HERMITE SERIES SOLUTIONS OF LINEAR FREDHOLM INTEGRAL EQUATIONS

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1 Mhemcl nd Compuonl Applcons, Vol 6, o, pp 97-56, Assocon fo Scenfc Resech ERMITE SERIES SOLUTIOS OF LIEAR FREDOLM ITEGRAL EQUATIOS Slh Ylçınbş nd Müge Angül Depmen of Mhemcs, Fcul of Scence nd As, Cell B Unves, 57 Mude, Mns, Tuke slcn@fefsduedu mugengul@bedu Absc- A m mehod fo ppomel solvng lne Fedholm negl equons of he second knd s pesened The soluon nvolves unced eme sees ppomon The mehod s bsed on fs kng he unced eme sees epnsons of he funcons n equon nd hen subsung he m foms no he equon Theeb he equon educes o m equon, whch coesponds o lne ssem of lgebc equons wh unknown eme coeffcens In ddon, some equons consdeed b ohe uhos e solved n ems of eme polnomls nd he esuls e comped Kewods- eme sees, Lne Fedholm negl equons ITRODUCTIO Fedholm negl equons e wdel used fo modellng nd foecsng n lmos ll es of scence nd engneeng Fedholm negl equons e usull dffcul o solve nlcll so s equed o obn n effcen ppome soluon As we know, much wok hs been done on developng nd nlzng numecl mehods fo solvng lne Fedholm negl equons [--] One of he ohogonl polnomls s eme polnomls,,,, whch e ohogonl on,, so eme polnomls hve compson dvnges wh ohe ohogonl polnomls The subjec of he pesened ppe s o ppl he eme mehod fo solvng lne Fedholm negl equons In hs ppe, we consde he Fedholm negl equons of he second knd f K, d whee s he funcon o be deemned The consn, he kenel funcon K, nd he funcon f e gven We ssume h he neesed domn of he vbles s, The soluon of Eq s epessed s he unced eme sees whee whee s he eme polnoml of degee [6], o n he m fom [ ] A

2 98 S Ylçınbş nd M Angül nd,,,, [ A [ ] T e coeffcens o be deemned ] METOD FOR SOLUTIO To obn he soluon of Eq n he fom of epesson we cn fs deduce he followng m ppomons coespondng o he eme sees epnsons of he funcons f, K, nd Le he funcon f be ppomed b unced eme sees f f Then we cn pu sees n he m fom [ f ] F 5 whee T F [ f f f ] We now consde he kenel funcon K, If he funcon K, cn be ppomed b double eme sees of degee n boh nd of he fom [,7] s K, k 6, s s hen we cn pu sees 6 n he m fom [ K, ] K T whee ] 7 [ k k k k k k K k k k On he ohe hnd, fo he unknown funcon n negnd, we we fom epessons nd [ ] A 8 Subsung he m foms, 5, 7 nd 8 coespondng o he funcons, f, K, nd, especvel, no Eq, nd hen smplfng he esul equon, we hve he m equon o befl whee A F K T da I KQ A F 9

3 eme Sees Soluons of Lne Fedholm Inegl Equons 99 T Q d [ qs ],, s,,, nd I s he un m; he elemens of he fed m Q e sml b [,] In Eq 9, f D I KQ we ge q s s d A I KQ F, Thus he unknown coeffcens,,,, e unquel deemned b equon nd heeb he negl Eq hs unque soluon Ths soluon s gven b he unced eme sees ACCURACY OF SOLUTIO We cn esl check he ccuc of he mehod Snce he unced eme sees n s n ppome soluon of Eq, mus be ppomel ssfed hs equon, o If Then fo ech E E f K, d k k s n posve nege m k k k s n posve nege s pescbed, hen he uncon lm s ncesed unl he dffeence E ech k of he pons becomes smlle hn he pescbed On he ohe hnd, he eo funcon cn be esmed b f E K, d [6] UMERICAL ILLUSTRATIOS We show he effcenc of he pesened mehod usng he followng emples Emple Le us fs consde he lne Fedholm negl equon [,] nd seek he soluon so h d n eme sees f K,

4 S Ylçınbş nd M Angül 5 B usng he epnsons fo he powes n ems of he eme polnomls [6], we esl fnd he epesenons f nd 6 8 8, K nd hence, fom elons 5 nd 7, he mces / / F, 8 K If we use epesson fo,,, s, we obn he fed m 56 /5 8/ Q e, we subsue hese mces no Eq nd hen smplf o obn / / 9 / 5/8 9 / 9 / The soluon of hs equon s 6 5 / /8 B subsung he obned coeffcens n he soluon of becomes o whch s he ec soluon Emple We cn sud he followng lne Fedholm negl equon [5,8] 5 9 d If we use epesson fo,,, s, we obn Q m e subsue hese mces n Eq nd hen smpl obn whch s he ec soluon Emple Le us now consde [] d e e

5 eme Sees Soluons of Lne Fedholm Inegl Equons 5 The compson of soluons fo 7 wh ec soluon nd Fgue e s gven n Tble Tble Compng he soluons nd eo nlss whch hs been found fo 7 Emple Pesen mehod : eme Mehod Ec Soluon e E E E E E- 8 8 E- 8 6 E-5 6 E E E- 979 E E E E E E E- 587 E E E E E E E E- 6 8 E- E- 55 E E E- 557 E E E- 957 E E E E E E E E E E- 789 E-5 eme Mehod E Emple We consde he poblem =7 =8 =9 = Fgue The bsolue eos of Emple fo 7 We gve numecl nlss fo 7 e e d n Tble nd Fgue Emple 5 Le us consde he Fedholm negl equon of second knd The esuls obned fo e e d wh vous vlues e pesened n Tble

