APPROXIMATE SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH UNCERTAINTY ZHEN-GUO DENG ad GUO-CHENG WU 2, 3 * School of Mahemaics ad Iformaio Sciece, Guagi Uiversiy, Naig 534, PR Chia 2 Key Laboraory of Numerical Simulaio of Sichua Provice, Neijiag, Sichua 642, PR Chia 3 College of Mahemaics ad Iformaio Sciece, Neijiag Normal Uiversiy, Neijiag, Sichua 642, PR Chia *E-mail: wuguocheg22@yahoo.com.c Received July 2, 2 Fracioal differeial equaios have bee caugh much aeio durig he pas decades. I his sudy, ieraio formulae of a fracioal differeial equaio wih uceraiy are proposed ad he approimae soluios for a simple case are derived via a fracioal variaioal ieraio mehod. Key words: Modified Riema-Liouville derivaive; Fracioal Variaioal Ieraio Mehod; Fracioal differeial equaios. INTRODUCTION I he las decades, scieiss ad applied mahemaicias have foud fracioal differeial equaios useful i various fields: rheology, quaiaive biology, elecrochemisry, scaerig heory, diffusio, raspor heory, probabiliy poeial heory ad elasiciy. For deails, see he moographs of Kilbas e al. [], Kiryakova [2], Lakshmikaham ad Vasala [3], Miller ad Ross [4], ad Podluby [5]. Recely, Agarwal proposed he cocep of soluio for fracioal differeial equaios wih uceraiy [6]. Oe kid of differeial equaios wih uceraiy of he ype is govered by d ( ) = f (, ( )), < T, () d where f (, ()) E E is coiuous, ad () = E. If f (, ( )) R R ad R, he Eq. () reduces o a fracioal differeial equaio. Rom. Jour. Phys., Vol. 56, Nos. 7 8, P. 868 872, Buchares, 2
2 Approimae soluio of fracioal differeial equaios 869 If =, he Eq. () is a fuzzy differeial equaio. Thus, Eq. () is a ew dyamic sysem called fuzzy differeial equaios of fracioal order. Agarwal e al. iroduced he cocep of soluio for such sysem. I his sudy, we adop he modified Riema-Liouveille derivaive [7] hrough his paper. The fracioal derivaive has bee successfully used i fracioal Lagrage mechaics [8], fracioal variaioal approach [9, ] ad fracioal Lie group mehod []. Wih a fracioal variaioal ieraio mehod [2, 3], approimae soluios of Eq. () are give. 2. PROPERTIES OF MODIFIED RIEMANN-LIOUVILLE DERIVATIVE Comparig wih he classical Capuo derivaive, he defiiio of modified Reima-Liouville derivaive is o required o saisfy higher ieger-order h derivaive ha. Secodly, derivaive of a cosa is zero. Now we iroduce some properies of he fracioal derivaive. Assume f : R R, f( ) deoe a coiuous (bu o ecessarily differeiable) fucio i he ierval [, ]. Through he fracioal famous Riema Liouville iegral I f f Γ( ) ( )= ( ξ) (ξ)dξ, >, (2) he modified Riema-Liouville derivaive is defied as [7] d D f( )= ( ) ( f( ) f())d, Γ( ) d ξ ξ ξ (3) where [,] ad < <. I he e secios, we will use he iegraio wih respec o (d ) (Lemma 2. of [4]) I f f f Γ( ) Γ ( + ) ( ) = ( ξ) ( ξ)d ξ = ( )(d ), <. ξ ξ (4) 3. FRACTIONAL VARIATIONAL ITERATION METHOD The variaioal ieraio mehod for differeial equaios has bee eesively worked ou for may years by umerous auhors. I his mehod, he equaios are iiially approimaed wih possible ukows. A correcio fucioal is esablished by he geeral Lagrage muliplier which ca be ideified opimally via he variaioal heory. Besides, he VIM has o resricios or
