NUMERICAL SOLUTIONS OF SOME KINDS OF FRACTIONAL DIFFERENTIAL EQUATIONS

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1 Jourl of Al-Nhr Uersty Vol.1 (), Deceber, 009, pp Scece NUMERICAL SOLUTIONS OF SOME KINDS OF FRACTIONAL DIFFERENTIAL EQUATIONS L. N. M. Tfq I. I. Gorl Deprtet of Mthetcs, Ib Al Hth College Eucto, Bgh Uersty. Abstrct I ths pper e preset scuss lgorth for the uercl soluto of soe s of frctol fferetl equtos,.e. of orer +, 3, orer, 3. The lgorth for the uercl soluto of these equtos s bse o Neto pproch. The stblty coergece of the frctol orer uercl etho re escrbe. Flly, soe uercl eples re proe to sho tht the uercl etho for solg the frctol fferetl equto s effecte soluto etho. Itroucto I recet yers, the cocept of frctol opertor hs bee estgte etesely y egeerg sceces pplctos. I the reserch re of frctol clculus, the teger orer of erte f () of fucto f() s geerlze to frctol orer f (), here s o- teger uber. Dfferetl equtos olg erte of o teger orer he sho to be equte oels for y fels of sceces egeerg [1,, 3]. Bsc Cocepts I ths secto, soe fuetl cocepts ecessry for ths or re trouce. Mthetcs cosere the proble of efg frctol fferetto tegrto otce s the follog []: For eery fucto f(z), z = +y of suffcetly e clss eery uber, rrtol, frctol or cople fucto D c zf (z) g(z) or D c f() g() shoul be ssge subect to the follog crter: 1. If f(z) s lytc the D c zf (z) s lytc of z.. D c f () ust prouce the se result s orry fferetto he s poste teger. If s egte teger, sy, the D c f () ust prouce the se result s orry -fol tegrto D c f () ust sh log th ts 1ertes t =c. 3. D 0 c f () f (). 13.Lerty: c D f () bg() cd f () bcd g() Defto hch fulflls these crter s the follog: Nshoto Defto for Derte of Orer, []: If f(z) s lytc fucto t hs o brch pot o se C C, C : ( 1) f (t) C f C f (z) t 1 (t z) C C here t z, rg( t z), z. ( 1) f (t) C f C f (z) t 1 (t z) here t z, 0 rg(t z), z. C C, C, the: Cf L Cf If s poste tegers, f Nshhoto frctol tegrl of orer tht s the erte of frctol orer ( 0, s the frctol tegrl of orer ) f f ests. I geerl: erte, for Re() 0 f s orgl, for 0 t egrl, for Re() 0 Proe tht f ests. No, e trouce the follog theore hch s eough to efe hyperbolc fuctos. Theore 1. [] e e, for 0, z C. z

2 L. N. M. Tfq Theore. [] e e e, here 0, z z C. Estece Uqueess Estece of solutos of frctol fferetl equtos re of gret portce prctcl pplctos of frctol fferel equtos. No, coser the frctol fferetl equto: D ( ) y() F, y()...(1) D ( ) Where y() 0, (0,1 ), here represet the frctol erte opertor for of orer suppose tht Ω s ope subset of R C[ 0, ] f : Ω R be cotuous boue fucto, the for y ( 0,0) Ω, there est soluto to the frctol fferetl equto (1) hch psses through ( 0, 0). Uqueess s copleetry ts to the spect of estece of solutos fferetl equtos geerl frctol fferetl equto prtculr. No, e scuss the uqueess of solutos of frctol fferetl equtos: Let Ω be ope subset of R C[ 0, ] suppose tht f: Ω R be cotuous fucto F(,0) be Lpschtz th respect to seco rguet eery copct subset of Ω. If ( 0,0)Ω, the the frctol fferetl equto hs uque soluto hch psses through ( 0,0). Frctol Dfferetl Equtos As s etoe secto to, e efe: f s erte, for Re() 0 orgl, for 0 t egrl, for Re() 0 I ths secto, e f the uercl soluto of the fors: 1. +-th orer frctol fferetl equtos: () b 0, here 0, 0,, tegers, gc(,)=1, Z.. -th orer frctol fferetl equtos: () () b 0,,, Z, gc(,)= th orer frctol fferetl equtos: () 0, 0,,, Z, 0 gc(,)=1 3 Z. 1. Frctol Dfferetl Equtos of Orer + Coser the hoogeous frctol fferetl equto of orer + of the for: () b 0...() 0 No, to f 's tht stsfes e of ths equto, the: e e, 0 z Substtutg (), e obt: e b be b 0 Tg the -th poer ses: soluto b 0 Whch s lgebrc equto of egree + the uo. Solg ths to get the roots 1,,, +, but soete e c ot sole ths sply, therefore e use uercl etho such Neto-Rphso etho to get these roots the soluto of () s: 0 c e, here c ' s re rbtrry costts. Eple 1 : Coser frctol fferetl equtos of () the for: b 0 such tht 0 =1,, 1 =-1, 0 =1 b=1, eps=

