Collocation Method for Nonlinear Volterra-Fredholm Integral Equations

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1 Ope Joural of Appled Sees 5- do:436/oapps6 Publshed Ole Jue (hp://wwwsrporg/oural/oapps) Colloao Mehod for olear Volerra-Fredhol Iegral Equaos Jafar Ahad Shal Parvz Daraa Al Asgar Jodayree Akbarfa Depare of Maheas ad Copuer See Uversy of abrz abrz Ira Depare of Maheas Fauly of See Ura Uversy Ura Ira Eal: {_ahadshal Reeved Aprl 8 ; revsed May 4 ; aeped May 5 ABSRAC A fully dsree verso of a peewse polyoal olloao ehod based o ew olloao pos s osrued o solve olear Volerra-Fredhol egral equaos I hs paper we oba exsee ad uqueess resuls ad aalyze he overgee properes of he olloao ehod whe used o approxae sooh soluos of Volerra- Fredhol egral equaos Keywords: Colloao Mehod; olear Volerra-Fredhol Iegral Equaos; Covergee Aalyss; Chelyshkov Polyoals Iroduo We shall osder he olear Volerra-Fredhol egral equao y g y Fy I () he Volerra egral operaors gve by : C I C I y k s ys CD ad d s where k D s : s ad Fredhol egral operaors gve by F : C I C I Fy k s y s d s where r r deoes (real or oplex) paraeers ad k CII ad le gci be a gve fuo he eoed equaos are haraerzed by he presee of a lear fuoal argue ad play a pora role explag ay dffere pheoea I parular hey ur ou o be fudaeal whe ordary dffereal equaos based odel fal hese equaos arse dusral applaos ad sudes based o bology eooy orol ad elero-dya Colloao ehod s a wdely popular ueral ehque solvg egral equaos dffereal equaos e Whe olloao ehod s used o solve oplaed egeerg probles has several dsadvaages ha s low effey ll-odoed e hus () (3) dffere ypes of ehques were proposed o prove he opuaoal perforae of olloao ehod Reely Chelyshkov has rodued sequees of polyoals [] whh are orhogoal over he erval wh he wegh fuo hese polyoals are explly defed by k k k k k k P k (4) he polyoals Pk have properes whh are aalogous o he properes of he lassal orhogoal polyoals hese polyoals a also be oeed o a fxed se of Jaob polyoals P Presely k k k k k P P Ivesgag ore o (4) we dedue ha he faly of orhogoal polyoals Pk have k ulple k zeros ad k ds real zeros he erval Hee for every he polyoal P has exaly sple roos Followg [] a be show ha he sequee of polyoals P geerae a faly of orhogoal polyoals o whh possesses all he properes of oher lass orhogoal polyoals eg Legedre or Chebyshev polyoals herefore f he roos of P are hose as olloao pos he we a oba a aurae ueral quadraure I he prese paper we furher develop he works arred ou [-6] Copyrgh SRes

2 6 J A SHALI E AL We dsuss exsee ad uqueess resuls ad aalyze he overgee properes of he olloao ehod whe used o approxae sooh soluos of lear Volerra-Fredhol egral equaos ad fally soe ueral resuls are preseed he fal seo whh suppor he heoreal resuls obaed hs paper Exsee ad Uqueess Resuls Le CI deoe he Baah spae ouous realvalued fuos suh ha g CI wh g ax g (5) I Lea Assue H s a oepy losed se a Baah spae V ad ha : H H s ouous Suppose s a orao for soe posve eger he has a uque fxed-po H Proof: For proof see [7] Here egral Equao () we assue ha k for soe osas M sasfes a Lpshz odo wh respe o s hrd argue k s yk s y M y y (6) s y heore Assue g ad k sasfy he odo (6) ad gve fuos g k are ouous o her doas Moreover assue M (7) he he egral Equao () has a uque soluo y CI Proof: We defe he olear egral operaor : C I C I y g y Fy I (8) Le us show ha for suffely large he operaor s a orao o C I For y y C I he Se y y k s y s k s y s ds k s y s k s y s ds y y M y s y s ds M M y y (9) M y s y s d s () y y d k s y s k s y s s () k s y s k s y s d s we ge M y y M y y! () By a aheaal duo we oba hus y y M M y y! y y M M y y! Se M he l M (3) (4) ad M l! he operaor s a orao o C I whe s hose suffely large By he Lea he operaor has a uque fxed-po C I 3 Colloao Mehod Le h paro for I ad le : he esh defe a ufor : s osraed he followg sese: h wh a gve esh we assoae he se of s eror pos Z : : For a fxed ad for gve egers d ad he d peewse polyoal spae S dz s defed by d : : d S Z u C I ; u d d where π d deoes he se of (real) polyoals of a degree o exeedg d he deso of hs d spae s gve by d S Z d d Copyrgh SRes

