Fracional Mehod of Characerisics for Fracional Parial Differenial Equaions Guo-cheng Wu* Modern Teile Insiue, Donghua Universiy, 188 Yan-an ilu Road, Shanghai 51, PR China Absrac The mehod of characerisics has played a very imporan role in mahemaical physics. Preciously, i was used o solve he iniial value problem for parial differenial equaions of firs order. In his paper, we propose a fracional mehod of characerisics and use i o solve some fracional parial differenial equaions. Keywor: Modified Riemann-Liouville Derivaive; Fracional Mehod of Characerisics; Fracional Parial Differenial Equaions 1 Inroducion In he pas cenuries, many meho of mahemaical physics have been developed o solve he parial differenial equaions (PDEs) [1 - ], among which he mehod of characerisics is an efficien echnique for PDEs [3]. Fracional parial differenial equaions have araced many researchers ineress. Then one of he quesions may be naurally proposed: would i be possible o derive he eac soluions of parial differenial equaions (FPDEs) using a fracional mehod of characerisics? Recenly, wih he modified Riemann-Liouville derivaive [4], G. Jumaire ever proposed a Lagrange characerisic mehod [5] which can solve he FPDEs u(, ) u(, ) a(, ) b(, ) c(, ),, 1, where. However, in mos cases, he wo fracional order parameers and may no be equivalen. In his leer, we presen a more general fracional mehod of characerisics and consider he case. (1) * Corresponding auhor, E-mail address: wuguocheng@yahoo.com.cn (G.C. Wu). 1
Properies of Modified Riemann-Liouville derivaive and Inegral Through his paper, we adop he fracional derivaive in modified Riemann-Liouville sense. The modified Riemann-Liouville derivaive has been successfully applied in successfully applied in he probabiliy calculus [6], fracional Laplace problems [7], fracional variaional calculus [8] and fracional variaional ieraion mehod [9]. Firsly, we inroduce he definiions and properies of he fracional calculaion. Assume f : R R : f ( ) denoe a coninuous (bu no necessarily differeniable) funcion and le he pariion h > of an inerval [, 1]. Through he fracional Riemann Liouville inegral 1 1 I f ( ) = ( ) f ( ) d, >. ( ) () The modified Riemann-Liouville derivaive is defined as 1 d D f ( ) = ( ) ( f ( ) f ()) d, (1 ) d (3) where [,1], 1. One also can obained Eq. (3) hrough consequence of a more basic definiion, a local one, in erms of a fracional finie difference [1] k = ( FW 1) f ( ) = ( 1) f( ( k h) ) k (4) where FWf ( ) = f ( h). Then he fracional derivaive is defined as he following limi f ( ) f( ) = lim. h h (5) lised Some properies for he proposed modified Riemann-Liouville derivaive are (a) Fracional Leibniz produc law [1] ( ) ( ) ( ) D ( uv) = u v uv, (6) where ( ) ( ) u D ( u). Much more generally, we can have
n ( n ) k ( k ) (( nk ) ) n k D ( uv) = C u v. (7) (b) Fracional Leibniz Formulaion [1] I D f ( ) = f ( ) f (), < 1. (8) Noe ha he properies (a) and (b) lead o he inegraion by pars: I u v = ( uv) / I uv. (9) ( ) b ( ) a b a a b (c) Fracional Jumarie-Taylor series [1] k h ( k ) f ( h) = f ( ). (1) ( k)! k = (d) Inegraion wih respec o ( d) Assume f( ) denoe a coninuous R R funcion. We use he following equliiy for he inegral w.r. ( d) [1] 1 1 ( ) ( 1) 1 I f ( ) = ( ) f ( ) d f ( )( d), 1. (e) Some useful formulas (11) df f d (1 ) D (1 ) ( ) ( ) ([ ( )]) ( ); ( ) ( d) ; (1 ) d d. ; (1) 3 Fracional Mehod of Characerisics I is well known ha he mehod of characerisics has played a very imporan role in mahemaical physics. Preciously, he mehod of characerisics is used o solve he iniial value problem for general firs order. Consider he following firs order equaion, u(, ) u(, ) a(, ) b(, ) c(, ). (13) 3
