Modeling and simulation of the fractional space-time diffusion equation

Commun Nonlinear Sci Numer Simulat 30 (2016) 115–127

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Modeling and simulation of the fractional space-time diffusion equation J.F. Gómez-Aguilar a,b,∗, M. Miranda-Hernández c, M.G. López-López b, V.M. Alvarado-Martínez b, D. Baleanu d,e a

Cátedras CONACYT, Mexico Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico c Instituto de Energías Renovables, Universidad Nacional Autónoma de México (IER-UNAM), A.P. 34, Temixco, Morelos, C.P. 62580, Mexico d Faculty of Art and Sciences, Department of Mathematics and Computer Sciences, Cancaya University, Balgat 0630, Ankara, Turkey e Institute of Space Sciences, P.O. Box, MG-23, Magurele-Bucharest, R 76900, Romania b

a r t i c l e

i n f o

Article history: Received 22 September 2014 Revised 10 April 2015 Accepted 15 June 2015 Available online 22 June 2015 Keywords: Fractional diffusion Transmission lines Caputo derivative Anomalous diffusion Subdiffusion Superdiffusion

a b s t r a c t In this paper, the space-time fractional diffusion equation related to the electromagnetic transient phenomena in transmission lines is studied, three cases are presented; the diffusion equation with fractional spatial derivative, with fractional temporal derivative and the case with fractional space-time derivatives. For the study cases, the order of the spatial and temporal fractional derivatives are 0 < β , γ ≤ 2, respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation. The general solutions of the proposed equations are expressed in terms of the multivariate Mittag-Leffler functions; these functions depend only on the parameters β and γ and preserve the appropriated physical units for any value of the fractional derivative exponent. Furthermore, an analysis of the fractional time constant was made in order to indicate the change of the medium properties and the presence of dissipation mechanisms. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, irreversible thermodynamics, quantum optics or turbulent diffusion, thermal stresses, models of porous electrodes, the description of gel solvents and anomalous complex processes. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Fractional calculus (FC) is the investigation and treatment of mathematical models in terms of derivatives and integrals of arbitrary order [1–4]. In the last thirty years, the interest in the field has considerably increased due to many practical potential applications [5–10]. The equation of diffusion describes a non-equilibrium situation which is moving towards equilibrium at a rate governed by its distance from equilibrium, so that it reaches equilibrium in a time which is theoretically infinite, diffusion processes involve the transport of mass in diffusion, the transport of momentum in friction or viscosity and the transport of energy in thermal conductivity. The fractional representation of the standard diffusion equation in space-time has been ∗

Corresponding author. Tel.: +52 (777) 3627770. E-mail address: [email protected], [email protected] (J.F. Gómez-Aguilar).

http://dx.doi.org/10.1016/j.cnsns.2015.06.014 1007-5704/© 2015 Elsevier B.V. All rights reserved.