3 J A SHALI E AL 7 For egral equao we have d olloao spae wll be S Z Le for all we have u S Z r r u u sh L s u h r hee he (5) Fro (5) we see ha a elee u S Z s well defed whe we kow he oeffes u h for all I order o opue hese oeffes we osder he se of olloao paraeers X : r where ad defe he se of olloao pos by : h he olloao soluo u S Z ered by posg he odo ha egral Equao () o he fe se X u g u Fu X wll be deu sasfes he (6) hus for h he olloao Equao (6) assues he for k s usd s u g k s u s ds 7) Fro hs equao ad afer soe opuaos we oba u g h k sh u sh ds h k sh u sh ds h k sh u sh ds (8) ow by usg he loal Lagrage bass fuos s r L s l (9) r rl l r for approxag he egral ers we use he Lagrage erpolag polyoal o approxae k s us ad k s us we oba l l l u g h k h u h L s ds h k hu h L s ds l l l l h k hu h L s d s () Defg he quadraure weghs wl : L d l s s l () ad w l : L d l s s l () he fully dsrezed olloao equao orrespodg o ()-() s hus gve by u l l l g h wk hu h l h w k h u h l l l l l l l l u S Z h wk h u h (3) oe ha ad Equao (3) represe for eah a reursve syse of olear algebra equaos wh he ukows u 4 Global Covergee Le u S Z deoe he (exa) olloao soluo o () defed by (6) I our overgee aalyss we exae he lear es equao y g k s y s ds where k CD k CI I s o he speru F k sysd s I (4) We wll assue ha of he Fredhol egral operaor F A oe of he overgee resuls o he olear Equao () a be foud a he ed of hs seo heore 4 Assue ha he gve fuo (4) sasfy g C k C D k C I I he for all suffely sall h he osraed esh olloao soluo u S Z o (4) for all sasfes C M h (5) Copyrgh SRes

4 8 J A SHALI E AL where C are posve osas o depedg o h hs esae holds for all olloao paraeers wh Proof: I eah erval he exa soluo y of (4) s es ouously dffereable hs follows fro he soohess hypoheses we have posed o g k k ad fro he expressos for y Fro hs s obvous ha boh he lef ad rgh ls of y as eds o h exs ad are fe We wll prove he esae (5) by usg he Peao s heore o wre ad r r y sh L s y h h R s Here we have r : s (6) R s K s z y zh d z (7) Ks z sz Lk sk z! k z (8) hus follows fro (5) ha he olloao error : y u possesses o he loal represeao sh L s h R s s(] r r r wh y h u h equao k s s ds k s s d s (9) ad sasfes he (3) By subsug he (9) he (3) ad afer soe opuaos we oba h k sh Lr s ds r r h k d sh R s s h k d sh Lr s s r r h k d sh R s s h k d sh Lr s s r r h k d sh R s s (3) Defe he ares L k d sh Lr s s k d sh Lr s s r k sh Lr s ds 3 r ad he veors by A k sh R s d s d A k sh R s s 3 d A k sh R s s (3) (33) (34) (35) (36) (37) (38) by subsug he Equaos (3)-(38) Equao (3) we oba h h A h h A h h A 3 3 (39) hs lear algebra syse ay be wre ore osely as h h 3 h h A h A h A3 ow le I hb I hb I hb (4) (4) Copyrgh SRes