The goal of he mehod of characerisics is o change coordinaes from (, ) o a new coordinae sysem (, s ) in which he PDE becomes an ordinary differenial equaion (ODE) along cerain curves in he plane. The cures are called he characerisic curves which read du, c(, ) d, a(, ) d. b(, ) Wih he modified Riemann-Liouville derivaive, G. Jumaire ever gave a Lagrange characerisic mehod [5] which can solve some classes of fracional parial differenial equaions. In his paper, we presen a more generalized fracional mehod of characerisics and use i o solve linear fracional parial equaions. equaions More generally, we eend his mehod o linear space-ime fracional differenial u(, ) u(, ) a(, ) b(, ) c(, ),, 1. (14) Epand u as he fracional Jumarie-Taylor s series of mulivariae funcions [1], u(, ) u(, ) du ( d) ( d),, 1. (1 ) (1 ) The oal derivaive here is more generalized. The funcion u here is (15) h order differeniable wih respecer o and h order differeniable o, respecively. Similarly, noe ha he generalized characerisic curves can be presened by du c(, ), ( d) a(, ), (1 ) ( d) b(, ). (1 ) (16) 4
Eq. (16) can be simplified as Jumaire s Lagrange mehod of characerisics (See. A. 1 in Ref. [5]) if we assume ( d) ( d) (1 ) du d u. (17) a(, ) b(, ) c(, ) c(, ) Obviously, if 1 in Eq. (19), we can ge he characerisic curve for Eq. (13) u(, ) u(, ) a(, ) b(, ) c(, ). 4 Applicaion of Fracional Mehod of Characerisics Eample 1. As he firs eample, we consider space-ime fracional equaions for he ranspor equaion in porous media, u(, ) u(, ) c,, 1. (18) Assume Eq. () subjecs o he iniial value u(,) ( ). The generalized characerisic curves saisfy du, ( d) c(1 ), ( d) (1 ). (19) Then we can obain cs C1, (1 ) sc (1 ) u C. 3, () where C 1, C and C 3 are inegral consans. Eliminaing he parameer s, we find c ha he fracional curves and u is a consan along he (1 ) (1 ) fracional curves. Then we can direcly derive he eac soluion of Eq. (18) has he following form 5
u c f( ), (1 ) (1 ) (1) where f( ) ( ). (1 ) c By seing u(, ) f ( ), (1 ) (1 ) we direcly have an eac soluion for he iniial-value problem here. equaion, Eample.. As he second eample, we invesigae he more complicaed u(, ) u(, ),, 1. (1 ) (1 ) () We can have he generalized cure equaions du, ( d), (1 ) (1 ) ( d). (1 ) (1 ) (3) Wih he properies (1), direc calculaion lea o ce 1 (1 ) ce (1 ) u c. Then we can find Eq. () has he following soluions 3 s, s, (4) u f( / ), (1 ) (1 ) (5) where he funcion f is arbirary. Assume /. Wih he properies (1), we noe ha (1 ) (1 ) u (, ) ( ) / f f, (1 ) ( (1 )) (6) and 6
As a resul, we can proof u(, ) ( ) f f /. (1 ) (9) u(, ) u(, ) (1 ) (1 ) f. / f / (1 ) (1 ) (1 ) (1 ) 5. Conclusion The classical mehod of characerisics is an efficien echnique for solving parial differenial equaions. In his paper, fracional mehod of characerisics is under consideraion for some classes of fracional parial differenial equaions and wo eamples are illusraed o show is efficiency. Besides, he presened mehod provides a poenial ool o solve fracional symmery equaions in Lie group mehod. We will discuss such work in fuure. 6. Acknowledgemen The auhor feels graeful o Prof. G. Jumarie (Deparmen of Mahemaics, Universiy of Quebec a Monreal, Canada) for his sincere help in preparing his paper. Reference [1] R. Couran, D. Hilber, Meho of Mahemaical Physics, Parial Differenial Equaions (Volume ) Inerscience, John Wiley & Sons, 196. [] Harold Jeffreys, Berha Jeffreys, Meho of Mahemaical Physics, Cambridge Universiy Press, 3rd ediion,. [3] Delgado, Manuel, The Lagrange-Charpi Mehod, SIAM Rev. 39 (1997) 98 34. [4] G. Jumarie, Sochasic differenial equaions wih fracional Brownian moion inpu, In. J. Sysems Sci. (6) (1993) 1113 113. 7
[5] G. Jumarie, Lagrange characerisic mehod for solving a class of nonlinear parial differenial equaions of fracional order, Appl. Mah. Le. 19 (6) 873 88. [6] G. Jumarie, New sochasic fracional models for Malhusian growh, he Poissonian birh process and opimal managemen of populaions, Mah. Compu. Model. 44 (6) 31-54. [7] G. Jumarie, Laplaces ransform of fracional order via he Miag-Leffler funcion and modified Riemann-Liouville derivaive, Appl. Mah. Le. (9) 1659-1664. [8] G. Jumarie, Lagrangian mechanics of fracional order, Hamilon Jacobi fracional PDE and Taylor s series of nondiffereniable funcions, Chaos. Solion. Fracal. 3 (3) (7) 969-987. [9] G.C. Wu, E. W. M. Lee, Fracional Variaional Ieraion mehod and Is Appliaion, Phys. Le. A 374 (1) 56-59. [1] G. Jumarie, Modified Riemann-Liouville derivaive and fracional Taylor series of non-differeniable funcions furher resuls, Compu. Mah. Appl. 51 (6) 1367-1376. 8