Itertiol Jourl of Mthemtics d Computtiol Sciece Vol., No. 5, 5, pp. 5-54 http://www.isciece.org/jourl/ijmcs Power Series Solutios to Geerlized Abel Itegrl Equtios Rufi Abdulli * Deprtmet of Physics d Mthemtics, Sterlitm Brch of the Bshir Stte Uiversity, Sterlitm, Russi Abstrct Eve though they hve rther specilized structure, Abel equtios form importt clss of itegrl equtios i pplictios. This hppes becuse completely idepedet problems led to the solutio of such equtios. I this pper we cosider the geerlized Abel itegrl equtio of the first d secod id. Authors hve bee proposed ew method for costructig solutios of Abel by power series. Keywords Geerlized Abel Itegrl Equtios, Itegrl Equtio, Power Series Received: My, 5 / Accepted: My 7, 5 / Published olie: Jue 3, 5 @ 5 The Authors. Published by Americ Istitute of Sciece. This Ope Access rticle is uder the CC BY-NC licese. http://cretivecommos.org/liceses/by-c/4./. Itroductio The rel world problems i scietific fields such s solid stte physics, plsm physics, fluid mechics, chemicl ietics d mthemticl biology re olier i geerl whe formulted s prtil differetil equtios or itegrl equtios. Abel s itegrl equtio occurs i my brches of scietific fields [], [] such s microscopy, seismology, rdio stroomy, electro emissio, tomic sctterig, rdr rgig, plsm digostics, X-ry rdiogrphy, d opticl fiber evlutio. Abel s itegrl equtio is the erliest emple of itegrl equtio [5], [9]. I this pper, we use the method of geerlized power series, to solve lier Volterr itegrl equtios of the first d secod id. This power series re udetermied coefficiets method, or method bsed o the pplictio of the Tylor series. The result obtied i the form of geerlized power series solutio further coverted to the iversio formul of the itegrl equtio. Oe such method is the represettio of the solutio of the equtio i the form of power series [8], []. Moreover, the bsic theorems of this sectio re give without proof.. Mi Results Cosider the geerlized Abel itegrl equtio of the first id: φ() t f ( ), () ( t) where < < rbitrry rel costt, f( ):[, b] R is give fuctio. Let s ssume tht the fuctio f( ) c be represeted s follows: f c c c () ( ) ( ) ( )... c( )... We shll see for solutio equtio i the form of the followig geerlized power series: φ() t ( t ) ( t ) ( t )... ( t )..., (3) where the uow coefficiets tht must be determied. Substitutig the power series (), (3) ito equtio (), we * Correspodig uthor E-mil ddress: rufi.bdulli94@mil.ru
Itertiol Jourl of Mthemtics d Computtiol Sciece Vol., No. 5, 5, pp. 5-54 5 obti: ( ( ) ( )... ( )...) c c( )... c ( )... m m t t t ( t) ( t ) ( t) m t ( ) z ( ) dz t ( )( z) z z dz t ( ) z t z t z m m m( ) B(, m ) cm ( ). m m m m m( ) ( ) m (4) Β Euler bet fuctio, ( ) Here (, b) fuctio. Γ Euler gmm Let, the, equtig the terms with the sme power of i (4) yields: cm mb(, m ) cm m. B(, m ) If we substitute obtied coefficiets i (3), the by simple clcultios we obti: c( ) Γ ( ) c B(, ) Γ( ) Γ ( ) φ( ) ( )! si! c( ) ( ) ( )( ) ( ) c( ) Γ Γ si! c( ) si Γ( ) Γ ( ) c( ) ( ) Γ ( ) t z, dz, si m t с( ) ( z) z dz z, z t, t z t. ( ( )... ( )... ) si c c t c t si f( t) ( t) ( t) We re thus led to the solutio: si () φ( ) t f t. (5) ( ) This solutio is ideticl to the solutio tht is obtied i [4]. Theorem. If f ( t ) C [, b], the there eists uique solutio of equtio Abel of first id, which is epressed i the form (5). The theorem is proved by direct verifictio tht the formul (5) is solutio of equtio (). Let s ow cosider geerlized Abel itegrl equtio of the secod id: ( ) y( ) y t f( ). (6) t The solutio of this equtio will be sought i the form of sum of two geerlized power series: b (7) y( )
5 Rufi Abdulli: Power Series Solutios to Geerlized Abel Itegrl Equtios Similrly to the first cse, substitutig the power series (), (7) ito equtio (6), performig the clcultios, obti b t c b t z b dz b t z t t t z t z ( ) ( ) b z z dz b z z dz 3 b Β, b Β, The by equtig the terms with ideticl powers of we get: m bmβ, m, b c, 3 3 b Β, m c b Β, m Β, m c m m m m m p! Γ ( ) Γ( p ) p! ( p )! Γ ( p ) p p p p p b c c p p, p p c Β (, p ) p ( p )! p p!, c c Β (, ) ( )!! p p p p c! p p 3 p Γ ( ) ( p Γ Γ ) Γ p p c p 3 ( p )! 3 Γ Γ p p p 3 p c Β, Β, p p p ( p )! 3 3 3 c Β, c Β, Β,! d this implies tht our solutio c be writte s 3 3 3 y( ) c Β, c Β, Β,!
