SOLUTION OF TWO DIMENSIONAL FRACTIONAL ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

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1 Jourl of Al-Nhr Uversty Vol. (4), Deeber, 009, See SOLUTION OF TWO DIMENSIONAL FRACTIONAL ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS Mh A. Mohed * d Fdhel S. Fdhel ** * Dertet of Mthets, Ib-Al-Hth College of Eduto, Uversty of Bghdd. ** Dertet of Mthets, College of See, Uversty of Al-Nhr. Abstrt I ths er, our s to study vrtol forulto d solutos of -desol tegrodfferetl eutos of frtol order. We wll gve suery of reresetto to the vrtol forulto of ler ohoogeous -desol Volterr tegro-dfferetl eutos of the seod kd wth frtol order. A ele wll be dsussed d solved by usg the MthCAD softwre kge whe t s eeded..itroduto The tegrl euto s euto whh the ukow geerlly y futo of oe or ore vrbles, ours uder tegrl sg, [6]. Also, the oe-desol tegrl euto s tegrl whh the ukow futo deeds oly o oe deedet vrble, [5]. Whle the ult-desol tegrl euto s eteso of the oedesol tegrl eutos. My reserhs studed the ult-desol rtl tegro-dfferetl eutos, sy Beker 999 lso Vder d Soejer 996, [] d []. Ths er ossts of four setos, seto oe, troduto bout tegrl eutos. I seto two, we gve soe deftos of the rtl frtol dervtves d frtol tegrto. I seto three, we del wth the vrtol ethod to solve ler roble. I seto four, we gve the vrtol forulto to solve the twodesol frtol tegro-dfferetl eutos wth ele el ths roh.. The Vrto Aroh The roble lulus of vrto s to fd the u or u vlues of gve futol F(u), ths eessry odto s lled the Euler-Lgrge euto d the soluto of ths roble s lled the dret roble of lulus of vrto, [], [3] d [8]. Soe ortt deftos d oets whh re eeded to uderstd the vrtol roh wll be gve [7] d []. The ost ortt dffulty of the subjet of lulus of vrto s to fd the vrtol forulto, whh orresods to the ler oertor euto L u = f... () Where f deotes slr-vetor vlued futol d L deotes ler oertor. Theore (.): If the gve ler oertor L s syetr wth reset to the hose o-degeerte bler for < u,v >. But f the ler oertor L s ot syetr wth reset to the hose bler for <u,v >, d the roble s to fd the vrtol forulto, the ths ould be doe the trsforto: (u,v) = < u,lv >, where v V d u D(L)... () The bler for () kes the gve ler oertor syetr se: (Lu,u ) = < Lu,Lu > = < Lu,Lu > = (Lu,u ) Therefore, geerl we wll use the bler for () to fd vrtol forulto beuse of the syetry of L. Se L s syetr d by usg theore (.), the soluto of euto () s rtl ot to the futol: F(u) = (Lu,u) (f,u) 85

2 Mh A. Mohed = < Lu,Lu > < f,lu >...(3) The futol (3) s vrtol forulto for the ler euto Lu = f, [9]. 3.Frtol Clulus 3. Itroduto to Frtol Clulus I reet yers, there hs bee growg terest the feld of frtol lulus. Euto of frtol order hs ered ore d ore dfferet reserh felds d egeerg ltos. Frtol lulus s the feld of thetl lyss whh dels wth the vestgto d ltos of tegrls d dervtves of rbtrry order. The frtol lulus y be osdered s old d yet ovel to, tully, t s old to se strtg fro soe setru of Lebz ( ) d Euler (730) who sd "whe s teger, the rto d, s futo of, to d lwys be eressed lgebrlly. Now, t s sked: wht kd of rto be de f s frto? ", t hs bee develoed u to owdys. I ft, the de of geerlzg the oto of dervtve to o-teger order, rtulr to the order of (whh s lled setegerl or se-dervtve) s foud the orresodee of Lebz d Beroull, L Hotl d Wlls. Euler took the frst ste by observg tht the result of the dervtve evluto of the ower futo hs eg for o-teger order thks to hs G futo, [4]. 3. Re Louvlle Prtl Frtol Dervtves The frtol dervtve of order of f(,y) s () ( ) f (, y) ( t) f (t, y)dt...(4) d () yf (, y) (y t) f (, t)dt..(5) ( ) resetvely se s egtve frtol uber, d: d f (, y) ( t) f (t, y)dt d ( )... (6) d yf (, y) (y t) f (, t)dt d ( )... (7) resetvely se s ostve frtol uber, [9]. 3.3 G Futo () The G futo () lys ortt role the theory of dfferetto. The defto for the () s gve by () y e( y) dy, 0 0 It s ore useful th other deftos of (), [0]. 4.Vrtol Forulto for Frtol Clulus Cosder the -desol ler frtol order tegro-dfferetl euto of Volterr tye: f (, y) g(, y) k(, y,z,)f (z,)dzd... (8) where b() d y d(y). whh hve tegrl oertor of the for L k(, y,z,)dzd where s rtl frtol oertor of, s frto uber d g(,y) s gve futo, 0 < <. The oertor L s ler se t s esly see tht: L(w f w f ) w f (, y) w w f (, y) w k(, y, z, )f (z, )dz d k(, y, z, )f (z, )dz d wlf w Lf Therefore, the oertor L s ler. Now, defe the bler for 86

