Research Article  Open Access
Nonlocal Transport Processes and the Fractional CattaneoVernotte Equation
Abstract
The CattaneoVernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the meansquare displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the CattaneoVernotte equation using the fractional calculus approach; the spatialtime derivatives of fractional order are approximated using the Caputotype derivative in the range . In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional spacetime derivatives into the fractional CattaneoVernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional CattaneoVernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to .
1. Introduction
Fourier’s law satisfies the heat conduction induced by a small temperature gradient in steady state. In steady state, the heat transfer through a material is proportional to the negative gradient of the temperature and to the area. However, there are some cases in which the Fourier equation is not adequate to describe the heat conduction process. More precisely, Fourier law is diffusive and cannot predict the finite temperature propagation speed in transient situations, in this context, the CattaneoVernotte equation corrects the nonphysical property of infinite propagation of the Fourier and Fickian theory of the diffusion of heat, and this equation also known as the telegraph equation for the temperature is a generalization of the heat diffusion (Fourier’s law) and particle diffusion (Fick’s laws) equations. Processes where the traditional Fourier heat equation leads to inaccurate temperature and heat flux profiles are known as nonFourier type processes [1]; these processes can be Markovian or nonMarkovian [2]. In the Markovian processes case, the meansquare displacement of the diffusing particle is proportional to time, while, in the disordered systems or nonMarkovian process, the diffusion can be anomalous; in this case the meansquare displacement is proportional to a fractional power of time not equal to one; when anomalous diffusion occurs, the probability density for a diffusing particle is not the usual Gaussian distribution. The mechanism of diffusion is Brownian motion, and this motion is the simplest continuoustime stochastic process. Continuoustime random walks can be coupled with Brownian motion and fractional calculus (FC) to provide an improved estimator in the modeling of anomalous diffusion. A random walk is a mathematical formalization of a path that consists of a succession of random steps [3]. A Lévy flight, also referred to as Lévy motion, is a random walk in which the steplengths have a heavytailed probability distribution. When a random walk is defined as a walk in a space of dimension greater than one, the steps are defined in terms of a probability distribution, and steps move with isotropic random directions [4], and continuoustime random walk schemes are considered in the derivation of timefractional differential equations. Recently, the subject of FC has attracted interest of researches; this mathematical concept involves nonlocal operators which can be applied in physical systems yielding new information about its behavior, fractional derivatives with respect to coordinates describe powerlaw nonlocal properties of the distributed system, and there are several papers about the recent history of the FC; see [5–7]. Several approaches have been used for investigating anomalous diffusion, Langevin equations [8, 9], random walks [10, 11], or fractional derivatives, based on FC several works connected to anomalous diffusion processes which may be found in [12–30]. Using phenomenological arguments, Compte and Metzler [31] generalize the CattaneoVernotte equation by introducing fractional derivatives with a continuoustime random walk scheme. The authors of the works presented in [32–34] studied the generalized CattaneoVernotte equation with fractional spacetime derivatives, and the order of the spatial and temporal fractional derivatives are . Lewandowska and Kosztołowicz in [35] investigate the subdiffusive impedance phenomena of a spatially limited sample for large pulsation of electric field. Tarasov in [36] based on the Liouville equation obtained the fractional analogues of the classical kinetic and transport equations. Qi and Jiang in [37] derived the exact solution of the CattaneoVernotte equation by joint Laplace and Fourier transforms. Other applications of FC to CattaneoVernotte equation are given in [38–41].
The aim of this work is to contribute to the development of a new version of fractional fundamental CattaneoVernotte equation applying the idea proposed in the work [42]; the order considered is for the fractional equation in spacetime domain; this representation preserves the dimensionality of the equation 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 CattaneoVernotte equation and give conclusions in Section 4.
2. Basic Definitions of Fractional Calculus
The most commonly used definitions in FC are RiemannLiouville (RL), GrünwaldLetnikov (GL), Caputo fractional derivative (CFD), and Riesz fractional derivative () [43–46].
The RL definition of the fractional derivative for is
For function the CFD is given bywhere is a CFD with respect to , is the order of the fractional derivative, , and represents Euler’s gamma function.
In the present paper, we would use the CFD definition, since the former is more popular in real applications. For the CFD definition we need to specify additional conditions in order to produce a unique solution, these additional conditions are expressed in terms of integerorder derivatives [46], and this definition is used mainly for the problem with memories. In the case of the RL definition there exist physically unacceptable initial conditions [47].
The Laplace transform of Caputo’s derivative (2) has the form [48]where is the Laplace transform of the function and . From this expression we have two particular cases:
The MittagLeffler function has gained extensive interest among physicists due its vast potential of applications in the solution of fractional differential equations [48]:when and , from (6) we obtain the exponential function.
denotes the complementary error function [48] and it is defined as
Some common MittagLeffler functions are described in [48]:
The MittagLeffler function is defined by Miller in [49]:where , , , . For the calculation of generalized MittagLeffler functions at arbitrary precision, see [50, 51].
The Riesz fractional derivative for is [43–46]where is a Riesz fractional derivative with respect to , is the order of the fractional derivative, , and represents Euler’s gamma function.