6 5 S Ylçınbş nd M Angül Tble Compng he soluons nd eo nlss whch hs been found fo 7 Emple Ec Soluon e Pesen mehod : eme Mehod E E E E E E E E E-6 9 E E E E E E E E E-7 67 E E E E E E-9 5 E E-7 E E E-5 7 E-7 E-7 6 E E E-7 98 E E E E-7 8 E E E-6 55 E E E E E E E-8 Tble Ec, ppome soluon nd eo nlss of Emple 5 fo he vlues Pesen mehod : eme Mehod Ec Soluon 8 9 e E E E E- 5 E E E-5 65 E E E E-5 88 E E- 669 E E E- E- 7 7 E E E E E- 9 E- 9 7 E E- 589 E E E- 67 E E E- 888 E E E- E E E- 5 E- 8 7 E E- 99 E E E- 8 E- 8 7 E E- 565 E E E- 789 E E E- E- 9 8 E E- 55 E E E E-5 95 E E E E E E E-5

7 eme Sees Soluons of Lne Fedholm Inegl Equons 5 eme Mehod 5 E =7 =8 =9 = Fgue The bsolue eos of Emple fo 7 Emple 6 Consde he followng he negl equon [] d e e We gve numecl nlss fo vous vlues n Tble nd Fgue eme Mehod E =7 =8 =9 = Fgue The bsolue eos of he soluons b usng he pesen mehod fo dffeen vlues of

8 5 S Ylçınbş nd M Angül Tble Compng he soluons nd eo nlss whch hs been found fo 7 Emple 6 Pesen mehod : eme Mehod Ec Soluon e E E E E 8 E-5 8 E-6 6 E-7 E E-5 57 E-6 57 E E E-5 8 E-6 5 E-7 9 E E E E E E E-6 98 E E E E E E E E E E E E E E E E E E E E E E E E E E-8 Emple 7 Ou ls emple s he negl equon [] 5 e e d We gve numecl nlss fo vous vlues n Tble 5 nd Fgue Ec Soluon BPF Mehod TF Mehod Legende Mehod =8 eme Mehod =8 Fgue The gphcs of he Ec Soluon, BPF mehod, TF Mehod, Legende Mehod nd Pesen mehod clculed fo he vlues of he nevl [,]

9 eme Sees Soluons of Lne Fedholm Inegl Equons 55 Tble 5 Eo nlss of Emple 7 fo he vlues nd compson of pesen mehod, ec nd he mehods n [] 6,7,8 Pesen mehod :eme Mehod Ec Soluon e E E E E- 5 5 E E-5 7 E- 7 E- 7 E E- 9 7 E E E E E E E E E E E E- 8 7 E- E E E E- 9 E E E- 69 E E E E E E- Legende Mehod n [] BPF Mehod n [] Block Pulse Fnc 8 m TF Mehod n [] Tngul Fnc COCLUSIOS AD DISCUSSIOS In hs ppe, he usefulness of he mehod pesened fo he ppome soluon of Fedholm negl equon s demonsed To show he ccuc of he mehod, fve negl equons e chosen A consdeble dvnge of he mehod s h he soluon s epessed s unced eme sees Ths mens h, fe clculon of he eme coeffcens, he soluon cn be esl evlued fo b vlues of low compuon effo If he funcons f nd K, cn be epnded o he eme sees n,, hen hee ess he soluon ; ohewse, he mehod cnno be used n On he ohe hnd, would ppe h ou mehod shows o bes dvnge when he known funcons f nd K, hve Tlo sees bou he ogn whch convege pdl To ge he bes ppomng soluon of he equon, we mus ke moe ems fom he eme epnsons of funcons,

10 56 S Ylçınbş nd M Angül especll when he convege slowl Befl, fo compuonl effcenc, he uncon lm mus be chosen suffcenl lge 6 REFERECES L Fo, I Pke, Chebshev polnomls n umecl Anlss, Clendon Pess, Ofod, 968 M Seze, S Dogn, Chebshev sees soluons of Fedholm negl equons, In J Mh Educ Sc Technol, 7, 5, , 996 R P Knwl, nd K C Lu, A Tlo epnson ppoch fo solvng negl equons, In J Mh Educ Sc Technol,, -, 989 E Bboln, R Mzbn, M Slmn, Usng ngul ohogonl funcons fo solvng Fedholm negl equons of he second knd, App Mh nd Compu,, 5 6, 8 5 R Fnoosh, M Ebhm, Mone Clo mehod fo solvng Fedholm negl equons of he second knd, App Mh nd Compu, 95, 9 5, 8 6 S Ylçınbş, M Seze, Sokun, Legende polnoml soluons of hgh-ode lne Fedholm nego-dffeenl equons, App Mh nd Compu,,, - 9, 9 7 K Bsu, SIAM J ume Anl,, 96-55, 97 8 K Mleknejd, M Km, Usng he WPG mehod fo solvng negl equons of he second knd, App Mh nd Compu, 66,, 5 9 K Mleknejd, Y Mhmoud, umecl soluon of lne Fedholm negl equon b usng hbd Tlo nd Block-Pulse funcons, App Mh nd Compu, 9, , S Ylçınbş, M Angül, T Akk, Legende sees soluons of Fedholm negl equons, Mh nd Compu App, 5,, 7-8, S Ylçınbş, T Akk, M Angül,, Lguee sees soluons of Fedholm negl equons, Eces Unves Jounl of Scence nd Technolog, 6,, -,