Abstract
The Cattaneo or telegrapher’s equation describes the crossover from initial ballistic to normal diffusion. Here we study and survey time-fractional generalisations of this equation that are shown to produce the crossover of the mean squared displacement from superdiffusion to subdiffusion. Conditional solutions are derived in terms of Fox H-functions and the δth-order moments as well as the diffusive flux of the different models are derived. Moreover, the concept of the distribution-like is proposed as an alternative to the probability density function.
1 Introduction
Brownian diffusion processes [18, 81, 37] are characterised by the linear growth in time of the mean squared displacement (MSD),
here used in dimensionless units, respectively representing the constitutive equation (Fick’s first law) and the continuity equation, where J(x, t) is the diffusive flux and ϱ(x, t) is the distribution of the diffusing substance at position x and time t. Combining both equations, we obtain the diffusion equation (Fick’s second law)
In this formulation Brownian motion is overdamped, i.e., any inertial effects of the diffusing tracer particle are considered to have decayed. In the original Langevin equation formulation, the particle motion at times shorter than the relaxation time (given in terms of the particle mobility) is inertia-dominated and thus ballistic [37, 34]. One way to incorporate the crossover from initial ballistic motion 〈x2(t)〉 ∼ t2 to final diffusive motion 〈x2(t)〉 ∼ t was achieved in the discussion of heat transport [33] in the form of the Cattaneo or telegrapher’s equation discussed in Section 2. Today, this field has adopted more general stochastic processes to account for the ever better resolved patterns involved in heat transport, [17]. In this paper we study and survey the mathematical properties of generalised Cattaneo-type equations, in which the integer scaling exponents of ballistic and diffusive motion are replaced by given real-valued exponents.
Already in 1926 Richardson reported a process which deviates from equations (1.1) in the form of the cubic time dependence of the relative MSD of two particles in a turbulent flow [68]. In a wide variety of systems, the MSD of the diffusing particles follows the power-law form [6, 48, 49]
distinguishing subdiffusive processes with 0 < β < 1, the normal Brownian process with β = 1, superdiffusive processes with 1 < β < 2, the ballistic process with β = 2, and superballistic processes, β > 2.
To describe such anomalous diffusion processes, fractional kinetic equations as proposed by Schneider and Wyss [88, 79], Zaslavsky [90], and Nigmatullin [61], are found to be outstanding tools. The time-fractional diffusion equation, for instance, given in the natural (Caputo) form [40, 41, 42]
or in the modified (Riemann-Liouville) form
capture well the subdiffusive regime for ϱ(x, 0+) = δ(x), x ∈ ℝ, and lim|x|→∞ϱ(x, t) = 0. Here
or in the modified form
0 < β < 1, cover the superdiffusive regime when ϱ(x, 0+) = δ(x), x ∈ ℝ, and lim|x|→∞ϱ(x, t) = 0.
Several early studies suggested that the scaling behaviour in terms of a single exponent, equation (1.2), is unable to describe many physical processes, such as the Sinai diffusion model [80], truncated Lévy-flights [44], or subdiffusive motion in velocity fields [50]. By now, the mono-scaling behaviour has been challenged to describe the temporal variation of the MSD in biological cells [4, 29, 63, 57], in protein surface water [84], phospholipids and cholesterols in a lipid bilayer [31, 32, 54], and in viscoelastic flows [89], see also the reviews [55, 47] and the references therein. These works were the basis for a string of papers [11, 12, 13, 14, 82, 15, 72] in which fractional order derivatives in the fractional diffusion equations (1.3) and (1.4) are replaced by derivatives of distributed-order [9]. Distributed-order time or space fractional diffusion equations are shown to be a versatile tool for the mathematical description of physical processes that become less anomalous in the course of time (accelerating subdiffusion and decelerating superdiffusion) or more anomalous in the course of time (retarding subdiffusion and accelerating superdiffusion).
