Modified Taylor's Method and Nonlinear Mixed Integral Equation

Transcription

1 Uversl Jourl of Iegrl quos 4 (6), 9 wwwpperscecescom Modfed Tylor's Mehod d oler Mxed Iegrl quo R T Moog Fculy of Appled Scece, Umm Al Qurh Uversy Mkh, Kgdom of Sud Ar rmoog_777@yhoocom Asrc I hs pper, oler mxed egrl equo (MI) of ype Hmmerse Volerr egrl equo (H VIS) of he secod kd, uder cer codos, re cosdered The Hmmerse egrl erm s cosdered vrle spce wh couous kerel; whle he Volerr erm me A qudrc umercl mehod s used, o o sysem of Hmmerse egrl equos (SHIs) of he secod kd I ddo, he modfed Tylor's mehod s ppled o o oler lgerc sysem (AS) Moreover, he AS s solved umerclly d he error esme, ech cse s compued Mhemcs Sujec Clssfco : 45B5, 45G Key Word d Phrses oler Iegrl quo, Hmmerse (Fredholm (F))Volerr Iegrl quo, Modfed Tylor's Mehod (MTM), oler Algerc Sysem Iroduco Cosder he I of he secod kd: ( x, ) f ( x, ) k( x, y) (, y, ( y, )) dy F(, ) ( x, ) d () The ove formul () s clled H VI of he secod kd The Hmmerse egrl erm s cosdered vrle spce wh couous kerel k( x, y), whle he Volerr egrl erm s cosdered poso wh couous kerel F (, ), [, T]; T The cos defes he kd of egrl equo, whle s cos, my e complex, whch hs physcl meg The wo fucos f (x, ) d (, x, ( x, )) re kow d couous wh s dervves, whle he fuco (x, ) s ukow If () (, x, ( x,)) ( x,), we hve he FredholmVolerr egrl equo (FVI) The dscusso of FVI of he frs kd oe, wo d hree dmesos, wh s pplcos he coc prolems, ws sed [] I [], [3], he uhor used sympoc umercl mehods o o he soluo of FVI of he secod kd I [4], he relo ewee he hree dmesol coc prolem, he heory of elscy, d FVI ws cosdered d he soluo of he I ws oed I [5], he specrl reloshps of he FVI of he frs kd d VFI, whe he kerel of poso kes geerlzed poel form, re dscussed d oed More formo for he physcl meg of he kerel F(, ) () d he followg VFI ( x, ) f ( x, ) k( x, y) F(, ) ( x, ) dyd, () c e foud [4], [5] I rece yers, umerous works hve ee focusg o he developme of more dvced d effce mehods for he ler versos of () d () The sple colloco mehod [6] d he erve mehod [7] were roduced for og he pproxmo soluo I [8], echque sed o Homoopy lyss mehod, s used, for solvg VHIs of he frs kd, whle Adou e l [9] reduced VFI of he secod kd, wh dscouous kerel o sysem of FIs usg Toeplz mrx mehod d Produc ysröm mehod I [], he wodmesol Berse

