Results for a fractional diffusion equation with a nonlocal term in spherical symmetry

Physics Letters A 372 (2008) 6121–6124

Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

Results for a fractional diffusion equation with a nonlocal term in spherical symmetry E.K. Lenzi a,∗ , M.K. Lenzi b , R. Rossato a , L.R. Evangelista a , L.R. da Silva c a b c

Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790 – 87020-900 Maringá, Paraná, Brazil Departamento de Engenharia Química, Universidade Federal do Paraná, Setor de Tecnologia – Jardim das Américas, Caixa Postal 19011, 81531-990 Curitiba – Paraná, Brazil Departamento de Física, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 30 May 2008 Received in revised form 9 August 2008 Accepted 14 August 2008 Available online 22 August 2008 Communicated by A.R. Bishop

We obtain time dependent solutions for a fractional diffusion equation containing a nonlocal term by considering the spherical symmetry and using the Green function approach. The nonlocal term incorporated in the diffusion equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution. In addition, a rich class of anomalous diffusion processes may be described from the results obtained here. © 2008 Elsevier B.V. All rights reserved.

PACS: 66.10.Cb 05.60.-k 05.40.Fb Keywords: Fractional diffusion equation Anomalous diffusion

1. Introduction Recently the diffusion equations which extend the usual one have been successfully investigated due to the broadness of applications, in particular, to anomalous diffusion [1–9]. This variety of applications has motivated the study of some formal aspects of these equations. In fact, in [10] a fractional Fokker–Planck equation is derived from a generalized master equation; in [11], the behavior of fractional diffusion at the origin is analyzed; in [12], a fractional Kramers equation is introduced; in [13–15], fractional reaction diffusion equations are analyzed, and in [16–25] solutions for fractional diffusion equations are obtained. In this Letter, we analyze the fractional diffusion equation:

∂γ ρ (r, t ) = D∇ 2 ρ (r, t ) − ∂tγ

t

dt¯



dr K(r − r , t − t¯)ρ (r , t¯)

0

for spherical symmetry, i.e., r = (r , θ, φ) and

 2π

π



dr ≡

∞ 0

(1) dr r 2 ×

0 dφ 0 dθ sin θ , where the fractional time derivative considered here is the Caputo’s one [30], with 0 < γ  1 and the solution

*

Corresponding author. Tel.: +5504432614330; fax: +5504432634623. E-mail address: [email protected] (E.K. Lenzi).

0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.08.029

is subjected to the boundary condition lim|r|→∞ ρ (r, t ) = 0. The last part of Eq. (1) represents a nonlocal term acting on the system which quantifies the effect the neighbouring ρ (r , t¯) on ρ (r, t ) and can be useful to incorporate long ranges effects present in the system [26]. In this direction, performing a suitable choice for this term, we may investigate, for example, anomalous transport in plasma [27], tumor development [9], drug absorption [28], and irreversible reaction [29]. It can also be related to the time or spatial fractional derivatives [30] depending on the choice of K(r, t ) and, in particular, when the nonlocal term is connected  to a spatial fractional derivative it is possible to verify that dr ρ (r, t ) is constant. Other interesting aspect manifested by Eq. (1) is the presence of different regimes of diffusion depending on the choice of the kernel K(r, t ) present in the nonlocal term. In particular, for K(r, t ) ∝ δ(t )/|r|3+μ the solution has two regimes, one of them is asymptotically governed by a power law behavior like a Lévy distribution for long time and for small time the behavior is dominated by a stretched exponential. It is worth mentioning in this regard that situations characterized by two regimes have been reported in several contexts as, for instance, long range interaction systems [31] and enhanced diffusion in active intracellular transport [32]. Other possibility is to relate this nonlocal term with a reaction diffusion process [33,34] or fractional diffusion equation of distributed order.