In [16] a generalised Cattaneo-type equation was proposed, yielding the dynamic crossover behaviour
where 0 < α < 1. When 0 < α < 0.5, equation (1.7) represents a retarded subdiffusion process, which can be considered a special case of the double-order time fractional diffusion equation of the natural type (β2 = 2β1) [11], and when 0.5 < α < 1, it represents a transition from the superdiffusive to subdiffusive regime, see also [65, 45, 2]. In the case α = 1 we recover the characteristic property of the classical Cattaneo model, see equation (2.2) below, the crossover from ballistic to normal diffusion. The reader is also referred to generalisations of diffusion, Fokker-Planck, and diffusion-wave equations by introduction of a generalised memory kernel [73, 74, 75]. However, recent numerical simulations have shown different transitions from superdiffusion to subdiffusion which are not, in all cases, covered by equation (1.7), see e.g., quasiperiodic interacting systems [38] and dewetting-spreading-wetting transitions of self-propelled (run-and-tumble) clustered disks in a substrate with randomly placed pinning sites [76].
In this work, we present different forms of generalised Cattaneo-type equations, based on single-order and distributed-order time-fractional derivatives, which provide a variety of transitions from superdiffusive to subdiffusive motion. We first outline these forms in Section 2. Conditional solutions for the distributions of the proposed models are derived in terms of Fox H-functions, and the corresponding δth-order moments are discussed in Section 3. In Section 4 we consider the diffusive flux, an important measurable quantity. Moreover, we introduce and study the so-called distribution-like, a quantity similar to the physical distribution, from which we obtain alternative formulations for the diffusion, fractional diffusion, and fractional Cattaneo-type diffusion equations. The subordination of time-fractional Cattaneo-type diffusion to the Gaussian process is also discussed. Finally, we summarise our results, and draw our conclusions in Section 5. We give a brief outline of the mathematical preliminaries in Appendix A and show the methods of deriving our solutions by solving four problems in Appendix B.
2 Generalised Cattaneo equations
Several representative models were introduced during the last seven decades in order to address the shortcomings of the Fourier-Fickian law in modelling the necessary finite propagation speed in heat and signal transport. One of the most prominent is the Cattaneo model, also known as Maxwell-Cattaneo or telegrapher’s equation [10, 58], as well as the phonon scattering model [25], and the parabolic and hyperbolic two-step models [70, 71]. The monograph [86] contains a comprehensive historical background on these developments, see also [3].
The Cattaneo equation in dimensionless form reads [10]
which encodes the characteristic ballistic-diffusive crossover property of the MSD [65, 46, 87]
with an exact solution for ϱ(x, t) in terms of modified Bessel functions [30]. The first fractional Cattaneo equation was proposed by Nonnenmacher and Nonnenmacher [62]. The time-fractional Cattaneo equation (TFCE) has two familiar forms [16, 2]. The first form (natural type), uses the Caputo fractional derivative,
with ϱ(x, 0+) = δ(x), JC(x, 0+) = 0, x ∈ ℝ, lim|x|→∞ϱ(x, t) = 0, and α, β ∈ (0, 1]. The second form of the time-fractional Cattaneo equation uses the Riemann-Liouville fractional derivative and is given by
