Green s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary Value Problem

Transcription

1 IOSR Jourl of Mthemtics (IOSR-JM) e-issn: , p-issn: X. Volume 3, Issue 3 Ver. IV (My - Jue 7), PP -8 Gree s Fuctio Approch to Solve Nolier Secod Order Four Poit Directiol Boudry Vlue Prolem Goteti V R L Srm, Awet Merhtu, Merhtom Sehtu 3 Deprtmet of Mthemtics, Eritre Istitute of Techology, Mi-efhi, Eritre. Astrct: I this rticle four poit oudry vlue prolem ssocited with secod order differetil eutio ivolvig directiol derivtive oudry coditios is proposed. The its solutio is developed with the help of the Gree s fuctio ssocited with the homogeeous eutio. Usig this ide d Itertio method is proposed to solve the correspodig olier prolem. Key words: Gree s fuctio, Schuder fixed poit theorem, Vitli s covergece theorem. I. Itroductio No locl oudry vlue prolems rise much ttetio ecuse of its ility to ccommodte more oudry poits th their correspodig order of differetil eutios [5], [8]. Cosiderle studies were mde y Bi d Fg [], Gupt [4] d We [9]. This reserch rticle is cocered with the existece d uiueess of solutios for the secod order four poit oudry vlue prolem with directiol type oudry coditios u" f ( t, u), t [, ], (.) u( ) ku( ), u( ) ku( ); (.) where is give fuctio d k, k R The Gree s fuctio plys importt role i solvig oudry vlue prolems of differetil eutios. The exct expressios of the solutios for some lier ODEs oudry vlue prolems c e expressed y the correspodig Gree s fuctios of the prolems. The Gree s fuctio method will e used to oti iitil estimte for shootig method. The Grees fuctio method for solvig the oudry vlue prolem is effect tools i umericl experimets. Some BVPs for olier itegrl eutios the kerels of which re the Gree s fuctios of correspodig lier differetil eutios. The udetermied prmetric method we use i this pper is uiversl method, the Gree s fuctios of my oudry vlue prolems for ODEs c e otied y similr method. I (8), Zho discussed the solutios d Gree s fuctios for o locl lier secod-order Three-poit oudry vlue prolems. suject to oe of the followig oudry vlue coditios: i. ii. iii. iv. where k ws the give umer d is give poit. I (3), Mohmed ivestigte the positive solutios to sigulr secod order oudry vlue prolem with more geerlized oudry coditios. He cosider the Sturm-Liouville oudry vlue prolem with the oudry coditios, where re ll costts, is positive prmeter d is sigulr t. Also the existece of positive solutios of sigulr oudry vlue prolems of ordiry differetil eutios hs ee studied y my reserchers such s Agrwl d Stek estlished the existece criteri for positive solutios sigulr oudry vlue prolems for olier secod order ordiry d dely differetil eutios usig the Vitli s covergece theorem. Gticl et l proved the existece of positive solutio of the prolem with the oudry coditios, usig the itertive techiue d fixed poit theorem for coe for decresig mppigs. Recetly Goteti V.R.L. Srm et l., studied the solvility of four poit oudry vlue prolem with ordiry oudry coditios u f ( t), t [, ] stisfyig the oudry coditios u( ) k u( ), u( ) k u( ); where re rel costts. DOI:.979/ Pge

2 Gree s Fuctio Approch to Solve Nolier Secod Order Four Poit Directiol Boudry This rticle is orgized s follows: I sectio we costruct the Gree s fuctio to the homogeeous BVP correspodig to (.) stisfyig (.) d the usig this we proved the existece d uiueess of the solutio of the oudry vlue prolem (.) stisfyig the coditio (.). I sectio 3 we preset the itertive method of solutio to the correspodig o lier oudry vlue prolem. We illustrted our results y costructig suitle exmple. II. The Gree s fuctio: We hve the followig coclusios: Theorem. Assume k k. The the Gree s fuctio for the secod-order four-poit lier oudry vlue prolem (.), (.) is give y k k t k k t G( t, s) K( t, s) k k where ( s )( t), s t K( t, s) ( s)( t ), t s t K (, s) K (, s) Proof: It is well kow tht the Gree s fuctio is K(t, s) s i (.) for the secod-order two-poit lier oudry vlue prolem u f ( t), t [, ], u( ), u( ) Ad the solutio of (.3) is give y d t (.) (.) (.3) w( t) K( t, s) f ( s) ds, (.4) w( ), w( ), w( ) K(, s) f ( s) ds. (.5) The four-poit oudry vlue prolem (.), (.) c e otied from replcig u() =, u() = y u( ) k u ( ) d u( ) k u ( ) i (.3). Thus, we suppose the solutio of the four-poit oudry vlue prolem (.), (.) c e expressed y u( t) w( t) c dt w ( ) w ( ) (.6) where c d d re costts tht will e determied. From (.5), (.6) we kow tht u( ) c d w ( ) w ( ) u( ) c d w ( ) w ( ) u( ) w( ) c d w( ) w( ) u( ) w( ) c d w( ) w( ) Puttig this ito (.) yields c w( ) w( ) d k w( ) w( ) kw( ) cw( ) w( ) d k w( ) w( ) k w ( ) k k, solvig the system of lier eutios o the ukow umers c, d, usig Crmer s Sice rule we oti DOI:.979/ Pge