3. Fractional CattaneoVernotte Equation
In previous studies of the fractional CattaneoVernotte equation the authors did not consider the physical dimensionality of the solutions. The authors of the work [42] proposed a systematic way to construct fractional differential equations for the physical systems. To keep the dimensionality of the fractional differential equations a new parameter was introduced in the following way:where is an arbitrary parameter which represents the order of the derivative, has dimension of length, and has the dimension of time. These new parameters maintain the dimensionality of the equation invariant and characterizes the fractional space or fractional temporal structures (components that show an intermediate behavior between a conservative system and dissipative one) [42]; when the expressions (15) and (16) reduce to the ordinary derivative. In the following we will apply this idea to generalize the case of the fractional CattaneoVernotte equation.
In this work, we consider generalized CattaneoVernotte equation in the direction of the form [32–35]where is a characteristic relaxation time constant (or the nonFourier character of the material) and is the generalized thermal diffusivity; (17) is a hyperbolic diffusion equation; when the parameter , (17) recovers a parabolic form; in this limit, one has to replace CattaneoVernotte equation by Fourier’s heat transfer equation.
Considering the CFD (2) and (15) and (16), the fractional representation of (17) isThe order of the derivative considered is for the fractional CattaneoVernotte equation in spacetime domain.
3.1. Fractional Space CattaneoVernotte Equation
Considering (18) and assuming that the space derivative is fractional equation (15) and the time derivative is ordinary, the spatial fractional equation isSuppose the solutionsubstituting (20) into (19) we obtainwhereis the dispersion relation in the direction andis the fractional dispersion relation; from the fractional dispersion relation (23), we can expect the fractional wave number in the direction to have real and imaginary parts, and , respectively. Let us writesubstituting (24) into (23) we havewheresolving for we obtainand for substituting (28) into (27) we have
Now the fractional wave number is , where and are given by (28) and (29), respectively,equation (30) describes the real and imaginary part of the fractional wave number in terms of the frequency , the relaxation time , and the generalized thermal diffusivity , in presence of fractional space components .
Considering (23), (21) givesthe solution of (31) can be obtained applying direct and inverse Laplace transform [47], and the solution of the above equation is given bywhere is the MittagLeffler function.
Therefore the general solution of (21) is given by
Next, we will analyze the case when takes different values.
Case 1. When , we haveand equation (34) represents the fractional wave number in presence of fractional space components .
In this case, (33) is written as follows:where is given by (12) and solution (35) is
Case 2. When , we haveand equation (37) represents the fractional wave number in presence of fractional space components .
In this case, (33) is written as follows:where is given by (11) and solution (38) is
Case 3. When , we have and equation (40) represents the classical wave number . From (33) we havewhereand, in (42), , indicates the real part, and is wave number (40); substituting in (42) we haveEquation (43) represents the classical case for the space CattaneoVernotte equation. The first exponential gives the usual planewave variation of the thermal field with position and time . The second exponential gives and exponential decay in the amplitude of the thermal wave.
Case 4. When , from (33) we havewhere isand equation (45) represents the fractional wave number in presence of fractional space components .
The solution for (44) iswhere indicates the real part.
Case 5. When , from (33) we havewhere isand equation (48) represents the fractional wave number in presence of fractional space components .
In this case, (33) is written as follows:where is given by (8) and solution (49) is denotes the error function defined in (8). Equation (50) represents the space evolution of the temperature and the amplitude exhibits an algebraic decay for .
For this case there exists a physical relation between the auxiliary parameter and the wave number given by the order of the fractional differential equationwhere is the wavelength. We can use this relation in order to write (33) aswhere is a dimensionless parameter. Figures 1(a) and 1(b) show the simulation of (52) for values and , respectively.
(a)
(b)
Table 1 shows the different solutions of (52). The order of the fractional differential equation is , , , , and .

3.2. Fractional Time CattaneoVernotte Equation
Considering (18) and assuming that the time derivative is fractional equation (16) and the space derivative is ordinary, the temporal fractional equation issuppose the solutionwhere is the wave number in the direction. Substituting (54) into (53) we obtainthe solution of (55) can be obtained applying direct and inverse Laplace transform [47]. Taking solution (54) we haveand solution (56) represents a temporal nonlocal thermal equation interpreted as an existence of memory effects which correspond to intrinsic dissipation characterized by the exponent of the fractional derivative in the system.
For underdamped case, we have , is the undamped natural frequency expressed in radians per second, and is the damping factor expressed in meters per second. Next, we will analyze the case when takes different values.
Case 1. When , from (56) we havewhere is given by (10) and by (12); in this case solution (57) is
Case 2. When , from (56) we havewhere is given by (13) and by (11); in this case solution (59) iswhere , , , .
Case 3. When , from (56) we haveand (61) represents the classic case and the wellknown result; from (61) we see that there is a relation between and given byThen solution (56) for the underdamped case takes the formwhere is a dimensionless parameter.
Due to the condition we can choose an exampleSo, solution (56) takes its final form:
Case 4. When , from (56) we haveand denotes the error function defined in (8). Equation (66) represents the time evolution of the temperature and the amplitude exhibits an algebraic decay for . Plots for different values of are shown in Figures 2(a) and 2(b).
(a)
(b)
Table 2 shows the different solutions of (65). The order of the fractional differential equation is , , , and . The change of the order of the derivative describes the crossover from ballistic transport to the diffusion behavior.

In the overdamped case, or , the solution of (56) has the form
Next, we will analyze the case when takes different values.
Case 1. When , we have