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Solutions for a mass transfer process governed by fractional diffusion equations with reaction terms
Commun Nonlinear Sci Numer Simulat 48 (2017) 307–317
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Research paper
Solutions for a mass transfer process governed by fractional diffusion equations with reaction terms E.K. Lenzi a,b,∗, M.A.F. dos Santos a, M.K. Lenzi c, R. Menechini Neto a a
Departamento de Física, Universidade Estadual de Ponta Grossa, Av. General Carlos Cavalcanti, 4748 - Ponta Grossa, PR 87030-900, Brazil b National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150 Rio de Janeiro, RJ 22290-180, Brazil c Departamento de Engenharia Química, Universidade Federal do Paraná, Av. Cel. Francisco H. dos Santos, 210 - Jardim das Americas Curitiba, PR 81531 - 990, Brazil
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 14 June 2016 Revised 2 January 2017 Accepted 6 January 2017 Available online 7 January 2017
We investigate the behavior of a mass transfer process governed by a set of fractional diffusion equations coupled by appropriate reaction terms. The presence of memory effects in the diffusive term is also considered. For this set of equations, we obtain solutions and analyze the influence of the reaction terms on the spreading of these solutions. Particularly, we observe that for reversible reaction processes the reaction terms play an important role for intermediate times and for long times the processes are essentially governed by the bulk equations. These results show a rich class of behaviors which can be connected to sub- or superdiffusive regime.
Keywords: Anomalous diffusion Fractional derivative Lévy distribution
© 2017 Elsevier B.V. All rights reserved.
1. Introduction Many situations of interest existing in physics [1–3], chemistry [4], and biology [5–7] have shown that the anomalous diffusion phenomenon [1] plays an important role with direct consequences on the dynamic behavior of these systems. One of the most important characteristics of the anomalous diffusion, when the central limit theorem is satisfied, remains on the nonlinear dependence on the mean square displacement, e.g., (x − x )2 ∼ t α with α = 1 , which is usually related to the non-Markovian features manifested by these systems. On the other hand, there are situations characterized by the Lévy distributions, asymptotically characterized by power-laws, which do not satisfy the central limit theorem and consequently have a divergent behavior of the mean square displacement. In order to describe these systems, several approaches have been previously reported in the literature, such as the generalized Langevin equations [8,9], nonlinear diffusion equations [10–13], master equations, and the fractional diffusion equations [14–22]. In this context, reaction diffusion problems have also been investigated [23–29], due to the importance of chemical reactions in scientific and industrial applications. Toward this, we devote this manuscript to analyze the behavior of a mass transfer process [30] governed of a set of fractional diffusion equations coupled by appropriate reaction terms. More precisely, we investigate the following set of diffusion equations, which has been used to describe a subdiffusion limited reaction system:
∂γ ∂2 ρ1 (x, t ) = 2 γ ∂t ∂x ∗
∞ −∞
dx D1 (x − x )ρ1 (x , t ) − r1 ρ1 (x, t ) + r2 ρ2 (x, t ) ,
Corresponding author. E-mail address: [email protected] (E.K. Lenzi).
http://dx.doi.org/10.1016/j.cnsns.2017.01.009 1007-5704/© 2017 Elsevier B.V. All rights reserved.
(1)
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∂γ ∂2 ρ (x, t ) = 2 ∂tγ 2 ∂x
∞
−∞
dx D2 (x − x )ρ2 (x , t ) + r1 ρ1 (x, t ) − r2 ρ2 (x, t ) ,
(2)
where 0 < γ < 1. In these equations, ρ 1 (x, t) and ρ 2 (x, t) represent two different diffusing systems (e.g., substances, particles or species), D1 (x ) and D2 (x ) are the diffusion coefficients for each species and the reaction rates, and r1 and r2 , are connected with the reaction process, which in this case can be represented by the reversible reaction 1 2 or an irreversible process, e.g., 1 → 2. The fractional time operator considered is the Caputo’s one [31]. These equations may be obtained from a random walk by considering the following coupled balance equations
ρ1 (x, t ) = (t )ρ1 (x, 0 ) + −
t 0
−
t 0
t dx dt (x − x , t − t )ρ (x , t )
∞
−∞
0
dt
r1 (t − t )ρ1 (x, t ) +
ρ2 (x, t ) = (t )ρ2 (x, 0 ) +
t
0
dt r2 (t − t )ρ (x, t ),
(3)
t dx dt (x − x , t − t )ρ2 (x , t )
∞
−∞
0
dt
r1 (t − t )ρ1 (x, t ) −
t
0
dt r2 (t − t )ρ2 (x, t ),
(4)
t ∞ where (t ) = 1 − 0 −∞ (x, t )dt dx, r1 (t ) = r1 (t ), and r2 (t ) = r2 (t ). In Eqs. (3) and (4), (x, t ) represents a probability density function from which the waiting time distribution and jumping probability can be obtained, i.e., ω (t ) = ∞ ∞ −∞ (x, t )dx and λ (x ) = 0 (x, t )dt . It is worth mentioning that both, the fractional time derivative and the memory effect present in the diffusive term, change the properties of the waiting time and the jumping probability distributions. For Eqs. (1) and (2), we analyze the mean square displacement for the case D1 (x ) = 0 with D2 (x ) = 0, when the limit central theorem is verified. We also analyze the spreading of the system and obtain exact solutions for the cases D1 (x ) ∝ 1/|x|μ1 −1 with D2 (x ) = 0 and D1 (x ) ∝ 1/|x|μ1 −1 and D2 (x ) ∝ 1/|x|μ2 −1 (1 < μ1 , μ2 ≤ 2). Note that these choices for the kernels present in the diffusive term lead to situations characterized by distributions asymptotically governed by power – laws such as the Lévy distributions, which may be connected to a random walk with long tailed jump probability density. In addition, for μ1 = 2 and μ2 = 2 we have an interplay among different regimes during the time evolution of the solutions obtained for these species. These developments are performed in Sections 2 and in 3, we present our conclusions. 2. Diffusion equation and reaction terms Let us analyze the behavior of the previous set of fractional diffusion equations with linear reaction terms by firstly considering the case characterized by D1 (x ) = 0 with D2 (x ) = 0 and afterwards, the case where D1 (x ) = 0 with D2 (x ) = 0. In the first case, one can assume that one of the species remains somehow immobile in the bulk, while in the second case both species can diffuse in the bulk. For this case, the Eqs. (1) and (2) can be rewritten as
∂γ ∂2 ρ1 (x, t ) = 2 γ ∂t ∂x
∞
−∞
dx D1 (x − x )ρ1 (x , t ) − r1 ρ1 (x, t ) + r2 ρ2 (x, t ) ,
∂γ ρ (x, t ) = r1 ρ1 (x, t ) − r2 ρ2 (x, t ) , ∂tγ 2
(5)
(6)