with ϱ(x, 0+) = δ(x), x ∈ ℝ, lim|x|→∞ϱ(x, t) = 0, and α, β ∈ (0, 1]. The time-fractional Cattaneo equations (2.3) and (2.4) are termed GCE-I with two fractional parameters, where the equivalence between them for the distribution ϱ(x, t) was shown in [2]. Note that we decorate the flux in (1.3) and (2.3) with C, and in (1.4) and (2.4) with RL to point out that these equations are not necessarily equivalent for the flux. Note also that both equations (2.3) and (2.4) reduce to (2.1) in the limiting case α = β = 1, and the generalised version of the Cattaneo-type equation of [16] is recovered whenever α = β. If we only set α = 1 in equations (2.3) and (2.4), we get
with ϱ(x, 0+) = δ(x), JRL(x, 0+) = 0, x ∈ ℝ, lim|x|→∞ϱ(x, t) = 0, and β ∈ (0, 1]. Alternatively,
with ϱ(x, 0+) = δ(x), JC(x, 0+) = 0, x ∈ ℝ, lim|x|→∞ϱ(x, t) = 0, and β ∈ (0, 1], which are known in the literature as GCE-III. Conversely, if we set β = 1 in (2.3) and (2.4), we obtain the GCE-IV
with ϱ(x, 0+) = δ(x), x ∈ ℝ, lim|x|→∞ϱ(x, t) = 0, and α ∈ (0, 1], where
Equations (2.5) to (2.7) can be further generalised using the distri-buted-order derivatives to take the more general forms
with ϱ(x, 0+) = δ(x), JRL(x, 0+) = 0, x ∈ ℝ, lim|x|→∞ϱ(x, t) = 0, and sup(p) ⊆ [0, 1],
with ϱ(x, 0+) = δ(x), JC(x, 0+) = 0, x ∈ ℝ, lim|x|→∞ϱ(x, t) = 0, and sup(p) ⊆ [0, 1],
with ϱ(x, 0+) = δ(x), x ∈ ℝ, lim|x|→∞ϱ(x, t) = 0, and sup(p) ⊆ [0, 1], where
and p(ν) is a probability density function, namely, p(ν) ≥ 0∀ν ∈ sup(p), and ∈ tsup(p)p(ν)dν = 1. The non-negativity of the distribution ϱ(x, t) of equations (2.3) to (2.10) was discussed in detail in [2]. Equation (2.10) was proposed in [1] and analysed in [91] using the multi-term assumption [39].
In this work, we are interested in the case
where 0 < α < β ≤ 1 and p1 + p2 = 1. It is salient that when α = β, or p1 = 1, p2 = 0, the distributed-order versions of the time-fractional Cattaneo-type equations (2.8) to (2.10) respectively reduce to the generalised Cattaneo equations of type III and IV, equation (2.5) to (2.7). Let us call equations (2.8) to (2.10) with (2.11) and (2.12) the double-order time-fractional Cattaneo equations of types I, II, and III. The generic form of the analytical solution for the above Cattaneo-type equations in Laplace-Fourier space, (A.5) and (A.6), can be written as
where η(s) is the characteristic function to be defined below, and in Laplace space as
with δth-order moment
and thus
3 Analytical solution and mean squared displacement
Here we provide the exact solutions of the Cattaneo-type equations featured in the previous section, in addition to their corresponding MSDs.
3.1 Generalised Cattaneo equation of type I
In the case of GCE-I with two fractional parameters (TFCE), equations (2.3) and (2.4), the characteristic function η(s) defined in (2.13) is given by
Thus, we have
Referring to Problem B.1 in appendix B the solution of equation (3.2) can be extracted from (B.1) by setting χ0,1 = 1, and γ = α + β, so that we get
Three subcases of (3.3) can be deduced: the GCE-III (α = 1), the GCE-IV (β = 1), and GCE-I with single fractional parameter (α = β). We refer to the solution of the space-time fractional Cattaneo equation obtained in [69] which can be considered a special case of (3.3) if the space-fractality is disregarded. Further properties on the space-time Cattaneo equation can be found in [8] and [85].
