Solutions For A Generalized Fractional Anomalous Diffusion Equation Long-Jin Lv,1,∗ Jian-Bin Xiao,1 Lin Zhang,1 and Lei Gao1 1School of Science, Hangzhou Dianzi University, Hangzhou 310037, China In this paper, we investigate the solutions for a generalized fractional diffusion equation that 2 extendssomeknowndiffusionequationsbytakingaspatialtime-dependentdiffusioncoefficientand 1 an external force into account, which subjects to the natural boundaries and the generic initial 0 condition. We obtain explicit analytical expressions for the probability distribution and study the 2 relation between our solutions and those obtained within the maximum entropy principle by using n the Tsallis entropy. a J 1 1 Keywords: Anomalous diffusion; Fractional diffusion; Green function; Fox function ] h p I. INTRODUCTION - h t a m Recently,anomalousdiffusionequationshavebeenextensivelyinvestigatedduetothebroadnessoftheir [ physicalapplications. Infact, fractionaldiffusionequationsandthe non-linearfractionaldiffusion equations 1 have been successfully applied to several physical situations such as percolation of gases through porous v 6 media[1],thinsaturatedregionsinporousmedia[2],inthetransportoffluidinporousmediaandinviscous 6 fingering[3], thin liquid films spreading under gravity [4], modeling of non-Markviandynamical processes in 2 2 protein folding [5], relaxation to equilibrium in system (such as polymer chains and membranes) with long . 1 temporalmemory[6],anomaloustransportindisorderedsystems[7],diffusiononfractals[8],andthe multi- 0 2 physical transportin porous media, such as electroosmosis[9-10]. Note that the physical systems mentioned 1 above essentially concern anomalous diffusion of the correlated type (both sub and super-diffusion; see [11] : v and references therein) or of the Lévy type (see [12] and references therein). The anomalous correlated i X diffusion usually has a finite second moment x2 tσ (σ > 1,σ = 1 and 0 < σ < 1 correspond to super- r h i ∝ a diffusion, normal diffusion and sub-diffusion, respectively; σ = 0 corresponds basically to localization). Due to the broadness of the problems involving anomalous diffusion, one needs to apply different kinds of theoretical approaches such as nonlinear Fokker-Planck equation (or modified porous media equation), fractionalFokker-Planckequation,Fokker-Planckequationwith spatialdependent diffusioncoefficient, and generalizedLangevinequations. Thepropertiesconcerningtheseequationshavebeenintensivelyinvestigated [13-17] and the lattice Boltzmann method was used to get the numerical solutions for this equations which governthe multi-physical transfort in porous media. Inordertocovertheabovesituations,weemployaspatialtime-dependentdiffusioncoefficient,inother words,our work is aimed at