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
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
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
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
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
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: 5 8 4 3 f ( x, ) [( ) x ( ) x ( )] 6 5 5 3 3 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
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 3 3 3 4 4 4 9999999 9999999 99999998 9999997 9999996 9999995 8 4 6 8 9 99999 99999 999997 999995 999993 99999 99999 99999 99999 999997 999995 999993 99999 99999 9999995 999999 999997 9999957 999993 999998 999993 9999994 999999 999997 9999956 999999 999997 999993 9999994 999999 9999969 9999956 999998 999997 99999 9999993 9999989 9999998 9999955 999997 999996 99999 9999993 9999988 9999967 9999955 999997 999995 99999 9999993 9999987 9999966 9999953 999996 999995 99999 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 3 3 3 4 4 4 9 9 9 9 9 9 8 8 7 6 6 6 5 4 4 99 98 98 97 96 96 5 9 89 89 88 87 86 85 84 6 79 79 78 77 76 75 74 74 8 7 69 68 67 66 65 64 63 9 58 57 56 55 54 53 5 5 7
RT Moog 8 8 6 6 4 S 4 4 6 8 Fg 4 6 8 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), 43 446 3 Adou M A, Fredholm Volerr egrl equo wh sgulr kerel, Appl Mh Compu, 7 (3), 3 43 4 Adou M A, Musf O L, Fredholm Volerr egrl equo coc prolem, Appl Mh Compu, 38 (3), 99 5 5 Adou M A, Fredholm Volerr egrl equo d geerlzed poel kerel, Appl Mh Compu, 3 (), 894 6 Bruer H, O he umercl soluo of he oler VolerrFredholm egrl equos y colloco mehods, Sm J umer Al, 7 (99), 987 7 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), 63637 8 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), 57538 Lrdy L J, A vro of ysröm's mehod for Hmmerse egrl equos, Mh Compu, 48 (98), 43 6 3 Kumr S, A dscree colloco ype mehod for Hmmerse equos, SIAM, J umer Al, 5 (988), 38 34 4 Hc L, Approxme soluos of he Hmmerse egrl equos, Fsccul Mhemc,8 (988), 4 48 5 Hc L, Solvg oler egrl equos y projeco ero mehods, ol V Prol, 5 (993), 35 4 6 Keko H, Xu Yu, Degeere kerel mehod for Hmmerse equos, Mh Compu, 56 (99), 4 48 8
RT Moog 7 Bs J, Iegrle soluo of Hmmerse d Urysoh egrl equos, J Aus Mh Soc A, 46 (989), 6 68 8 mmuele G, Iegrle soluos of fucol egrl equos, J I q Appl, 4 (99), 89 94 9 mmuelle G, Iegrle soluos of Hmmerse equos, Appl Al, 5, (993), 77 84 Bugjewsk D, O BV soluos of some oler egrl equo, Iegrl q Oper Theory, 46 (3), 387 398 Adou M A, l Bor M M, lkojk M M, Toeplz mrx mehod d oler egrl equo of Hmmerse ype, J Cm Appl Mh, 3 (9), 765776 Wg K, Wg Q Tylor polyoml mehod d error esmo for kd of mxed Volerr Fredholm egrl equos, Appl Mh Compu, 9 (4), 5359 3 Prpour M, Kmyr M, umercl soluo of oler VolerrFredholm egrl equos y usg ew ss fucos, Commu umer Al, 3 (3), 4 Husse SA, Shhmord S, Tl F, A mrx sed mehod for wo dmesol oler Volerr Fredholm egrl equos, umer Algor, 4 (4),86 5 Ghsem M, Kj MT, Dvr A, umercl soluo of wodmesol oler dfferel equo y homoopy peruro mehod, Appl Mh Compu, 89 (7),34345 6 Dog C, Che Z, JgW, A modfed homoopy peruro mehod for solvg he oler mxed VolerrFredholm egrl equo, J Compu Appl Mh, 39 (3), 359366 7 Adou M A, O he soluo of ler d oler egrl equo, Appl Mh Compu, 46 (3), 857 87 8 Adou M A, lsyed W G, Dees I, A soluo of oler egrl equo, Appl Mh Compu, 6 (5), 4 9 Ldopoulos G, oler egrodfferel equos sdwch ples sress lyss, Mech Res Commu, (994), 95 3 Ldopoulos G, oler sgulr egrl represeo for usedy vscd flowfelds of D rfols, Mech Res Commu, (995), 5 34 3 Ldopoulos G, oler muldmesol sgulr egrl equos dmesol flud mechcs lyss, I JoL Mech, 35 (), 7 78 3 Ldopoulos G d Zss VA, xsece d uqueess for oler sgulr egrl equos used flud mechcs, Appl Mh, 4 (997), 345 367 33 Delves L M, Mohmed J L, Compuol Mehods for Iegrl quos, Cmrdge Uv Press, Cmrdge, 985 34 Golerg M A ed, umercl Soluo for Iegrl quos, Pleum Press, ew York, 99 35 Kwl R P, Lu K C, A Tylor expso pproch for solvg egrl equos, I J Mh duc Sc Techol, (989), 44 36 Sezer M, Tylor polyoml soluo of Volerr egrl equos, I J Mh duc Sc Techol 5 (994), 6563 37 Sezer M, A mehod for he pproxme soluo of he secodorder ler dfferel equos erms of Tylor polyomls, I J Mh duc Sc Techol, 7 (996), 889 9