3 Gree s Fuctio Approch to Solve Nolier Secod Order Four Poit Directiol Boudry c d kw( ) k kw( ) k k k w( ) w( ) kw( ) kw( ) k k w( ) w( ) Hece, the solutio of (.) with the oudry coditio (.) is u( t) w( t) ( c dt) w( ) w( ) k( k t) w( ) k( k t) w( ) wt () k k This together with (.4) implies tht ( ) ( ) ( ) k k t (, ) ( ) (, ) ( ) (, ) ( ) k k t u t K t s f s ds Kt s f s ds Kt s f s ds k k k k Coseuetly, the Gree s fuctio G(t, s) for the oudry vlue prolem (.), (.) is s descried i (.). From Theorem. we oti the followig corollry. k k, the the secod-order four-poit lier oudry vlue prolem Corollry.. If u f ( t), t [, ], u( ) k u ( ), u( ) k u ( ) hs uiue solutio u( t) G( t, s) f ( s) ds where G(t, s) is s i (.). Proof: Assume tht the secod-order four-poit lier oudry vlue prolem (.7) hs two solutios u(t) d v(t), tht is Let u f ( t), t [, ], u( ) k u ( ), u( ) k u ( ) v f ( t), t [, ], v( ) k v ( ), v( ) k v ( ) (.7) d DOI:.979/ Pge (.8) z( t) v( t) u( t), t, (.9) The (.7) d (.8) z ( t) v ( t) u ( t), t,, therefore z( t) Ct C, d z( t) C. (.) where C d C re udetermied costts. From (.7), (.8) d (.9) we hve z( ) v( ) u( ) k z( ), (.) z( ) v( ) u( ) k z( ). (.) Usig (.) we oti z( ) C C, (.3) z( ) C C, (.4) z( ) C, (.5) z( ) C, (.6)

4 Gree s Fuctio Approch to Solve Nolier Secod Order Four Poit Directiol Boudry From (.), (.3) d (.5) we kow tht C k C (.7), d from (.), (.4) d (.6) we kow tht C k C (.8), Solvig the system of eutios (.7) d (.8), we get C d C. Therefore z( t), t,, so u( t) v( t), t,, tht is uiueess of the solutio. Corollry.. Suppose the olier fuctio g(t, u) is cotiuous o olier four-poit oudry vlue prolem u g( t, u), t [, ], u( ) k u ( ), u( ) k u ( ) is euivlet to the olier itegrl eutio,, the if k k, the u( t) G( t, s) g s, u( s) ds where G(t, s) s i (.) If the edpoits of the itervl re, i the oudry coditio, from Theorem., Corollries. d. we oti the followig corollry. Corollry.3. If k k the the Gree s fuctio for the secod-order four-poit lier oudry vlue prolem is where u f ( t), t [,], u() k u ( ), u() k u ( ) k k t Bt(, s) k k t Bt(, s) G( t, s) B( t, s) k k s( t), s t B( t, s) ( s) t, t s Hece the prolem (.9) hs uiue solutio If g(t, u) is cotiuous o u( t) G( t, s) f s ds.,, the the olier four-poit oudry vlue prolem u g( t, u), t [,], u() k u ( ), u() k u ( ) is euivlet to the olier itegrl eutio u( t) G( t, s) g s, u( s) ds. Exmple: Cosider the secod-order four-poit oudry vlue prolem: u cost, t [,], u() u, u() u (.9) (.) (.) Sice 4 k d k re ot eul, from (.), the Gree s fuctio is: 3 3 DOI:.979/ Pge

