Diffusion equations and different spatial fractional derivatives/Equacoes de difusao e diferentes derivadas espaciais fracionarias.
Actually, the fractional calculus represents an important tool which has been successfully applied to several contexts (DAS; MAHARATNA, 2013; DEBNATH, 2003; MACHADO et al., 2014; GLOCKLE; NONNENMACHER, 1995; HILFER, 2000; SHLESINGER et al., 1994). For example, electrical response (LENZI et al., 2013; SANTORO et al., 2011), biological systems (CASPI et al., 2000; PLOTKIN; WOLYNES, 1998), finance, viscoelasticity (GLOCKLE; NONNENMACHER, 1991) and anomalous diffusion (PEKALSKI; SZNAJDWERON, 1999). In particular, the last point has received much attention since that the usual approach (RISKEN, 1989; GARDINER, 2009) does not provide a suitable description of the experimental results and, consequently, requires extensions. In this sense, by using fractional calculus, the diffusion equation (or Fokker--Planck equation) and the Langevin equation have been extended and, consequently, used to investigate several situations such as the ones present in Refs. (HILFER, 2000; METZLER; KLAFTER, 2000; LENZI et al., 2009; METZLER et al., 1994; SOKOLOV, 2012). However, there is more than one definition of the fractional (or noninteger) differential operators which have been used to investigate these situations. In this sense, our goal is to investigate the differences manifested by the solutions when three representative fractional operators are incorporated to in the diffusive term, i.e., the usual spatial derivative is replaced by a fractional differential operator. The operators analyzed here are the Riemann--Liouville (PODLUBNY, 1999), Caputo (PODLUBNY, 1999), and the one proposed by Qianqian et al. (2010) which reminds us the usual case. For these operators, we consider the situations characterized by a limited and no limited regions in order to establish the differences. Analysis is performed next, followed by discussion and conclusion.
Material and methods
Diffusion equations and different fractional operators
Let us start our analysis about the differences of these operators by investigating the behavior of the solutions when these fractional differential operators are incorporated in the diffusive term. The first spatial differential operator considered here is the Riemann--Livouville, defined as (PODLUBNY, 1999)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
n is a integer and n - 1 < [alpha] < n. Using this definition for the spatial derivative the usual diffusion equation can be extended to
[partial derivative]/[partial derivative]t [rho](x, t) = [K.sub.[alpha],a] [sup.Rl][sub.a][D.sup.a.sub.x] + [K.sub.[alpha],b] [sup.RL][sub.x][D.sup.[alpha].sub.b])[rho](x, t) (2)
to describe, for simplicity, a system defined in the interval [a, b] and subjected to boundary and initial conditions. The usual diffusion equation can also be extended by incorporating the fractional differential operators in the Caputo (PODLUBNY, 1999) sense. In this case, the diffusion equation is modified to
[partial derivative]/[partial derivative]t [rho](x,t) = ([K.sub.[alpha],a] [sup.C][sub.a][D.sup.a.sub.x] + [K.sub.[alpha],b] [sup.C][sub.x][D.sup.[alpha].sub.b])[rho](x, t) (3)
which is very similar to Equation (2), however the differential operator, to be considered in this case, is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
with n - 1 < [alpha] < n where n is an integer number and [[rho].sup.(n)] (x, t) is the nth--spatial derivative of the distribution [rho](x, t). Note that for [alpha] integer Equation (2) and (3) recover the usual diffusion equation and the presence of the constants [K.sub.[alpha],a] and [K.sub.[alpha],b] permit us, from the form point of view, to work out the operators present in the diffusive term in the asymmetric and symmetric form. In addition, it is interesting to mention that the presence, in the diffusive term, of these differential operators can related to a long-tailed behavior of the jump probabilities which characterize the diffusive process manifested by the system.
For these operators when the system is defined in a infinite interval, i.e., (-[infinity], [infinity]), both equations lead us to the same solution since that the operators in this limit are equivalent as discussed in Ref. (PODLUBNY, 1999). This feature may be observed from the numerical from point of view. In fact, by using the discrete form of these operators known as the L2 method (QIANQIAN et al., 2010), we obtain, for the Grunwald-Letnikov fractional differential operator,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
with the interval [-L, L] divided in N parts, h = L/N, such that -N < i < N and [x.sub.i] = ih . Whereas, the discrete form of the Caputo's operator, using the same procedure, can be written as follows
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Notice that equations for [sup.GL][sub.-L][D.sup.a.sub.x]u([x.sub.i]) and [sup.C][sub.L][D.sup.[alpha].sub.x]u([x.sub.i]) only differ by the first two terms before the sum, and these terms are inversely proportional to L. Therefore their discrepancies become smaller if we increase the size of the system, as illustrated in Figure 1.