The δth-order moments can be derived by substituting (3.1) into (2.16) and inverting the Laplace transform using (A.19),
where
Both forms of the MSD, (3.4) and (3.5), should be equivalent. This equivalence of (3.4) and (3.5) can be verified by the aid of Legendre’s duplication rule and the recurrence relation (A.21). Both forms also have the same limiting behaviour, see (A.17),
and when δ = 2, we find the asymptotic behaviour of the MSD,
The MSD (3.7) generalises the result of GCE-I with one fractional parameter (1.7). It can also represent a retarded subdiffusion process if 0 < α + β < 1, and a generic crossover from superdiffusion to subdiffusion if 1 < α + β < 2. The case 0 < α + β < 1 is analogous to the finding of Chechkin and colleagues [11, 14, 82, 15, 72]. On the other hand, the case 1 < α +β < 2 can depict many phases in [76] for certain pinning forces at low disk activity. The probability density (3.3) and the MSD (3.7) are presented in Figures 1a and 1b respectively. In Figure 1a, the subdiffusive transition regime in the short-time limiting included in the Cattaneo-type equation is clear in the blue curve, where 0 < α +β < 1. In the red and green curves, the superdiffusion process is dominant since 1 < α + β < 2. In Figure 1b the blue curve, i.e., the case corresponding to α = 0.5, β = 0.2, the retarded subdiffusion process is salient in the course of time. In the red and green curves of Figure 1b, which consider the choices α = 0.5, β = 0.8 and α = 0.8, β = 0.8, namely 1 < α + β < 2, respectively, the crossover from superdiffusion to subdiffusion in both curves is obvious.
(a) Spatial evolution of the probability density ϱGCE-I of GCE-I at short time t = 0.1 for different fractional parameters. (b) Temporal evolution of the MSD of GCE-I. The normal and ballistic diffusion limits are shown in the light grey curves.
Remark 3.1
In spite of the generality of the limiting behaviour (3.7) covering a wide range of transitions over the subdiffusive regime or from the superdiffusive to the subdiffusive regime, it is still constrained by the fact that the parameter β occurs in both the short-time and the long-time behaviours. In essence, the presence of the parameter β in the short-time behaviour causes an exemplary transition such as
impossible to be captured by the GCE-I (and thus its special cases; GCE-III and GCE-IV) with two fractional parameters.
Remark 3.2
It is noteworthy mentioning that GCE-I (3.2) can be written in terms of the generalised diffusion-wave equation with two power-law memory kernels [75], namely,
where
where ϱ(x, 0+) = δ(x), ∂tϱ(x, 0+) = 0 lim|x|→∞ϱ(x, t) = 0, and provided that 0 < α, β < 1 and 1 < α + β < 2.
Remark 3.3
The normalisation of (3.3) can be verified through the Mellin transform,
3.2 Double-order Cattaneo equation of type I
The double-order time-fractional Cattaneo equation of type I (DTFCE-I) (2.8), (2.11) and (2.12), which is distinguished from the double-order time-fractional diffusion equation of the modified form by the presence of a wave motion in the short-time behaviour, possesses the characteristic function
where 0 < α < β ≤ 1 and p1 + p2 = 1. For sufficiently large values of time, it can be shown that η(s) ∼ (p1s−α + p2s−β)−1. The solution of DTFCE-I in the Laplace-Fourier space is thus given by (2.13) and (3.10),
which is a special case of Problem B.2, when χ0,1 = 1, and γ = α + 1. Then, one can deduce that
where
where p1 ≥ p2. It can be verified that the distribution (3.12) is normalised as follows
Since 1/Γ(0) = 0, the above series has non-zero value only at m = 0, then
The δ th- order moments of DTFCE-I can be determined from equations (2.16) and (3.10),
where the expansion
Note that the δth-order moments (3.13) represents the exact formula at short and intermediate times time t (since Re(s) > 1).