with r1 = 0 and r2 = 0 characterizing a reversible first order reaction process present in the bulk. This set of equations can be related to an intermittent motion where the reaction terms can be related to the rate of switching the particles from the diffusive mode to the resting mode r1 or switching them from the resting to the movement r2 . It can also be regarded as a problem in diffusion in which some of the diffusing substances become immobilized as the diffusion proceeds, or a problem in chemical kinetics in which the rate of reaction depends on the rate of supply of one of the reactants by diffusion. Before obtaining the solutions for ρ 1 (x, t) and ρ 2 (x, t) from the previous set of diffusion equations, we analyze the behavior of the second moment and, consequently, the time dependence exhibited by the mean square displacement. In order to perform this analysis, we consider that the distributions obtained from these equations satisfy the central limit ∞ theorem. For Eqs. (5) and (6), typical situations which may verify this theorem are usually characterized by −∞ dxD1(2 ) (x ) = const. By taking into account this requirement and the boundary conditions ρ1 (±∞, t ) = 0 and ρ2 (±∞, t ) = 0, it is possible to show that the dynamic of the second moment for both species is governed by the following equations:
∂γ 2 x 1 = 2ID1 S1 (t ) − r1 x2 1 + r2 x2 2 , ∂tγ
(7)
∂γ 2 x 2 = r 1 x2 1 − r2 x2 2 , ∂tγ
(8)
E.K. Lenzi et al. / Commun Nonlinear Sci Numer Simulat 48 (2017) 307–317
309
∞ ∞ ∞ with x2 1(2 ) = −∞ dxx2 ρ1(2 ) (x, t ), ID1 = −∞ dxD1 (x ), and S1(2 ) (t ) = −∞ dxρ1(2 ) (x, t ) representing the survival probability or the quantity of each specie present in the bulk. We observe that the reaction terms are responsible for coupling these equations and, consequently, each distribution is influenced by the other during the spreading process. The analytical solution obtained by solving Eqs. (7) and (8) is given by
x2 1 (t ) = 2ID1
0
t
S1 (t − t )t γ −1 Eγ ,γ −r12 t γ + r2 t γ Eγ ,2γ −r12 t γ
dt
+r2 t γ Eγ ,1+γ (−r12 t γ )x2 2 (0 ) + Eγ (−r12 t γ ) + r2 t γ Eγ ,1+γ (−r12 t γ ) x2 1 (0 )
(9)
for the specie 1, where r12 = r1 + r2 , and
t x2 2 (t ) = 2ID1 r1 S1 (t − t )t 2γ −1 Eγ ,2γ −r12t γ dt + r1t γ Eγ ,1+γ (−r12t γ )x2 1 (0 ) 0 + Eγ (−r21 t γ ) + r1 t γ Eγ ,1+γ (−r2 t γ ) x2 2 (0 )
(10)
for the specie 2. We can also obtain from the Eqs. (5) and (6) the first moment and the survival probability. After some calculations, it is possible to show that they are given by
x1 (t ) = Eγ (−r12t γ ) + r2t γ Eγ ,1+γ (−r12t γ ) x1 (0 ) + r2t γ Eγ ,1+γ (r12t γ )x2 (0 )
S1 (t ) = Eγ (−r12 t γ ) + r2 t γ Eγ ,1+γ (−r12 t γ ) S1 (0 ) + r2 t γ Eγ ,1+γ (−r12 t γ )S2 (0 )
(11)
(12)
for the specie 1 and
x2 (t ) = Eγ (−r12t γ ) + r1t γ Eγ ,1+γ (−rt t γ ) x2 (0 ) + r1t γ Eγ ,1+γ (−r12t γ )x2 1 (0 )
S2 (t ) = Eγ (−r12 t γ ) + r1 t γ Eγ ,1+γ (−rt t γ ) S2 (0 ) + r1 t γ Eγ ,1+γ (−r12 t γ )S1 (0 )
(13)
(14)
for the specie 2. By using the above results, we may obtain the mean square displacement and show the influence of the reaction terms. In this sense, Fig. 1 illustrates the behavior of the mean square displacement, ( 1(2 ) x )2 = (x − x1(2 ) )2 1(2 ) by considering different values for the reaction rates r1 and r2 . For r1 = 0 and r2 = 0 , we observe that the system for intermediate time exhibits a regime essentially influenced by the reaction terms, with the asymptotic behavior (t → ∞) governed by the diffusive term. Thus, the asymptotic behavior manifests the same time dependence for both species as shown in Fig. 1. This figure also shows the behavior of the mean square displacement for two different values of γ . A direct consequence of the γ = 1 is verified in the limit of long times, leading us to a subdiffusive behavior of both species if r1 = 0 and r2 = 0. This feature can be formally verified, for simplicity, for the initial condition ρ1(2 ) (x, 0 ) = ρ1(2 ) δ (x ) with
ρ1 + ρ2 = 1, which leads to ( 1 x )2 ≈ 2r22 ID1 t γ /[(r1 + r2 )2 (1 + γ )] and ( 2 x )2 ≈ 2r2 r1 ID1 t γ /[(r1 + r2 )2 (1 + γ )] for long
times. On the other hand, for small times (t → 0) the behavior manifested by the mean square displacement for the specie 2 is strongly influenced by the initial condition and may manifest ballistic, superdiffusive or usual behavior depending on the choice of γ . For an irreversible reaction process, e.g., r1 = 0 and r2 = 0 , Fig. 1 shows that the specie 1 is totally consumed to produce the specie 2 and this goes to a steady-state situation due to the absence of the diffusive term. In order to obtain the solution for Eqs. (5) and (6), we may use the Laplace in time (L{ρ (x, t )} = ρ (x, s ) and L−1 {ρ (x, s )} = ρ (x, t )) and Fourier in space (F {ρ (x, t )} = ρ˜ (k, t ) and the F −1 {ρ˜ (k, t )} = ρ (x, t )) transforms. Applying these integral transforms in Eqs. (5) and (6), with these equations subjected to the boundary conditions ρ1 (±∞, t ) = 0 and ρ2 (±∞, t ) = 0 and the initial conditions ρ1 (x, 0 ) = ϕ1 (x ) and ρ2 (x, 0 ) = ϕ2 (x ), it is possible to show that ρ˜¯ 1 (k, s ) and ρ˜¯ 2 (k, s ) are given by
ρ˜¯ 1 (k, s ) =
ρ˜¯ 2 (k, s ) =
(sγ + r2 )sγ −1
(sγ + r2 )(sγ + D˜ 1 (k )k2 ) + sγ r1 r1 sγ −1 (sγ + r2 )(sγ + D˜ 1 (k )k2 ) + sγ r1
sγ −1 + γ ϕ˜ 2 (k ) s + r2
ϕ˜ 1 (k ) +
ϕ˜ 1 (k ) +
r2 sγ + r2 r2 sγ + r2
ϕ˜ 2 (k )
ϕ˜ 2 (k )
(15)
(16)
Now, we perform the inverse Fourier and Laplace transform, for the case characterized by D˜ 1 (k ) = D1 |k|μ1 −2 (1 < μ1 ≤ 2 and D1 = const) which interpolates the situations with short (μ1 = 2) and long tailed behaviors (μ1 = 2). This choice for the diffusion coefficient can also be connected with spatial fractional derivatives [31]. After some calculations, it is possible
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Fig. 1. Behavior of the mean square displacements ( 1 x)2 and ( 2 x)2 versus t for γ = 1/2 and γ = 1, by considering the initial condition ρ1 (x, 0 ) = (9/10 )δ (x − 1 ) and ρ2 (x, 0 ) = (1/10 )δ (x + 1 ) with D1 (x ) = D1 δ (x ) (D1 = 1). The black solid lines represent the case r1 = r2 = 1, the black dashed lines correspond to r1 = 1 with r2 = 0. The green lines were included as a guide for the behavior in the asymptotic limit. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
to show that
ρ1 (x, t ) =
∞ −∞
dx ϕ1 (x )(x − x , t )
+r2
ρ2 (x, t ) = r1
∞ −∞
t 0
dx ϕ2 (x )
t
0
dt (x − x , t − t )t γ −1 Eγ ,γ −r2 t γ ,
dt ρ1 (x, t )t γ −1 Eγ ,γ −r2 t γ + Eγ (−r2 t γ )ϕ2 (x ) ,
(17)
(18)
with
(x, t ) = Gγ(1,μ) 1 (x, t ) + × ×
tn
dtn−1
0
t2
dt1 0
∞
(−1 )n
n=1 ∞
−∞ ∞
−∞
t
dtn 0
∞ −∞
d xn ϒ ( x − xn , t − tn )
dxn−1 ϒ (xn − xn−1 , tn − tn−1 ) · · ·
t3
dt2 0
∞
−∞
) dx1 ϒ (x2 − x1 , t2 − t1 )Gγ(1,μ ( x1 , t1 ), 1
ϒ (x, t ) = r1 Gγ(2,μ) 1 (x, t ) − r1 r2
t 0
d x2 ϒ ( x3 − x2 , t3 − t2 ) (19)
) Gγ(2,μ (x, t − t )t γ −1 Eγ ,γ −r2 t γ dt , 1
(20)
E.K. Lenzi et al. / Commun Nonlinear Sci Numer Simulat 48 (2017) 307–317
and
) Gγ(1,μ (x, t ) 1
=
1
μ1 | x |
H32,,31
|x| 1, μ11 , 1, μ γ1 ,(1, 12 ) , μ1 (1,1), 1, 1 ,(1, 1 ) μ1 2 D tγ 1
) Gγ(2,μ (x, t ) = 1
μ1 | x |
a,A
m,n where H p,q x ((b, B))
(21)
1
1
311
t γ −1 H32,,31
|x| 1, μ11 , γ , μγ1 ,(1, 12 ) , μ1 (1,1), 1, 1 ,(1, 1 ) μ1 2 D tγ 1
(22)
1
is the Fox H function [31] and Eα , β (x) is the generalized Mittag–Leffler function [31]. Note that in
) the asymptotic limit, i.e., |x| → ∞, with γ = 1 and μ1 = 2, it is possible to show that Gγ(1,μ (x, t ) ∼ D1 t γ / μ1 |x|1+μ1 1 ) and Gγ(2,μ (x, t ) ∼ D1 t 2γ −1 / μ1 |x|1+μ1 . Others choices for the parameters γ and μ1 may lead to different behaviors for 1 Eqs. (21) and (22). In particular, for the case γ = 1 with μ1 = 2, we have that
1
Gγ(1,2) (x, t )
=
Gγ(2,2) (x, t )
1 γ −1 1,0 = t H1,1 2|x|
H11,,10 4D1 t γ
|x| 1− γ , γ ((0,1)2 2 ) γ
,
D1 t
|x| γ , γ ((1,1)2 ) γ
(23)
.