2 RT Moog operol mrces mehod, s ppled o solve (), whle [], Dsjerd e l used he rdl ss fuco pproxmo for umercl soluo of mxed VFIs Le, q (), F (, ),, o ge ( x) g (x) k(x, y) (y, (y))dy, ( x,) (x), f (x,) g (x), (, y, (y,)) (y, (y) (3) The egrl equo (3) s clled HI of he secod kd For he couous kerel of HI d x [,], we follow he work of some uhors who solved () umerclly Lrdy, [], used produc ysröm mehod; Kumr, [3], used dscree colloco ype mehod Hc, [4], [5] dscussed he exsece d uqueess of soluo of SHIs Bch spce d used projeco ero mehod o solve (3), respecvely I ddo, Keko d Xu, [6], used he degeere kerel mehod o dscuss he soluo of (3) Bs d mmuelle [7] d [8], [9], respecvely, suded he HI of he secod kd L [,], where her lyss depeded o he echque of ocompcess I [], Bugjewsk proved he uqueess heorems for ouded vro soluo d couous ouded vro of Hmmerse d V HIs Bch spce Adou e l [], used Toeplz mrx mehod o o umerclly he soluo of HI wh dscouous kerel I ddo, I [], Tylor polyoml mehod hs ee ppled o o he pproxme soluo of VFIs O he oher hd, [3] ppled ew ss fucos for pproxmg he soluo of oler VFIs v drec mehod s ppled I ddo, he mrx sed mehod, he homoopy peruro mehod d he modfed homoopy peruro mehod hve ee ppled for pproxmg he soluo of oler VFIs [4], [5] d [6], respecvely The relo ewee he HVI d he coc prolem, he heory of elscy ws dscussed [7], [8] d my pplcos were sed More formo for some dffere mehods o solve egrl equos, umerclly d s pplcos egeerg c e foud Ldopoulos [9][3] I order o guree he exsece of uque soluo of (), we ssume he followg codos: () The kerel k(x, y) C([,] [,]) hs m, dervves wh respec o x, y, respecvely () The kerel of me F(, ) C[, T], T, s posve couous fuco, d ssfes F (, ) M, M s cos,, [,T] () The gve fuco f (x, ) wh s prl dervves wh respec o poso d me re couous he spce L [, ] C[,T], d s orm s defed s: f ( x, ) L [, ] C[, T ] mx T f ( x, ) dx d H, ( H s cos) (v) The kow couous fuco (, x, (x, ) ) ssfes, for he coss A A d A A, he followg codos: () T x x dx d A x mx (,, (, ) (, ) (), x, (x, )) (, x, (x, )) (, x) (x) (x), ( where (, x) mx ( x, ) dx d A L, C, T T

3 RT Moog I he remder pr of hs pper, umercl mehod s used, () o o SHIs of he secod kd poso Moreover, he modfed Tylor's mehod s used o o he umercl soluo of SHIs, where AS s oed The exsece d uqueess of soluo of he AS re dscussed d proved Flly, wo exmples re sed o expl he mehod Sysem of oler Iegrl quos (3) (4) To o SHIs, we dvde he ervl T s T, whe,,,,, The Volerr egrl erm of (), hs cse, fer usg he qudrure rule formul (more formo s foud [33, 34]), kes he form: F( P, ) (x, ) d u j F(, j) (x, j) O ( ), (, p ) j where, mx hj, hj j j j The vlues of he wo coss u j () d p deped o he umer of dervves of he fuco 3 F(, ) wh respec o For exmple, f F(, ) C [,T], he we hve p 3, p d P u h, u h, u j h j ( j, ) Here, O( ) s he order of sum errors of he umercl mehod of dvdg he ervl [, T], d he dfferece ewee he egro d summo, where he error s deermed y: j j j F(, ) ( x, ) d u F(, ) ( x, j ) () Usg q () () d eglecg O( P ), we ge: ( x) k(x, y) (y, (y))dy g (x), (,,,, ) (3) Here, we used he followg oos: u F, ( x, ) ( x), g ( x) f ( x) u F ( x), f ( x, ) f ( x);, j, j j j ( x, ( x)) (, x, ( x, )) (4) Hece, he formul (3) represes SHIs of fe ukow fucos ( x) correspodg o he me ervl T, d depedg o he umer of dervves of F(, ) [,T] wh respec o me for ll vlues of The recurrece relos c e used o o he soluo of he sysem (3) For hs, = we hve: ( x) k (x, y) (y, (y))dy g (x), (g (x) f (x)) (5) 3 Modfed Tylor's Mehod I hs seco, we develop he Tylor expso mehod o o umerclly he soluo of (5) The Tylor expso for solvg I hd ee preseed y Kwl d Lu [35] d he hs hd ee exeded, y Sezer, o VI d Volerr dfferel equos; see [36] d [37], respecvely Here, he echque s sed o dffereg oh sdes of he I mes The, susug he Tylor polyoml for he ukow fuco he resulg equo d ler, rsformg o 3