The reader can analogously deduce a complementary exact formula for long times by assuming Re(s) < 1, and expanding then (1 + s)−δ/2 instead of (1 + 1/s)−δ/2. The asymptotic behaviour of all the δth-order moments is
with MSD limiting behaviours
for 0 < α < β ≤ 1, which can be generally viewed as a modified transition from superdiffusion to subdiffusion. It is also included in the wide range of crossovers covered by the generic form (3.7), however, the temporal path on which they evolve is distinct. In the case 0.9 < β ≤ 1, it represents processes switching from superdiffusion to normal diffusion, whereas the case 0 < α ≤ 0.1 describes processes crossing over from normal diffusion to subdiffusion. This type of transition can be found in the transport of electrons in an interacting system subject to quasiperiodic potential [38], specifically, when different interacting strengths U are considered. In Figures 2a and 2b the graphical representations of the distribution (3.12) and MSD of DTFCE-I, 〈x2(t)〉I, are shown. In the course of time, the transition from superdiffusion at small times to subdiffusion at longer times is notable in Figure 2a. The apparent warp in the MSD temporal path of Figure 2b seems almost like the result of changing the interacting strength with fixed quasiperiodic potential amplitude, see Figure 2 of [38]. The curvature of this warp can be increased by decreasing the second constant, e.g., p1 = 0.99 and p1 = 0.01.
(a) Spatial distribution of DTFCE-I at different instants of time for fractional parameters α = 0.3, β = 0.8 and constants p1 = p2 = 0.5. (b) MSD of the distribution DTFCE-I versus time for fractional parameters α = 0.3, β = 0.8 and different values of the model constants; p1 = 0.9, p2 = 0.1 (blue curve) and p1 = 0.1, p2 = 0.9 (red curve). The case p1 = p2 = 0.5, lying between blue and red curve, is omitted.
3.3 Double-order Cattaneo equation of type II
We consider here the double-order time-fractional Cattaneo equation of type II (DTFCE-II), (2.9), (2.11) and (2.12). It reduces to the double-order time-fractional diffusion equation of the natural form in the relatively long temporal domain which typifies subdiffusive processes that become more anomalous with time progress. The characteristic function η(s) is given as
where 0 < α < β ≤ 1 and p1 + p2 = 1. It appears from (3.16) that η(s) ∼ p1sα + p2sβ for relatively large values of time which coincides the characteristic function of the double-order time-fractional diffusion equation of the natural form. In the long time limit, η(s) ∼ p1sα + p2sβ ∼ p1sα proviso p1 ≤ p2. Upon setting η(s) ∼ p1sα in (2.13), one can get 〈x2(t)〉 ∼ tα, and vice versa in the short time limit. This result was analytically shown in [11, 82, 15, 72]. The distribution of DTFCE-II is a direct consequence of using of (3.16) in (2.13),
A similar case to (3.17) has been processed in Problem B.3. Thus, setting χ0,1 = 1, γ1 = α + 1 and γ2 = β + 1 in (B.3) leads to the solution of DTFCE-II,
where p1 < p2. The distribution (3.18) is normalised over ℝ. Indeed,
which can be attained by expansion for n = 0, 1, 2, 3. We find that all terms cancel each other except the first term of 1/Γ(1 + m + (β − α)(n − m + ℓ)).
Following similar procedures to the δth-order moments of DTFCE-I, we obtain
for short- and intermediate- times, with the limiting behaviours
and MSD asymptotes
where 0 < α < β ≤ 1 and p1 + p2 = 1. The limiting behaviour (3.21) proves that the DTFCE-II provides a second modified version of the transitions from superdiffusion to subdiffusion. On the other hand, for 0.9 < β ≤ 1 and 0 < α ≤ 0.9, we have a crossover from ballistic behaviour to subdiffusion, which essentially characterises the DTFCE-II. This crossover could not be performed by the generic behaviour (3.7) nor the specific behaviour (3.15). The distribution (3.18) and the MSD of the DTFCE-II are drawn in figures 3a and 3b, respectively. In accord with the condition (p1 < p2), see (3.18) below, we set p1 = 0.4, p2 = 0.6 in Figure 3a. Also, we choose α = 0.3, β = 0.8 for closer comparisons with Figure 2a. The DTFCE-II curve remarkably records values for the distribution ϱII(x, t) higher than those recorded in figure 2a for DTFCE-I. In the course of time, the transition from superdiffusion at small times to subdiffusion at long times is notable here as well. The presence of p1 and p2, where p1 + p2 = 1, in the MSD covers a significant range in the vicinity of the points tβ+1 and tα, see Figure 3b.