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successfully used for modeling physical relevant processes [11–14]. These representations arise in continuous-time random walks. A random walk is a mathematical formalization that consists of a succession of random steps related to the diffusion models, by introducing the fractional approach it is possible to include external fields [11]. A Lévy flight, also referred as Lévy motion, is a random walk in which the step-lengths have a heavy-tailed probability distribution. When defined as a walk in a space of dimension greater than one, the steps are defined in terms of a probability distribution, and move with isotropic random directions [15]. Recently, the fundamental space-time fractional diffusion equation obtained from the standard diffusion equation was discussed [16]. Gorenflo and Mainardi reported the time fractional diffusion equation obtained from a fractional Fick law, the fundamental obtained solution is interpreted as a probability density of a non-Markovian stochastic process and is related to a phenomenon of slow anomalous diffusion [17]. Lin proposed the numerical resolution based on a finite difference scheme in time and Legendre spectral methods, in a space of a time-fractional diffusion equation in a cantor set [18]. Based on the classical Crank–Nicholson method, Tadjeran in [19] obtained temporally and spatially second-order accurate numerical estimates of the fractional order diffusion equations. In this context the diffusion, diffusion-advection, Fokker–Planck equations and a Cauchy problem for the time-fractional diffusion equation were suggested based on local FC theory [20,21]. In recent papers of Luchko [22–24], the generalized time-fractional diffusion equation with variable coefficients was considered. The author shows the existence and uniqueness of the solution for the initial boundary value problem of the generalized time-fractional diffusion equation; the Fourier method was used to construct a formal solution. Other applications of fractional calculus in anomalous diffusion are given in [25–33]. Frequently, authors replace the integer derivative by another derivative of fractional order on a purely mathematical form. However, from the physical point of view that is not totally correct because the physical parameters contained in the differential equation have not the dimensionality measured in the laboratory [34], as a consequence, some dimensional corrections in the fractional equation are necessary. In this paper we employ the idea suggested by [34] to the analyze the diffusion phenomena related with the electromagnetic transient phenomenon in transmission lines. This paper discusses the diffusion wave relating the voltage or current wave in transmission lines using the Caputo-type fractional derivative. The order of the considered fractional diffusion equation is 0 < β , γ ≤ 2 in space-time domain, respectively. With the aim to introduce dimensional correction in the fractional equation, two parameters are introduced σ x and σ t ; these parameters characterize the existence of the fractional spatial and temporal structures, hence, the representation preserves the physical units of the system for any value taken by the exponent of the fractional derivative. The paper is structured as follows, in section 2 we explain the basic concepts of the fractional calculus, in section 3 we present the fractional diffusion equation and give conclusions in section 4. 2. Basic definitions Caputo fractional derivative (CFD) was introduced by M. Caputo in 1967, with this fractional derivative definition, the derivative of a constant is zero and the initial conditions for the fractional order differential equations can be given in the same manner as for the ordinary differential equations with a known physical interpretation. The left CFD for a function f(t) is given by C φ a Dt

f (t ) =

1  (n − φ)



t a

f (n) (η) dη, (−η + t )φ −n+1

n − 1 < φ ≤ n.

(1)

The right CFD has the following form C φ t Db

f (t ) =

(−1)n  (n − φ)



b a

f (n) (η) dη, (η − t )φ −n+1

n − 1 < φ ≤ n,

(2)

where n = 1, 2, . . . ∈ N and  (·) represents the Euler’s gamma function. If φ is an integer, n − 1 < φ ≤ n, the fractional derivatives become the classical ones, named

 d φ

C φ a Dt

f (t ) =

C φ t Db

f (t ) = −

and

dt

f (t ),

 d φ dt

(3)

f (t ).

(4)

Another definition which will be used is the Mittag-Leffler function. This function which plays an important role in the theory of fractional differential equations due to its vast potential of applications describing realistic physical systems with memory and delay, it is defined by the series expansion as [3]

Eφ ,ϕ (t ) =

∞  m=0

tm ,  (φ m + ϕ)

(φ > 0), (ϕ > 0).

Some common Mittag-Leffler functions are [35]

(5)

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i(x,t)

Rdx

Cdx

V(x,t)

i(x+dx,t)

A

Ldx

117

Gdx

V(x+dx,t)

x Fig. 1. Equivalent circuit for the transmission line considerer losses.

E1/2,1 (±α) = eα [1 ± erfc(α)], 2

E1,1 (±α) = e±α , E2,1 (−α 2 ) = cos (α), E3,1 (α) =

1 2

E4,1 (α) =

1 2



eα

1/3

+ 2e−(1/2)α

1/3

cos

√

3 2

α 1/3



(6)

( cos (α 1/4 ) + cosh (α 1/4 )).