4 RT Moog AS The exsece d uqueess of soluo of he AS re cosdered, d he he soluo of he sysem wll e oed Cosder he soluo of (5) kes he form: r ( x) (c )( x c ) ; ( x,c (,)) (3)! Ths s Tylor polyoml of degree r x c, where ( c ),,,,,r, re coeffces o e deermed To o he soluo of (5) he expresso form (3), we frs dfferee oh sdes of (5), mes wh respec o x, o ge: The, we pu where: ( ) k ( x, y) ( ) x c (3), d susue Tylor expso for ( x) ( y) dy g ( x); ( y) ( y, ( y )) (3) x (y), o hve: r ( m c m ( ) ( m) ( ) c ) k ( c ) g ( ), (33) k x y m ( ) ; (,,, ) x c (, ) km y c dy m m! x ( ) The ques m ( c), (34) q (33), wh he d of [3] c e foud he form: (h) ( ) (m) m! (c ) (c ) (c ) (c )!!!!!! p m p p h where,,, re posve egers d zeros p (c, (c )), If we ke, m,,,,, he (33) represes AS of ( ) equos for he (+) ( ) () () ukows (c ), (c ),, ( c ), s followg: m (m) (c ) k (c ) g (c ), (,,,, ) (36) m I order o guree he exsece of uque soluo of he AS (36) frsly, we he lemm: Lemm If he kerel k (x, y) of (5) possess couous prl dervves of ll order wh respec o he vrle x ope eghorhood ( c,c ) (, ), he here exss smll cos such h: () k m, m where, ( ) Proof I vew of he formul (35) we hve: k m k (x, y) x xc dy k(x, y) m (y c ) dy m! x xc m m m! ; p (35) 4

5 RT Moog Applyg CuchySchwrz equly d summg from m, o m,, he ove equly for ech (c,c ) c e dped he form: y m k ( x, y) km dy m x x c m m! ( ) (37) Theorem Uder he codo () of Lemm, d he followg codos: ( ) () g ( c) G, ( G s cos ) (3) The kow fucos c, (c )) for he coss Q Q, Q ssfy ( Q (3) (c, (c )) Q (c ), ( ) ( ) ( ) ( ) (3) ( c, ( c)) ( c, ( c)) Q ( c) ( c) he AS (36) hs uque soluo Proof To prove he heorem, we wre he AS (36) he operor form: ( ) ( ) ( ) L ( c) g ( c) L ( c), (38) Where: ( ) ( m) L ( c) km ( c, ( c)), m (,,,, ), (39) Lemm Uder he codos () (3), he operor L defed y (36) mps he spce o self Proof Le e he se of ll fucos c ), such h orm of he operor L y: ( Defe he L L c From (38) d (39), we ge : ( ) ( ) ( ) g ( ) ( c) ( m) ( ) m (, ( )) m L c k c c Applyg CuchySchwrz equly, he usg he codos (3), d summg from, we o: ( ) ( ) g c ( ) ( m ) L ( c) Q k m ( c) m m (3) o 5

6 RT Moog The ove equly, fer pplyg CuchySchwrz equly, usg he codos (), (), d leg, kes he form: G L,( Q ) (3) I vew of he equly (3), he operor L mps he se o self, where: G (3) [ Q ] Sce, G, he I ddo, from he equly (3), we c deduce he oudedess of he operor L, where: L (33) Moreover, he equles (3) d (33) volve he oudedess of he operor L Lemm 3 Uder he wo codos () d (3), L s corco operor he spce Proof Le ( c ) d ( c ) e y fucos he spce, he he lgh of formuls (38) d (39), we o: (m) (m) V (c ) V (c ) k m (c, (c )) (c, (c )) m Applyg CuchySchwrz equly, he summg from o, d usg he codos (), (3), he ove equly kes he form: m m ( ) ( ) ( ) ( ) L ( c) L ( c) ( c) ( c) m Flly, s, he ls equly reduces o L L (34) Iequly (34) shows he couy of he operor L he spce, he L s corco operor uder he codo Hece, y Bch Fxed Po Theorem L hs uque fxed po whch s he uque soluo of he AS (36) I s ovous h s, he sysem of Is (33) s equvle o he egrl equo (), d cosequely he soluo s he sme 4 Applcos I hs seco, we pply he MTM o solve he I of HVI of he secod kd xmple Cosder he HVI of he secod kd: ( x, ) ( x y x y ) ( y, ) dy ( x, ) d f ( x, ) { ( x, ) x } (4) The free erm f ( x, ) fer usg he exc soluo, yelds: f ( x, ) [( ) x ( ) x ( )] I Tle, for x,9,,4, he umercl compuol resuls of he exc d pproxme soluo of (4) re clculed I ddo, Fgure, he relo ewee he exc x,, [,) s compued soluo d he umerc soluo for ll 6