D1 t
(24)
Eqs. (22) and (24) can be approximated, in the asymptotic limit, by the following equations
Gγ(1,2) (x, t ) ∼
1 4π D1 t γ
⎛
× exp ⎝−
1 2−γ
12−−γγ 2
γ
γ 2 − γ γ 2−γ 2
2
1 t γ −1 Gγ(2,2) (x, t ) ∼ 2|x| π ( 2 − γ )
⎛
2−γ × exp ⎝− 2
γ 12−−γγ
γ 2−γ γ 2
|x|
D1 t γ
2−2γ ⎞
⎠,
D1 t γ
2
γ2−−1γ
|x|
32−2−γγ
|x| D1 t γ
|x| D1 t γ
(25)
2−2γ ⎞ ⎠.
(26)
Another case is obtained by considering μ1 = 2 with γ = 1. In this case, Eqs. (21) and (22) can be simplified to
G1(1,μ) 1 (x, t ) =
1
μ1 | x |
H21,,21
|x| 1, μ11 ,(1, 12 ) , μ1 (1,1),(1, 12 ) D t 1
(27)
1
with G1(1,μ) (x, t ) = G1(2,μ) (x, t ). Eq. (27) is a Lévy distribution [15] and, consequently, the asymptotic limit is characterized by 1
1
a power – law, i.e., it is given by G1(1,μ) (x, t ) ∼ D1 t/ μ1 |x|1+μ1 . 1 Let us consider that both species of substances are diffusing, i.e., D1 (x ) = 0 with D2 (x ) = 0. Similar to the case worked out above, we may use the Fourier and Laplace transform to find a formal solution for this case. After applying the Fourier transform, we obtain that
ρ˜¯ 1 (k, s ) =
(sγ + D˜ 2 (k )k2 + r2 )sγ −1 (sγ + D˜ 2 (k )k2 + r2 )(sγ + D˜ 1 (k )k2 ) + (sγ + D˜ 2 (k )k2 )r1
r2 × ϕ˜ 1 (k ) + ϕ˜ 2 (k ) γ ˜ s + D2 ( k )k2 + r2
,
(28)
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E.K. Lenzi et al. / Commun Nonlinear Sci Numer Simulat 48 (2017) 307–317
Fig. 2. Behavior of the distributions 1/ρ12 (0, t ) and 1/ρ22 (0, t ) obtained from Eqs. (30) and (31) versus t. In Fig. 2a, the black dashed and solid lines represent the case with μ1 = 3/2 and μ2 = 2 with r1 = r2 /4 = 1. In Fig. 2b, the black dashed and solid represent the case with μ1 = 3/2 and μ2 = 2 with r1 = r2 /2 = 1. We consider, for simplicity, D = 1, ϕ1 (x ) = (1/2 )δ (x ), and ϕ2 (x ) = (1/2 )δ (x ). The green lines were included as a guide for the behavior in the asymptotic limit. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ρ˜¯ 2 (k, s ) =
r1 sγ −1
(sγ + D˜ 2 (k )k2 + r2 )(sγ + D˜ 1 (k )k2 ) + (sγ + D˜ 2 (k )k2 )r1
× ϕ˜ 1 (k ) +
r2 ϕ˜ 2 (k ) γ ˜ s + D2 ( k )k2 + r2
+
sγ −1 sγ + D˜ 2 (k )k2 + r2
ϕ˜ 2 (k ) .
(29)
In order to obtain the inverse Fourier and Laplace transform, we consider the case characterized by D˜ 1 (k ) = D1 |k|μ1 −2 (1 < μ1 ≤ 2 and D1 = const) and D˜ 2 (k ) = D2 |k|μ2 −2 (1 < μ2 ≤ 2 and D2 = const). These choices for D˜ 1 (k ) and D˜ 2 (k ) enable us to describe situations characterized by distributions with short (μ1 = μ2 = 2) and long tailed (μ1 = 2 and μ2 = 2) behaviors for the species 1 and 2. Particularly, the case μ1 = μ2 leads us to an interplay between different diffusive regimes with the asymptotic limit governed by the distribution characterized by the lower value of the parameters μ1 and μ2 . After performing some calculations,Eqs. (22) and (23) can be written as
ρ1 (x, t ) =
∞ −∞
×
ρ2 (x, t ) = r1 +
dx ϕ1 (x )1,2 (x − x , t ) + r2 t
0
∞ −∞
d x
∞ −∞
dx ϕ2 (x − x )
) dt 1,2 (x − x , t − t )Gγ(4,μ (x , t ; r2 ) 2
∞
−∞ ∞ −∞
d x
0
t
(30)
) dt ρ1 (x, t − t )Gγ(4,μ ( x − x , t − t ; r2 ) 2
) dx Gγ(3,μ ( x − x , t )ϕ2 ( x ) 2
(31)
E.K. Lenzi et al. / Commun Nonlinear Sci Numer Simulat 48 (2017) 307–317
313
Fig. 3. This figure shows the behavior of the distributions ρ 1 (x, t) and ρ 2 (x, t) obtained from Eq. (40) and (41) by considering, for simplicity, t = 0.1 and D = 1. The black dashed–dotted and solid lines represent the cases γ = 1 with μ = 3/2 and γ = 1/2 with μ = 2. The green dashed line is the case γ = 1/2 with μ = 3/2. We consider, for simplicity, ϕ1 (x ) = δ (x ), ϕ2 (x ) = 0, r1 = 2, and r2 = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
with
1,2 (x, t )=Gγ(1,μ) 1 (x, t ) + × ×
tn 0
dtn−1
t2
dt1 0
∞
n=1
t
dtn 0
∞ −∞
d xn I ( x − xn , t − tn )
∞
dxn−1 I (xn − xn−1 , tn − tn−1 )· · ·
−∞ ∞
−∞
t3
dt2 0
∞ −∞
d x2 I ( x3 − x2 , t3 − t2 )
) dx1 I (x2 − x1 , t2 − t1 )Gγ(1,μ ( x1 , t1 ) 1
where ) I (x, t ) = r1 Gγ(2,μ (x, t ) − r1 r2 1
and
(−1 )n
(32)
t 0
) dt¯Gγ(2,μ (x, t − t¯ )Gγ(4,μ) 2 (x, t¯; r2 ) 1
(33)
) Gγ(3,μ (x, t; r2 ) 2
∞
(−r2 t γ )n 2,1 |x| 1, μ12 , 1+nγ , μγ2 ,(1, 12 ) = H3,3 1 1 1 μ2 |x| n=0 (1 + n ) (D2 t γ ) μ2 (1,1), 1+n, μ2 ,(1, 2 )
) Gγ(4,μ (x, t; r2 ) 2
∞ t γ −1 (−r2 t γ )n 2,1 |x| 1, μ12 , (1+n) γ , μγ2 ,(1, 12 ) = H3,3 1 1 1 μ2 |x| n=0 (1 + n ) (D2 t γ ) μ2 (1,1), 1+n, μ2 ,(1, 2 )
1
,
(34)
.
(35)
Fig. 2 illustrates the time dependent behavior of 1/ρ12 (0, t ) and 1/ρ22 (0, t ) as a measurement of the spreading of the species 1 and 2 in the bulk by taking into account different situations. It shows the asymptotic limit is governed by the diffusive term related to the long tailed distribution obtained from the lower value of the parameters μ1 and μ2 .
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Fig. 4. Behavior of the distributions ρ 1 (x, t) and ρ 2 (x, t) obtained from Eqs. (40) and (41) by considering different times for D˜ 1 (k ) = D˜ 2 (k ) = 1/(1 + Cδ |k|δ ) with Cδ = 1 and δ = 3/2. We consider, for simplicity, ϕ1 (x ) = δ (x ), ϕ2 (x ) = 0, r1 = 2, and r2 = 1.