The erfc (α ) denotes the complementary error function [35] and it is defined as

2 erfc(α) = √

π

 α

e−t dt. 2

(7)

0

3. Fractional diffusion equation and the classical transmission line model with losses The equivalent circuit for the transmission line with losses is represented in Fig. 1. Applying the Kirchhoff circuit laws in the node A we have the equations of the transmission lines with losses for the voltage and current, respectively

∂2 ∂2 ∂ V (x, t ) − LC 2 V (x, t ) − (RC + GL) V (x, t ) − GRV (x, t ) = 0, 2 ∂t ∂x ∂t

(8)

∂2 ∂2 ∂ I ( x, t ) − LC I(x, t ) − (RC + GL) I(x, t ) − GRI(x, t ) = 0, ∂t ∂ x2 ∂t2

(9)

where R denotes the resistance of the conductors, L the inductance due to the magnetic field around the wires, C the capacitance between the two conductors and G the conductance of the dielectric material separating the conductors. Considering the Eq. (8) and L = G = 0 (the transmission line has negligible inductance and shunt conductance), we have the cable model for lossy telegraph lines

∂ 2V (x, t ) ∂ V (x, t ) = 0, − RC ∂t ∂ x2

(10)

changing the cable model for lossy telegraph lines (8) into diffusion equation, the Eq. (10) describes the ordinary diffusion, for 1 represents the reciprocal of the time constant of the system or diffusion this case only R and C determine its properties, D = RC coefficient. The velocity of the concentration wave through a medium is determined by the inertia and the elasticity of the medium. These two properties are capable of storing wave energy. The introduction of a resistive or loss element in the transmission line, Fig. 1, produces an exponential decay with distance along the transmission line in exactly the same way as an oscillator is damped with time, for the fractional case, the damping is intrinsic to the equation of motion and not by introducing an additional force as in the case of an ordinary damping harmonic oscillator, therefore this decay should be considered as an ensemble average of harmonic oscillators [34]. Such a loss mechanism, resistive, viscous, frictional or diffusive, will always result in energy loss from the propagating wave. These are all examples of random collision processes which operate in only one direction in the sense that they are thermodynamically irreversible. Usually these dissipation is know as internal friction. In previous studies, the diffusion equation is considered with fractional derivatives, but these equations are not acceptable physically due to the dimensional incompatibility of the solutions. To be consistent with dimensionality and following [34] we introduce a parameter σ x (dimensions of length) and σ t (dimensions of time) in the following way

1 ∂ ∂β → 1−β · β , n − 1 < β ≤ n, ∂x ∂x σx

(11)

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1 ∂ ∂γ → 1−γ · γ , ∂t ∂t σt

n − 1 < γ ≤ n,

(12)

where n is an integer, when β = 1 and γ = 1, the expressions (11) and (12) become an ordinary derivative. The existence of fractional structures (components that show an intermediate behavior between a system conservative and dissipative) in the system are characterized by the parameters σ x and σ t , these parameters related to equation results in a fractal space-time geometry presented an entire new family of solutions for the diffusion equation. From now on, we will apply this idea to generalize the case of the fractional diffusion equation. Considerer the Eqs. (11) and (12), the fractional representation of (10) is

1 ∂ γ C (x, t ) ∂ 2β C (x, t ) − = 0, σt1−γ ∂ t γ σx2(1−β) ∂ x2β D

(13)

the order of the derivative is 0 < β , γ ≤ 2, V (x, t ) = C (x, t ) represents the potential related to concentration. The fractional diffusion Eq. (13) might be useful for investigating the mechanism of anomalous diffusion usually found in electrochemical phenomena, propagation of energy in dissipative systems, models of porous electrodes and transport processes through complex and/or disordered systems including fractal media. Now two cases will be analyzed: the first case occurs when β (the order of the space derivative) is fractional and the time derivative is ordinary; the second case takes place when γ (the order of the time derivative) is fractional and the space derivative is ordinary. 3.1. Fractional space diffusion equation For the first study case, considering the Eq. (13) and assuming that the space derivative is fractional and the time derivative is ordinary, the spatial fractional equation is

∂ 2β C (x, t ) σx2(1−β) ∂ C (x, t ) =0 − D ∂t ∂ x2β

0<β ≤2

(14)

the Eq. (14) represents a random walk type Lévy flight, there, the step-lengths has a probability distribution that is heavy-tailed [31]. A particular solution of the Eq. (14) may be found in the form

