Chebyshev Polynomial Solution of Nonlinear Fredholm-Volterra Integro- Differential Equations
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1 Çny Ünvee Fen-Edeby Füle Jounl of A nd Scence Sy : 5 y 6 Chebyhev Polynol Soluon of onlne Fedhol-Vole Inego- Dffeenl Equon Hndn ÇERDİK-YASA nd Ayşegül AKYÜZ-DAŞCIOĞU Abc In h ppe Chebyhev collocon ehod [] developed o fnd n ppoe oluon fo nonlne Fedhol-Vole nego dffeenl equon. h ehod nfo he nonlne Fedhol-Vole nego dffeenl equon no he equon wh he help of Chebyhev collocon pon. he equon coepond o ye of nonlne lgebc equon wh he unnown Chebyhev coeffcen. Fnlly oe nuecl eple e peened o llue he ccucy of he ehod. Keywod: onlne nego-dffeenl equon; Chebyhev ee; Colllocon ehod. Öze Bu çl şd lnee olyn Fedhol-Vole nego dfenyel denlelen ylş çözülen bul çn Chebyhev l yöne [] gelşlş. Bu yöne lnee olyn Fedhol-Vole nego dfenyel denlen l nol n ulln denlene dönüşüü. Bu denle e blneyen Chebyhev y l oln lnee olyn cebel denle ene ş l gel. Çl şn n onund yönen doğuluğunu göee çn bz y l önele unuluşu. Anh Kelele: nee Olyn nego-dfenyel denlele; Chebyhev ele; S l Yöne. Pule Ünvee Fen Edeby Füle e Bölüü K n l Denzl. URKEY. yegulyuz@yhoo.co 89
2 Chebyhev Polynol Soluon of onlne Fedhol-Vole Inego-Dffeenl Equon. IRODUCIO Conde he followng nonlne Fedhol Vole nego-dffeenl equon P y g λ F y d λ K y d unde he ed condon [ y b y c y c ] µ c whee y n unnown funcon he funcon g P F nd K e defned on nevl nd b c λ λ µ e conn. e u ee he oluon of epeed n e of Chebyhev polynol y 3 whee e unnown Chebyhev coeffcen nd choen ny pove nege uch h. ' denoe u whoe f e hlved denoe he Chebyhev polynol of he f nd of degee. he Chebyhev collocon pon defned by π co... 4 e ued n he followng econ.. FUDAEA REAIOS e u we Eq. n he fo D g λi λ J 5 whee he dffeenl p D P y Fedhol negl p 6 9
3 Hndn ÇERDİK-YASA Ayşegül AKYÜZ-DAŞCIOĞU I F y d 7 nd Vole negl p J K y d 8.. Repeenon fo Dffeenl P e u ue h he h devve of he funcon 3 wh epec o h he unced Chebyhev ee epnon by y ' whee... e Chebyhev coeffcen. hen he oluon epeed by 3 nd devve cn be wen n he fo epecvely y A 9 nd y A I well nown fo [6] h he elon beween he Chebyhev coeffcen A of y nd he Chebyhev coeffcen A of y gven whee A A hen he epeon becoe y A [... ] A... 9
4 9 Chebyhev Polynol Soluon of onlne Fedhol-Vole Inego-Dffeenl Equon / 5/ 3/ / O fo odd nd / 3/ / O fo even Subung he Chebyhev collocon pon no Eq.6 nd ung he epeenon of D cn be gven by P D A.. Repeenon fo Fedhol Inegl P e u ubue he Chebyhev collocon pon no Eq. 7 o obn he elon of I nd ue h fo ech F nd F epnded o he Chebyhev ee n he fo f F '' whee uon ybol wh double pe denoe u wh f nd l e hlved nd Chebyhev coeffcen f e deened by en of he elon F f '' π co... hen he epeenon of F becoe
5 93 Hndn ÇERDİK-YASA Ayşegül AKYÜZ-DAŞCIOĞU F F 3 whee f f f f F Bede y funcon cn be wen n he fo [5] B y 4 n whch [ ]... b b b... B nd he eleen b of he colun B con of nd - follow: fo odd fo even b When he elon 9 3 nd 4 e ubued n I we hve B Z A Z I F F 5 whee Z d [ ] z Z d z [ ] nd d fo odd fo even z
6 94 Chebyhev Polynol Soluon of onlne Fedhol-Vole Inego-Dffeenl Equon. 3. Peenon fo Vole Inegl P Fly he Chebyhev collocon pon e ubued no 8. Slly he pevou econ uppoed h he enel funcon K cn be epnded o unvee Chebyhev ee wh epec o. hen he fo of he enel funcon K... K 6 whee K Subung he elon 9 4 nd 6 n J he epeenon of J obned A B J Z K Z K 7 whee Z d z [ ] I Z d z [ ] I nd z d odd fo even fo fo fo 4 fo -
7 Hndn ÇERDİK-YASA Ayşegül AKYÜZ-DAŞCIOĞU. 4. Repeenon fo he Condon Ung he elon he fo of he condon defned n cn be wen b c c A µ e u defne U U [ ] b c c u u... u hu he fo of condon becoe U A µ 8 3. EHOD OF SOUIO o conuc he fundenl equon coepondng o Eq. he Chebyhev collocon pon e ubued n 5 nd hen ung he elon 5 nd 7 obned fo P A g λ λ F Z A F Z B K Z A K Z B heeby he fundenl gned of he fo P λfz λkz A- λfz λk Z B G 9 whee P K n P P Kn O Kn O P Kn g Fn Z n g G Fn Fn Z n Z n fo n g F Z n n 95