Similarly to Eqs. (21) and (22), we can obtain approximated expressions for Eqs. (34) and (35) in the asymptotic limit of |x| → ∞. In particular, the case μ2 = 2 yields ) Gγ(3,μ (x, t; r2 ) ∼ 2
∞
∞
(−r2 t γ )n μ2 | x | (1 + n )
1
n=0 m=1
×
) Gγ(4,μ (x, t; r2 ) ∼ 2
D t γ m (1 + μ2 m ) (1 + n + m ) 2 − μ |x| 2 (1 + m ) (1 + (n + m )γ ) 1 + μ22 m − μ22 m 1
μ2 | x | ×
t γ −1
(36)
∞
∞
(−r2 t γ )n (1 + n ) n=0 m=1
D t γ m (1 + μ2 m ) (1 + n + m ) μ2 − 2 μ2 , μ2 |x| (1 + m ) ((1 + n )γ + mγ ) 1 + 2 m − 2 m
(37)
and the case μ2 = 2 leads us to obtain
Gγ(3,2) (x, t; r2 ) ∼
2−1γ 1−γ 2−γ
⎛
γ γ γ−2
1 r2 |x| 2 exp ⎝− t γ γ γ 2 2 D t 2|x| π ( 2 − γ ) 2 ⎛ 2−2γ ⎞ γ 2−γ γ 2 − γ | x| ⎠, × exp ⎝− 2
2
D2 t γ
|x| D2 t γ
2(21−−γγ ) ⎞ ⎠
(38)
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315
Fig. 5. Fig. 5a illustrates 1/ρ12 (0, t ) and 1/ρ22 (0, t ) obtained from Eqs. (40) and (41) versus t for D˜ 1 (k ) = D˜ 2 (k ) = 1/(1 + Cδ |k|δ ). In Fig. 5b, we compare the distributions ρ 1 (x, t) and ρ 2 (x, t) obtained for D˜ 1 (k ) = D˜ 2 (k ) = 1/(1 + Cδ |k|δ ) (colored lines) with the case D˜ 1 (k ) = D˜ 2 (k ) = 1 (black dotted lines) and observe a transition from a Lévy – like distribution to a Gaussian one for t → ∞. We consider, for simplicity, Cδ = 1, δ = 3/2, ϕ1 (x ) = δ (x ), and ϕ2 (x ) = 0. The green line was included as a guide for the behavior in the asymptotic limit, which in this case characterizes the usual diffusion. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Gγ(4,2) (x, t; r2 ) ∼
γ 12−−γγ
t γ −1
32−2−γγ
⎛
γ γ γ−2
r2 |x| exp ⎝− t γ γ 2 2 2 D t 2|x| π ( 2 − γ ) 2 ⎛ 2−2γ ⎞ γ 2−γ γ 2 − γ | x| ⎠. × exp ⎝− 2
2
D2 t γ
2(21−−γγ ) ⎞ |x| ⎠ D2 t γ
(39)
These asymptotic limits obtained for Eqs. (34) and (35) in different conditions show that the case μ2 = 2 leads to long tailed distributions characterized by power – laws, in contrast to the case μ2 = 2 where the distribution is essentially governed by stretched like exponential. For the particular case D˜ 1 (k ) = D˜ 2 (k ) = D|k|μ−2 (1 < μ ≤ 2 and D = const), the previous solutions can be simplified to
ρ1 (x, t ) =
ρ2 (x, t ) =
∞ r2 ) dx Gγ(1,μ ( x − x , t ) ϕ1 ( x ) + ϕ2 ( x ) r1 + r2 −∞ ∞ 1 ) + dx Gγ(3,μ (x − x , t; r1 + r2 ) r1 ϕ1 (x ) + r2 ϕ2 (x ) r1 + r2 −∞
∞ r1 ) dx Gγ(1,μ ( x − x , t ) ϕ1 ( x ) + ϕ2 ( x ) r1 + r2 −∞ ∞ 1 ) − dx Gγ(3,μ (x − x , t; r1 + r2 ) r1 ϕ1 (x ) + r2 ϕ2 (x ) . r1 + r2 −∞
(40)
(41)
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Fig. 3 illustrates the behavior of Eqs. (40) and (41) for different values of γ with μ. Note that for γ = 1/2 with μ = 2 (black solid line) the distribution may be essentially characterized by an stretched exponential like behavior, in contrast to the case γ = 1 with μ = 3/2 (black dashed – dotted line) which may be asymptotically governed by a power – law. The case γ = 1/2 with μ = 3/2 is mixing of these cases (green dashed line) asymptotically characterized by stretched like exponential and power – laws. Fig. 4 illustrates the case D˜ 1 (k ) = D˜ 2 (k ) = 1/(1 + Cδ |k|δ ), which can be related to a truncated Lévy distribution [32]. For this case, we have a transition, in the limit t → ∞, from a Lévy like distribution to a Gaussian one (see Fig. 5). Similar situation in absence of reaction terms is discussed in Ref. [32]. In addition, Fig. 5 evidences a normal diffusion in this limit and, consequently, that the distributions may be approximated to the situations characterized by D˜ 1 (k ) ≈ D1 = const and D˜ 2 (k ) ≈ D2 = const. 3. Discussion and conclusions We have analyzed the solutions for a mass transfer process governed by a set of fractional diffusion equation coupled by reaction terms. For this set of equations, we have first considered the situation characterized by one of the species mobile and the second immobile in the bulk, i.e., D1 = 0 with D2 = 0. For this case, we have investigated the behavior of the mean square displacement when the central limit theorem is satisfied. In this scenario, Fig. 1 illustrates the mean square displacement for both species by considering the cases γ = 1 and γ = 1/2. We have observed for these cases that the behavior of the specie 1 is essentially governed by the bulk equation for small and long times when r1 = 0 and r2 = 0. The reaction terms have played an important role for intermediate times. Similar features were observed for the mean square displacement connected to specie 2, for intermediate and long times. For small times, the specie 2 presented a rich variety of behaviors depending on the γ value covering sub- and superdiffusive situations due to the absence of the diffusive term. We obtained exact solutions, for the case D˜ 1 (k ) = D1 |k|μ1 −2 with D˜ 2 (k ) = 0, in terms of the Fox H functions and the generalized Mittag - Leffler functions. These results can be connected with a diffusion problem in which some of the diffusing substance becomes immobilized as the diffusion proceeds or a problem in chemical kinetics in which the rate of reaction depends on the rate of supply of one of the reactants by diffusion. After this analysis, we considered the situation where both species can diffuse in the bulk, i.e., D˜ 1 (k ) = D1 |k|μ1 −2 with D˜ 2 (k ) = D2 |k|μ2 −2 . For this case, we found the solutions and observed that depending on the values of γ and μ, as shown in Fig. 3, the distributions may exhibit an asymptotic behavior characterized by a stretched exponential or a power - law. Figs. 4 and 5 illustrate a truncated Lévy distribution, where a transition from a Lévy like behavior to a gaussian one is observed in the asymptotic limit of t → ∞. In this sense, Fig. 5a shows that the behavior is initially anomalous and, after some time, goes to an usual diffusion. Finally, we hope that the results presented here may be useful to investigate mass transfer processes where nonusual diffusion processes are present. Acknowledgment We thank the CNPq for partial financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Pekalski A, Sznajd-Weron K. Anomalous diffusion: from basics to applications. Springer, Heidelberg, Germany; 1998. Mendez V, Campos D, Bartumeus F. Stochastic foundations in movement ecology. Springer, Heidelberg, Germany; 2014. ben Avraham D, Havlin S. Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge, UK; 2005. Avnir D. The fractal approach to heterogeneous chemistry. John Wiley and Sons Inc., New York, USA; 1990. Banks DS, Fradin C. Anomalous diffusion of proteins due to molecular crowding. Biophys J 2005;89:2960–71. Iyiola OS, Zaman FD. A fractional diffusion equation model for cancer tumor. AIP Adv 2014;4:107–21. Hofling F, Franosch T. Anomalous transport in the crowded world of biological cells. Rep Prog Phys 2013;76:046602. Coffey WT, Kalmykov YP, Waldron JT. The langevin equation: with applications to stochastic problems in physics, chemistry and electrical engineering. World Scientific Publishing Company, Singapore, Singapore; 2004. Sandev T, Metzler R, Tomovski Z. Velocity and displacement correlation functions for fractional generalized langevin equations. Fract Calc Appl Anal 2012;15:426–50. Lenzi EK, Mendes GA, SMendes R, da Silva LR, Lucena LS. Exact solutions to nonlinear nonautonomous space-fractional diffusion equations with absorption. Phys Rev E 2003;67:051109. DFrank T. Nonlinear Fokker-Planck equations. Germany: Springer, Heidelberg; 2005. Plastino AR, Plastino A. Non-extensive statistical mechanics and generalized fokker-planck equation. Physica A 1995;222:347–54. Giordano C, Plastino AR, Casas M, Plastino A. Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy. Eur Phys J B 2001;22:361–8. Lenzi EK, dos Santos MAF, Lenzi MK, Vieira DS, Silva LRd. Solutions for a fractional diffusion equation: anomalous diffusion and adsorption-desorption processes. J King Saud Univ Sci 2016;28:3–6. Meztler R, Klafter J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 20 0 0;339:1–77. KLenzi E, dos Santos MAF, Vieira DS, Zola RS, Ribeiro HV. Solutions for a sorption process governed by a fractional diffusion equation. Physica A 2016;443:32–41. KLenzi E, Vieira DS, Lenzi MK, alves GGA, Leitoles DP. Solutions for a fractional diffusion equation with radial symmetry and integro-differential boundary conditions. Therm Sci 2015;19:S1–6. Klafter J, Sokolov IM. First steps in random walks: from tools to applications. UK: Oxford University Press, Oxford; 2011. Rocca MC, Plastino AR, Plastino A, Ferrie GL, de Paolia A. General solution of a fractional diffusion-advection equation for solar cosmic-ray transport. Physica A 2016;447:402–10. Tarasov VE, Stat J. Fractional diffusion equations for lattice and continuum: grnwald-letnikov differences and derivatives approach. Int J Stat Mech 2014:873529.