(a) Spatial distribution of DTFCE-II at different instants of time for fractional parameters α = 0.3, β = 0.8 and constants p1 = 0.4, p2 = 0.6. (b) MSD 〈x2〉II of DTFCE-II as a function of time for fractional parameters α = 0.3, β = 0.8 and different values of model constants; p1 = 0.9, p2 = 0.1 (blue curve) and p1 = 0.1, p2 = 0.9 (red curve).
3.4 Double-order Cattaneo equation of type III
In this subsection, we study the effect of distributed-order fractional derivatives on the generalised Cattaneo equation of type IV. The resulting model is termed double-order time-fractional Cattaneo equation of type III (DTFCE-III), having the characteristic function
where 0 < α < β ≤ 1 and p1 + p2 = 1. The long time behaviour of (3.22) is distinctly η(s) ∼ s, which in combination with (2.13) conducts as, the most prominent, the Gaussian distribution, i.e. 〈x2(t)〉 ∼ t. Conversely for short times (3.22) behaves as η(s) ∼ s(p1sα + p2sβ). This form resembles the characteristic function of the double-order time-fractional wave equation of natural type [2, 75, 23], which offers decelerating superdiffusive behaviour in the intermediate short-time domain, see Section 3.3 in [75]. Thus, the study of the characteristic function suggests the progress 〈x2(t)〉 ∼ tβ+1 → t in the course of time.
The characteristic function (3.22) with (2.13) produces the (k − s) form
which has been considered in a generic form in Problem B.4. Consequently, invoking equation (B.4) after setting χ0,1 = 1, γ1 = α + 1 and γ2 = β + 1, helps in getting the exact (conditional) form
where p1 ∈ [0, 0.3], p2 ∈ [0.7, 1], p1 + p2 = 1, β ∈ [0.4, 1], 0 < α < β. The normalisation of (3.24) can be checked through the expansion of the series
The scaling properties for the DTFCE-III can be obtained through its δth-order moments
for short- and intermediate-time values, with short- and long-time behaviours
Hence, the MSD of the DTFCE-III reads
which confirms the predictions of the characteristic functions (3.22). The domination of the largest exponent in the short-time domain is evident. The behaviour (3.26) represents generally a transition from superdiffusion to strictly normal diffusion. What distinguishes this model from the GCE-IV, in essence, is the presence of another smaller fractional parameter α, which unfortunately disappears from (3.26), and the constant p2 in the short-time limit. This parameter, however, seems to be activated in the short-time domain, which may lead to a retardation in this time scale. In Figure 4a we represent graphically the probability density function ϱIII(x, t) of DTFCE-III at different values of time for the fractional parameters α = 0.3, β = 0.8 and model constants p1 = 0.3, p2 = 0.7, depending on the conditions imposed on the closed-form solution (3.24), and its MSD for fractional parameters α = 0.3, β = 0.8 and different values of model constants in Figure 4b. Due to the resemblance between the MSD of DTFCE-II and DTFCE-III in the short-time behaviour, they record higher values in this time domain compared with the DTFCE-I where the smallest exponent α dominates.
(a) Spatial distribution of DTFCE-III at different instants of time for fractional parameters α = 0.3, β = 0.8 and constants p1 = 0.3, p2 = 0.7. (b) MSD 〈x2〉III of the DTFCE-III as a function of time for fractional parameters α = 0.3, β = 0.8 and different values of model constants; p1 = 0.9, p2 = 0.1 (blue curve) and p1 = 0.1, p2 = 0.9 (red curve).
4 Diffusive flux and distribution-like formulation
In this section we provide an alternative formulation for the Green’s function of diffusion (1.1), time-fractional diffusion (1.3) and (1.4), and time-fractional Cattaneo-type diffusion (2.3) and (2.4), by introducing a field variable that behaves like the distribution of the diffusing substance through having the same MSD of ϱ(x, t). The exact formulas for the diffusive flux are also derived.