6122

E.K. Lenzi et al. / Physics Letters A 372 (2008) 6121–6124

2. Diffusion equation and solutions

with

Let us start our investigation about the solutions of Eq. (1) by considering the boundary condition mentioned above and the  arbitrary initial condition ρ (r, 0) with dr ρ (r, 0) = 1. To pro ceed, we use the Fourier transform (F {· · ·} = dr e −ik·r · · · and

G¯ (r, t ) =

F −1 {· · ·}



=

dk e ik·r · · ·) (2π )3

in Eq. (1) in order to obtain the inte-

gral equation

t

dγ dt

ρ (k, t ) = −D|k|2 ρ (k, t ) − γ

dt¯ K(k, t − t¯)ρ (k, t¯).

(2)

0

This integral equation may  ∞ also be simplified by using the Laplace transform (L{· · ·} = 0 dt e −st · · · and L−1 {· · ·} = 1/(2π i ) ×

 i ∞+c

−i ∞+c ds e

st

· · ·), leading us to the following algebraic equation:

sγ ρ (k, s) − sγ −1 ρ (k, 0) = −D |k|2 ρ (k, s) − K(k, s)ρ (k, s) whose solution may be written as

(3)

ρ (k, s) = ρ (k, 0)G (k, s) with

sγ −1

G (k, s) = γ s + D |k|2 + K(k, s)

(4)

where ρ (k, 0) is the Fourier transform of the initial condition and G (k, s) is the Green’s function [35] of Eq. (1) in the Fourier–Laplace space. Note that Eq. (4) recovers a free case worked in [3], i.e., the Green function for the fractional diffusion equation in absence of external force, for K(k, s) = 0. Using Eq. (4), we investigate ¯ (k) (K(r, t ) = K ¯ (r)δ(t )) which by a suitthe cases: (i) K(k, s) = K able choice may be related to the spatial fractional derivative and (ii) K(k, s) (K(r, t )) arbitrary. The last case extends the first one and is useful to comprehend the situations where a spatial and time dependent nonlocal term can be considered as a perturbation on the system. By substituting the first choice for K(r, t ), which in the Fourier ¯ (k), in Eq. (4) we obtain and Laplace spaces is given by K(k, s) = K

G (k, s) =

sγ −1

¯ (k) s γ + D |k|2 + K



1 4π |r|

1, 0 H 3 1, 1

(1, γ )

|r| 2 √ Dt γ (2,1)

(8)

(1+nγ , γ )

2 |r| √ Dt γ (2,1)

(9)

and

G˜ (r, t ) =



1 4π |r|

1, 0 H 3 1, 1

(a1 , A 1 ),...,(a p , A p )

n where Hm p q [x|(b1 , B 1 ),...,(bq , B q ) ] is the Fox H function [36]. The presence of this function in the above Green function indicates that the fractional time derivative and nonlocal term incorporated in Eq. (1) produce an anomalous spreading of the solution. In order to clarify the influence of this nonlocal term on the solution of the fractional diffusion equation, we consider the illustrative situa¯ (r) ∝ 1/|r|3+μ (K ¯ (k) = K¯ |k|μ ) with 0 < γ  1. This tion given by K ¯ choice for K(r), which is related to the spatial fractional derivatives, has a long range behavior and may found applications, for example, in biologic systems [26]. By substituting this choice for K¯ (r) in Eq. (7), we obtain

 ∞  1 K (1− μ )γ n 2 t √ 3 Dμ π 2 |r|3 n=1 n! (1− nμ ,1),(1+(γ − μ )n,γ )

 2 2 |r|2 2, 1 × H2,3 γ 4Dt ( 3 ,1),(1+(1− μ )n,1),(1,1)

G (r, t ) = G¯ (r, t ) +

1

2

(10)

2

(see Fig. 1). Note that for γ = 1, as shown in Fig. 1, we have a divergent behavior at the origin which is in agreement with the results reported in [38] for the fractional diffusion equations. By using this result a remarkable aspect found is the presence of two regimes; one of them is characterized by an stretch exponential and the other one is dominated asymptotically by a power law. In fact, it is possible to verify that, for small time μ (t (D 2 /K)2/(γ (2−μ)) ),

(5)

.