C˜(x, t ) = C˜0 · e−ωt u(x),

(15)

substituting (15) into (14) we obtain

d2β u(x) ω 2(1−β) + σx u(x) = 0, D dx2β

(16)

where ω is the angular frequency and D is the diffusion coefficient, we can define

k2 =

ω

,

(17)

2(1−β) , k˜ 2 = k2 σx

(18)

D

and

where k˜ 2 is the fractional wave number and k is the classical wave number. Substituting (18) into (16) we obtain

d2β u(x) ˜ 2 + k u(x) = 0, dx2β

(19)

the solution of the Eq. (19) may be found in the form of the power series [22]. The solution is written as

u(x) = E2β (−k˜ 2 x2β ).

(20)

The particular solution of the Eq. (20) is

C˜(x, t ) = C˜0 · e−ωt · E2β (−k˜ 2 x2β ),

(21)

where E2β (−k˜ 2 x2β ) is the Mittag-Leffler function. Now the case when β takes different values will be analyzed. When β = 1/2, from the Eq. (21) we have

C˜(x, t ) = C˜0 · e−ωt · E1 (−k˜ 2 x),

(22)

where k˜ 2 = ω D σx . The solution of the Eq. (21) is ˜2 C˜(x, t ) = C˜0 · e−(ωt+k x) ,

the Eq. (23) represents the diffusion with spatial-decaying amplitude respecting to space x.

(23)

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119

Fig. 2. Simulation of the Eq. (31), β = 1, classical Markovian–Lévy flights.

When β = 1, from the Eq. (21) we have

C˜(x, t ) = C˜0 · e−ωt · E2 (−k˜ 2 x2 ),

(24)

where k˜ 2 = k2 . The solution of the Eq. (21) is

C˜(x, t ) = C˜0 · e−ωt cos (kx),

(25)

the solution (25) represents the classical case for the spatial diffusion equation. When β = 3/2, from the Eq. (21) we have

C˜(x, t ) = C˜0 · e−ωt · E3 (−k˜ 2 x3 ),

(26)

−1 where k˜ 2 = ω D σx and E3 is given by (6), the solution of the Eq. (21) is

C˜(x, t ) =

  √3  C˜0 · e−ωt k˜ 2/3 ˜ 2/3 · e−k x + 2e 2 x · cos − k˜ 2/3 x . 2 2

(27)

When β = 2, from the Eq. (21) we have

C˜(x, t ) = C˜0 · e−ωt · E4 (−k˜ 2 x4 ),

(28)

−2 where k˜ 2 = ω D σx and E4 is given by (6), the solution of the Eq. (21) is

C˜(x, t ) =

C˜0 · e−ωt · [cos (−k˜ 1/2 x) + cosh (−k˜ 1/2 x)]. 2

(29)

In this case there exists a physical relation between the wave number k, the parameter σ x and the wave length λ given by the order β of the fractional differential equation

β = kσx =

σx , 0 < σx ≤ λ. λ

(30)

We can use this relation in order to write the Eq. (21) as

˜ t ) = C0 · e−ωt · E2β (−β 2(1−β) x˜2β ), C (x,

(31)

where, x˜ = λ , is a dimensionless parameter. The Fig. 2 shows the simulation of the Eq. (31) for β = 1 and the Fig. 3 shows the simulation of the Eq. (31) for 0 < β ≤ 2, β values were arbitrarily chosen. For the cases 0 < β ≤ 1 we observe the non-Markovian–Lévy flights and in the cases 1 < β ≤ 2, we observe the Markovian– Lévy flights [20]. x

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Diffusion 1 =1/2 =1 =3/2 =2

0.8

C(˜ x, t)

0.6

0.4

0.2

0 0

1

2

3

x ˜

4

5

6

7

Fig. 3. Concentration in the spatial case. Simulation of the Eq. (31) for 0 < β ≤ 2.