3 Jourl of Al-Nhr Uersty Vol.1 (), Deceber, 009, pp Scece The the frctol fferetl equto: 0 ( ) () b No, to f 's tht stsfes ths equto e, the: e soluto of No, to sole ths equto there s o lytc soluto, the e use Neto-Rphso etho t 1< < f ( ) f 1 ( ), = 1,, The the results the follog tble: The uercl soluto of eple (1) by usg the Neto-Rphso etho f ( ) f '( ) Stop coto = < eps f ( ) = < eps The s pprote 1 st root. No, f e clculte e he f ( ) 0 the s ect 1 st root. So, the se y, e f other roots for the equto therefore the geerl soluto s: z z 3z (z) c e 1 c e c e c e 1 3 z z 7z c e e c7 z ce here c ' s re rbtrry costts, 1,...,7 s the -th pprote root of the equto.. Frctol Dfferetl Equtos of Orer It's geerl for: () () b 0,...(3),, Z, gc (,)=1 To f 's tht stsfes e soluto of ths equto, the: e e b e 0 z z b 0 No, e use uercl ethos such Neto Rphso etho to f roots ( 's) of ths equtos. The, the geerl soluto of (3) s: 1 z (z) c e 0 here c re rbtrry costts. Eple : Coser frctol fferetl equtos () () of the for: b 0, such tht 1, b=3 3, eps= The the frctol fferetl equto: 1. 1 () () b No, to f 's tht stsfes e soluto of ths equto e, the: Use Neto-Rphso etho t 0 < < 1 f ( ) f 1 ( ), = 1,, The the results the follog tble: The uercl soluto of eple () by usg the Neto-Rphso etho f ( ) f '( ) Stop coto: 3 = E-8 < eps f ( ) = < eps The 0.1 s pprote 1 st root. As the se y e f reg roots for the equto therefore the geerl soluto s: 1z z (z) c e c e 1 1

4 L. N. M. Tfq Where c' s re rbtrry costts, 1,. s the -th pprote root of the equto. 3. Frctol Dfferetl Equtos of Orer, 3 Ths s the geerlzto of tht secto. ts geerl for s 0 () 0, 0...() To f ts soluto, e ust f 's tht stsfes e, the: 0 e 0 Sce e 0, the e get the lgebrc equto: 0 0 usg Neto Rphso ethos to f ts roots, the ef the geerl soluto of (). Eple 3: Coser frctol fferetl equtos of () the for: 0 0 such tht =3, 3, 0 = -, 1 = 1, = 1 3 = -1, eps=10-1 The the frctol fferetl equto: 0 () No, to f 's tht stsfes e, the: Use Neto-Rphso etho t 1 < < to f 's f ( ) f 1 ( ), =1,, The the results the follog tble: The uercl soluto of eple (3) by usg the Neto-Rphso etho f ( ) f '( ) Stop coto: = 0 < eps The 9 8 f ( 9 ) = < eps s pprote1 st root. So, the se y, e f other roots for the equto therefore the geerl soluto s: z z 3z (z) c e 1 c e c e c e z 1 3 z z 7z 8z 9z e c e e c7e c8e c9 c Where c ' s re rbtrry costts, 1,..., 9 s the -th pprote root of the equto. M Propertes of the Algorth We shll o escrbe the propertes of the lgorth. I prtculr, e f tht, th respect to the ost portt questos, the behor of the etho s epeet of the preter tht t behes y tht s ery slr to the clsscl Neto- Rphso etho. Moreoer, t s o proble to ete the cocept to clue ore frctol orer equtos, ee f the orer of the fferetl opertors ole res fro equto to equto. 1. Stblty The ssue of stblty s ery portt he pleetg the etho o coputer fte-precso rthetc becuse e ust te to ccout the effects trouce by roug errors. It s o [8] tht the clsscl Neto-Rphso etho s resoble prctclly useful coprose the sese tht ts stblty propertes llo for sfe pplcto to lly stff equtos thout uue 1

5 Jourl of Al-Nhr Uersty Vol.1 (), Deceber, 009, pp Scece propgto of roug errors, heres the pleetto oes ot requre etreely te cosug eleets. Fro the results of [7] e c see tht these propertes re uchge he e loo t the frctol erso of the lgorth ste of the clsscl oe, therefore t s lso cler tht the behor oes ot epe o the orer of the fferetl opertors ole. Coergece Of course, stblty loe s ot suffcet prctce to e sure tht the uercl soluto s goo pproto to the ect soluto. We ust lso ress the proble of error esttes,.e. the questo of coergece. I ths cotet, e c use soe of the str lyss techques to ere tht [ssug suffcet soothess of the fuctos ole] the coergece orer of the schee s,.e. e he error bou of the for: M ( ) O(h )...() here h M ( ) here e 1 ssue tht ll, 0 0 th soe fe *>0. Seco orer coergece y see soe ht slos, but oe ust eep tht ost of the preters of our oel (clug the orer of the fferetl equto tself), thus lso the put lues of our lgorth, re terl costts tht re usully o oly up to ery lte ccurcy. Thus t oes ot e sese to pply hgh orer etho, especlly he e ber tht hgh orer ethos frequetly sho feror stblty propertes he copre to lo-orer ethos. [3] K. Dethel, Alyss of Frctol Dfferetl Equtos, Mthetcs, Uersty of Mchester, Egl,1999. [] B. Ross, Frctol Clculus Its Applctos, Sprger-Verlg, Berl, 197. [] R. Ngtull, Frctol Clculus, Itegtos Dfferettos of Arbtrry Orer, Descrt-Press Co., Kory, 198. [] S. N, Soe Results Frctol Clculus, Ph.D. Thess, Ib Al-Hth, Uersty of Bgh, 00. [7] C. Lubch, Dscretze Frctol Clculus, SIAM.J. Mth. Al. Vol.17, 198, PP [8] P. Lz, Alytcl Nuercl Metho for Volterr Equtos, SIAM, Phlelph, PA, 198., 3, 3 Refereces [1] A. Scht L. Gul, FE Ipleetto of Vsoelstc Costtute Stress-Str Reltos Iolg Frctol Te Dertes, Jourl of ler Dycs, Vol. 1, 000, pp.3-0. [] L.Yu O. Agrl, A Nuercl Schee for Dyc Systes Cotg Frctol Dertes, Proceegs of DEC98, ASME Desg Egeerg, Techcl Coferece, Septeber, 1998, pp.13-1, Atlt, Georg. 17