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[21] Hristov J. A unified nonlinear fractional equation of the diffusion-controlled surfactant adsorption: reappraisal and new solution of the ward-tordai problem. J King Saud Univ Sci 2016;28:7–13. [22] West B, Bologna M, Grigolini P. Physics of fractal operators springer. New York, United States; 2003. [23] Lin R, Liu F, Anh V, Turner I. Stability and convergence of a new explicit finite–difference approximation for the variable-order nonlinear fractional diffusion equation. App Math Comp 2009;212:435–45. [24] Gafiychuk V, Datsko B. Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems. Comput Math Appl 2010;59:1101–7. [25] del Castillo-Negrete D, Carreras BA, Lynch VE. Front dynamics in reaction-diffusion systems with levy flights: a fractional diffusion approach. Phys Rev Lett 2003;91:018302. [26] Volpert VA, Nec Y, Nepomnyashchy AA. Fronts in anomalous diffusion - reaction systems, philos. T R Soc A 2015;371:20120179. [27] Campos D, Fedotov S, Méndez V. Anomalous reaction-transport processes: the dynamics beyond the law of mass action. Phys Rev E 2008;77:061130. [28] Nepomnyashchy AA. Mathematical modelling of subdiffusion-reaction systems. Math Model Nat Phenom 2016;11:26–36. [29] del Castillo-Negrete D, Carreras BA, Lynch VE. Front dynamics in reaction-diffusion systems with levy flights: a fractional diffusion approach. Phys Rev Lett 2003;91:018302. [30] Cussler EL. Diffusion: mass transfer in fluid systems. Cambridge, UK: Cambridge University Press; 1997. [31] Mathai AM, Saxena RK, Haubold HJ. The H-function: theory and applications. New York, USA: Springer; 2009. [32] Sokolov IM, Chechkin AV, Klafter J. Fractional diffusion equation for a power-law-truncated lévy process. Physica A 2004;336:245–51.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Research paper
Solutions for a mass transfer process governed by fractional diffusion equations with reaction terms E.K. Lenzi a,b,∗, M.A.F. dos Santos a, M.K. Lenzi c, R. Menechini Neto a a
Departamento de Física, Universidade Estadual de Ponta Grossa, Av. General Carlos Cavalcanti, 4748 - Ponta Grossa, PR 87030-900, Brazil b National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150 Rio de Janeiro, RJ 22290-180, Brazil c Departamento de Engenharia Química, Universidade Federal do Paraná, Av. Cel. Francisco H. dos Santos, 210 - Jardim das Americas Curitiba, PR 81531 - 990, Brazil
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 14 June 2016 Revised 2 January 2017 Accepted 6 January 2017 Available online 7 January 2017
We investigate the behavior of a mass transfer process governed by a set of fractional diffusion equations coupled by appropriate reaction terms. The presence of memory effects in the diffusive term is also considered. For this set of equations, we obtain solutions and analyze the influence of the reaction terms on the spreading of these solutions. Particularly, we observe that for reversible reaction processes the reaction terms play an important role for intermediate times and for long times the processes are essentially governed by the bulk equations. These results show a rich class of behaviors which can be connected to sub- or superdiffusive regime.
Keywords: Anomalous diffusion Fractional derivative Lévy distribution
© 2017 Elsevier B.V. All rights reserved.
1. Introduction Many situations of interest existing in physics [1–3], chemistry [4], and biology [5–7] have shown that the anomalous diffusion phenomenon [1] plays an important role with direct consequences on the dynamic behavior of these systems. One of the most important characteristics of the anomalous diffusion, when the central limit theorem is satisfied, remains on the nonlinear dependence on the mean square displacement, e.g., (x − x )2 ∼ t α with α = 1 , which is usually related to the non-Markovian features manifested by these systems. On the other hand, there are situations characterized by the Lévy distributions, asymptotically characterized by power-laws, which do not satisfy the central limit theorem and consequently have a divergent behavior of the mean square displacement. In order to describe these systems, several approaches have been previously reported in the literature, such as the generalized Langevin equations [8,9], nonlinear diffusion equations [10–13], master equations, and the fractional diffusion equations [14–22]. In this context, reaction diffusion problems have also been investigated [23–29], due to the importance of chemical reactions in scientific and industrial applications. Toward this, we devote this manuscript to analyze the behavior of a mass transfer process [30] governed of a set of fractional diffusion equations coupled by appropriate reaction terms. More precisely, we investigate the following set of diffusion equations, which has been used to describe a subdiffusion limited reaction system:
∂γ ∂2 ρ1 (x, t ) = 2 γ ∂t ∂x ∗
∞ −∞
dx D1 (x − x )ρ1 (x , t ) − r1 ρ1 (x, t ) + r2 ρ2 (x, t ) ,
Corresponding author. E-mail address: [email protected] (E.K. Lenzi).
http://dx.doi.org/10.1016/j.cnsns.2017.01.009 1007-5704/© 2017 Elsevier B.V. All rights reserved.
(1)
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∂γ ∂2 ρ (x, t ) = 2 ∂tγ 2 ∂x
∞
−∞
dx D2 (x − x )ρ2 (x , t ) + r1 ρ1 (x, t ) − r2 ρ2 (x, t ) ,
(2)
where 0 < γ < 1. In these equations, ρ 1 (x, t) and ρ 2 (x, t) represent two different diffusing systems (e.g., substances, particles or species), D1 (x ) and D2 (x ) are the diffusion coefficients for each species and the reaction rates, and r1 and r2 , are connected with the reaction process, which in this case can be represented by the reversible reaction 1 2 or an irreversible process, e.g., 1 → 2. The fractional time operator considered is the Caputo’s one [31]. These equations may be obtained from a random walk by considering the following coupled balance equations
ρ1 (x, t ) = (t )ρ1 (x, 0 ) + −
t 0
−
t 0
t dx dt (x − x , t − t )ρ (x , t )
∞
−∞
0
dt
r1 (t − t )ρ1 (x, t ) +
ρ2 (x, t ) = (t )ρ2 (x, 0 ) +
t
0
dt r2 (t − t )ρ (x, t ),
(3)
t dx dt (x − x , t − t )ρ2 (x , t )
∞
−∞
0
dt
r1 (t − t )ρ1 (x, t ) −
t
0
dt r2 (t − t )ρ2 (x, t ),
(4)