the investigationof solutions for a fractionaldiffusion equation taking a spatial time-dependence on the diffusion coefficient and an external force (drift) into account. More precisely,we ∗Electronicaddress: Lojin@stu.hdu.edu.cn 2 focus our attention on the following equation: ∂γ t ∂ ∂µ−1 ∂ ρ(x,t)= dt′ D(x,t t′) [ρ(x,t)]ν F(x)ρ(x,t) , (1) ∂tγ Z ∂x{ − ∂xµ−1 }− ∂x{ } 0 where 0 <γ 1,µ,ν R, the diffusion coefficient is given by D(x,t) = D(t)x−θ, which is a spatial time- ≤ ∈ | | dependent diffusion coefficient,and F(x) = ∂V(x)is an external force (drift) associated with the potential − ∂x V(x). Here, we use the Caputo operator [18] for the fractional derivative with respect to time t, and we work with the positive spatial variable x. Later on, we will extend the results to the entire real x-axis by the use of symmetry (in other words, we are working with ∂/∂ x and ∂µ−1/∂ xµ−1). Also, we employ, | | | | in general, the initial condition ρ(x,0) = ρ˜(x) (ρ˜(x) is a given function), and the boundary condition +∞ ρ(x ,t) 0. For Eq.(1), one can prove that dxρ(x,t) is time independent (hence, if ρ(x,t) is →±∞ → −∞ normalized at t=0,it will remain so forever). IndeedR, if we write Eq.(1) as ∂γ = ∂ J, and, for simplicity, t − x +∞ assumetheboundaryconditionsJ( )=0,itcanbeshownthat dxρ(x,t)isaconstantofmotion(see ±∞ −∞ R [19]andreferencestherein). Note thatwhen(µ,γ,ν)=(2,1,1),Eq.(1)recoversthe standardFokker-Planck equationinthepresenceofadrifttakingmemoryeffectsintoaccount. TheparticularcaseF(x)=0(nodrift) and D(x,t) = Dδ(t) with (µ,γ) = (2,1) has been considered by spohn [20]. The case D(x,t) = Dδ(t)x−θ | | with (µ,ν) = (2,1) and the case D(x,t) = D(t) with (µ,ν) = (2,1) have been investigated in [21] and [22], respectively. Explicitsolutionsplayanimportantroleinanalyzingphysicalsituations,sincetheycontain,inprinciple, precise information about the system. In particular, they can be used as an useful guide to control the accuracy of numerical solutions. For these reasons, we dedicated to this work to investigate the solutions for Eq.(1) in some particular situations. In all the particular cases, Eq.(1) satisfies the initial condition ρ(x,0) = ρ˜(x) (ρ˜(x) is a given function), and the boundary condition ρ( ,t) = 0. The remainder of this ±∞ paper goes as follow. In Sec.2, we obtain the exact solutions for the special cases. In Sec.3, we present our conclusions. II. EXACT SOLUTIONS FOR DIFFERENT CASE In this section, we start our discussion by considering Eq.(1) in the absence of external force with D(t)=Dtα−1/Γ(α) ( D(t)=Dδ(t)),(µ,ν)=(2,1) and γ, θ arbitrary. For this case, Eq.(1) reads ∂γ t ∂ ∂ ρ(x,t)= dt′D(t t′) x−θ ρ(x,t) . (2) ∂tγ Z − ∂x{| | ∂x } 0 Here, we use the Caputo operator[18]for the fractionalderivative with respectto time t. By employingthe Laplace transform in Eq.