Solution of Impulsive Differential Equations with Boundary Conditions in Terms of Integral Equations
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1 Joural of aheacs ad copuer Scece ( Soluo of Ipulsve Dffereal Equaos wh Boudary Codos Ters of Iegral Equaos Arcle hsory: Receved Ocober 3 Acceped February 4 Avalable ole July 4 ohse Rabba Depare of aheacs, Sar Brach, Islac Azad Uversy, Sar, Ira rabba@us.ac.r Absrac I hs arcle, we droduce soluo of pulsve dffereal equaos wh boudary codos by usg vareaoal erao ehod (VI ers of egral equaos. For fdg he above soluo, a frs we oba a solve for dffereal equaos wh boudary codos. Keywords: Ipulsve, Dffereal, Equao, Iegral. Iroduco Ipulsve probles are dscuased [4,8]. I hs paper we use varaoal erao ehod o oba soluo of pulsve proble. I order o, s eed for a revewg o VI. I [7] Iou proposed a geeral Lagrage ulpler ehod for solvg olear dffereal equaos he followg for: Lu NU = g, ( where L s a lear operaor, N s a olear operaor, g ( s a ow aalyc fuco ad u s a uow fuco ha o be deered. Also o he supposo ha u s he soluo of LU =, soe of he specal po such as =, oher words, u cor Lu Nu g, ( ( = u( d where s a geeral lagrage ulpler ad opu value. The Iou ehod s odfed by He as follows, correspodg auhor 39
2 ohse Rabba / J. ah. Copuer Sc. ( u ( = u ( Lu N u g ds, (3 where u s a al approao ad u ~ s a resrced varao, such ha u = (see [5,6], he above egral Eq.(3 s called a correco fuco ad de of deoes he h approao. Also Eq.(3 s called varaoal erao ehod (VI. I [5,6]hs ehod s used for solvg olear probles. The varaoal erao ehod s effecve ad easy for lear proble because eac soluo ca be gve by oly oe erao. I he above process Eq.(3 s wre followg for afer he s obaed. Lu ( s Nu ( s g( s. u ( = u( ds (4 I Eq.(4 by usg u ( as a al approao, we oba a sequece of approaos of eac soluo of Eq.(. For llusrag of effecvely, easly ad accuraely a large class of olear probles wh approaos whch coverge qucly. we gve uber of applcao of varaoal erao ehod. Ths ehod s used o solve Burger ad coupled Burger equaos [], (VIs used for solvg Foer-Plac equaos [3]. Also [], (VI s appled for solvg olear syse of ordary dffereal equaos. Thus, we ca say varaoal erao ehod s a well ow ehod o solve olear equaos. 3. Soluo of dffereal equaos wh boudary codos We wll solve he followg probles; f (,, J [,] ( (, ( (, (5 where f C[ J E, E]. g(,,, J [,], ( (, ( (. (6 where g C[ J E E, E]. y, J [,];, y C[ J, E] ( (, ( (. (7 To solve Eqs.(5-6, a frs we solve Eq.(7, where ( s uow fuco, s cosa ad y ( s ow. we proof soluo of Eq.(7 s gve he followg for, 3
3 ohse Rabba / J. ah. Copuer Sc. ( * ( (, (, G s y s ds (8 Such ha C J E (,, also ( s ( s e e, * s G s e ( s ( s (,,. e e, s (9 Proof: Accordg o (VI Eqs.(3-4, for fdg opal value of, we cosruc corrco fucoal by effec o he boh sdes of (4, so we have '' ( ( (, ( ( ( s s s y s ds '' ( (, ( (, ( (, (, s s ds s s ds s y s ds ( By cosderg ys ( ad egrag by pars saoary codos, we have ' S ' ( (, ( S (, ( (, ( S s s s s ds s s ds ' ' ' ( (, ( (, ( (, ( (, ( S s s ds s s ds ' ' ' ( (, ( (, ( (, ( ( S s s ds s s ds ' because (, we ca wre,, (, ' s ( (, ( S (, ( s (, ( (, ( ss s s s s ds s s ds ' ( (, ( s (, ( s(, ( (, ( (, ( ss s s ds s s ds ' s (, ( (, ( (, s (, s ( s ds, ss (, sce (, so accordg o rgh sde of ( we have a dffereal equaos as follows; ss (, s (, s, s (,, (,. ( Easley we ca fd he soluo of Eq.(, as; 3
4 ohse Rabba / J. ah. Copuer Sc. ( ( s ( s (, s e e, ( By subsug ( s, (4 we wll ge a sequece of approae of eac soluo, ( s ( s '' ( ( e e ( ( ( s s y s ds ( s ( s '' ( e e ( s ds ( ( s s ( s ( s e e ( (, s ds e e y s ds s ( s ( s ( s ( s ( e e ( s e e ( s ds s ( ( s s ( s ( s e e ( (, s ds e e y s ds Also, we have ( ( ( e e ( e e ( s ( s ( s ( s ( s e e ( s e e ( s ds s ( ( s s ( s ( s e e ( (, s ds e e y s ds ' ( e e ( ( e e ( ( s ( s e e y ( s ds, A las, we oba ' ( e e ( e e ( ( s ( s e e y ( s ds. (3 I Eq. (3, o fd ( ad (, we use boudary codo of Eq.(7, 3