t ∞ where (t ) = 1 − 0 −∞ (x, t )dt dx, r1 (t ) = r1 (t ), and r2 (t ) = r2 (t ). In Eqs. (3) and (4), (x, t ) represents a probability density function from which the waiting time distribution and jumping probability can be obtained, i.e., ω (t ) = ∞ ∞ −∞ (x, t )dx and λ (x ) = 0 (x, t )dt . It is worth mentioning that both, the fractional time derivative and the memory effect present in the diffusive term, change the properties of the waiting time and the jumping probability distributions. For Eqs. (1) and (2), we analyze the mean square displacement for the case D1 (x ) = 0 with D2 (x ) = 0, when the limit central theorem is verified. We also analyze the spreading of the system and obtain exact solutions for the cases D1 (x ) ∝ 1/|x|μ1 −1 with D2 (x ) = 0 and D1 (x ) ∝ 1/|x|μ1 −1 and D2 (x ) ∝ 1/|x|μ2 −1 (1 < μ1 , μ2 ≤ 2). Note that these choices for the kernels present in the diffusive term lead to situations characterized by distributions asymptotically governed by power – laws such as the Lévy distributions, which may be connected to a random walk with long tailed jump probability density. In addition, for μ1 = 2 and μ2 = 2 we have an interplay among different regimes during the time evolution of the solutions obtained for these species. These developments are performed in Sections 2 and in 3, we present our conclusions. 2. Diffusion equation and reaction terms Let us analyze the behavior of the previous set of fractional diffusion equations with linear reaction terms by firstly considering the case characterized by D1 (x ) = 0 with D2 (x ) = 0 and afterwards, the case where D1 (x ) = 0 with D2 (x ) = 0. In the first case, one can assume that one of the species remains somehow immobile in the bulk, while in the second case both species can diffuse in the bulk. For this case, the Eqs. (1) and (2) can be rewritten as
∂γ ∂2 ρ1 (x, t ) = 2 γ ∂t ∂x
∞
−∞
dx D1 (x − x )ρ1 (x , t ) − r1 ρ1 (x, t ) + r2 ρ2 (x, t ) ,
∂γ ρ (x, t ) = r1 ρ1 (x, t ) − r2 ρ2 (x, t ) , ∂tγ 2
(5)
(6)
with r1 = 0 and r2 = 0 characterizing a reversible first order reaction process present in the bulk. This set of equations can be related to an intermittent motion where the reaction terms can be related to the rate of switching the particles from the diffusive mode to the resting mode r1 or switching them from the resting to the movement r2 . It can also be regarded as a problem in diffusion in which some of the diffusing substances become immobilized as the diffusion proceeds, or a problem in chemical kinetics in which the rate of reaction depends on the rate of supply of one of the reactants by diffusion. Before obtaining the solutions for ρ 1 (x, t) and ρ 2 (x, t) from the previous set of diffusion equations, we analyze the behavior of the second moment and, consequently, the time dependence exhibited by the mean square displacement. In order to perform this analysis, we consider that the distributions obtained from these equations satisfy the central limit ∞ theorem. For Eqs. (5) and (6), typical situations which may verify this theorem are usually characterized by −∞ dxD1(2 ) (x ) = const. By taking into account this requirement and the boundary conditions ρ1 (±∞, t ) = 0 and ρ2 (±∞, t ) = 0, it is possible to show that the dynamic of the second moment for both species is governed by the following equations:
∂γ 2 x 1 = 2ID1 S1 (t ) − r1 x2 1 + r2 x2 2 , ∂tγ
(7)
∂γ 2 x 2 = r 1 x2 1 − r2 x2 2 , ∂tγ
(8)
E.K. Lenzi et al. / Commun Nonlinear Sci Numer Simulat 48 (2017) 307–317
309
∞ ∞ ∞ with x2 1(2 ) = −∞ dxx2 ρ1(2 ) (x, t ), ID1 = −∞ dxD1 (x ), and S1(2 ) (t ) = −∞ dxρ1(2 ) (x, t ) representing the survival probability or the quantity of each specie present in the bulk. We observe that the reaction terms are responsible for coupling these equations and, consequently, each distribution is influenced by the other during the spreading process. The analytical solution obtained by solving Eqs. (7) and (8) is given by
x2 1 (t ) = 2ID1
0
t
S1 (t − t )t γ −1 Eγ ,γ −r12 t γ + r2 t γ Eγ ,2γ −r12 t γ
dt
+r2 t γ Eγ ,1+γ (−r12 t γ )x2 2 (0 ) + Eγ (−r12 t γ ) + r2 t γ Eγ ,1+γ (−r12 t γ ) x2 1 (0 )
(9)
for the specie 1, where r12 = r1 + r2 , and
t x2 2 (t ) = 2ID1 r1 S1 (t − t )t 2γ −1 Eγ ,2γ −r12t γ dt + r1t γ Eγ ,1+γ (−r12t γ )x2 1 (0 ) 0 + Eγ (−r21 t γ ) + r1 t γ Eγ ,1+γ (−r2 t γ ) x2 2 (0 )
(10)
for the specie 2. We can also obtain from the Eqs. (5) and (6) the first moment and the survival probability. After some calculations, it is possible to show that they are given by
x1 (t ) = Eγ (−r12t γ ) + r2t γ Eγ ,1+γ (−r12t γ ) x1 (0 ) + r2t γ Eγ ,1+γ (r12t γ )x2 (0 )
S1 (t ) = Eγ (−r12 t γ ) + r2 t γ Eγ ,1+γ (−r12 t γ ) S1 (0 ) + r2 t γ Eγ ,1+γ (−r12 t γ )S2 (0 )
(11)
(12)
for the specie 1 and
x2 (t ) = Eγ (−r12t γ ) + r1t γ Eγ ,1+γ (−rt t γ ) x2 (0 ) + r1t γ Eγ ,1+γ (−r12t γ )x2 1 (0 )
S2 (t ) = Eγ (−r12 t γ ) + r1 t γ Eγ ,1+γ (−rt t γ ) S2 (0 ) + r1 t γ Eγ ,1+γ (−r12 t γ )S1 (0 )
(13)
(14)
for the specie 2. By using the above results, we may obtain the mean square displacement and show the influence of the reaction terms. In this sense, Fig. 1 illustrates the behavior of the mean square displacement, ( 1(2 ) x )2 = (x − x1(2 ) )2 1(2 ) by considering different values for the reaction rates r1 and r2 . For r1 = 0 and r2 = 0 , we observe that the system for intermediate time exhibits a regime essentially influenced by the reaction terms, with the asymptotic behavior (t → ∞) governed by the diffusive term. Thus, the asymptotic behavior manifests the same time dependence for both species as shown in Fig. 1. This figure also shows the behavior of the mean square displacement for two different values of γ . A direct consequence of the γ = 1 is verified in the limit of long times, leading us to a subdiffusive behavior of both species if r1 = 0 and r2 = 0. This feature can be formally verified, for simplicity, for the initial condition ρ1(2 ) (x, 0 ) = ρ1(2 ) δ (x ) with
ρ1 + ρ2 = 1, which leads to ( 1 x )2 ≈ 2r22 ID1 t γ /[(r1 + r2 )2 (1 + γ )] and ( 2 x )2 ≈ 2r2 r1 ID1 t γ /[(r1 + r2 )2 (1 + γ )] for long
times. On the other hand, for small times (t → 0) the behavior manifested by the mean square displacement for the specie 2 is strongly influenced by the initial condition and may manifest ballistic, superdiffusive or usual behavior depending on the choice of γ . For an irreversible reaction process, e.g., r1 = 0 and r2 = 0 , Fig. 1 shows that the specie 1 is totally consumed to produce the specie 2 and this goes to a steady-state situation due to the absence of the diffusive term. In order to obtain the solution for Eqs. (5) and (6), we may use the Laplace in time (L{ρ (x, t )} = ρ (x, s ) and L−1 {ρ (x, s )} = ρ (x, t )) and Fourier in space (F {ρ (x, t )} = ρ˜ (k, t ) and the F −1 {ρ˜ (k, t )} = ρ (x, t )) transforms. Applying these integral transforms in Eqs. (5) and (6), with these equations subjected to the boundary conditions ρ1 (±∞, t ) = 0 and ρ2 (±∞, t ) = 0 and the initial conditions ρ1 (x, 0 ) = ϕ1 (x ) and ρ2 (x, 0 ) = ϕ2 (x ), it is possible to show that ρ˜¯ 1 (k, s ) and ρ˜¯ 2 (k, s ) are given by
ρ˜¯ 1 (k, s ) =
ρ˜¯ 2 (k, s ) =
(sγ + r2 )sγ −1
(sγ + r2 )(sγ + D˜ 1 (k )k2 ) + sγ r1 r1 sγ −1 (sγ + r2 )(sγ + D˜ 1 (k )k2 ) + sγ r1
sγ −1 + γ ϕ˜ 2 (k ) s + r2