5 Gree s Fuctio Approch to Solve Nolier Secod Order Four Poit Directiol Boudry k k t Bt(, s) k k t Bt(, s) G( t, s) B( t, s) k k where k,, k, t t G( t, s) B( t, s) Bt, s Bt, s where 6 5 t s, s t B( t, s) ( s) t, t s s, s t Bt ( t, s) which implies ( s), t s s, s s, s Bt (, s) d Bt (, s) s, s s, s Hece the solutio of secod-order four-poit oudry vlue prolem is: u( t) G( t, s) f sds B( t, s) 4 3 t B, s t B, s f sds 6 5 3(t ) 3t 4 t 6t 8 cos( t) cos() si si 6 5 III. Applictio to olier prolem: I this sectio, we study the itertive solutios for the followig olier four-poit oudry vlue prolem u f ( t, u), t (,), u() k u( ), u() k u( ) (3.) with,,,. k k Let J I,,,,,, D xc( I) M m, such tht m t x( t) M t, t I. x x x x Cocerig the fuctio f we impose the followig hypotheses: f ( t, u) is oegtive cotiuous o J, f ( t, u) is mootoe icresig o u, for fixed t J, (,) such tht f ( t, ru) r f ( t, u), r, ( t, u) J. (3.) Oviously, from (3.) we oti (, ) f t u f ( t, u),, ( t, u) J. (3.3) We c see tht if, ( t) re oegtive cotiuous o J, for i =,,,,m, the i i m i f ( t, u) ( t) u stisfy the coditio (3.). i i DOI:.979/ Pge

6 Gree s Fuctio Approch to Solve Nolier Secod Order Four Poit Directiol Boudry Cocerig the oudry vlue prolem (3.), we hve followig coclusios. Theorem 3.. Suppose the fuctio f(t,u) stisfy the coditio (3.), it my e sigulr t t= d/or t=, d f ( t, t) dt. (3.4) The olier sigulr oudry vlue prolem (3.) hs uiue solutio w(t) i C( I) C ( J). Costructig successively the seuece of fuctios h( t) G( t, s) f ( s, h ( s)) ds,,,... (3.5) for y iitil fuctio h ( t ) ( ), t I of covergece is the h t () must coverge to w(t) uiformly o I d the rte mx ( ) ( ) h t w t O N, (3.6) ti where < N <, which depeds o the iitil fuctio h ( ), (, ) t G t s s i (.). Proof. Let P x( t) x( t) C I, x( t), Fx( t) G( t, s) f ( s, x( s)) ds, x( t) D. It is esy tht the opertor : (3.7) F D P is icresig; From Corollry.3 we kow tht if u D u( t) Fu( t), t I, (3.8) the u C I C J is solutio of (3.). For y x D, there exist positive umers m M such tht By (.) d (.) we hve m ( s) x( s) M ( s), s I, x x mx f ( s, s) f ( s, x( s)) M x f ( s, s), s J. k k k k t x (3.9) G( t, s) B( t, s) x k k( k t) Bt(, s) k( k t) Bt(, s) G( t, s) B( t, s) k k stisfies t B (, s) B (, s) t k t k( k t) kbt(, s) Bt(, s) t G( t, s) ( t) k k kbt(, s) k( k t) Bt(, s) G( t, s) t( t) k k k( k t) kbt(, s) Bt(, s) t G( t, s) ( t) k k Usig (3.7), (3.3) d (3.9)-(3.) d the coditios (3.), we oti (3.) (3.) DOI:.979/ Pge

7 Gree s Fuctio Approch to Solve Nolier Secod Order Four Poit Directiol Boudry Fx( t) G( t, s) f ( s, x( s)) ds t t k ( k t) k B (, s) B (, s) t ( t) mx f ( s, s) ds k k k( k t) k t ( t) m x Bt(, s) f ( s, s) ds Bt(, s) f ( s, s) ds, t I k k (3.) Fx( t) G( t, s) f ( s, x( s)) ds k( k t) kbt(, s) Bt(, s) t ( t) M x f ( s, s) ds k k k k kk B(, s) k kk B(, s) t ( t) M x f ( s, s) ds, t I k k k k (3.3) By (3.4), (3.) d (3.3) we oti For y o F : D D. h D, we let l sup l lh ( t) Fh ( t), t I, ho o o L if L Lh ( t) Fh ( t), t I, ho o o m mi, lh, M mx, L o h o u ( t) mh ( t), u ( t) Fu ( t) v ( t) Mh ( t), v ( t) Fv ( t),,,,... (3.4) (3.5) Sice the opertor F is icresig, from (3.), (3.4) d (3.5) we kow tht u ( t) u ( t) u ( t) v ( t) v ( t) v ( t), t I. (3.6) m For t, from (3.), (3.7) d (3.5), it c otied y iductio tht M u ( t) t v ( t), t I,,,, (3.7) From (3.6) d (3.7) we kow tht u ( t) u ( t) v ( t) u ( t) t Mh ( t),, p (3.8) p so tht there exist fuctio w() t D such tht DOI:.979/ Pge