3.2. Fractional time diffusion equation Second case, considering the Eq. (13) and assuming that the time derivative is fractional and the space derivative is ordinary, the temporal fractional equation is 2 ∂ γ C (x, t ) 1−γ ∂ C (x, t ) − Dσt = 0, 0 < γ ≤ 2 γ ∂t ∂ x2

(32)

the Eq. (32) represents a stochastic scheme (continuous time random walk) [33]. A particular solution of the Eq. (32) may be found in the form

C˜(x, t ) = C˜0 · e−ikx u(t ),

(33)

substituting (33) into (32) we obtain

dγ u(t ) 1−γ + Dk2 σt u(t ) = 0, dt γ

(34)

where k is the wave number and D is the diffusion coefficient, we can define

ω = Dk2 ,

(35)

ω˜ = ωσt1−γ ,

(36)

and

where ω ˜ is the angular frequency in the medium in presence of fractional time components and ω is the angular frequency without its presence. Substituting (36) into (34) we obtain

dγ u(t ) +ω ˜ u(t ) = 0, dt γ

(37)

the solution of the Eq. (37) may be found in the form of the power series. The solution is written as

u(t ) = Eγ (−ω ˜ t γ ).

(38)

The particular solution of the Eq. (38) is

˜ t γ ), C˜(x, t ) = C˜0 · e−ikx · Eγ (−ω

(39)

where Eγ (−ω ˜ t γ ) is the Mittag-Leffler function. Now will be analyzed the case when γ takes different values. When γ = 1/2, from the Eq. (39) we have

˜ t 1/2 ), C˜(x, t ) = C˜0 · e−ikx · E1/2 (−ω

(40)

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121

where ω ˜ = ωσt1/2 , E1/2 is given by (6), substituting (6) into (40) leads to the solution 2 ˜ t 1/2 ), C˜(x, t ) = C˜0 · e−ikx · eω˜ t erfc(ω

(41)

the erfc (α ) denotes the error function defined in (7). The Eq. (41) represents the time evolution of the concentration and the amplitude which exhibits an algebraic decay for t → ∞. When γ = 1, from the Eq. (39) we have

˜ t ), C˜(x, t ) = C˜0 · e−ikx · E1 (−ω

(42)

where ω ˜ = ω. The solution (39) is written as

C˜(x, t ) = C˜0 · e−(ωt+ikx) ,

(43)

the Eq. (43) represents the classical case for the time diffusion equation. When γ = 3/2, from the Eq. (39) we have

˜ t 3/2 ), C˜(x, t ) = C˜0 · e−ikx · E3/2 (−ω

(44)

ωσt−1/2 .

where ω ˜ = The Mittag-Leffler function Es/2, r is defined by Miller in the paper [36]

Es/2,r (z) =

s−1 2n−1  z2κ(1−r)  1−(s/2+r) α j z2κ zk αj (e )(α s/2 + erfc(α 1/2 zκ )) − z−2n , j j s  (sk/2 + μ) j=0

(45)

k=0

˜ t 3/2 . where κ = 1/s, r = ns + μ, n = 0, 1, 2, 3, . . . , μ = 1, 2, 3, . . . , in this case, z = −ω When γ = 2, from the Eq. (39) we have

˜ t 2 ), C˜(x, t ) = C˜0 · e−ikx · E2 (−ω where ω ˜ =

ωσt−1 .

(46)

The solution (39) is written as

√ ˜ t ). C˜(x, t ) = C˜0 · e−ikx cos ( ω

(47)

In this case there exists a physical relation between the natural frequency ω, the parameter σ t and the period T0 given by the order γ of the fractional differential equation

γ = ωσt =

σt

T0

0 < σt ≤ T0 .