7 RT Moog xmple : Cosder he H VI of he secod kd 3 3 x e 3 x ( x, ) ( y, ) dy ( ) ( x, ) d ( e ) e { ( x, ) e } (4) 3 The exc d pproxme soluos of (4) re oed umerclly Tle, d Fgure, x,9,,4 I c oserve from he le h: The error s wh dffere vlues of for x The pproxme soluo s erly cocde wh he exc soluo for ech vlue of x he le Tle x k I Tle we clcule he lycl d pproxme soluo dffere mes The relo ewee he exc soluo d he umerc soluo s oed I Tle, d Fg, he lyc d pproxme soluo re compued me =, =, =3 d =4 for x[,9] Tle x k

8 RT Moog S Fg Fg 5 Coclusos From he ove dscusso d resuls, we c deduce he followg: We cosder geerl mxed egrl equo he oler form, me d poso Mos he egrl equos he refereces re cosdered specl cses of hs pper 3 The modfed Tylor's mehod s cosdered s he es mehods o o he soluo of he MI wh couous kerel, umerclly Fuure work I fuure work, he soluo of he geerl form of MI he oler form wll e cosdered, especlly, whe he kerel of poso kes he dscouous form Refereces Adou M A, Fredholm Volerr egrl equo of he frs kd coc prolem, Appl Mh Compu, 5 (), 7793 Adou M A, O sympoc mehod for Fredholm Volerr egrl equo of he secod kd coc prolem, J Com Appl Mh, 54 (3), Adou M A, Fredholm Volerr egrl equo wh sgulr kerel, Appl Mh Compu, 7 (3), Adou M A, Musf O L, Fredholm Volerr egrl equo coc prolem, Appl Mh Compu, 38 (3), Adou M A, Fredholm Volerr egrl equo d geerlzed poel kerel, Appl Mh Compu, 3 (), Bruer H, O he umercl soluo of he oler VolerrFredholm egrl equos y colloco mehods, Sm J umer Al, 7 (99), Wg K, Wg O, Gu K, Ierve mehod d covergece lyss for kd of mxed oler Volerr Fredholm egrl equo, Appl Mh Compue, 5 (3), Behzd Sh S, Homoopy pproxmo echque for solvg oler VolerrFredholm egrl equos of he frs kd, I J Id Mh, 6 (4), 6 9 Adou MA, lkll IL, AlBugm AM, umercl soluo for VolerrFerdholm egrl equo wh geerlzed sgulr kerel, J Mod Meh umer Mh, (), 5 Shekr FH, Mlekejd K, zz R, Applco of wodmesol Berse polyomls for solvg mxed VolerrFredholm egrl equos, Afr M, 6 (4), 7 Dsjerd HL, Gh FMM, Hdzdeh M, A meshless pproxme soluo of mxed Volerr Fredholm egrl equos, Ier J Compue Mh, 9 (3), Lrdy L J, A vro of ysröm's mehod for Hmmerse egrl equos, Mh Compu, 48 (98), Kumr S, A dscree colloco ype mehod for Hmmerse equos, SIAM, J umer Al, 5 (988), Hc L, Approxme soluos of he Hmmerse egrl equos, Fsccul Mhemc,8 (988), Hc L, Solvg oler egrl equos y projeco ero mehods, ol V Prol, 5 (993), Keko H, Xu Yu, Degeere kerel mehod for Hmmerse equos, Mh Compu, 56 (99),

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