A Fractional Lie Group Method For Anomalous Diffusion Equations

Comm un. F rac. Calc. 1 (2010) 27 - 31 A F ractional Lie Group Metho d F or Ano malous Diffusion Equation s Guo-c heng W u * Mo dern T extile Institute, Dongh ua Univ ersit y , 1882 Y an’an Xilu Road, Shanghai 200051 , Chin a Receiv ed 20 Ma y 2010; accepted 13 July 2010 Abstract The Lie group metho d provides an efficient to ol to solve nonlinear pa rtial differential equa tions. This pa per suggests a fractional partner for fractional partial differential equations. A space-time fractional diffusion equation is used a s an example to illustrate the effectiveness of the Lie group metho d. Keyw ords: Lie group metho d; Anon ymous diffusion e q uation; F r actional characteristic method 1 In tro duction In the last three decades, researchers h av e found fractional differen tial equations (FDEs) useful in v arious fields: rheology , quantita tiv e biology , electrochemistry , s cattering theory , diffusion, transp ort theory , pr obabilit y p oten tial t heory and elasticit y [1], for details, see the monographs of Kilbas et al. [2], Kir y ak o v a [3], Lakshmik an tham and V atsala [4], Miller and R oss [5], and P o dlubny [6]. O n the other hand, findin g accurate and efficien t method s for solving FDEs has b een an activ e researc h undertaking. Since S oph us Lie’s w ork on group analysis, more th an 100 yea rs ago, Lie group theory has b ecome more and more p erv asiv e in its influence on other mathematical disciplines [7, 8]. T hen a question ma y naturally arise: is there a fractional Lie group metho d for f ractional different ial equations? Up to n o w, only a f ew works can b e found in the literature. F or example, Buc kw arand and L u c hk o deriv ed scaling transform ations [9] for the fractional diffusion equation in Riemann-Liouville sense ∂ α u ( x, t ) ∂ t α = D ∂ 2 u ( x, t ) ∂ x 2 , 0 < α, 0 < x, 0 < t, 0 < D . (1) Gazizo v et al. find symm etry prop erties of fractional diffusion equ ations of Cap u to d eriv ativ e [10] ∂ α u ( x, t ) ∂ t α = k ∂ ( k ( u ) u x ( x, t )) ∂ x , 0 < α, 0 < x, 0 < t, 0 < k . (2) Djordjevic a nd A tanac ko vic [11] o btained some similarit y solutions for the time-frac tional heat diffusion ∂ α T ( x, t ) ∂ t α = k ∂ 2 ( T ( x, t )) ∂ x 2 , 0 < α, 0 < x, 0 < t. (3) In this study , w e in v estigate anonymous diffusion [12] ∂ α u ( x, t ) ∂ t α = ∂ 2 β u ( x, t ) ∂ x 2 β , 0 < α, β ≤ 1 , 0 < x, 0 < t, (4) ∗ Corresponding author, E-mail: wuguo c heng2002@yahoo.com.cn. (G.C. W u) Cop yright@2009 Asia Academic Publishor Limited. All righ ts reserved. 28 G.C. W u, Comm un. F rac. Calc. V ol.1., 2010 with a fr actional Lie group metho d , and derive its classification of solutions. Here the fractional deriv ativ e is in th e mo dified Riemann-Liouville sense [13] and ∂ 2 β u ( x,t ) ∂ x 2 β is defin ed by ∂ β ∂ x β ( ∂ β u ( x,t ) ∂ x β ) . 2 Characteristic Method for F ractional Diff eren tial Equations Through this pap er, we adopt th e fractional deriv ativ e in mo dified Riemann-Liouville sense [13]. Firstly , we introd uce some pr op erties of the fractional calculus that we will use in this study . (I) Inte gration with resp ect to ( dx ) α (Lemma 2.1 of [14]) 0 I α x f ( x ) = 1 Γ( α ) Z x 0 ( x − ξ ) α − 1 f ( ξ ) dξ = 1 Γ( α + 1) Z x 0 f ( ξ )( dξ ) α , 0 < α ≤ 1 . (5) (I I) Some other usefu l formulas f ([ x ( t )]) ( α ) = d f dx x ( α ) ( t ) , 0 D α x x β = Γ(1 + β ) Γ(1 + β − α ) x β − α . (6) The prop erties of J u marie’s deriv ativ e w ere summarized in [13]. The extension of Ju maire’s fractional d eriv ativ e and in tegral to v ariational approac h of sev