,

(48)

We can use this relation in order to write to Eq. (39) as

C˜(x, t˜) = C˜0 · e−ikx · Eγ (−γ 1−γ t˜γ ),

(49)

where, t˜ = ωt is a dimensionless parameter. The Fig. 4 shows the simulation of the Eq. (49) for γ = 1 and the Fig. 5 shows the simulation of the Eq. (49) for 0 < γ ≤ 2, γ values were arbitrarily chosen. For the cases 0 < γ ≤ 1, we observe the subdiffusion phenomena and the case γ = 1 the phenomena of Brownian diffusion, in the cases 1 < γ ≤ 2, we observe the sub-wave phenomena (superdiffusion) and the case γ = 2 the ballistic diffusion [20]. Analysis of the fractional time constant Now we consider two cases in accordance with the election of time constant of system or diffusion coefficient. 1−γ 1 First case. Considerer, ω ˜ = Dk2 σt and D = RC the reciprocal of the time constant or diffusion coefficient, we have

ω˜ = k2

 σ 1−γ  t

RC

,

(50)

ω˜ is the angular frequency in the medium in presence of fractional time components, k is the wave number and RC is the time constant of the system. Substituting (50) in (39) we obtain

 k2  tγ , τγ

C˜(x, t ) = C˜0 · e−ikx · Eγ −

(51)

γ −1

where τγ = RC σt it can be called fractional time constant due to its dimensionality, sγ . When γ = 1, we have the well known time constant τ1 = RC measured in seconds. Second case. Considering the Eq. (8) and C = R = 0, we have

∂ 2V (x, t ) ∂ V (x, t ) = 0, − GL ∂t ∂ x2

(52)

the fractional representation of (52) considerer the Eqs. (11) and (12) is

1

σ

2(1−β) x

GL ∂ γ C (x, t ) ∂ 2β C (x, t ) − 1−γ = 0, 2 β ∂tγ ∂x σt

(53)

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Fig. 4. Simulation of the Eq. (49), γ = 1, phenomena of Brownian diffusion.

Diffusion 1 =1/2 =1 =3/2 =2

˜ t˜) C(x,

0.5

0

−0.5 0

1

2

3

t˜

4

5

6

7

Fig. 5. Diffusion in the temporal case. Simulation of the Eq. (49) for 0 < γ ≤ 2. In the case γ = 1, we have the phenomena of Brownian diffusion and in the case γ = 2, the ballistic diffusion.

1−γ

the order of the derivative is 0 < β , γ ≤ 1 and 1 < β , γ ≤ 2, C(x, t) represents the concentration, G = 1/R, ω ˜ = Dk2 σt R D = L stand for the reciprocal of the time constant of the system or diffusion coefficient

ω˜ = k2

R L

 σt1−γ ,

and

(54)

ω˜ is the angular frequency in the medium in presence of fractional time components, k is the wave number and L/R is the time constant of the system. Substituting (54) in (39) we obtain

 k2  tγ , τγ

C˜(x, t ) = C˜0 · e−ikx · Eγ −

(55)

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123

Diffusion 1 =1 =0.95 =0.9 =0.85

0.9 0.8 0.7

˜ t) C(x,

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1 Constant time (τ )

1.5

2

Fig. 6. Diffusion vs. Constant time, exponent γ = 1, τ = 0.63 located in t = 1 s, γ = 0.95, τ = 0.63 located in t = 1.02 s, γ = 0.9, τ = 0.63 located in t = 1.06 s and γ = 0.85, τ = 0.63 located in t = 1.1 s. Table 1 Diffusion vs. constant time.

γ

Constant time (τ )

Diffusion (C)