(2), we obtain ∂ ∂ D˜(s) x−θ ρ˜(x,s) sγρ˜(x,s)= sγ−1ρ(x,0), (3) ∂x{| | ∂x }− − whereρ˜(x,s)=L ρ(x,t) , D˜(s)=L D(t) ,andL f(t) = ∞dte−stf(t)denotesthe Laplacetransform { } { } { } 0 R of the function f. This equation can be solved by Green function method [23]. By substituting ρ˜(x,s)= dx′˜(x x′,s)ρ˜(x′), (4) Z G − 3 into Eq.(3) which yields ∂ ∂ D˜(s) x−θ ˜(x,s) sγ˜(x,s)= sγ−1 (x,0). (5) ∂x{| | ∂xG }− G − G where (x,t) subjects to the initial condition (x,0)=δ(x) and the boundary condition ( ,t)=0. G G G ±∞ In order to solve Eq.(5), it is convenient to perform the transform [24] y =A(s)xv, (x,s)=yδZ(y) (6) G to translate Eq.(5) into the second-order Bessel equation as ∂2Z ∂Z y2−δ y y2 +y (λ2+y2)Z(y)= δ(( )v1) (7) ∂y2 ∂y − − s A(s) with parameter λ2 under the following conditions: 2+θ 1 sγ 1+θ 1+θ 1 v = ,A(s)= [ ]2,λ= ,δ = . (8) 2 v D˜(s) 2+θ 2+θ Since Eq.(5) should fit the boundary condition ( ,t) = 0, i.e. ( ,s) = 0, we get the solution of G ±∞ G ±∞ Eq.(5) (x,s)=C(s)yδK (y), (9) λ G whereK (x)isthemodifiedBesselfunctionofsecondkind;andC(s)canbedeterminedbythenormalization n of (x,t), i.e. ∞dx˜(x,s)= 1 . After some calculations, we obtain G 0 G 2s R G˜(x,s)= Γ(2+1 θ)s(2+1 θ(D˜s(γs))12)23++θθ|x|1+2θK21++θθ(2+2 θ(D˜s(γs))12|x|2+2θ), (10) 2+θ where, we used the formula ∞ 1+v+λ 1+v λ dy yvK (ay)=2v−1a−v−1Γ( )Γ( − ). (11) Z · λ 2 2 0 Case 1. D(t)=Dδ(t), i.e. D˜(s)=D. Since K (x) = 1H2 0[x2 ], we can get the Laplace inverse of ˜(x,s) by applying the λ 2 0 2 4 |(−λ/2,1)(λ/2,1) G property of the Laplace inverse of Fox function, which yields G(x,t)= 2Γ2(+1θ )((2+θ1)2Dtγ)2+1θH12 20[(2+|xθ|2)+2θDtγ|((01,−1)2,+γ(θ1+,γθ),1)], (12) 2+θ 2+θ where Hm n[x(a1,A1),...,(ap,Ap)] is the FOX function [25]. Thus, we can find the solution by substituting p q |(b1,B1),...,(bq,Bq) Eq.(10) into Eq.(4), which yields ρ(x,t)= 2Γ2(+1θ )((2+θ1)2Dtγ)2+1θ Z +∞dx′ρ˜(x′)H12 20[(|2x+−θx)′2|2D+tθγ|((01,−1)2,+γ(θ1+,γθ),1)]. (13) 2+θ −∞ 2+θ Infig.1,weshowthebehavioroftheaboveequationbyconsideringtypicalvaluesofγ andθwithρ˜(x)=δ(x). At this point, it is interesting to analyze the asymptotic behavior of Eq.(13). For simplicity, we consider ρ˜(x)=δ(x), so ρ(x,t)= (x,t); and the asymptotic behavior of ρ(x,t) is G ρ(x,t) 2+θ (2 γ)−21γ(2+θ)γ(2−γ)−21( 1 )(2+θ)1(2−γ) x 2γ−−γ1 ∼ 2Γ(1/(2+θ)) − (2+θ)2Dtγ | | γ x2+θ 1 exp( (2 γ)γ2−γ( | | )2−γ). (14) × − − (2+θ)2Dtγ 4 1.5 ——Γ=1(cid:144)4,Θ=0 1.25 --Γ=1(cid:144)2,Θ=1(cid:144)2 L ,t 1 x H G L 0.75 Γ , Θ H C 0.5 0.25 0 -4 -2 0 2 4 ÈxÈ2+Θ(cid:144)H2+ΘL2DtΓ FIG.1: ThebehaviorofgreenfunctionG(x,t)inEq.(12)isillustratedbyconsideringC(θ,γ)G(x,t)versus |x|2+θ (2+θ)2Dtγ for typical values of γ and θ. HereC(θ,γ)= 2Γ(1/(2+θ))((2+θ)2Dtγ)1/(2+θ). 2+θ In this direction,Eq.