5 ohse Rabba / J. ah. Copuer Sc. ( ( ( ( (, ( ( ( (, (4 By dffereag of Eq. (3 ad usg Eq. (4, we have a syse he followg for, ( s ( s e e ( e e ( e e y ( s ds ( s ( s e e ( e e ( e e y ( s ds. (5 By assue e, he (5 syse have a soluo as follows; ( s s ( (, e e y s ds ( s s ( (. e e y s ds (6 By subsug Eq.(6 Eq.(3, we oba ( s s ( e e e e y ( s ds ( s s e e e e ( y s ds ( s ( s e e y ( s ds, By cosderg o e, he, hs way we have, e ( s ( s ( s ( s (7 ( e e y ( s ds e e y ( s ds. So, * ( (, (, G s y s ds Where 33
6 ohse Rabba / J. ah. Copuer Sc. ( * G (, s ( s ( s e e, S ( s ( s e e, S By cosderg of Eq. (7, we ca wre ( s y ( s ( s, ad for obag soluo of Eqs. (5-6, we use ( s he followg for; * ( (,, ( (, G s f s s s ds (8 ad * ( (,, (, ( (. G s g s s s s ds (9 4. Soluo of pulsve dffereal equaos wh boudary codos We cosder o a pulsve (D.E as follows; f (,,, (,,..., I (,, ( (, ( Where,..., f C[ JXP, P], J [,], f (,, I C [ P, P], I ( (,,...,, P s a coe E ad J J \,,...,, such ha,,..., are f PC J P C J E [, ] [, ] be a soluo of (, he s he followg for; ( C (, s f s, ( s ds I ( where, s G (, s {, s}. s, s ( Proof: For showg ( o ( Eq for we use varaoal erao ehod so, ( ( (, s ( s f s, ( s ds, ( by effec o he boh sdes of ( Eq, we have, 34
7 ohse Rabba / J. ah. Copuer Sc. ( ( ( (, s ( s f s, ( s ds ( (, s ( s ds (, s f s, ( s ds, ( (, s ( s ds, by assupo ( s ad ( s are couous, so, we ca wre, s s f s s ds s s ds ( ( (, (, ( (, ( (, s ( s ds s s ds ( (, ( (, s s s s ds s ( ( (, ( (, ( s ( (, (, (, ( (, ( (, ( ( (, (, (, (, s (, (, ea of ( s s (, ( s s s ss s s ds s ss (, s ( s ds l ( s l ( s ( ( s s ( (, ( (, (... (, ( (, ( (, (... (, ( s (, ( s (, (... s (, ( s (, ( s (, (... s (, ( order o we have, ss (, s ( s ds... ss (, s ( s ds 35
8 ohse Rabba / J. ah. Copuer Sc. ( ( (, ( (... (, ( ( s (, s ( (... s (, ( ( s (, ( (, ( ss (, s ( s ds ( (,... (, (, ( So, we have, s (, s ( (... s (, ( ( s (, ( ss (, s ( s ds ( s (, ( ss (, s ( s ds s (, ( Slarly ( Eq we ca wre, ss (, s, s (,, s (,. so (, s ( s. (3 by subsug (3 Eq ( Eq, we oba, ( ( ( s ( s f s, ( s ds ( ( (... ( ( ( ( ( s f s, ( s ds f C JP, P ( ( ( (, ( ( s s ds s s ds s s ds s s ds s f s s ds s ( ( ( (, ( s where ( s ds ( ( so, s s s s f s s ds 36
9 ohse Rabba / J. ah. Copuer Sc. ( ( ( ( ( ( ( ( ( s f s, ( s ds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (... ( ( ( s f s, ( s ds ( ( ( ( (, ( ( ( I ( ( s f s, ( s ds a las, we oba s f s s ds ( ( s f s, ( s ds I ( (4 (4 Eq s equal o (. 4 Cocluso I hs arcle we raduce sple ad drece ehod for obag soluo of pulsve dfferaal equaos ers of egral equaos such ha soe reffereces such as [8] he above resul s fd by he cople ad dfcul ehod. Refereces [].A, Abdou, A.A. Sola, varaoal erao ehod for solvg Burger's ad coupled Burger's equaos, J.copu.Appl.ah, 8 ( [] J. Bazar, H. Ghazv, he's varaoal erao ehod for solvg lear ad olear syses of ordary dffereal equaos, Appled aheacs ad copuao, 9 ( [3]. Dehgha,. Taar, he use of He's varaoal erao ehod for solvg he Foer- Plac equao,phys.scrpa,74 ( [4] D.Guo ad X. Lu, Ereal soluos of olear pulsve egro-dffereal equaos Baach Spaces, J. ah. Appl. 77 (993, [5] J.H. He, varaoal erao ehod for olear ad 's applcaos, echacs ad pracce,, ( ( ( chese 37
10 ohse Rabba / J. ah. Copuer Sc. ( [6] J.H. He, varaoal erao ehod - a d of olear aalycal echque:soe eaples, I.Joural of Nolear echacs, 34 ( [7]. Iou, geeral use of he Lagrage ulpler olear aheacal physcs,: S.Nea-asser(Ed., Varaoal ehod echacs of solds, Progao press, oford,( [8] X. Lu, oooe erave echque for pulsve dffereal equaos a Baach space. J. ah. Phy. Sc. 4(99,
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