1 0.95 0.9 0.85

1 1.02 1.06 1.09

0.63 0.63 0.63 0.63

γ −1

where τγ = RL σt it can be called fractional time constant due to its dimensionality, sγ . When γ = 1, we have the well known time constant τ1 = L/R measured in seconds. It can be seen that the solutions (51) and (55) have the same mathematical structure. Then, we can analyzed the behavior of either. 1 and R = 1 k, L = 1 mH for D = RL we simulate the Eq. (51) or (55). The values Given the values, R = 1 M, C =1 μF for D = RC for the concentration are shown in the Fig. 6 for the fractional exponents γ = 1, γ = 0.95, γ = 0.9 and γ = 0.85, respectively, Table 1. In assessing the fractional exponent shows that the constant time tends to move forward in time as this exponent γ changes from γ = 1, γ = 0.95, γ = 0.9 to γ = 0.85, respectively; that is, the diffusion occurs in more time than the ordinary diffusion. This phenomenon indicates the change of the medium properties, different from the ideal properties presented in the Eq. (51) or (55), an intermediate behavior occurs, it remains between a conservative system (capacitor) and a dissipative system (resistor), namely anomalous diffusion (subdiffusion). In this case the crosswise capacitor of ordinary diffusion is replaced by a phase constant element (CPE), this electrical component models the behavior of a double layer, that is an imperfect capacitor. For the Eq. (55), when γ is less to 1, the resistor R is replaced by a CPE and L represents a dispersive inductor [28]. 3.3. Fractional space-time diffusion equation Now considering the Eq. (13) and assuming that the space and time derivative are fractional, the order of the space-time fractional differential equation is 0 < β , γ ≤ 2, σ x has dimension of length and σ t of time. Applied the Fourier method of the variable separation, the full solution of the Eq. (13) is

˜ t˜) = A · E2β ( − β 2(1−β) x˜2β ) · Eγ ( − γ 1−γ t˜γ ), C˜(x,

(56)

where, x˜ = λx and t˜ = ωt are dimensionless parameters and A is a constant. Figs. 7 and 8 show simulations where the fractional time derivative and the spatial fractional derivative are taken at the same time for different particular cases of β and γ .

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Diffusion 3

2

C(x, t)

1

0 =0.85; =0.85 =1.1, =1.1 =1.2, =1.2 =1.3; =1.3

−1

−2 0

2

4

6

x, t

8

10

Fig. 7. Simulation of the Eq. (56) for 0 < β , γ ≤ 2, when (β = 0.85, γ = 0.85), (β = 1.1, γ = 1.1), (β = 1.2, γ = 1.2) and (β = 1.3, γ = 1.3), we have the subdiffusion and the superdiffusion.

Diffusion 80

=1.5, =1.5 =1.6, =1.6 =1.7, =1.7

60 40

C(x, t)

20 0 −20 −40 −60 −80 0

2

4

6

x, t

8

10

Fig. 8. Simulation of the Eq. (56) for 1 < β , γ ≤ 2, when (β = 1.5, γ = 1.5), (β = 1.6, γ = 1.6) and (β = 1.7, γ = 1.7), we have the sub-wave phenomena diffusion (superdiffusion).

Illustrative example. Consider the fractional diffusion equation described by

∂ γ C (x, t ) 1 ∂ 2β C (x, t ) = · , 0 < γ,β ≤ 1 ∂tγ 2 ∂ x2β

(57)

with C (0, t ) = 0, C (10, t ) = 0 and C (x, 0) = 5 sin ( π2 x) + 2 sin ( 25π x). Applying the Fourier method of the variable separation, the full solution of the Eq. (57) is

 π2

C (x, t ) = 5Eγ −

8

tγ





·  Ei β

π 2

xβ



+ 2Eγ



− 0.08π 2 t γ



· [Eiβ (0.4π xβ )].

(58)

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125

Diffusion 3 =1, =1 =0.9, =0.9 =0.8, =0.8

2.5

2

C(x, t)

1.5

1

0.5

0

−0.5

0

2

4

x, t

6

8

10

Fig. 9. Simulation of the Eq. (58) for 0 < β , γ ≤ 1, when (β = 1, γ = 1), we have the classical diffusion, when (β = 0.9, γ = 0.9) and (β = 0.8, γ = 0.8), we have the space-time fractional diffusion (subdiffusion).

where  indicates the imaginary part. In the case when β = 1 and γ = 1, we have the classical solution

C (x, t ) = 5 exp(−

π2 t) 8

sin

π  2

2 x + 2 exp(−0.08π t ) sin (0.4π x).