8 d Gree s Fuctio Approch to Solve Nolier Secod Order Four Poit Directiol Boudry u ( t) w( t), v ( t) w( t), (uiformly o I), (3.9) u ( t) w( t) v ( t), t I,,,, (3.) From the opertor F is icresig d (3.5) we hve u ( t) Fu ( t) Fw( t) Fv ( t) v ( t),,,, This together with (3.9) d uiueess of the limit imply tht wt () stisfy (3.8), hece w() t C I C J is solutio of (3.). From (3.5) d (3.5) d the opertor F is icresig, we oti u ( t) h ( t) v ( t), t I,,,, (3.) thus, from (3.8), (3.) d (3.) we kow h ( t) w( t) h ( t) u ( t) u ( t) w( t) so tht v ( t) u ( t) t M h ( t), mx h ( t) w( t) t M mx h ( t). ti ti So tht (3.6) holds. From h () t which is ritrry i D we kow tht w(t) is the uiue solutio of the E. (3.8) i D. Suppose () w t is C I C J z( t) w ( t) Fw( t), t I. solutio of oudry vlue prolem (3.). Let Similr to the proof of (.9) i sectio we oti w ( t) w( t), hece wt () is the uiue solutio of E. (3.) i C I C J. Remrk: If f t, u is cotiuous o I X R + uiue solutio wt C J. the it is uite evidet tht the coditio (3.4) holds. Hece the Refereces [] Al-Hy, W. (7). A domi Decompositio method with Gree s fuctios for solvig Twelfth-order of oudry vlue prolems. Applied Mthemticl scieces, Vol. 9, 5, o. 8, [] Beder, C. M. d S. A. Orzg (999). Advced mthemticl methods for scietists d Egieers; Asymptotic methods d perturtio theory, ACM3. [3] Bechohr, M. et l Secod-Order oudry vlue prolem with itegrl oudry coditios. Boudry vlue prolems rticle ID 639 vol. [4] Dr. Risighi, M. D. (3), Itegrl eutios d oudry vlue prolems sixth editio; S. Chd & Compy PVT. LTD. [5] Greegrd, L. d V. Kokhli (99). O the umericl solutio of two-poit oudry vlue prolems. Commuictios o pure d pplied mthemtics vol. XLIV, 49-45(99) [6] Goteti V R L Srm, Merhtom Sehtu d Awet Merhtu, Solvility Of Four Poit Nolier Boudry Vlue Prolem Itl Jourl of Egieerig Reserch d Applictios. Volume 7, Issue ( Prt 4) Ferury 7 Pp 8. [7] Herro, I. H. Solvig sigulr oudry vlue prolems for ordiry differetil eutios. Cri. J. Mth. Comput. Sci. 5, 3, - 3. [8] Kumli, P. (3/4), A ote o ordiry differetil eutios; TMA4/MAN 67 Fuctiol Alysis. Mthemtics Chlmers & GU [9] Liu, Z., Kg, S. M d J. S. Ume (9). Triple positive solutios of olier third order oudry vlue prolems. Tiwese Jourl of Mthemtics Vol. 3, o. 3 pp [] Mohmed, M. & W. A. W. Azmi (3), positive solutios to solutios to sigulr secod order oudry vlue prolems. It. Jourl of mth. Alysis, Vol.7, 3, o. 4, 5-7. [] Risighi, M. D. (), Itegrl eutios d oudry vlue prolems sixth editio; S. Chd & Compy PVT. LTD. New Delthi- 55. [] Teteri, A. O. (3), The Gree s fuctio method for solutios of fourth order olier oudry vlue prolem. The uiversity of Teessee, Koxville [3] Yg, C. & P. Wg (7). Gree s fuctio d positive solutios for oudry vlue prolems of third order differetil eutios. Computers d mthemtics with pplictios 54(7) [4] Zho, Z. (7) positive solutios for sigulr three-poit oudry vlue prolems. Electroic Jourl of Differetil eutios Vol. 7(7), o. 56, pp. -8. ISSN [5] Zho, Z. (7), Solutio d gree s fuctios for lier secod order three-poit oudry vlue prolems; Computers d mthemtics with pplictios 56(8)4-3. DOI:.979/ Pge