Taylor series expansion of nonlinear integrodifferential equations
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1 AMERCA JOURAL OF SCEFC AD DUSRAL RESEARCH 2, Sciece Huβ, SS: X doi:.525/jsir yor series expsio of oier itegrodiffereti equtios Eke A.. d 2 Jckreece P. C. Deprtet of Mthetics, Uiversity of igeri, suk 2 Deprtet of Mthetics/Sttistics, Uiversity of Port Hrcourt, Cho Port Hrcourt ABSRAC this study yor s ethod is deveoped to fid pproxite soutio for iiti vue proe for oier itegro-differeti equtios of the Fredho type. he ethod trsfors the oier itegro-differeti equtio to trix equtio which correspods to syste of oier equtios with ukow coefficiets. eywords: tegro-differeti equtios, yor s pproxitio, Fredho equtio. RODUCO Fidig exct soutios of oier itegrodiffereti equtios re usuy difficut to sove yticy. Sice y physic proes re odeed y itegr d itegro-differeti equtios, the ueric soutios of such equtios hve ee highy studied y y uthors (see, Avudiyg d Vi (2, Godfie (977, Rshed (23, Sezer (994, ythe d Puri (22, w d Liu (989. uerous works hve ee focusig o the deveopet of ore dvced d efficiet ethods for sovig itegr d itegrodiffereti equtios. E-Syed d Ade (23 did coprtive study of the pproxite soutio of itegro-differeti equtio usig Adoi decopositio ethod which is sei-ytic techique d Wveet-Gerki s ethod. Avudiyy d Vi (2, Mhoudi (25, the ueric soutio of the oier itegr equtio ws coputed usig Wveet- Gerki ethod. Here the cotiuous Legedre wveets costructed o the iterv [, ] is used to sove the oier Voterr d itegr equtio of the secod kid. he oier prt of the itegr is pproxited y Legedre wveets, d the syste of itegr equtio is reduced to syste of oier equtios. recet yers there hs ee icresig iterest i yor s series soutio of itegr d itegrodiffereti equtios. cis (22, Mekejd d Mhoudi (23, w d Liu (989, Dri d Edi (26, Sezer (994, deveoped yor s expsio pproch to fid the pproxite soutio for oier itegr d itegro-differeti equtios. hey trsfored the oier itegr d itegro-differeti equtio to trix which correspods to syste of oier equtios. However Rshed (23, used the Lgrge iterpotio ethod to copute the ueric soutios of differeti d itegro-differeti equtio whie i Hosseii d Shhor (23, ethod is used to fid the pproxite soutio of Fredho itegrodiffereti equtio with ritrry poyoi ses. A these ethods require ore efforts to chieve the resut d re usuy deveoped for speci types of itegro-differeti equtios. this pper we cosider sovig oier itegrodiffereti equtio ( y doptig the sic ides of the works of Mekejd d Arzhg (26, Dri d Edi (26, Mekejd d Mhoudi (23. Cosider the oier itegrodiffereti syste of the for ( x y ( x f ( x + ( x t, g( t y( t u,, dt x ( with iiti coditio y y Where, re costts, u x, f x, x, t, g t, y t re kow fuctios ( d y( x is the soutio to e deteried. We ssue tht the fuctios u ( x, f ( x, ( x, t, g( t, y( t re cotiuous d re + tie cotiuousy differetie o the iterv [, ], so g ( t y( t, ( x, t [, ] d u( x, f ( x L [, ],
2 A. J. Sci. d. Res., 2, 2(3: Let the soutio of ( e expressed i ters of yor poyoi s ( + y x y x,! which is yor poyoi of degree + t the ( poit of expsio x d y re the coefficiets to e deteried. Mtrix represettio of copoets: Let us rewrite ( s i, Mekejd d Arzhg (26, Mekejd d Mhoudu (23 s ( x ( x D (2 where D ( x u( x y ( x d is ced the differeti prt ( x f ( x ( x t g( t y( t +,, dt (3 s ced the itegr prt of equtio (. We eed to d the itegr prt covert the differeti prt D( x ( x to trix for. Differetitig (2 -ties w.r.t x we oti D ( x ( x (4.2.. MARX REPRESEAO OF HE DFFEREAL PAR he differeti prt D ( x c e writte s ( (,,..., ( (,,..., ( 5 D u x y x ud y + C Du. D ( y + C D u. D ( y C D u. Dy + y D u ( D u x y x ud y C Du D y C D u D y C D u D y Dy D u Usig Leiitz s rue, for the j th ter ( j k k+ uj ( x y ( x U x k k (6 for,,...,,,,...,,, the ( + ( + ( ( x [ u x y x ],[ u x y ],...,[ u x y x ] U where U is trix d is otis s ( ( U ( U U ( ( i i ( i U U ( U ( ( ( ( U... (... U U U of the copoets of (5 t x we oti (7 377