(14)canbe consideredasan extensionofthe asymptotic behavior ofhomogeneousand isotropic random walk models [26]. Case 2. D(t)= Dtα−1, i.e.D˜(s)=Ds−α. Γ(α) By using the same method as in case 1, we obtain G(x,t)= 2Γ2(+1θ )((2+θ)12Dtγ+α)2+1θH12 20[(2+|θx)|22+Dθtγ+α|((01,−1)γ2,++(1αθ+,θγ,+1α))], (15) 2+θ 2+θ and ρ(x,t)= 2Γ2(+1θ )((2+θ)12Dtγ+α)2+1θ Z +∞dx′ρ˜(x′)H12 20[(2|x+−θ)x2D′|2t+γθ+α|((01,−1)γ2,++(1αθ+,θγ+,1α))]. (16) 2+θ −∞ 2+θ Let us go back to Eq.(1), and consider the external force F(x) ∝ xxα−1, D(x,t) = Dδ(t)x−θ and | | | | µ = 2, ν = 1. In this case ,analytical solution can not easily be obtained for a generic α, θ. However, for θ = 0, and α+θ+1 = 0. By following the same procedure as in the above case, an exact solution can be 6 obtained and it is given by ρ(x,t)= 2Γ( 12++θ K )((2+θ1)2Dtγ)2+1θH12 20[(2+|xθ|2)+2θDtγ|((1−K2+γθ,,γ1)),(1+θ,1)], (17) 2+θ (2+θ)D (2+θ)D 2+θ where, for simplicity, we are considering the initial condition ρ(x,0) = δ(x), and the external force (drift) F(x)= xα. Thesecondmomentisgivenby∝t22+γθ,whichscaleswiththeexponent 2γ andclearly K 2+θ depends only on γ and θ. So, when 2γ <1, =1 and >1, the system is sub-diffusion, normal diffusion and 2+θ supper-diffusion respectively. The presence of the external force in Eq.(1) is now changed into F(x) = k x+k x−1−θ. In order 1 2 − to obtain the solution of Eq.(1), we expand ρ(x,t) in terms of the eigenfunctions, i.e. we employ ρ(x,t) = 5 ∞ φ (t)ψ (x) with ψ (x) determined by the spatial equation n=0 n n n P ∂ ∂ ∂ λ ψ (x)=D x−θ ψ (x) + (k x+k x−1−θ)ψ (x) (18) n n n 1 2 n − ∂x{ ∂x } ∂x{ } and φ (t) determined by the time equation n ∂γ φ (t)= λ φ (t). (19) ∂tγ n − n n The solution for the time equation is given by in terms of the Mittag-Leffter function φ(t) E ( λ tγ). (20) γ n ∝ − In order to get the solution for Eq.(18), we perform the transform ψ (x)=e−yyδZ(y), y =Axv (21) n to translate Eq.(18) into the associated Laguerre equation yZ′′(y)+(α+1 y)Z′(y)+nZ(y)=0 (22) − with parameter α under the following condition: k2 1 θ k k α= D − − ,v =2+θ,A= 2 ,δ = 2 ,λ =(2+θ)nk . (23) n 1 2+θ (2+θ)D (2+θ)D Then, using the Green function methods and after some calculations, it is possible to show that ∞ ρ(x,t)= dx ρ˜(x ) (x,x ,t), (24) Z 0 0 G 0 −∞ ∞ G(x,x0,t)=((2+k1θ)D)(k22++θ)DD|x|kD2e−(2+kθ1)D|x|2+θX (Γ2(+n+θ)Γk(2n++D1))Eγ(−λntγ) n=0 (2+θ)D k k L(α)( 1 x2+θ)L(α)( 1 x 2+θ), (25) × n (2+θ)D| | n (2+θ)D| 0| where L(α)(x) is the associated Laguerre polynomial and it is the solution for Eq.(22). Here, we used the n formula ∞ Γ(n+α+1) dyyαe−yL(α)(y)L(α)(y)= . (26) Z n n Γ+1 0 Notice that Eq.(25) contains the usual Ornstein-Uhlenbeck process [27] and the usual Rayleigh process [28] asparticularcasesanditextends the resultsobtainedin[21]. Inthis context,inthe presenceofanconstant absorbentforce,i.e. α˜ρ(x,t), we canobtain the solution for Eq.