Itertiol Jourl of Mthemtics d Computtiol Sciece Vol., No. 5, 5, pp. 5-54 53! c c Β (, ) F( ) c c Β, f( ) ct ( t) ct ( ) f( ) f t f( t) t 3 I c 3 3 (, ),,! c! Β Β Β 3 c t t c Β t t 3 ( ), ( )!! t ( t) ( t) 3 c c t Β,!! c t e c t Β ( t) ( t) 3,! ( ) ( t) t [ ] e f( t) F( t) f( t) e F( t) Ad s fil result: f( t) ( t) ( ) ( ) ( ) y f e F t. (8) t This solutio is ideticl to the solutio tht is obtied i [4]. Theorem. If f ( t ) C [, b], the there eist uique solutio of equtio Abel of secod id, which is epressed i the form (7). As i first cse, the theorem is proved by direct verifictio tht the formul (8) is solutio of equtio (6). 3. Coclusios I this pper, solutio is obtied by power series method. This my be used i more combitoril wy to obti solutio of higher degree o-lier itegrl equtios. Acowledgmets My sicere ths go to Prof. Adrey Aimov from the Sterlitm Brch of the Bshir Stte Uiversity for his help d dvice. Refereces [] R. Goreflo d S. Vessell, Abel itegrl Equtios: Alysis d Applictio, Spriger-Verlg, Berli-New Yor, 99. [] Jerri, A Itroductio to Itegrl Equtios with Applictios. Wiley, New Yor, 999. [3] Dvis, H. T. Itroductio to Nolier Differetil d Itegrl Equtios, st ed., Dover Publictios, Ic., New Yor, 96. [4] A. V. Mzhirov d A. D. Polyi, Hdboo of Itegrl Equtios: Solutio Methods [i Russi], Fctoril Press, Moscow,. [5] Deutsch, M, Beimiy, I Derivtive-free Iversio of AbeVs Itegrl Equtio. Applied Physics Letters 4: pp. 7-8, 98. [6] Abel, NH (86) Auflosug eier Mechische Aufgbe. Jourl für Reie Agewe Mthemti : pp. 53-57, 86. [7] Adersse, RS Stble Procedures for the Iversio of AbeVs Equtio. Jourl of the Istitute of Mthemtics d its Applictios 7: pp. 39-34, 976. [8] Mierbo, GN, Levy, ME Iversio of AbeVs Itegrl Equtio by Mes of Orthogol Polyomils. SIAM Jourl o Numericl Alysis 6: pp. 598-66, 969. [9] J. D. Tmri, O itegrble solutios of Abel s itegrl equtio, Als of Mthemtics, vol. 3, o., pp. 9 9, 93.
54 Rufi Abdulli: Power Series Solutios to Geerlized Abel Itegrl Equtios [] A.C. Pipi, A Course o Itegrl Equtios, Spriger Verlg, New Yor, 99.