4.1 Normal diffusion
At first, let us start with the normal diffusion equation (1.1) which can be converted using the transform technique to
where ϱ(x, t) and J(x, t) have the same partial differential (diffusion) equation
ensures the non-negativity of ϱ(x, t) and J(x, t). It is clear that ϱ͠(x, λ) is a completely monotone function for λ > 0, see Section A.5 in Appendix A, as a product of two completely monotone functions, and
which shows the Gaussian behaviour for particle distribution with
Let us consider the quantity
which takes the dimensional form
We note that Q̃(x, λ) is a completely monotone function for λ > 0, see definitions (A.1) and (A.4), and
namely, Q encodes a behaviour similar to the classical distribution ϱ(x, t) although it possesses a different form, and we call it the “distribution-like”.
Using the concept of this distribution-like, we can summarise the main result of this subsection as follows: the classical Fourier-Fick equation (1.1) subject to a Dirac delta initial distribution and unit doublet for the initial flux can be rewritten in the sense of the distribution-like as
subject to the initial conditions Q(x, 0+) = − U1(x) and diffusive flux J(x, 0+) = −u1(x), and boundary condition lim|x|→∞Q(x, t) = 0, where U1(x) is the modified unit doublet function, see equations (A.27) and (A.29).
4.2 Time-fractional diffusion
To describe the anomalous diffusive behaviour
Among the single-parameter forms of the time-fractional diffusion equations we pay attention to the two forms (1.3) and (1.4) which result in the same behaviour of the MSD (1.2), 0 < β ≤ 1, however, they produce two different partial integrodifferential equations for the diffusive flux,
which requires the initial condition
that works well with the conventional initial condition J(x, 0+) = − u1(x).
Remark 4.1
It is worth remarking that the RL form (1.4) can be recovered from the form
where
Let us now turn our interest to the properties offered by the forms (1.3) and (4.10), where we disregard equation (1.4) because of its equivalence with (59). They possess the transformed distribution
which remarkably can be retrieved from (3.2) if we consider the long time behaviour of the characteristic function (3.1), η(s) = sβ(sα + 1) ∼ sβ, with equation (2.13), and the transformed fluxes
By taking the inverse Fourier transform of (4.11) and (4.12) we have
It was established that the distribution ϱ(x, t) is a PDF [88] with MSD
with its transform
One can easily check that Q̃(x, λ) ∈ 𝓒𝓜𝓕 for λ > 0, because it is a product of two completely monotone functions as equation (4.16) explains. Also,
which likens its counterpart in the normal diffusion case, equation (4.6).
Utilising the definition of distribution-like (4.15), we can rewrite expressions (1.3) and (4.10) in the form
subject to the initial conditions Q(x, 0+) = − U1(x), the diffusive flux
4.3 Time-fractional finite-speed diffusion
The two forms of the time-fractional Cattaneo equation describing the finite-speed diffusion were introduced in equations (2.3) and (2.4) in the sense of Caputo and Riemann-Liouville fractional derivatives respectively. It was shown that the distribution ϱ(x, t) resulting from (2.3) exactly coincides with the corresponding one based on (2.4), see [2]. Nevertheless, this coincidence is lost when the diffusive flux is considered. In point of fact, in order to mathematically utilise Cattaneo-type equations in studying the itinerant (non-scaling) anomalous behaviour, the initial conditions should be imposed on the rate of the diffusive flux, not the diffusing flux itself. Loosely speaking, equation (2.1) should be subject to ∂t J(x, 0+) = − u1(x), equation (2.3) subject to
Remark 4.2
The following Cattaneo-type equation
with ϱ(x, 0+) = δ(x),
Employing the transform technique to equations (2.3) and (4.20) we obtain
in the Laplace-Fourier domain, and