eral v ariables is done by Almeida et al. [ 15]. F rac tional v ariational in teractional metho d is prop osed for fractional d ifferen tial equations [16]. It is well known that the metho d of c haracteristics h as pla y ed a v ery imp ortant role in mathemat- ical physics. Preciously , the m etho d of c haracteristics is used to solv e the in itial v alue pr oblem for general fi rst order. With the mo difi ed Riemann-Liouville deriv ativ e, Jumaire eve r ga v e a Lagrange c haracteristic metho d [17]. W e p resen t a more generalized fr actional metho d of c haracteristics and use it to solv e linear fractional p artial equations. Consider the follo wing fir st order equation, a ( x, t ) ∂ u ( x, t ) ∂ x + b ( x, t ) ∂ u ( x, t ) ∂ t = c ( x, t ) . (7) The goal of th e metho d of c haracteristics is to c hange co ord inates from ( x, t ) to a new co ord inate system ( x 0 , s ) in whic h the PDE b eco mes an ord inary differen tial equation along certain curves in the x − t plane. The curve s are called the charac teristic curv es. More generally , we consid er to extend this metho d to linear sp ace-time fractional differenti al equations a ( x, t ) ∂ β u ( x, t ) ∂ x β + b ( x, t ) ∂ α u ( x, t ) ∂ t α = c ( x, t ) , 0 < α, β ≤ 1 . (8) With the fractional T aylo r’s series in t w o v ariables [13] du = ∂ β u ( x, t ) Γ(1 + β ) ∂ x β ( dx ) β + ∂ α u ( x, t ) Γ(1 + α ) ∂ t α ( dt ) α , 0 < α, β ≤ 1 . (9) Similarly , we deriv e the generalized c haracteristic curv es du ds = c ( x, t ) , (10) ( dx ) β Γ(1 + β ) ds = a ( x, t ) , (11) ( dt ) α Γ(1 + α ) ds = b ( x, t ) . (12) Eqs. (10)-(12 ) can b e reduced to J umarie’s result if α = β in [17]. G.C. W u, Comm un. F rac. Calc. V ol.1., 2010 29 As an example, we consid er the fractional equation x β Γ(1 + β ) ∂ β u ( x, t ) ∂ x β + 2 t α Γ(1 + α ) ∂ α u ( x, t ) ∂ t α = 0 , 0 < α, β ≤ 1 . (13) W e can hav e the fractional scaling transform ation u = u ( x 2 β Γ 2 (1 + β ) / 2 t α Γ(1 + α ) ) . (14) Note that when α = β = 1 , as is well kno wn, x 2 2 t is one in v ariant of the line differential equation x ∂ u ( x, t ) ∂ x + 2 t ∂ u ( x, t ) ∂ t = 0 . (15) 3 Lie Gr oup metho d for F ractional diffusion equa tion With the prop osed fractional metho d of c haracteristics, no w w e can consider a fractional Lie Group method for the fr actional diffu sion equation, whic h are the generalizatio ns of the classical diffusion equations treating the s u p er-diffu s iv e fl o w pro cesses. These equations arise in con tin uous- time random walks, mo deling of anomalous diffusive and su b-diffusive systems, unification of d iffusion and w a v e prop agation ph enomenon [18 - 23]. W e assum e the one-paramete r Lie group of transformations in ( x, t, u ) give n by ˜ x β Γ(1+ β ) = x β Γ(1+ β ) + εξ ( x, t, u ) + O ( ε ) , ˜ t α Γ(1+ α ) = t α Γ(1+ α ) + ετ ( x, t, u ) + O ( ε ) , ˜ u = u + εφ ( x, t, u ) + O ( ε ) , (16) where ε is the group parameter. W e start f rom the set of fractional v ector fields in s tead of using the one of integ er order [9 - 11] V = ξ ( x, t, u ) D β x + τ ( x, t, u ) D α t + φ ( x, t, u ) D u . (17) The fracti onal second ord er p rolongation P r (2 β ) V of the infi nitesimal generators can be represen ted as P r (2 β ) V = V + φ [ t ] ∂ φ ∂ D α t u + φ [ x ] ∂ φ ∂ D β x u + φ [ tt ] ∂ φ ∂ D 2 α t u + φ [ xx ] ∂ φ ∂ D 2 β x u + φ [ xt ] ∂ φ ∂ D β x D α t u . (18) As a result, we can h a v e P r (2 β ) V (∆[ u ]) = 0 , (19) where ∆[ u ] = ∂ α u ( x,t ) ∂ t α − ∂ 2 β u ( x,t ) ∂ x 2 β . Eq. (19) can b e r ewritten in the form ( φ [ t ] − φ [ xx ] )    ∆[ u ]=0 = 0 . (20) The generalized prolongation v ect or fields are