Eq. (5) may be related to the continuous time random walk (CTRW) formalism [3] by considering as Laplace transform of the waiting time distribution ω(s) = 1/(1 + (τ s)γ ) and as an approximation for Fourier transform of the jumping probability density ¯ (k)), where τ is a characterisfunction λ(k) ≈ 1 − τ γ (D |k|2 + K tic waiting time. The result obtained for ω(s) and λ(k) indicates that the previous choice for the nonlocal term directly change the jumping probability density function. In particular, depending on ¯ (r) the solution may manifest, for example, a long the choice of K tailed behavior in the asymptotic limit of large argument for long time and a second moment divergent. By performing the inverse of Laplace transform, Eq. (5) may be written as ∞     (−t γ K¯ (k))n (n)  G (k, t ) = Eγ −D|k|2 t γ + Eγ −D |k|2 t γ (1 + n)

(6)

n=1

∞

where Eγ (x) = n=0 xn / (1 + γ n) is the Mittag–Leffler function (n) and Eγ (x) is the nth-derivative of the Mittag–Leffler function, i.e., (n)

n

d Eγ (x) ≡ dx n Eγ (x) [30]. Employing the inverse Fourier transform and the convolution theorem in Eq. (6), we obtain that

G (r, t ) = G¯ (r, t ) +  ×

 ∞  (−t γ )n n=1

n!

¯ (r − rn ) · · · drn K

¯ (r2 − r1 )G˜ (r1 , t ) dr1 K

(7)

Fig. 1. This figure illustrates the behavior of versus (4π Dt γ )3/2 G (r , t ) versus r 2 /(4Dt γ ) for Eq. (10) by considering typical values of γ and μ with t = 1.

E.K. Lenzi et al. / Physics Letters A 372 (2008) 6121–6124

G (r, t ) ∼

 2(23−γ γ ) − 12

γ

1

2



2−γ (4π |r|2 ) 2 2   2−γ γ  2−2γ

|r| γ × exp − √ 2 Dt γ 3

which for the particular case

|r| √ Dt γ

6123

2−3γ

(11)

γ = 1 recovers the Gaussian behavior, μ

whereas for large time (t  (D 2 /K)2/(γ (2−μ)) )

G (r, t ) ∼



1 3

π 2 |r|3

2, 1 H2,3

(1,1),(1,γ )

|r|μ 2μ Kt γ ( 3 , μ ),(1,1),(1, μ ) 2

2

(12)

2

which is essentially a Lévy like distribution found in anomalous diffusion processes. Now, we address our discussion for the general case characterized by an arbitrary kernel K(r, t ) in the nonlocal term and assume that this kernel has the Fourier and Laplace transforms defined. By performing in Eq. (4) the inverse of Laplace and Fourier transforms taking this assumption into account, we obtain that

G (r, t ) = G¯ (r, t ) +

 ∞  (−1)n n=1

t2

 ×

dr1

n!

t drn

¯ (r − rn , t − tn ) · · · dtn K

0

¯ (r2 − r1 , t 2 − t 1 )G˜ (r1 , t 1 ) dt 1 t 1 K γn

(13)

0

with G¯ (r, t ) and G˜ (r, t ) given by Eqs. (8) and (9). Eq. (13) with

Fig. 2. This figure illustrates the behavior of versus (4π Dt γ )3/2 G (r , t ) versus r 2 /(4Dt γ ) for Eq. (14) by considering typical values of γ and η with t = 1.

the Green functions defined by Eqs. (8) and (9) extends results presented in [34,37] and similar to the previous results it may exhibit different diffusion regimes depending on the choice of the kernel K(r, t ). As an illustration of this case, we consider the time ˆ s−η ). After some dependent kernel K(r, t ) ∝ t η−1 δ(r) (K(k, s) = K calculation using Eq. (13), it is possible to show that is given by ∞   1 ˆ t γ +η n −K 3 π 2 |r|3 n=1 n! (1+(γ +η)n,γ )

 |r|2 × H12,,02 4Dt γ 3

G (r, t ) = G¯ (r, t ) +

1

(14)

