrtte s m( x f f m y d y Where m d f re computed y the followg reltos d re oted from () m 4 '' '' t l cos cos m xt x t l f m '' m x t wwwertorg 995

7 For d x cos A summto symol wth doule prmes deotes sum frst d lst terms hled Also re computed esly ths cse sce s (see prt () secto ) l l dt t ( ) t ( ) l dt Where dt d degree d odd for odd degree hece l l re Cheyshe polyoml of degree d l respectely But these polyomls re ee for ee ; f odd l ; f l ee ( ) ( ) l l All other elemets re computed s Secto he sme pplcto wll deftely dopted for the other Polyomls whe the recurse formule were troduced to rre t the pproxmte soluto of the rous polyoml ses fucto 5 NUMERICAL EXAMPLES I ths secto we cosder some exmples demostrtg the ccurcy of the method d effecteess of the Cheyshe polyoml ss fucto compred wth the other rous polyoml ses fuctos Hece error of exmples d re lso compred wth [9] Exmple polyomls hs exmple ws cosdered y Ylcs d Sezer [9] for the method of soluto terms of ylor '' ' x t y xy xy( x) e s s e y( dt x y ( ) ' y () he exct soluto s y x e wwwertorg 996

8 le : les of errors whe N = 4 X CHEBYSHEV LEGENDARY LAGUERRE HERMIE FIBONACCI E- 35E- 665E E- 996E- 8E- 889E E- 47E- 75E- 675E E- 668E- 463E- 5E E- 776E- 95E- 48E-3 E+ E+ 7E- 875E E-4 6E+ 548E- 4E E- 39E+ 3E- 3E E- 6E+ 55E- 75E E- 78E+ 6E- 348E E- 9E+ 75E+ 45E- Ylcs et l [9] le : les of errors whe N = 5 X CHEBYSHEV LEGENDARY LAGUERRE HERMIE FIBONACCI Ylcs et l [9] - E-8 54E-3 4E-3 58E-3 44E-7-8 4E-9 43E-3 69E-4 886E-4 59E-7 964E E-9 36E-3 74E-3 995E-3 357E-6 636E-7-4 6E-9 475E-3 833E-3 96E-3 483E-6 46E-8-3E-9 74E-4 855E-3 98E-4 E-6 753E-7 E+ E+ E+ E+ 494E-6 E+ E-9 477E-6 58E- 485E-4 493E-5 76E-6 4 8E-9 595E-4 63E- 5E-3 346E-5 47E-6 6 E+ 495E-3 53E- 7E-3 458E-5 88E-6 8 8E-9 55E-3 664E- 578E-3 765E-5 98E-7 8E-8 48E-3 78E- 34E-3 53E-5 83E-6 From the umercl results d le of error (see les ) I le t s edet tht etter performce proded y the rous method here proposed compred wth the results of Ylcs d Sezer [9]Hece t ws osered tht the Legedre d Hermte solutos he sme errors oer the terl he error of the Focc soluto shows tedecy to crese rpdly s N creses Exmple (see [9]) '' y xty dt x s y ( ) ' y () x wwwertorg 997

9 he exct soluto s y s For umercl results see tle 3 d 4 le 3: les of errors whe = 4 Ylcs et l X CHEBYSHEV LEGENDARY LAGUERRE HERMIE FIBONACCI 879E-5 375E-5 53E-4 E-4 5E-3 44E-4 43E-4 998E- 54E-3 98E- 456E-3 964E-4 9E- 99E- 395E- 96E- 856E- 36E E- 9E- 654E- 9E- 745E- 546E E- 38E- 563E- 384E- 46E- 753E E- 469E- 83E- 473E- 3E- E E- 55E- 8E- 558E- 97E- 76E E- 68E- 854E- 636E- 455E- 47E E- 698E- 348E- 78E- 484E- 885E E- 76E- 3E- 773E- 78E- 98E-3 668E- 84E- 375E- 83E- 589E- 83E-3 π/ 9E- 938E- 934E- 977E- 68E- E-3 le : les of errors whe = 5 X CHEBYSHEV LEGENDARY LAGUERRE HERMIE FIBONACCI 93E-6 E+ E-4 E+ 996E-5 544E-6 54E-6 E-5 4E-4 66E-5 46E-5 96E-6 539E-5 33E-5 39E-5 693E-5 886E-5 636E E-5 5E-4 6E-5 53E-4 675E-4 546E E-6 5E-4 3E-5 4E-5 546E-4 475E E-5 86E-5 E-5 863E-5 43E-4 98E E-7 48E-4 8E-5 5E-4 597E-4 8E E-6 E-4 8E-5 7E-4 745E-4 547E E-5 966E-4 348E-4 963E-4 48E-4 588E E-6 585E-4 E-4 39E-6 74E-4 493E-6 68E-5 47E-4 738E-5 8E-4 659E-4 98E-5 π/ 37E-6 86E-4 793E-4 37E-3 6E-4 7E-6 Ylcs et l[8] he tle shows tht the results of the polyoml ses re umerclly stle 6 CONCLUSIONS Our results dcte tht the tu method wth rous polyoml ses c e regrded s structurlly smple lgorthm tht s coetolly pplcle to the umercl soluto of IDEs I ddto lthough we he restrcted our tteto to ler Fredholm IDEs we expect the method to e esly exteded to more geerl IDEs Despte the reltely low degrees used the umercl results show the superor performce of the u method prtculrly wth the Cheyshe d Legedre ses Neertheless the error of the u soluto shows tedecy to crese rpdly s N creses hs ehour lso dcted other polyoml/methods wwwertorg 998

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