(1)only need to changethe argumentof the Mittag-Leffler function present in Eq.(20) to λ +α˜. n Let us now discuss Eq.(1) by considering a mixing between the spatial and time fractional derivatives. Forsimplicity,weconsiderEq.(1)intheabsenceoftheexternalforcewith(ν,θ)=(1,0)andγ,µarbitraries. 6 Applying the FourierandLaplacetransformto Eq.(1)andemployingthe Riez representationforthe spatial fractional derivatives, we have sγρˆ˜(k,s) sr−1ρˆ(k,0)= D˜(s)k µρˆ˜(k,s), (27) − − | | where ρˆ(k,t)= ρ(x,t) = +∞ρ(x,t)e−ikxdx , so ρ(k,0)=1. Here, we consider the diffusion coefficient F{ } −∞ givenby D(t)= Dtα−1, i.e. D˜R(s)=Ds−α andρ(x,0)=δ(x). By using the inverseof Laplacetransform,we Γ(α) obtain ρ(k,t)=E ( D k µtγ+α) γ+α,1 − | | =H1 1[D k µtγ+α (0,1) ]. (28) 1 2 | | |(0,1),(0,γ+α) This solution recovers the usual one for (µ,γ)=(2,1) and for µ=2 it extends the results found in [29]. In 6 order to perform the inverse of Fourier transform, we employ the procedure presented in [30]. Then we can obtain 1 x ρ(x,t)= H2 1[ | | (1−1/µ,1/µ),(1−(γ+α)/µ,(γ+α)/µ)]. (29) 2µ√π(Dtγ+α)µ1 2 3 2(Dtγ+α)µ1 |(0,1/2),(1−1/µ,1/µ),(1/2,1/2) The stationary solution that emerges from this process is a Levy distribution. Now, we consider a particular case of Eq.(1) for γ = 1 and nonzero values of µ and θ, and consider a lineardrift,i.e. F(x)= x. Forsimplicity,weemployD(t)=Dδ(t)andtheinitialconditionρ(x,0)=δ(x), −K then Eq.(1) yields to ∂ ∂ ∂µ−1 ∂ ρ(x,t)=D x−θ [ρ(x,t)]ν + xρ(x,t) . (30) ∂t ∂x{| | ∂xµ−1 } ∂x{K } Let us investigated time dependent solutions for Eq.(30). We use similarity methods to reduce Eq.(30) to ordinary differential equations. The explicit form for these ordinary differential equations depends on the boundary conditions or restrictions in the form of conservation laws. In this direction, we restrict our analysis to find solution that can be expressed as a scaled function of the type [31] 1 x ρ(x,t)= ρ˜(z), z = . (31) φ(t) φ(t) Inserting Eq.(31) into Eq.(30), we obtain φ(˙t) ∂ D ∂ ∂µ−1 ( + K ) [zρ˜(z)]= [z−θ ρ˜(z)ν]. (32) − φ(t)2 φ(t) ∂z φ(t)θ+µ+ν ∂z ∂zµ−1 By choosing the ansatz ˙ φ(t) kD + K = , (33) φ(t)2 φ(t) φ(t)θ+µ+ν where k is an arbitrary constant which can be determined by the normalization condition. By solving Eq.(33), we have that Dk φ(t)=[(φ(0))θ+µ+ν−1e−(θ+µ+ν−1)Kt+ (1 e−(θ+µ+ν−1)Kt)]θ+µ+1ν−1. (34) − K 7 By substituting Eq.(33) into Eq.(32), we obtain ∂ ∂µ−1 ∂ [z−θ ρ˜(z)ν]= k [zρ˜(z)]. (35) ∂z ∂zµ−1 − ∂z Then, we perform an integration and the result is ∂µ−1 z−θ ρ˜(z)ν = kzρ˜(z)+ , (36) ∂zµ−1 − C where is another arbitrary constant. Also, we use the following generic result [32]: C Γ[α+1] Dδ[xα(a+bx)β]=aδ xα−δ(a+bx)β−δ (37) x Γ[α+1 δ] − withDxδ ≡dδ/dxδ andδ =α+β+1. Bydefiningg(x)≡xαν(a+bx)βν andλ≡α(1−ν1)−δ,andrearranging the indices, Eq.