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
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
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
9 Chebyhev Polynol Soluon of onlne Fedhol-Vole Inego-Dffeenl Equon 5 3 4 / 5/ 3/ / O fo odd nd 5 3 4 5/ 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
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
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 -
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
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
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 6 4 3 3 5 whee g nd y y. 5 3 5 3 b y y g y d y d 3 6 4 3 5 3 whee g nd y. 3 3 3 4 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: 4 4 8 6 3 3 5 3 5 3 5 4 5 4 4 3 3 5 3 8 4 3 nd condon equon e 97
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: 7 9 6 8 6 6 3 7 3 6 3 6 5 6 6 359 6 4 3 9 47 3 4 6 6 4 7 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
Hndn ÇERDİK-YASA Ayşegül AKYÜZ-DAŞCIOĞU y.65.5.5 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......65 -.65 -.65 -.65 -.65.5 -.5 -.498 -.498 -.498.875 -.874 -.874 -.874 -.874.5 -.497 -.4967 -.4967 -.4967.35 -.37 -.37 -.37 -.37.375 -.3734 -.37336 -.37336 -.37336.4375 -.4345 -.43446 -.43446 -.43446.5 -.4948 -.4948 -.4948 -.4948.565 -.554 -.5543 -.5543 -.5543.65 -.64 -.643 -.643 -.643.6875 -.669 -.6696 -.6697 -.6697.75 -.74 -.745 -.745 -.745.85 -.777 -.7779 -.7779 -.7779.875 -.877 -.8766 -.8767 -.8767.9375 -.8756 -.87557 -.87557 -.87557. -.95 -.947 -.947 -.948 99
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.788.788.5944.5944.85.85 3.4664.4664 4.689.689 5.49368.49369 6.396976.396976 7.367879.367879 6. 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.
Hndn ÇERDİK-YASA Ayşegül AKYÜZ-DAŞCIOĞU REFERECES. A. Ayüz. Seze A Chebyhev collocon ehod fo he oluon lne nego dffeenl equon J. Copu. h. 7 999 49-57.. A. Avudnyg C. Vn Wvele-Glen ehod fo nego-dffeenl equon Appled uecl hec 3 47-54. 3. K. lened Y. houd ylo polynol oluon of hgh-ode nonlne Vole- Fedhol nego-dffeenl equon Appl. h. Copu. 45 3 64-653. 4. B. Sepehn. Rzzgh Sngle-e Wlh ee ehod fo he Vole nego-dffeenl equon Engneeng Anly wh Boundy Eleen 8 4 35-39. 5.. Seze S. Doğn Chebyhev ee oluon of Fedhol Inegl equon In. J. h. Educ. Sc. echnol. 7 996 649-657. 6.. Seze. Kyn Chebyhev polynol oluon of lne dffeenl equon In. h. Educ. Sc. echnol. 7 996 67-6.