8 Chebyhev Polynol Soluon of onlne Fedhol-Vole Inego-Dffeenl Equon h equon coepond o ye of nonlne lgebc equon wh unnown Chebyhev coeffcen. Fnlly o obn he oluon of Eq. unde he ed condon equon n nonlne lgebc ye 9 e eplced wh equon n lne lgebc equon ye 8. heefoe Chebyhev coeffcen e deened by olvng he new nonlne lgebc ye. he ehod lo cn be developed fo he poble defned on he don [] P y g λ F y d λ K y d he oluon of h equon unde he ed condon found n e of hfed Chebyhev polynol of he fo y ' whee. I followed he pevou pocedue ung he collocon pon defned by π co... nd he elon whee A 4 A... A. hen we obn he fundenl equon fo 4 λ[ ] λ PA FZA FZB [ KZA KZB] G oeove he fo of he condon becoe 4 b c c A µ 96
9 Hndn ÇERDİK-YASA Ayşegül AKYÜZ-DAŞCIOĞU I ely een h Z n Z n nd Z of he popee of he Chebyhev polynol. n Z fo n becue n 5. UERICA EXPERIEAIOS he effcency of he peened ehod hown n followng hee eple. Reul wee copued ung he pog wen n hcd Pofeonl. Eple. e u conde wo eple of nonlne Fedhol- Vole nego-dffeenl equon. hee poble h been olved by ylo polynol fo 4 nd 5 epecvely n [3]. y y y g y d y d whee g nd y y b y y g y d y d whee g nd y e u e fo oluon of Eq. nd ee he oluon y unced Chebyhev ee y ' Fundenl equon of h poble defned n Secon 3 4 P P - F Z A K Z B G nd condon equon e A nd A h equon coepond o nonlne lgebc ye follow: nd condon equon e 97
10 Chebyhev Polynol Soluon of onlne Fedhol-Vole Inego-Dffeenl Equon 5 In ye 4 f nd econd equon e eplced by condon equon n 5 nd new lne lgebc ye obned. h ye olved ely o we hve y whch he ec oluon of Eq.. b e u conde oluon of Eq. 3 fo nd ee he oluon y unced Chebyhev ee y ' 6 he fundenl equon of h poble defned n Secon 3 4P P F Z A K Z B G nd fo condon equon A 4 4 he equon coepond o nonlne lgebc ye follow: nd condon equon 8 When he f equon he ye 7 eplced by Eq. 8 new nonlne lgebc ye obned. ng ng pon he oluon of he ye obned nd we hve 98
11 Hndn ÇERDİK-YASA Ayşegül AKYÜZ-DAŞCIOĞU y o y whch he ec oluon of Eq.3. Eple. Conde he nonlne Vole nego-dffeenl equon y y d y 9 Ung he ehod n Secon 3 Eq. 9 olved fo 6. he oluon of h eple cn be found nlyclly by educng o dffeenl equon bu he nlycl oluon no epeened by he eleeny funcon. Howeve cn be epeened by hypegeoec funcon. he nuecl oluon of Eq. 9 wee gven by Sepehn-Rzzgh [4] nd by Avudnyg-Vn []. A copon of hee oluon wh he peen oluon gven n ble. ble. uecl eul of Eple Wvele-Glen ehod Wlh See ehod 6 Peened ehod 6 Ec Soluon
12 Chebyhev Polynol Soluon of onlne Fedhol-Vole Inego-Dffeenl Equon n y y n y e e y d wh he condon y y y. e u uppoe h y ppoed by Chebyhev ee 7 y ' Ung he pocedue n Secon 3 we fnd he ppoe oluon of h equon. A copon of he obned oluon wh he ec oluon he collocon pon gven n ble. ble. uecl eul of Eple 3 Peened ehod Ec oluon e COCUSIOS In h wo Chebyhev collocon ehod h ppled o nonlne negodffeenl equon. he udy h howed h olvng Fedhol p ee hn Vole p. An neeng feue of h ehod h he nlycl oluon obned fo lle hown n he Eple. oeove h ehod gve bee ppoe oluon hn he ohe ehod hown n he Eple. One of he dvnge of h ehod h oluon epeed unced Chebyhev ee hen y cn be ely evlued fo by vlue of.
13 Hndn ÇERDİK-YASA Ayşegül AKYÜZ-DAŞCIOĞU REFERECES. A. Ayüz. Seze A Chebyhev collocon ehod fo he oluon lne nego dffeenl equon J. Copu. h A. Avudnyg C. Vn Wvele-Glen ehod fo nego-dffeenl equon Appled uecl hec K. lened Y. houd ylo polynol oluon of hgh-ode nonlne Vole- Fedhol nego-dffeenl equon Appl. h. Copu B. Sepehn. Rzzgh Sngle-e Wlh ee ehod fo he Vole nego-dffeenl equon Engneeng Anly wh Boundy Eleen Seze S. Doğn Chebyhev ee oluon of Fedhol Inegl equon In. J. h. Educ. Sc. echnol Seze. Kyn Chebyhev polynol oluon of lne dffeenl equon In. h. Educ. Sc. echnol
HERMITE SERIES SOLUTIONS OF LINEAR FREDHOLM INTEGRAL EQUATIONS
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