THE SOLUTION OF THE FRACTIONAL DIFFERENTIAL EQUATION WITH THE GENERALIZED TAYLOR COLLOCATIN METHOD

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1 IJRRAS August THE SOLUTIO OF THE FRACTIOAL DIFFERETIAL EQUATIO WITH THE GEERALIZED TAYLOR COLLOCATI METHOD Slih Ylçıbş Ali Kourlp D. Dömez Demir 3* H. Hilmi Soru 4 34 Cell Byr Uiversity Fculty of Art & Scieces Deprtmet of Mthemtics Murdiye Cmpus Mis Turey 4547 ABSTRACT I this pper we propose the geerlized Tylor colloctio method for solvig the vrible coefficiets frctiol differetil equtio of order for (] uder the give iitil or boudry coditios d give mtrix represettios of the problem. Additiolly lyticl form solutio of the problem is clculted by usig this techique. Keyword: Frctiol differetil equtio Tylor colloctio method Colloctio poits.. ITRODUCTIO I this pper we cosider the vrible coefficiets frctiol differetil equtio of order : ( y ( x : P ( x y L ( x P ( x y ( x P ( x y( x f ( x ( ( ( (] P P P d f re the rbitrry fuctios of x defied o the itervl [ hvig th ( order differetitio y for deotes the. order frctiol differetitio of y with respect to x such tht is rbitrry umber i the itervl (]. More th three hudreds yers the pplictios of the frctiol clculus tht re llowed the relted problems to be more uderstdble re improved d re exteded i lmost ll fields of mthemtics d the other scieces. Usig the frctiol differetil equtios modeled i my res the obtied costructios re eeded to be solved. The frctiol clculus is delt by my uthors i most cyosure fudmetl boos re delt he frctiol clculus is. For exmple the frctiol clculus o bioegieerig i [] the fudmetl solutio of the spce time frctiol diffusio equtio. Recetly frctiol clculus hs foud ew pplictios i ssorted fields such s egieerig physics fice chemistry bioegieerig [3-59-3] etc. d is still used i ew umericl simultios of the chotic systems [54] rel world pplictios [] cotrol processig []. Also the frctiol vritiol priciples hve developed d pplied to frctiol problems []. Durig the lst yers He s vritiol itertio method hs exteded to solve the frctiol differetil equtios [6-867]. I the cse i which belogs to the itervl (] by ot chgig the structure of the secod order differetil equtios with vrible coefficiets ( the pproximte solutios re obtied. For this purpose we will proceed the Tylor colloctio method tht is preseted by Sezer d Krmete for geerl form of the m P ( x y ( 96 ( x f ( x [5]. This method is itroduced for solvig itegrl equtios by Kwll ve Liu [8] d is developed by Sezer [93]. Recetly Çeesiz proposed method i order to pply Bgley-Torvi frctiol differetil equtio i [3]. I this study we lso propose Tylor colloctio method i frctiol sese to solve the differetil equtio ( which hs frctiol derivtive.. PRELIMIARIES AD OTATIOS The defiitios of frctiol derivtive re cosidered i my ppers. I most recetly [3] the defiitios of Riem-Liouville Cputo frctiol opertors re give i sese of right d left side. The defiitio of Riem- Liouville frctiol itegrl opertor d the Cputo frctiol differetil opertor re used durig our ivestigtios. (

2 IJRRAS August Defiitio.: Let [ ( b be fiite itervl o rel xis R. The Riem-Liouville frctiol itegrl of order R ( I f of fuctio f Cμ x ( I f ( x ( x t f ( t dt ( x ; ( d ( I f ( x f ( x for. For this opertor t most commo properties re i ( I I f ( x ( I f ( x ( (4 ii ( I I f ( x ( I I f ( x ( iii ( ( It f ( x x ( ( (5 (3 (6 Sice the Riem-Liouville defiitio for frctiol derivtive is usuitble for iitil vlue frctiol problems we shll give the defiitio of Cputo frctiol derivtive (s i [3]: C Defiitio.: The frctiol derivtive ( D f ( x of order R ( o [ x ( C f ( t ( D f ( x dt ( I ( D f x ( ( x t d dx D Lemm.3: If d d f ( x AC [ i [3]. d f C is defied by (7 the C ( D I f ( x f ( x (8 C ( ( I D f ( x f ( x f ( x! x. (9 The reso of usig Cputo frctiol derivtive is its scedcy th other defiitios of frctiol derivtives i pplyig to trditiol iitil d boudry problems. For more iformtio bout frctiol derivtives itegrls d theirs properties reders c cosult to [493]. It is lso used the followig defiitio of Geerlized Tylor s Formul tht hs lredy bee writte s forml versio i [6]: Theorem.4: Suppose tht D f ( x C[ for Tylor Series expsio bout x with x x ( the we hve the i ( ( x i ( D f ( ( ( i ( i (( f ( x D f ( ( x D D. D D ( -times. 3. TAYLOR COLLOCATIO METHOD I THE FRACTIOAL SESE We will ow geerlize the method i [593] i order to solve the frctiol differetil equtios. Let us first cosider the equtio ( s 97