ϕ˜ 1 (k ) +
ϕ˜ 1 (k ) +
r2 sγ + r2 r2 sγ + r2
ϕ˜ 2 (k )
ϕ˜ 2 (k )
(15)
(16)
Now, we perform the inverse Fourier and Laplace transform, for the case characterized by D˜ 1 (k ) = D1 |k|μ1 −2 (1 < μ1 ≤ 2 and D1 = const) which interpolates the situations with short (μ1 = 2) and long tailed behaviors (μ1 = 2). This choice for the diffusion coefficient can also be connected with spatial fractional derivatives [31]. After some calculations, it is possible
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E.K. Lenzi et al. / Commun Nonlinear Sci Numer Simulat 48 (2017) 307–317
Fig. 1. Behavior of the mean square displacements ( 1 x)2 and ( 2 x)2 versus t for γ = 1/2 and γ = 1, by considering the initial condition ρ1 (x, 0 ) = (9/10 )δ (x − 1 ) and ρ2 (x, 0 ) = (1/10 )δ (x + 1 ) with D1 (x ) = D1 δ (x ) (D1 = 1). The black solid lines represent the case r1 = r2 = 1, the black dashed lines correspond to r1 = 1 with r2 = 0. The green lines were included as a guide for the behavior in the asymptotic limit. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
to show that
ρ1 (x, t ) =
∞ −∞
dx ϕ1 (x )(x − x , t )
+r2
ρ2 (x, t ) = r1
∞ −∞
t 0
dx ϕ2 (x )
t
0
dt (x − x , t − t )t γ −1 Eγ ,γ −r2 t γ ,
dt ρ1 (x, t )t γ −1 Eγ ,γ −r2 t γ + Eγ (−r2 t γ )ϕ2 (x ) ,
(17)
(18)
with
(x, t ) = Gγ(1,μ) 1 (x, t ) + × ×
tn
dtn−1
0
t2
dt1 0
∞
(−1 )n
n=1 ∞
−∞ ∞
−∞
t
dtn 0
∞ −∞
d xn ϒ ( x − xn , t − tn )
dxn−1 ϒ (xn − xn−1 , tn − tn−1 ) · · ·
t3
dt2 0
∞
−∞
) dx1 ϒ (x2 − x1 , t2 − t1 )Gγ(1,μ ( x1 , t1 ), 1
ϒ (x, t ) = r1 Gγ(2,μ) 1 (x, t ) − r1 r2
t 0
d x2 ϒ ( x3 − x2 , t3 − t2 ) (19)
) Gγ(2,μ (x, t − t )t γ −1 Eγ ,γ −r2 t γ dt , 1
(20)
E.K. Lenzi et al. / Commun Nonlinear Sci Numer Simulat 48 (2017) 307–317
and
) Gγ(1,μ (x, t ) 1
=
1
μ1 | x |
H32,,31
|x| 1, μ11 , 1, μ γ1 ,(1, 12 ) , μ1 (1,1), 1, 1 ,(1, 1 ) μ1 2 D tγ 1
) Gγ(2,μ (x, t ) = 1
μ1 | x |
a,A
m,n where H p,q x ((b, B))
(21)
1
1
311
t γ −1 H32,,31
|x| 1, μ11 , γ , μγ1 ,(1, 12 ) , μ1 (1,1), 1, 1 ,(1, 1 ) μ1 2 D tγ 1
(22)
1
is the Fox H function [31] and Eα , β (x) is the generalized Mittag–Leffler function [31]. Note that in
) the asymptotic limit, i.e., |x| → ∞, with γ = 1 and μ1 = 2, it is possible to show that Gγ(1,μ (x, t ) ∼ D1 t γ / μ1 |x|1+μ1 1 ) and Gγ(2,μ (x, t ) ∼ D1 t 2γ −1 / μ1 |x|1+μ1 . Others choices for the parameters γ and μ1 may lead to different behaviors for 1 Eqs. (21) and (22). In particular, for the case γ = 1 with μ1 = 2, we have that
1
Gγ(1,2) (x, t )
=
Gγ(2,2) (x, t )
1 γ −1 1,0 = t H1,1 2|x|
H11,,10 4D1 t γ
|x| 1− γ , γ ((0,1)2 2 ) γ
,
D1 t
|x| γ , γ ((1,1)2 ) γ
(23)
.
D1 t
(24)
Eqs. (22) and (24) can be approximated, in the asymptotic limit, by the following equations
Gγ(1,2) (x, t ) ∼
1 4π D1 t γ
⎛
× exp ⎝−
1 2−γ
12−−γγ 2
γ
γ 2 − γ γ 2−γ 2
2
1 t γ −1 Gγ(2,2) (x, t ) ∼ 2|x| π ( 2 − γ )
⎛
2−γ × exp ⎝− 2
γ 12−−γγ
γ 2−γ γ 2
|x|
D1 t γ
2−2γ ⎞
⎠,
D1 t γ
2
γ2−−1γ
|x|
32−2−γγ
|x| D1 t γ
|x| D1 t γ
(25)
2−2γ ⎞ ⎠.
(26)
Another case is obtained by considering μ1 = 2 with γ = 1. In this case, Eqs. (21) and (22) can be simplified to
G1(1,μ) 1 (x, t ) =
1
μ1 | x |
H21,,21
|x| 1, μ11 ,(1, 12 ) , μ1 (1,1),(1, 12 ) D t 1
(27)
1
with G1(1,μ) (x, t ) = G1(2,μ) (x, t ). Eq. (27) is a Lévy distribution [15] and, consequently, the asymptotic limit is characterized by 1
1
a power – law, i.e., it is given by G1(1,μ) (x, t ) ∼ D1 t/ μ1 |x|1+μ1 . 1 Let us consider that both species of substances are diffusing, i.e., D1 (x ) = 0 with D2 (x ) = 0. Similar to the case worked out above, we may use the Fourier and Laplace transform to find a formal solution for this case. After applying the Fourier transform, we obtain that
ρ˜¯ 1 (k, s ) =
(sγ + D˜ 2 (k )k2 + r2 )sγ −1 (sγ + D˜ 2 (k )k2 + r2 )(sγ + D˜ 1 (k )k2 ) + (sγ + D˜ 2 (k )k2 )r1
r2 × ϕ˜ 1 (k ) + ϕ˜ 2 (k ) γ ˜ s + D2 ( k )k2 + r2
,
(28)
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Fig. 2. Behavior of the distributions 1/ρ12 (0, t ) and 1/ρ22 (0, t ) obtained from Eqs. (30) and (31) versus t. In Fig. 2a, the black dashed and solid lines represent the case with μ1 = 3/2 and μ2 = 2 with r1 = r2 /4 = 1. In Fig. 2b, the black dashed and solid represent the case with μ1 = 3/2 and μ2 = 2 with r1 = r2 /2 = 1. We consider, for simplicity, D = 1, ϕ1 (x ) = (1/2 )δ (x ), and ϕ2 (x ) = (1/2 )δ (x ). The green lines were included as a guide for the behavior in the asymptotic limit. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ρ˜¯ 2 (k, s ) =
r1 sγ −1
(sγ + D˜ 2 (k )k2 + r2 )(sγ + D˜ 1 (k )k2 ) + (sγ + D˜ 2 (k )k2 )r1
× ϕ˜ 1 (k ) +
r2 ϕ˜ 2 (k ) γ ˜ s + D2 ( k )k2 + r2
+
sγ −1 sγ + D˜ 2 (k )k2 + r2
ϕ˜ 2 (k ) .
(29)
In order to obtain the inverse Fourier and Laplace transform, we consider the case characterized by D˜ 1 (k ) = D1 |k|μ1 −2 (1 < μ1 ≤ 2 and D1 = const) and D˜ 2 (k ) = D2 |k|μ2 −2 (1 < μ2 ≤ 2 and D2 = const). These choices for D˜ 1 (k ) and D˜ 2 (k ) enable us to describe situations characterized by distributions with short (μ1 = μ2 = 2) and long tailed (μ1 = 2 and μ2 = 2) behaviors for the species 1 and 2. Particularly, the case μ1 = μ2 leads us to an interplay between different diffusive regimes with the asymptotic limit governed by the distribution characterized by the lower value of the parameters μ1 and μ2 . After performing some calculations,Eqs. (22) and (23) can be written as
ρ1 (x, t ) =
∞ −∞
×
ρ2 (x, t ) = r1 +
dx ϕ1 (x )1,2 (x − x , t ) + r2 t
0
∞ −∞
d x
∞ −∞
dx ϕ2 (x − x )
) dt 1,2 (x − x , t − t )Gγ(4,μ (x , t ; r2 ) 2
∞
−∞ ∞ −∞
d x
0
t
(30)
) dt ρ1 (x, t − t )Gγ(4,μ ( x − x , t − t ; r2 ) 2
) dx Gγ(3,μ ( x − x , t )ϕ2 ( x ) 2
(31)
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Fig. 3. This figure shows the behavior of the distributions ρ 1 (x, t) and ρ 2 (x, t) obtained from Eq. (40) and (41) by considering, for simplicity, t = 0.1 and D = 1. The black dashed–dotted and solid lines represent the cases γ = 1 with μ = 3/2 and γ = 1/2 with μ = 2. The green dashed line is the case γ = 1/2 with μ = 3/2. We consider, for simplicity, ϕ1 (x ) = δ (x ), ϕ2 (x ) = 0, r1 = 2, and r2 = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
with
1,2 (x, t )=Gγ(1,μ) 1 (x, t ) + × ×
tn 0
dtn−1
t2
dt1 0
∞
n=1
t
dtn 0
∞ −∞
d xn I ( x − xn , t − tn )
∞
dxn−1 I (xn − xn−1 , tn − tn−1 )· · ·
−∞ ∞
−∞
t3
dt2 0
∞ −∞
d x2 I ( x3 − x2 , t3 − t2 )
) dx1 I (x2 − x1 , t2 − t1 )Gγ(1,μ ( x1 , t1 ) 1
where ) I (x, t ) = r1 Gγ(2,μ (x, t ) − r1 r2 1
and
(−1 )n
(32)
t 0
) dt¯Gγ(2,μ (x, t − t¯ )Gγ(4,μ) 2 (x, t¯; r2 ) 1
(33)
) Gγ(3,μ (x, t; r2 ) 2
∞
(−r2 t γ )n 2,1 |x| 1, μ12 , 1+nγ , μγ2 ,(1, 12 ) = H3,3 1 1 1 μ2 |x| n=0 (1 + n ) (D2 t γ ) μ2 (1,1), 1+n, μ2 ,(1, 2 )
) Gγ(4,μ (x, t; r2 ) 2
∞ t γ −1 (−r2 t γ )n 2,1 |x| 1, μ12 , (1+n) γ , μγ2 ,(1, 12 ) = H3,3 1 1 1 μ2 |x| n=0 (1 + n ) (D2 t γ ) μ2 (1,1), 1+n, μ2 ,(1, 2 )
1
,
(34)
.