87 Zhe-Guo Deg, Guo-Cheg Wu 3 urealisic assumpios such as liearizaio or small parameers ha are used i he oliear operaors. We cosider he simple case of Eq. () o illusrae he fracioal variaioal ieraio mehod. Oe of he liear fracioal equaios is d ( ) + a() = d(), < T, <, a R. (5) d The we ca cosruc he followig ieraio form () = () + λ (, ){ + a () d()}(d), (6) + Γ( + ) where λ (, ) is ukow ad o be deermied. Wih he fracioal variaioal heory, we ca fid δ () δ + () =δ () + (, ){ + a() d()}(d) Γ( + ) λ ( ) = ( + λ ) δ( ) ( a ) ( )(d ). = + δ Γ( + ) λ λ λ (, ) mus saisfy + λ = ad As a resul, λ (, ) ca be ideified eplicily = ( ) a = (7) λ λ. (8) λ(, ) = E ( a( ) ), (9) where E ( a( ) ) is defied by he classical Miag-Leffler fucio k z E ( z) =. () Γ ( +k) Therefore, we ca obai he followig ieraio formulae for Eq. (5), k = () = () E ( a( ) ){ + a ( ) d()}(d). + Γ( + ) O he oher had, if () is hadled as a resriced variaio i Eq. (9), similarly, he Lagrage muliplier ca be ideified by ( ) + λ = ad λ. = =
4 Approimae soluio of fracioal differeial equaios 87 As a resul, we ca derive he geeralized muliplier λ (, ) =. If we assume d( ) = for simpliciy, we ca have he ieraio form () = () { a ()}(d). () + Γ( + ) Sar from () = (), we ca obai () = { a }(d) Γ( + ) a =, Γ( + ) () = () { a ( )}(d) 2 Γ( + ) 2 2 a a = +, Γ( + ) Γ( + 2 ) () = () { a ( )}(d) 2 3 2 2 Γ( + ) 2 2 3 3 a a a = + The ieraio process leads o he resul, Γ( + ) Γ( + 2 ) Γ( + 3 ) ( a) () = () = = E ( a ). k k (2) = k= Γ( + k) We ca check Eq. (2) is he eac soluio of he followig equaio d ( ) + a() =, < T, <, a R. d 4. CONCLUSIONS I his sudy, approimae soluio of a fracioal differeial equaio is ivesigaed by fracioal variaioal ieraio mehod. Compared wih he previous works via variaioal ieraio mehod, his paper esablishes a fracioal fucioal
872 Zhe-Guo Deg, Guo-Cheg Wu 5 ad derives a geeralized Lagrage muliplier for he give fracioal differeial equaio. This approach ca be used o solve oher fracioal oliear differeial equaios. REFERENCES. A.A. Kilbas, H.M. Srivasava, J.J. Trujillo, Theory ad Applicaios of Fracioal Differeial Equaios, Elsevier Sciece B.V, Amserdam, 26. 2. V. Kiryakova, Geeralized Fracioal Calculus ad Applicaios, Logma Scieific & Techical, Harlow, 994, copublished i he Uied Saes wih Joh Wiley & Sos, Ic., New York. 3. V. Lakshmikaham, A.S. Vasala, Basic heory of fracioal differeial equaios,noliear Aal., 69, 2677 2682 (28). 4. K.S. Miller, B. Ross, A Iroducio o he Fracioal Calculus ad DiffereialEquaios, Joh Wiley, New York, 993. 5. I. Podluby, Fracioal Differeial Equaio, Academic Press, Sa Diego, 999. 6. Ravi P. Agarwal, V. Lakshmikahama, Jua J. Nieo, O he cocep of soluio for fracioal differeial equaios wih uceraiy, Noliear Aal., 72, 2859 2862 (2). 7. G. Jumarie, Modified Riema-Liouville derivaive ad fracioal Taylor series of o-differeiable fucios furher resuls, Compu. Mah. Appl., 5, 367 376 (26). 8. G. Jumarie, Lagragia mechaics of fracioal order, Hamilo Jacobi fracioal PDE ad Taylor s series of odiffereiable fucios, 32, 969 987 (27). 9. R. Almeida, A.B. Maliowska ad D.F.M. Torres, A Fracioal Calculus of Variaios for Muliple Iegrals wih Applicaio o Vibraig Srig, J. Mah. Phys. 5, 3353 (2). A.B. Maliowska, M.R.S Ammi ad D.F.M Torres, Composiio fucioals i fracioal calculus of variaios, Commu. Frac. Calc., 4 47 (2). G.C. Wu, Variaioal Approach for Fracioal Parial Differeial Equaios, arxiv:6.4999v. G.C. Wu, Fracioal Lie Group Mehod for Aoymous Diffusio Equaios, Commu. Frac. Calc., 27 3 (2). 2. G.C. Wu, E.W.M. Lee, Fracioal Variaioal Ieraio Mehod ad Is Applicaio, Phys. Le. A, 374, 256 259 (2). 3. G.C. Wu, Fracioal Variaioal Ieraio Mehod for Fracioal Noliear Differeial Equaios, Comup. Mah. Appl., Acceped. 4. G. Jumarie, Laplace s rasform of fracioal order via he Miag Leffler fucio ad modified Riema Liouville derivaive, Appl. Mah. Le. 22, 659 664 (29).