5 J A SHALI E AL 9 I h3 h3 h 3 Q h3 h 3 I h3 he we have Q h h A h A h A 3 (4) (43) (44) Se he kerel K s ouous o her doas he elees of he arxes are all bouded By usg he eua Lea he verse of he arx I h exss wheever h for soe arx or hs learly holds wheever h s suffely sall I oher words here s a h so ha for ay esh wh h h eah arx has a uforly bouded verse herefore arx has a uforly bouded verse Also he verbly of he blok arx Q ow depeds o oly o h bu also o s guar- aeed f F where * (45) F : ax k s d s k k I assug ha * k s k ad he elees of he arxes Q are all bouded hus fro (44) we ge I (46) where ad Q (47) h h A (48) +h A h A3 I s lear ha arx has a uforly bouded verse ad he elees of he arxes are all bouded oe ha fro hese assupos ad Q here exss a osa D so ha for all esh d- aeers h h he ufor boud I P (49) holds Here for B L B deoes he arx (operaor) or dued by he l -or Assue ha ad B P for ad P Fro (46) ad (48) we have I P (5) h h A ad hee h A h A3 PP h P h km K h k M K h k M K (5) Mh (5) where PP h PK k k A k K M A k K M A k K M M : y 3 s[] I k : ax K s z d z : ax k s v d vk Also (53) l (54) l he fro (5) ad (54) we have Mh ow by usg he dsree Growall equaly we have M h where exp ow by usg he loal error represeao (9) hs yelds seg W : ax L sh W h KM CM h Copyrgh SRes

6 J A SHALI E AL uforly for s ad where C W K he s equvale o he esae C y h We olude hs seo wh a oe regardg he exeso of he resuls of heore o he olear Equao () Uder he assupo of he exsee of a (uque) soluo y o I he olear aalogue of he error Equao (3) s k s y s k s u s ds k s y s k s u s d s (55) k If he paral dervaves y are ouous ad bouded o D D wh D : y : y y s M si for soe M ad f h s suffely sall he (55) ay aga be wre he for (3) he roles of k are ow assued by k s z s H s: y where z s: ys us s Hee he above proof s easly adaped o deal wh he olear ase () ad so he overgee resuls of heore rea vald for olear Volerra-Fredhol egral equaos 5 Preseao of Resuls I hs seo we repor o he ueral resul of es proble solved by he proposed ehod of hs arle ypal fors of olloao paraeers are: Gauss pos: Zeros of P ; Radou I pos: Zeros of P P ; Radou II pos: Zeros of P P Chelyshkov pos: Zeros of P P ; where P ad P are Legedre ad Jaob polyoals respevely Exaple 5 he olear Volerra-Fredhol egral equao ; y y s ds 3 y s d s s (56) able Error for exaple Guass e Radau e Radau e Chelyshkov e has he followg aalyal soluo y herefore provdes a exaple o verfy he auray of hs ehod able shows he axu errors volved pre- seed ehod wh h alog wh he exa 4 8 soluo For opuaoal purposes he es proble dffere fors of kerels are osdered All he opuaos were arred ou wh Maple I eah ases of Exaple he obaed olear equaos was solved by he ewo s ehod he resul for olloao pos are preseed able whh daes ha he ueral soluos ob- aed fro (56) ad sep szes equal o 4 ad 8 are early deal hese resuls dae ha f we use he Chelyshkov pos he we oba he ueral soluos of u error 6 Coluso We have show ha he olloao ehod yelds a effe ad very aurae ueral ehod for he approxao of soluos o Volerra-Fredhol egral equaos Also we have show ha f he roos of P are hose as olloao pos he we a oba a aurae ueral quadraure 7 Akowledgees he auhors ruly appreae he oes ade by referees REFERECES [] V S Chelyshkov Alerave Orhogoal Polyoals ad Quadraures Elero rasaos o ueral Aalyss Vol 5 o 7 6 pp 7-6 [] H Bruer Colloao Mehods for Volerra Iegral ad Relaed Fuoal Equaos (Cabrdge Moographs o Appled ad Copuaoal Maheas) Vol 5 Cabrdge Uversy Press Cabrdge 4 [3] H Bruer Iplly Lear Colloao Mehods for olear Volerra Equaos Appled ueral Maheas Vol 9 o pp Copyrgh SRes

7 J A SHALI E AL do:6/68-974(9)98-9 [4] H Bruer Hgh-Order Colloao Mehods for Sgular Volerra Fuoal Equaos of eural ype Appled ueral Maheas Vol 57 o pp do:6/apu676 [5] H Bruer he ueral Soluo of Weakly Sgular Volerra Fuoal Iegro-Dffereal Equaos wh Varable Delays Couaos o Pure ad Appled Aalyss Vol 5 o 6 pp 6-76 do:3934/paa656 [6] A Dogo S MKee ad ag A Here-ype Colloao Mehod for he Soluo of a Iegral Equao wh a Cera Weakly Sgular Kerel IMA Joural of ueral Aalyss Vol o 4 99 pp do:93/au/4595 [7] K E Akso ad W Ha heoreal ueral Aalyss Sprger Berl 9 Copyrgh SRes