3 A. J. Sci. d. Res., 2, 2(3: Ad ( j ( j ( y (, y + (,..., y + (8 % d U % usig Let us defie ew trices (7 s U A U B [ ] Where A d B re ( ( j j j U i (9 + trices with zero eeets respectivey. Aso with (8 we defie ( ( ( % + ( +,,...,, (,..., ( y y y y y ow we c costruct the differeti prt of (2 t x i trix for ( ( ( D, D (,..., D % Where U (.2.2. MARX REPRESEAO OF HE EGRAL PAR Accordig to Dri d Edi (26 ( x e writte s ( x f ( x + F ( x where (2 c F ( x ( x, t g t, y t dt x We set g ( t, y( t G( y( t where G is kow sooth fuctio with G ( y( t oier i y ( t. he yor expsio of G( y( t t y( t is give s Let W i d! dy ( G( y y d y ( t he equtio ( c e writte s G ( y( t W ( t Sustitutio (3 ito (2 we otis ( x f ( x + W ( x, t x For, we hve ( t the yor expsio of ( t t t (4 ( t dt (5 d for > we oti s i Dri d Edi (26, i the foowig for ( t (! t (6 Aso if we sustitute equtio (6 ito equtio (5, we oti ( x, t x f + W x Which c we writte s (! ( ( ( x f + Where (,, x t, W( x t dt! (,,,,..., t dt (8 (7 i for,,..., i equtio (8 c e foud fro the coditio t d peruttio retio he qutities G ( y( t G( y y ( t d! dy y (3 378
4 A. J. Sci. d. Res., 2, 2(3: ( t ( t ( t (...,,... t + t t > tt 2... t (, 2 y y y (9! Where tt2... t t! t2!... t!, ( tt 2... t re positive itegers or zero As i Mekejd d Mhoudu (23, Mekejd d Arzhg (26, Dri d Edi (26, equtio (8 c e put i trix for s he trix F F F,, re defied y ( ( ( ( f f... f,,...,,,..., ,,..., (2... ( ( ( ( Fro this oier syste, the ukow yor ( + coefficiets y (,,2,..., re deteried d sustituted i (2, thus we oti the pproxite soutio of the equtio ( i the for ( y x y x (2! + COCLUSO oier itegrodiffereti equtios re usuy difficut to sove yticy. y cses, it is required to oti the pproxite soutio. this study vritio of yor poyoi pproch hs ee used to pproxite soutio of oier itegrodiffereti equtio of the fredho type. he preset ethod is efficiet ethod for the cses tht the kow fuctios hve eough derivtives withi the give iterv. he ethod trsfors oier itegrodiffereti equtio to trix equtio which correspods to syste of oier equtios with ukow coefficiets. Oe of the dvtges of this ethod is tht the soutio is expressed s tructed yor series t x c, the y( x c esiy e evuted for ritrry vues t ow coputtio effort. REFERECES Avudiyg, A d Vi, C (2, Wveet-Gerki ethod for itegro-differeti equtios, App. uer., 32,
5 A. J. Sci. d. Res., 2, 2(3: Dri,P d Edi, A. (26, Deveopet of the yor expsio pproch for oier itegrodiffereti equtios, t. J. Cotep. Mth. Sciece, Vo., o. 4, Esyed,S. M. d Ade-Azizi, M. R. (23, Acopriso decopositio ethod d wveet-gerki ethod for sovig itegro-differeti equtio, App. Mth. Coput. 36, Godfie, A (977, yor series ethods for the soutio of voterr itegr equtios d itegro-differeti equtios, Mthetics of coputtio, Vo. 3, Hosseii, S. M. d Shhorh, S. (23, u ueric soutio of Fredho itegro-differeti equtio with ritrry poyoi ses, App. Mth. Mode, 27, ythe, P..d Puri, P. (22, Coputtio ethods for ier itegr equtios, Uiversity of ew Ors, Lo 748, USA. w, R. P. d Liu,. C. (989, A yor expsio pproch for sovig itegr equtio, J. Mth. Edu. Sci. echo, 2, (3, Mekejd, d Mhoudu,. (23, yor poyoi soutio of high-order oier voterrfredho itegro-differeti equtios, Appied Mthetics d Coputtio, 45, Mekejd, d Arzhg, A (26, ueric soutio of the Fredho sigur itegro- differeti equtio with Cuchy kere y usig yor series expsio d Grerki ethed, Appied Mthetics d Coputtio, 82, Mhodi,. (25, Wveet- Gerki ethod for ueric soutio of oier itegr equtio, Appied Mthetics d Coputtio, 67, Rshed, M.. (23, Lgrge iterpotio to copute the ueric soutios of differeti d itegrodiffereti equtios, App. Mth. Coput. Vo. 43, (, Sezer, M. (994, yor poyoi soutio of voterr itegr equtios, t. J. Mth. Edu. Sci. echo, 25 (5, cis, S. (22, yor poyoi soutio of oier Voterr-Fredho itegr equtio, App. Mth. Coput. 27,
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