3 Jourl of Al-Nhr Uversty Vol. (4), Deeber, 009, See T T f,f f (, y)f (, y) d dy...(9) where T, y T. Thus, the vrtol forulto of the gve ler oertor L ould be foud s follows F(f) = < Lf,Lf > < g,lf > T T (Lf (, y)) d dy T T [ f (, y) T T g(, y) Lf (, y)d dy k(, y,z,)f (z,)dz d] d dy T T [g(, y)[ f (, y) k(, y,z,)f (z,)dz d]d dy...(0) To solve the bove vrtol forulto, oe ust rote the soluto f of euto (8) s ler obto of lerly deedet futos {Q (, y)}, (the dret Rtz ethod) suh tht; f (, y) Q (, y)...() where { } re the ukow reters tht ust be detered. The, fter substtutg ths roted soluto to the futol gve by euto (0), oe get: T T F(f ) [ f (, y) k(, y,z,)f (z,)dz d] d dy T T g(, y)[ f (, y) k(, y,z,)f (z,)dz d]d dy... () where f e s the et results ssoted wth the roble, whle f s the rote results of ths roble. The roble here s to fd the rtl ot of the bove futol F(f) whh ould be foud by usg the ethod of vrto. Flly, the vlues of { } re obted by solvg the syste of ler eutos F whh be obted by settg 0, =,,,. The oe solve ths syste by usg the Mth CAD softwer Bkge for euto (), we get tbulted results. To llustrte ths ethod, osder the followg ele. Ele: Cosder the followg Volterr frtol tegro-dffretl euto y 0.5 f (, y) g(, y) ( y z)f (z,)dz d where g(, y) 4y y y 3 the lytl soluto of ths roble s gve by: f e (,y) = y By usg the dret Rtz ethod d Re-Louvlle rtl frtol dervtves euto (6), the f (,y) = y y + 5 y...(3) Hee y f (, y) 7 3 y 4 5y 7...(4) The oertor s gve by: y 0.5 L ( y z)dz d Assue y 0.5 R (, y) f (, y) ( y z)f (z,)dz d...(5) Hee, the vrtol forulto wth (3), (4), d (5) s gve by 3 3 F(f ) (R (, y)) d dy g(, y)r (, y)d dy...(6) 87

4 Mh A. Mohed Flly, the vlues of { } re obted by solvg the syste of ler eutos F whh be obted by settg 0, =,,,. The oe solve ths syste by usg the Mth CAD softwer Bkge for euto (6), we get tbulted results. The roted solutos by usg the vrtol ethod. Bss futos Coffets of the Bss Aroted Solutos y y y Aother se, whe rtl frtol for y: Cosder the ler frtol order tegrodfferetl euto of Volterr tye: f (, y) g(, y) y k(, y,z,)f (z,)dzd...(7) where b() d y d(y). whh hve tegrl oertor of the for L k(, y,z,)dzd y where y s rtl frtol oertor of y, s frto uber d g(,y) s gve futo, 0 < <. We rove slrly s the frst se the lerly d syetry of the oertor L. Therefore, the vrtol forulto s gve by T T F(f ) [ yf (, y) k(, y,z,)f (z,)dz d] d dy T T g(, y)[ f (, y) y k(, y,z,)f (z,)dz d]d dy...(8) We use the slr roh se oe, to fd the rtl ot of the bove futol F(f). Colusos. The -desol frtol tegrodfferetl eutos re so dffult, ost ses, to be solved lytlly, therefore uerl ethods re reured.. Whe we solve the frtol tegrl eutos usg vrtol roh, the bsolute error ofte rohes zero betwee the et d rote results. Referees [] Al-A s, J.A., "Vrtol Methods for Boudry Vlue Probles", M.S. Thess, College of See, Bghdd Uversty, 987. [] Beker, P., "A New Perturbtve Tehue for Solvg Itegro-Prtl Dfferetl Eutos", Jourl of Mthetl d hyss, Volue 40, No. 0, 999. [3] El sgol, L.E., "Clulus of Vrtos", Pergo Press, L td., 96. [4] Goreflo, R. d Mrd, F., "Essetls of Frtol Clulus", Mhysto Ceter, Prelry verso, 8 Jury 000. [5] Hohstdt, H., "Itegrl Eutos", Sos, 973. [6] Lz, P., "Alytl d Nuerl Methods for Volterr Eutos", S Phldelh, 985. [7] Mgr, F., "Vrtol Forulto for Every Ler Euto", It. J. Egeg. S., Vol., ,

5 Jourl of Al-Nhr Uversty Vol. (4), Deeber, 009, See [8] Mysks, A.D., "Adve Mthets for Egeers", Sel Courses, MIR Publshers, Mosow, Eglsh Trslto, 975. [9] Nshoto, K., "Frtol Clulus: Itegrtos d Dfferettos of Arbtrry Order", Desrtes Press Coy Kory J, Vol. 3, 989. [0] Oldh, K.B. d Ser, J., "The Frtol Clulus: Theory d Altos of Dfferetto d Itegrto to Arbtrry Order", Ade Press, New York d Lodo, 974. [] Reddy, J., "Aled Futol Alyss d Vrtol Methods Egeerg", Megrw-Hll I., 986. [] Vder, P., d Soejer, B., "Euler- Chebyshev Methods for Solvg Itegro- Dfferetl Euto", NM-96, No. 30, 996. (MthCAD 00 Softwre Pkge) (Volterr) 89