defined as φ [ t ] = D α t φ − ( D α t ξ ) D β x u − ( D α t τ ) D α t u, (21) φ [ x ] = D β x φ − ( D x ξ β ) D β x u − ( D β x τ ) D α t u, (22) φ [ xx ] = D 2 β x φ − 2( D β x ξ ) D 2 β x u − ( D 2 β x ξ ) D β x u − 2( D β x τ ) D β x D α t u − ( D 2 β x τ ) D α t u t . (23) Substituting Eqs. (21)-( 23) into Eq. (20) and setting the co efficien ts to zero, w e can obtain some line fractional equations f rom whic h w e can derive 30 G.C. W u, Comm un. F rac. Calc. V ol.1., 2010 ξ ( x, t , u ) = c 1 + c 4 x β Γ(1+ β ) + 2 c 5 t α Γ(1+ α ) + 4 c 6 x β t α Γ(1+ β )Γ(1+ α ) , τ ( x, t, u ) = c 2 + 2 c 4 t α Γ(1+ α ) + 4 c 6 t 2 α Γ(1+2 α ) , φ ( x, t, u ) = ( c 3 − c 5 x β Γ(1+ β ) − 2 c 6 t α Γ(1+ α ) − c 6 x 2 β Γ(1+2 β ) ) u + a ( x, t ) , where c i ( i = 0 ... 6) are real constan ts and th e function a ( x, t ) satisfies ∂ α a ( x, t ) ∂ t α = ∂ 2 β a ( x, t ) ∂ x 2 β , 0 < α ≤ 1 , 0 < β ≤ 1 . (24) It is easy to chec k that the t w o v ector fi elds { V 1 , V 2 , V 3 , V 4 , V 5 , V s } are closed und er th e Lie brack et. Th us, a basis for the Lie algebra is { V 1 , V 2 , V 3 , V 4 , V 5 } , whic h consists of the four-dimensional sub- algebra { V 1 , V 2 , V 3 , V 4 } v 1 = ∂ β ∂ x β , v 2 = ∂ α ∂ t α , v 3 = ∂ ∂ u , v 4 = x β Γ(1+ β ) ∂ β ∂ x β + 2 t α Γ(1+ α ) ∂ α ∂ t α , v 5 = 2 t α Γ(1+ α ) ∂ β ∂ x β − ux β Γ(1+ β ) ∂ ∂ u , v 6 = 4 t α Γ(1+ α ) x β Γ(1+ β ) ∂ β ∂ x β + 4 t 2 α Γ(1+2 α ) ∂ α ∂ t α − ( x 2 β Γ(1+2 β ) + 2 t α Γ(1+ α ) ) u ∂ ∂ u , and one infinite-dimensional su b-algebra v 7 = a ( x, t ) ∂ ∂ u . (25) Assume u = f ( x β Γ(1+ α ) , t α Γ(1+ β ) ) is an exact solution of Eq. (4). Then with the pr op osed fractional metho d of c haracteristics, solving the ab o v e symmetry equations, we can deriv e u (1) = f ( x β Γ(1+ α ) − ε, t α Γ(1+ α ) ) , u (2) = f ( x β Γ(1+ β ) , t α Γ(1+ α ) − ε ) , u (3) = e ε f ( x β Γ(1+ β ) , t α Γ(1+ α ) ) , u (4) = f ( x β Γ(1+ β ) e − ε , t α Γ(1+ α ) e − 2 ε ) , u (5) = e t α ε 2 Γ(1+ α ) − x β ε Γ(1+ β ) f ( x β Γ(1+ β ) − 2 ε t α Γ(1+ α ) , t α Γ(1+ α ) ) , u (6) = 1 q 1+4 ε t α Γ(1+ α ) e − x 2 β ε Γ(1+ α ) Γ(1+2 β )Γ(1+ α )+4 εt α Γ(1+2 β ) × f ( Γ(1+ α ) x β Γ(1+ β )Γ(1+ α )+4 ε Γ(1+ α ) x β , t α Γ(1+ β )+4 ε Γ(1+ α ) t α ) , u (7) = f ( x β Γ(1+ α ) , t α Γ(1+ α ) ) + εa ( x, t ) , whic h are all the classification of solutions of Eq. (4). T ak e the solution u (5) as an example, u (5) = e t α ε 2 Γ(1+ α ) − x β ε Γ(1+ β ) f ( x β Γ(1 + β ) − 2 ε t α Γ(1 + α ) , t α Γ(1 + α ) ) . (26) Assume f ( x β Γ(1+ β ) − 2 ε t α Γ(1+ α ) , t α Γ(1+ α ) ) = c, which can b e set as the initial v alue of Eq. (4). No w we can c heck that u (5) 1 = ce t β ε 2 Γ(1+ β ) − x α ε Γ(1+ α ) is one o f the e xact solutions. If we make f ( x β Γ(1+ α ) , t α Γ(1+ α ) ) = u (5) 1 = ce x β ε 2 Γ(1+ β ) − t α ε Γ(1+ α ) , we can derive a n ew iteration s olution u (5) 2 . As a r esult, by similar manipulations, we can giv e u (5) 3 . . . u (5) n whic h are new exact solutions of Eq. (4). G.C. W u, Comm un. F rac. Calc. V ol.1., 2010 31 4 Conclusions F ractional different ial equations ha v e caught considerable atten tion du e to their v arious applica- tions in real ph ysical pr oblems. Ho w ev er, there is no systematic metho d to deriv e the exact solution. No w, the problem is partly solv ed in this pap er. Another problem may arise: can the Lie group metho d b e extended to fr actional different ial equations of fractional order 0 ∼ 2? W e will discuss suc h w ork in future. References 1 K.B. O ldham, J. Spanier, The fractional calculus, Academic Press, New Y ork (1999). 2 A.A. Kilbas, H.M. 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