( 2 ,1),(1+n,1)

(see Fig. 2). A necessary remark about Eq. (14) concerns the behavior at the origin which is not divergent for γ = 1 and −1 < η  0 (see Fig. 3), in contrast to the results found for the fractional diffusion equations [38] with dimension great than one. Note also that Eq. (14) extends results obtained in [34] by considering the three dimensional case with a spatial and time dependent kernel and ˆ = 0 and γ = 1. recovers the usual one when K 3. Summary and conclusions A fractional diffusion equation taking into account the presence of a nonlocal term in spherical symmetry was considered. We ¯ (r)δ(t ), where K¯ (r) is first consider the nonlocal term K(r, t ) = K an arbitrary kernel with well defined Fourier transform and after K(r, t ) arbitrary with well defined Fourier and Laplace transforms. For the first case, we have obtained a solution and worked out it ¯ |k|μ ) with ¯ (r) ∝ 1/|r|3+μ (K ¯ (k) = K for the illustrative situation K 0 < γ < 1. This illustration has as particular case several situations accounted for by the equations investigated in Refs. [3,34,39] and exhibits an anomalous spreading of the solution. In this direction, one of the remarkable characteristics of our results is the presence of different regimes of spreading of the solution. This feature ¯ (r) which, for short is evident for the illustration worked out to K time, has a stretched exponential spreading of the initial condition whereas for long time the spreading is characterized by a Fox H

Fig. 3. This figure illustrates the behavior of versus (4π Dt γ )3/2 G (r , t ) versus r 2 /(4Dt γ ) for Eq. (14) by considering γ = 1 and η = −1/2 for typical values of t.

function whose asymptotic behavior is a power law, as is found for the Lévy distributions. Subsequently, we have found a general solution for the case represented by an arbitrary spatial and time dependent kernel which has well defined Fourier and Laplace transforms. For this case, we also worked out an illustrative example which extends results found in [3,34]. Similarly to the previous case, this dependence of the nonlocal term on the time variable

6124

E.K. Lenzi et al. / Physics Letters A 372 (2008) 6121–6124

introduces different regimes for the solution which are not present in the first scenario characterized by a spatial dependent kernel and, consequently, in the situations analyzed in Refs. [3,34,39]. Another remarkable aspect concerning Eq. (1) is its relation to the fractional diffusion equation of distributed order [40] and truncated Lévy distribution [41] for a suitable choice of the nonlocal term. Finally, we expect that this work may be useful in providing insights to investigate situations where the anomalous diffusion is present. Acknowledgements We would like to thank CNPq, CAPES and Fundação Araucária for partial financial support.

[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

References [29] [1] R. Hilfer, R. Metzler, A. Blumen, J. Klafter, Strange Kinetics, Chemical Physics, vol. 284, Pergamon–Elsevier, Amsterdam, 2004, Numbers 1–2. [2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [3] R. Metzler, J. Klafter, Phys. Rep. 339 (2000) 1. [4] E. Heinsalu, M. Patriarca, I. Goychuk, P. Hänggi, Phys. Rev. Lett. 99 (2007) 120602. [5] T. Kosztolowicz, K. Dworecki, St. Mrowczynski, Phys. Rev. Lett. 94 (2005) 170602. [6] V.V. Uchaikin, R.T. Sibatov, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 715. [7] F.J. Valdes-Parada, J.A. Ochoa-Tapia, J. Alvarez-Ramirez, Physica A 373 (2007) 339. [8] K. Chukbar, V. Zaburdaev, Phys. Rev. E 71 (2005) 061105. [9] A. Iomin, J. Phys.: Conf. Ser. 7 (2005) 57; A. Iomin, Phys. Rev. E 73 (2006) 061918; S. Fedotov, A. Iomin, Phys. Rev. Lett. 98 (2007) 118101. [10] R. Metzler, E. Barkai, J. Klafter, Europhys. Lett. 46 (1999) 431. [11] Y.E. Ryabov, Phys. Rev. E 68 (2003) 030102(R). [12] Y.P. Kalmykov, W.T. Coffey, S.V. Titov, Phys. Rev. E 75 (2007) 031101. [13] K. Seki, M. Wojcik, M. Tachiya, J. Chem. Phys. 119 (2003) 2165. [14] S.B. Yuste, L. Acedo, K. Lindenberg, Phys. Rev. E 69 (2004) 036126. [15] I.M. Sokolov, M.G.W. Schmidt, F. Sagués, Phys. Rev. E 73 (2006) 031102.