Interesting characteristics of Equation (2) and (3) appear when they are considered in a finite interval where the surface effects may play an important role. In order to face this point, we investigate the differences between Equation (2) and (3) when the system is considered, without loss of generality, in a limited interval, i.e., [0,L], with [K.sub.[alpha],b] = 0. We also assume that [rho](x, t) is (n - 1) times continuously differentiable and that [[rho].sup.(n)] (x, t) is integrable in the interval [0, L] . Using these assumptions, it is possible to show that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
for 1 < [alpha] < 2. Equation 5 shows a connection between the Riemann--Liouville and Caputo differential operators which implies in the presence of the addition terms. These terms related to the boundary conditions, imposed by the problem under consideration, may behave as a source or sink by introducing or removing particles of the system such as adsorption and/or desorption process. Thus, the extensions of the diffusion equation given by Equation (2) and (3) have different solutions and, consequently, are not equivalent. This feature is also verified for a semi-infinity interval, i.e., [0, [infinity]) .
Now, let us consider the fractional differential operator proposed in Ref. . This fractional operator is defined as [[summation].sup.[alpha]] [[psi].sub.n] = [[lambda].sub.n] [[psi].sub.n] (x) with [[nabla].sup.[alpha]] = -[(-[[nabla].sup.2]).sup.[alpha]/2] which brings the usual differential operator raised to a noninteger exponent. The eigenfunctions and eigenvalues may be represented by [[psi].sub.n](x) and [[lambda].sub.n], respectively. Incorporating this operator in the usual diffusion equation, we obtain that
[partial derivative]/[partial derivative]t [rho](x, t) = [K.sub.[alpha]] [[nabla].sup.[alpha]] [rho](x, t). (6)
The solution for this equation can be formally written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
for a system defined in an interval [a, b] with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
and [[psi].sub.n](x) subjected to boundary conditions. From this previous development, we verify that the definition of fractional operator present in Ref.  is very interesting since that it allows us to use some the properties of the integer differential operator and introduces a noninteger index.
To investigate a possible equivalence with the previous fractional diffusion equations, we consider the solutions of Equation (6) in a limited region. In particular, we analyze the behavior of the solutions of these equations in two situations, when the system is subjected to absorbing boundary conditions ([rho](0,t) = [rho](L, t) = 0). The first of them is to consider, for simplicity, the initial condition [rho] (x, 0) = ([pi]/2) sin([pi]x) and [K.sub.a,b] = 0 for Equation. (2), (3), and (6). For this case, the results shown in Figure 2 illustrate the differences evidencing the nonequivalence of the fractional diffusion equations. In the second case, we restrict our analysis, without loss of generality, to Equation (3) and (6) by taking into account the initial condition [rho](x,0) = J x [E.sub.[alpha],2] (-k[x.sup.[alpha]]) with k obtained from the equation [E.sub.[alpha],2] (-k[x.sup.[alpha]])= 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By using this initial condition is possible to show that a solution of Equation (3), satisfying the required boundary condition, is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where [alpha] = 2 recovers the usual solution connected to the initial condition. Figure 3 illustrates Equation (9) for different values of [alpha] in order to illustrate the effect of the noninteger index of the differential operator and the time evolution of the solution.
The solution for Equation (6), by considering the previous absorbent boundary condition, can be written as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)
Equation (10) is illustrated in Figure 4 in order to show the effect of the index [alpha] on the solution and the time evolution. From Figures 3 and 4, we observe that solutions obtained from Equation (3) and (6), subject to the certain initial condition, are different when the system is defined in a limited region. This point is illustrated in Figure 5 and shows the nonequivalence of these spatial differential operators for the conditions considered here.
Results and discussion
We have investigated the solutions of the fractional diffusion Equation (2), (3), and (6) by considering different situations in a finite interval.
The results found for these equations shown that the surface effects play an important role on the solutions of these equations. This point is shown, for example, by Equation (5) which implies that Equation (2) is equivalent to Equation (3) if additional terms connected to the surface are considered. Similar situation is evidenced in Figure 2 when Equation (2), (3), and (6) are compared for an initial condition. The equivalence is only found when the system is defined in the interval (-[infinity], [infinity]) where the surface effects are absent.
The results presented here for a system subjected to a finite interval shows that the fractional operators are not equivalent. In this sense, it is interesting to note that the surface effects play an important role on these operators.
We thank CNPq and Fundacao Araucaria.
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Received on April 30, 2014.
Accepted on July 14, 2014.
License information: This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
Alexandre Fernandes Basanini Duarte (1), Jessica de Melo Gatti Pereira (1), Marcelo Kaminski Lenzi (2), Giane Goncalves (3), Roberto Rossato (4) and Ervin Kaminski Lenzi (1) *
(1) Departamento de Fisica, Universidade Estadual de Maringa, Av. Colombo, 5790, 87020-900, Maringa, Parana, Brazil. (2) Departamento de Engenharia Quimica, Universidade Federal do Parana, Setor de Tecnologia, Jardim das Americas, Curitiba, Parana, Brazil. (3) Departamento de Engenharia Quimica, Universidade Tecnologica Federal do Parana, Ponta Grossa, Parana, Brazil. (4) Departamento de Fisica, Universidade Tecnologica Federal do Parana, Apucarana, Parana, Brazil. *Author for correspondence. E-mail: firstname.lastname@example.org