3 IJRRAS August with iitil coditios C i Pi ( x D y( x f ( x ( x b ( i y( x is the uow fuctio the ow fuctios P( x d f ( x D d dx re iterested d / up to the erest iteger. is ordiry differetitio such tht ( y ( R ( i re defied o the domi which we is the vlue of to be rouded Suppose tht the solutio of bove problem ( is C i i y( x ( D y( ( x R ( x (3 i ( i is chose y positive iteger with d C ( ( D y( R ( x ( x (( ( for x x ( d C i C C D D D C D (i -times. 4. THE MATRIX REPRESETATIOS C 4. For the fuctio y( x d its Cputo Frctiol Derivtive ( D y( x Let us we hve the solutio (3 of the equtio ( tht c be writte i the mtrix form X (4 [ y( x] XM A (5 ( x [ ( x ( x ( x ] d A [( D y( ( D y( ( D y( ( D y( ] C C C C T M ( ( ( ( (6 To obti solutio (5 we propose the Tylor Colloctio method i the frctiol sese s follows. I this method it is computed the geerlized Tylor coefficiets by usig colloctio poits d it is foud the mtrix A cotiig the uow geerlized Tylor coefficiets. ow let us defie the colloctio poits s so tht b xi i ; i... (7 xi b. The we substitute the colloctio poits (7 ito ( to obti the system 98

4 IJRRAS August C P ( xi ( D y( xi f ( xi ; i... (8 tht c be writte i the mtrix form P P ( x P ( x P ( x ( P Y F (9 f ( x f ( x F f ( x Y ( D y( x ( ( C ( D y( x C C ( D y x Let us ssume tht the th derivtive of the fuctio i (3 with respect to x hs the tructed Tylor series expsio C D C ( y x i i ( D y x x b ( ( ( ( ( ( ; C C ( D y(... re the geerlized Tylor coefficiets d ( D y( y(. The substitutig the Tylor colloctio poits ito ( we get the mtrix forms C [( D y( x i ] X ( x i M A (... ( or. ( Y XMA ( X( x ( x ( x ( x X ( x ( x X X( x ( x ( x colum ( ( M (( which re ( ( mtrices for. 99

5 IJRRAS August 4. For the coditios I view of ( by substitutig C [( D y( ] X( M A ito ( the coditios c lso be writte i the mtrix form s d tig it c be writte or the ugmeted mtrices of them re X M A =... ( X M = C ( c c c C A = (3 ; c c c ; C. (4 5. THE PROCESS OF THE METHOD BY USIG THE MATRIX REPRESETATIOS By cosiderig ( we hve the mtrix equtio d we c lso write (5 i the form tht correspods to system of ( P XM A F (5 WA = F or [ W; F ] (6 lgebric equtios with the uow geerlized Tylor coefficiets W [ w ] P XM p q... pq. (7 To obti the solutio of ( subject to ( ow we hve the ew ugmeted mtrix by replcig the row mtrix (3 by the lst row of mtrix (6 w w w ; f ( x w w w ; f ( x ; W ; F w m w m w m ; f ( x m. (8 c c c ; ; cm cm cm ; m ~ ~ r W r [ W; F ~ ] i (8 the we c write A ( W F (9 ~ det ( W the there is o solutio d the method cot be used or If it c be uiquely determied. If we my obti the prticulr solutios by mes of the system. 6. THE ERROR OF THE GEERALIZED TAYLOR POLYOMIAL The ccurcy of the obtied solutios c be checed by usig (4 which is icresed whe the lrge is chose d is decresed s the vlue of x moves wy from the ceter [6]. The obtied polyomil expsio is 3

6 IJRRAS August pproximte solutio whe the fuctio y( x d its derivtive ( y ( x x x equtio must be stisfied pproximtely: tht is for the colloctio poits [ re substituted i Eq. ( the resultig i i.... or C i i i i E( x P ( x ( D y( x f ( x E ( x i i ( i is y positive iteger If i mx( ( is y positive iteger is prescribed the the tructio limit is icresed util the differece E( x i t ech of poits xi becomes smller th the prescribed [593]. 7. UMERICAL EXAMPLE Exmple Firstly we cosider the problem i [6] s the fuctios ( x.5 f ( x x (.5.5 x re te i (: D.5 y P P ( x P ( x (.5.5 ( x y( x x x with the coditio y (. I order to solve the problem by usig the proposed method i Sectio 3 d 4 C.5 cosiderig the colloctio poits for 6 we firstly defie the mtrix represettios of y (x D y( x f (x d iitil coditio tht re writte from ( F T P P X M ( ( ( (3 ( 4 (5 (6 d 3

7 IJRRAS August M ( ( ( ( 4 (3 d for the coditio the ugmeted mtrix form is [ C; ] [ ; ]. (5 The pproximte solutio of y( x by the geerlized Tylor polyomil i frctiol sese with tructio error is 6 C i i y( x ( D y( ( x i ( i d. The mtrix form of the problem is defied by P XM P XM A F. After the system of the ugmeted mtrices d the coditio re computed we obti the ew ugmeted mtrix i the form ; ; ; W ; F ; ; ; ; Oce proceeds the procedure i Sectio 4 this system hs the solutio A [ T ]. Therefore from (9 we fid the exct solutio y( x x. 8. COCLUSIO I this pper we obti the mtrix formultio of the geerlized Tylor colloctio method for frctiol differetil equtio ( of order. For this reso we use Tylor colloctio method i the frctiol sese d C we cosider Cputo frctiol derivtive s ( D y( x. for the frctiol derivtive. After obtiig the mtrix represettios it is cocluded tht the coefficiets of the geerlized Tylor method c be foud by usig the resultig equtio (9. With the specific exmple it is see tht the solutio is the exct oe. 3

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