(35)
Fig. 2 illustrates the time dependent behavior of 1/ρ12 (0, t ) and 1/ρ22 (0, t ) as a measurement of the spreading of the species 1 and 2 in the bulk by taking into account different situations. It shows the asymptotic limit is governed by the diffusive term related to the long tailed distribution obtained from the lower value of the parameters μ1 and μ2 .
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Fig. 4. Behavior of the distributions ρ 1 (x, t) and ρ 2 (x, t) obtained from Eqs. (40) and (41) by considering different times for D˜ 1 (k ) = D˜ 2 (k ) = 1/(1 + Cδ |k|δ ) with Cδ = 1 and δ = 3/2. We consider, for simplicity, ϕ1 (x ) = δ (x ), ϕ2 (x ) = 0, r1 = 2, and r2 = 1.
Similarly to Eqs. (21) and (22), we can obtain approximated expressions for Eqs. (34) and (35) in the asymptotic limit of |x| → ∞. In particular, the case μ2 = 2 yields ) Gγ(3,μ (x, t; r2 ) ∼ 2
∞
∞
(−r2 t γ )n μ2 | x | (1 + n )
1
n=0 m=1
×
) Gγ(4,μ (x, t; r2 ) ∼ 2
D t γ m (1 + μ2 m ) (1 + n + m ) 2 − μ |x| 2 (1 + m ) (1 + (n + m )γ ) 1 + μ22 m − μ22 m 1
μ2 | x | ×
t γ −1
(36)
∞
∞
(−r2 t γ )n (1 + n ) n=0 m=1
D t γ m (1 + μ2 m ) (1 + n + m ) μ2 − 2 μ2 , μ2 |x| (1 + m ) ((1 + n )γ + mγ ) 1 + 2 m − 2 m
(37)
and the case μ2 = 2 leads us to obtain
Gγ(3,2) (x, t; r2 ) ∼
2−1γ 1−γ 2−γ
⎛
γ γ γ−2
1 r2 |x| 2 exp ⎝− t γ γ γ 2 2 D t 2|x| π ( 2 − γ ) 2 ⎛ 2−2γ ⎞ γ 2−γ γ 2 − γ | x| ⎠, × exp ⎝− 2
2
D2 t γ
|x| D2 t γ
2(21−−γγ ) ⎞ ⎠
(38)
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Fig. 5. Fig. 5a illustrates 1/ρ12 (0, t ) and 1/ρ22 (0, t ) obtained from Eqs. (40) and (41) versus t for D˜ 1 (k ) = D˜ 2 (k ) = 1/(1 + Cδ |k|δ ). In Fig. 5b, we compare the distributions ρ 1 (x, t) and ρ 2 (x, t) obtained for D˜ 1 (k ) = D˜ 2 (k ) = 1/(1 + Cδ |k|δ ) (colored lines) with the case D˜ 1 (k ) = D˜ 2 (k ) = 1 (black dotted lines) and observe a transition from a Lévy – like distribution to a Gaussian one for t → ∞. We consider, for simplicity, Cδ = 1, δ = 3/2, ϕ1 (x ) = δ (x ), and ϕ2 (x ) = 0. The green line was included as a guide for the behavior in the asymptotic limit, which in this case characterizes the usual diffusion. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Gγ(4,2) (x, t; r2 ) ∼
γ 12−−γγ
t γ −1
32−2−γγ
⎛
γ γ γ−2
r2 |x| exp ⎝− t γ γ 2 2 2 D t 2|x| π ( 2 − γ ) 2 ⎛ 2−2γ ⎞ γ 2−γ γ 2 − γ | x| ⎠. × exp ⎝− 2
2
D2 t γ
2(21−−γγ ) ⎞ |x| ⎠ D2 t γ
(39)
These asymptotic limits obtained for Eqs. (34) and (35) in different conditions show that the case μ2 = 2 leads to long tailed distributions characterized by power – laws, in contrast to the case μ2 = 2 where the distribution is essentially governed by stretched like exponential. For the particular case D˜ 1 (k ) = D˜ 2 (k ) = D|k|μ−2 (1 < μ ≤ 2 and D = const), the previous solutions can be simplified to
ρ1 (x, t ) =
ρ2 (x, t ) =
∞ r2 ) dx Gγ(1,μ ( x − x , t ) ϕ1 ( x ) + ϕ2 ( x ) r1 + r2 −∞ ∞ 1 ) + dx Gγ(3,μ (x − x , t; r1 + r2 ) r1 ϕ1 (x ) + r2 ϕ2 (x ) r1 + r2 −∞
∞ r1 ) dx Gγ(1,μ ( x − x , t ) ϕ1 ( x ) + ϕ2 ( x ) r1 + r2 −∞ ∞ 1 ) − dx Gγ(3,μ (x − x , t; r1 + r2 ) r1 ϕ1 (x ) + r2 ϕ2 (x ) . r1 + r2 −∞
(40)
(41)
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Fig. 3 illustrates the behavior of Eqs. (40) and (41) for different values of γ with μ. Note that for γ = 1/2 with μ = 2 (black solid line) the distribution may be essentially characterized by an stretched exponential like behavior, in contrast to the case γ = 1 with μ = 3/2 (black dashed – dotted line) which may be asymptotically governed by a power – law. The case γ = 1/2 with μ = 3/2 is mixing of these cases (green dashed line) asymptotically characterized by stretched like exponential and power – laws. Fig. 4 illustrates the case D˜ 1 (k ) = D˜ 2 (k ) = 1/(1 + Cδ |k|δ ), which can be related to a truncated Lévy distribution [32]. For this case, we have a transition, in the limit t → ∞, from a Lévy like distribution to a Gaussian one (see Fig. 5). Similar situation in absence of reaction terms is discussed in Ref. [32]. In addition, Fig. 5 evidences a normal diffusion in this limit and, consequently, that the distributions may be approximated to the situations characterized by D˜ 1 (k ) ≈ D1 = const and D˜ 2 (k ) ≈ D2 = const. 3. Discussion and conclusions We have analyzed the solutions for a mass transfer process governed by a set of fractional diffusion equation coupled by reaction terms. For this set of equations, we have first considered the situation characterized by one of the species mobile and the second immobile in the bulk, i.e., D1 = 0 with D2 = 0. For this case, we have investigated the behavior of the mean square displacement when the central limit theorem is satisfied. In this scenario, Fig. 1 illustrates the mean square displacement for both species by considering the cases γ = 1 and γ = 1/2. We have observed for these cases that the behavior of the specie 1 is essentially governed by the bulk equation for small and long times when r1 = 0 and r2 = 0. The reaction terms have played an important role for intermediate times. Similar features were observed for the mean square displacement connected to specie 2, for intermediate and long times. For small times, the specie 2 presented a rich variety of behaviors depending on the γ value covering sub- and superdiffusive situations due to the absence of the diffusive term. We obtained exact solutions, for the case D˜ 1 (k ) = D1 |k|μ1 −2 with D˜ 2 (k ) = 0, in terms of the Fox H functions and the generalized Mittag - Leffler functions. These results can be connected with a diffusion problem in which some of the diffusing substance becomes immobilized as the diffusion proceeds or a problem in chemical kinetics in which the rate of reaction depends on the rate of supply of one of the reactants by diffusion. After this analysis, we considered the situation where both species can diffuse in the bulk, i.e., D˜ 1 (k ) = D1 |k|μ1 −2 with D˜ 2 (k ) = D2 |k|μ2 −2 . For this case, we found the solutions and observed that depending on the values of γ and μ, as shown in Fig. 3, the distributions may exhibit an asymptotic behavior characterized by a stretched exponential or a power - law. Figs. 4 and 5 illustrate a truncated Lévy distribution, where a transition from a Lévy like behavior to a gaussian one is observed in the asymptotic limit of t → ∞. In this sense, Fig. 5a shows that the behavior is initially anomalous and, after some time, goes to an usual diffusion. Finally, we hope that the results presented here may be useful to investigate mass transfer processes where nonusual diffusion processes are present. Acknowledgment We thank the CNPq for partial financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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