[30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

[41]

T. Srokowski, Phys. Rev. E 75 (2007) 051105. N. Krepysheva, L. Di Pietro, M.C. Neel, Phys. Rev. E 73 (2006) 021104. B.N.N. Achar, J.W. Hanneken, J. Mol. Liq. 114 (2004) 147. F.-Y. Ren, J.-R. Liang, W.-Y. Qiu, X.-T. Wang, Y. Xu, R.R. Nigmatullin, Phys. Lett. A 312 (2003) 187. O.M.P. Agrawal, Nonlinear Dynam. 29 (2002) 145. A. Hanyga, Proc. R. Soc. London A 458 (2002) 429. S.A. El-Wakil, A. Elhanbaly, M.A. Zahran, Chaos Solitons Fractals 12 (2001) 1035. E.K. Lenzi, R.S. Mendes, K.S. Fa, L.C. Malacarne, L.R. da Silva, J. Math. Phys. 44 (2003) 2179. E.K. Lenzi, R.S. Mendes, J.S. Andrade, L.R. da Silva, L.S. Lucena, Phys. Rev. E 71 (2005) 052109. P.C. Assis da Silva, R.P. de Souza, P.C. da Silva, L.R. da Silva, L.S. Lucena, E.K. Lenzi, Phys. Rev. E 73 (2006) 032101. J.D. Murray, Mathematical Biolog. I. An Introduction, Heidelberg, Springer, 2001. R. Balescu, Phys. Rev. E 51 (1995) 4807. P. Macheras, A. Iliadis, Modeling in Biopharmaceutics, Pharmacokinetics, and Pharmacodynamics, Springer, New York, 2006. D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems, Cambridge Univ. Press, New York, 2000. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. V. Latora, A. Rapisarda, S. Ruffo, Phys. Rev. Lett. 83 (1999) 2104; V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64 (2001) 056134. A. Caspi, R. Granek, M. Elbaum, Phys. Rev. Lett. 85 (2000) 5655; A. Caspi, R. Granek, M. Elbaum, Phys. Rev. E 66 (2002) 011916. B.I. Henry, T.A.M. Langlands, S.L. Wearne, Phys. Rev. E 74 (2006) 031116. A. Schot, M.K. Lenzi, L.R. Evangelista, L.C. Malacarne, R.S. Mendes, E.K. Lenzi, Phys. Lett. A 366 (2007) 346. M.P. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw–Hill, New York, 1953. A.M. Mathai, R.K. Saxena, The H-Function with Application in Statistics and other Disciplines, Wiley Eastern, New Delhi, 1978. E.K. Lenzi, B.F. Oliveira, L.R. da Silva, L.R. Evangelista, J. Math. Phys. 49 (2008) 032108. Y.E. Ryabov, Phys. Rev. E 68 (2003) 030102(R). L.C. Malacarne, R.S. Mendes, E.K. Lenzi, M.K. Lenzi, Phys. Rev. E 74 (2006) 042101. A.V. Chechkin, R. Gorenflo, I.M. Sokolov, Phys. Rev. E 66 (2002) 046129; I.M. Sokolov, A.V. Chechkin, J. Klafter, Acta Phys. Pol. B 35 (2004) 1323; F. Mainardi, G. Pagnini, J. Comput. Appl. Math. 207 (2007) 245. I.M. Sokolov, A.V. Chechkin, J. Klafter, Physica A 336 (2004) 245.