(59)

Fig. 9 shows simulations where the fractional time derivative and the spatial fractional derivative are taken at the same time for different particular cases of β and γ . 4. Conclusions In this work we have used the idea suggested in [34] to construct fractional differential equations, in particular the fractional diffusion equation in space-time domain, The order of the fractional derivatives considered were 0 < β , γ ≤ 2. The solutions (21) and (39) correspond the space-time generalized solutions of the diffusion, specifically, when β = 1/2 and γ = 1/2, the diffusion presents an exponential behavior with spatial-temporal-decaying amplitude and shows a dissipative nature. An important measure in diffusion processes is the time evolution of the concentration in the medium, the Eq. (41) represents the time evolution of the concentration and the amplitude which exhibits an algebraic decay when t → ∞. The classical cases of the space-time diffusion equation, β = 1 and γ = 1, are represented in the Eqs. (25) and (43). In the range 0 < β ≤ 1, we observe the non-Markovian–Lévy flights and in the range 0 < γ < 1, the subdiffusion phenomena (the diffusion is slower), this subdiffusion occurs in many physical systems, such as chaotic dynamics charge transport in amorphous semiconductors [20], porus media [37], NMR diffusometry in disordered materials [38,39], dynamics of a bead in a polymer network [40,41] or gel solvents [42]. When β = 3/2 and β = 2, the diffusion exhibits an increment in the amplitude and the behavior presents anomalous dispersion (the diffusion increases with increasing order of β ). In the range 1 < β ≤ 2 we observe the Markovian–Lévy flights [20], this behavior is represented by the solutions (27) and (29), respectively. For the case when γ = 3/2 and γ < 2, the diffusion exhibits an increment in the amplitude and presents the sub-wave phenomena (superdiffusion), this behavior is represented by the solutions (44) and (47), respectively, the superdiffusion occurs in many physical systems such as rotating flow [43], Richardson turbulent diffusion [44], diffusion of ultracold atoms in an optical lattice [45,46] or quantum optics [47,48], the case γ = 2 represents the ballistic diffusion [20]. The general solutions of the fractional differential equations are given in terms of the Mittag-Leffler functions depending only on a small number of parameters β and γ and related to equation results in a fractal space-time geometry preserving the physical units of the system for any value taken by the exponent of the fractional derivative. 1−γ 1−γ = (RC )t , the With respect to the analysis of the fractional time constant, when γ is less to 1, in the Eq. (50) we have σt crosswise capacitor of the ordinary diffusion is replaced by a CPE, which is an element representing the double layer capacitance or an imperfect capacitor and R is the charge-transfer resistance. In the Eq. (54) , G and L determine the medium properties, when γ is less than 1, σt1−γ = (L/R)t1−γ , the resistor R is replaced by a CPE and L represents a dispersive inductor. For both cases, the anomalous diffusion phenomena occurs, this model of anomalous diffusion do not introduce additional elements characterizing

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the change of the medium properties and may be represented as a classical transmission line model or by an electrical equivalent distributed circuit. In the case where both derivatives are considered, the Eq. (56) shows the full solution of the diffusion equation when the Fourier method of the variable separation is applied. This solution represents a non-local behavior interpreted as the existence of memory effects, which correspond to intrinsic dissipation characterized by the exponent of the fractional derivative β and γ in the system; and related to diffusion in a fractal space-time geometry presented an entire new family of solutions. Furthermore, this solution has a universal character related with the conditions (30) and (48) in our case. We consider that these results can be useful to gain understanding of the electrochemical phenomena, non-Fourier processes, models of porous electrodes and the description of anomalous complex processes. An interesting problems for further research, is to treat the two- and three-dimensional diffusion equations in terms to fractional variational calculus (see [49] and the references therein) with different initial and/or boundary conditions, of course, it would be interesting to consider the diffusion equations with fractional derivatives defined in different ways. Acknowledgments The authors appreciate the constructive remarks and suggestions of the anonymous referees that helped to improve the paper. We would like to thank to Mayra Martínez for the interesting discussions. The authors acknowledge to: PAPIIT- DGAPA-UNAM (IN112212) and CONACYT-0167485 for the economic support. José Francisco Gómez Aguilar acknowledges the support provided by CONACYT: cátedras CONACYT para jovenes investigadores 2014. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

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