(37) can be rewritten as follows: Γ[α+1] Dδ[g(x)]ν =aδ xλg(x). (38) x Γ[α+1 δ] − For this case, we consider the ansatz ρ˜(z) = zαν(1+bz)νβ. By using the property of Eq.(38) in Eq.(36) N and ,for simplicity, choosing =0, we find C (2 µ)(µ+θ) α= − , 1 2µ θ − − (µ 1)(µ 2) β = − − , (39) − 1 2µ θ − − 2 µ ν = − . 1+µ+θ In this case, we have z(µ+θ)(1+µ+θ) 1 ρ(x,t)= N [ ]1−2µ−θ, (40) φ(t) (1+bz)(1−µ)(1+µ+θ) where φ(t) is given above, = [ k Γ(−β) ]1µ−+2θµ+−1θ and b is an arbitrary constant (to be taken, later on, as N − Γ(α+1) 1 according to the specific solutions that are studied). Severalregions can be analyzed. For simplicity, we ± illustratetwoofthem: <µ< 1 θ withθ 0,and0<µ<1/2with0 θ <1/2 µ. Letusstartby −∞ − − ≥ ≤ − considering the region <µ< 1 θ. Without loss of generality, we choose b= 1. The normalization −∞ − − − condition implies(see Fig. 2) 1 z(µ+θ)(1+µ+θ) 1 [ ]1−2µ−θdz =1. (41) N Z (1+bz)(1−µ)(1+µ+θ) −1 So Γ[1 µ θ] = − − . (42) N 2Γ[µ2+µθ−2θ−2µ]Γ[1−µ+µ2+θ2+2µθ] 1−2µ−θ 1−2µ−θ 8 0.7 —Μ=-1.8,Θ=0.5 ---Μ=-3,Θ=0.5 0.6 0.5 Lx,t 0.4 HΡ HLΦt 0.3 0.2 0.1 0 -1 -0.5 0 0.5 1 x(cid:144)ΦHtL FIG. 2: Behavior of φ(t)ρ(x,t) versus x/φ(t), which illustrates Eq.(40) with typical values for µ and θ satisfying 0<µ<−1−θandθ≥0. Wenoticethatthedistributionvanishesattheabcissaequal±1,andremainszerooutside of this interval. 0.02 ——Μ=0.4,Θ=0.05 --- Μ=0.45,Θ=0.02 0.015 Lx,t HΡ 0.01 HLΦt 0.005 0 -20 -10 0 10 20 x(cid:144)ΦHtL FIG. 3: Behavior of φ(t)ρ(x,t) versus x/φ(t), which illustrates Eq.(40) with typical values for µ and θ satisfying 0<µ<1/2 and 0≤θ<1/2−µ. Wenotice that the distribution vanishes at theinfinity. Let us now analyze the region 0 < µ < 1/2 with 0 θ < 1/2 µ. Again without the loss of generality, we ≤ − choose b=1. The normalization condition implies(see Fig. 3) Γ[1+θ−µ2−µθ] 1−2µ−θ = . (43) N 2Γ[1−µ+µ2+θ2+2µθ]Γ[µ+θ] 1−2µ−θ Let us finally mention a connection between the results obtained here and the solutions that arise from the optimization of the nonextensive entropy [33]. These distributions do not coincide for arbitrary value of x. However, the comparison of the x asymptotic behaviors enables us to identify the type of Tails. | | → ∞ 9 By identifying the behavior exhibited in Eq.(40)with the asymptotic behaviors 1/x2/(q−1) that appears in | | [33] for the entropic problem, we obtain 3+µ+θ q = . (44) 1+µ+θ This relation recovers the situation for θ =0 . III. SUMMARY AND CONCLUSIONS WehaveanalyzedthegeneralizedfractionaldiffusionequationsbyconsideringanexternalforceF(x)∝ xxα−1 and a spatialtime-dependent diffusion coefficient D(x,t)=D(t)x−θ. 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Solutions For A Generalized Fractional Anomalous Diffusion Equation

Similar Solutions For A Generalized Fractional Anomalous Diffusion Equation