in the Laplace domain. Accordingly, we take the distribution-like of the time-fractional Cattaneo equation in the form
whose Laplace transform is
We now claim that Q(x, t) given by (4.23) is a probability density function if α + β ≤ 1, where α, β ∈ (0, 1]. To justify this claim, we first prove the normalisation of Q(x, t) by integrating (4.24) over ℝ,
Obviously, f1(λ) is completely monotone for β ∈ (0, 1] whilst f2(λ) is completely monotone for α + β ∈ (0, 1], refer to definitions (A.1) and (A.3). On the other hand, we have that λβ and 1 + λα are complete Bernstein functions α, β ∈ (0, 1] as definition (A.4) reports, note that 1 ∈ 𝓢𝓕 ∩ 𝓒𝓑𝓕. Therefore,
where G̃(u, s), given by
is the Laplace transform of the function G(u, t) providing the subordination transformation from the time scale t to the time scale u. It remains to prove that G(u, t) is a PDF with respect to u > 0 for any t > 0. Firstly, we note that G(u, t) is normalised with respect to u. Indeed, since
one concludes that f1(λ) and g1(u, λ) are completely monotone functions for λ > 0 and β ∈ (0, 1], whilst f2(λ) and g2(u, λ) require that λ > 0, α, β ∈ (0, 1] and α + β ≤ 1 to be complete monotone functions, where subsection A.5 of Appendix A has been invoked repeatedly. Therefore, G̃(u, λ), as a product of complete monotone functions, is in turn a completely monotone, which proves the non-negativity of G(u, t). In view of equation (4.4) and the Laplace inversion of (4.26), we can state that the time-fractional Cattaneo equation with two fractional derivatives of different orders α, β ∈ (0, 1] represents a mathematical model for a stochastic process subordinated to Gaussian diffusion provided that the fractional derivative orders satisfy α + β ≤ 1. It is worth mentioning that the same criterion can be implemented for the distribution ϱ(x, t) using the same transformation G(u, t).
The mathematical definition of the distribution-like (4.23) enables us to replace the time-fractional Cattaneo equation in the RL-sense (2.4) and (4.20) by
subject to the initial conditions Q(x, 0+) = − U1(x) and diffusive flux
subject to the initial conditions Q(x, 0+) = − U1(x) and diffusive flux
Lastly, we complete this section by providing analytical solutions for the diffusive fluxes JRL(x, t) and JC(x, t), and the distribution-like Q(x, t) for TFCE. The diffusive fluxes in RL and C sense are given by
where φRL(x, t) and φC(x, t) are given by
where χ0,1 ∈ {0, 1}, and γ ≥ β such that γ = β if χ0,1 = 0 and γ > β if χ0,1 = 1. By following the same procedures of solving Problem B.1, one derives
Then, substituting (4.33) into (4.31) and using (A.13), we arrive at
where the relation Γ(z/2)Γ(1 − z)/Γ (−z) = −2Γ(1 + z/2) has been tacitly utilised. We have two special cases included in equation (4.34): First, when χ0,1 = 0, equation (4.34) presents the flux of time-fractional diffusion equation (1.4) and (4.10),
and when β = 1, we get the physical flux of the normal diffusion (1.1),
which coincides with equation (4.3). Here, we have used the Mellin transform with equation (A.15). Secondly, when χ0,1 = 1, then equation (4.34) yields the flux of time-fractional Cattaneo equation (2.4) and (4.20),
Analogously, we can deduce an exact generic formula for JC(x, t),
which reduces to the flux of TFDE with time-fractality in the continuity equation (1.3) when χ0,1 = 0,
Note that
The diffusive fluxes in the RL sense (without fractality in the continuity equation) (4.36), and in the C sense (with temporal fractality in the continuity equation) (4.39) are drawn in figures 5a and 5b, respectively.
(a) Diffusive flux
Finally, the distribution-like can be directly derived from the unified nondimensional relation
together with the universal distribution (B.1), we get, upon using (A.12) and (A.13), the Mellin transform and the known relation of the Gamma function, Γ(z/2)Γ(1 − z)/Γ(−z) = −2Γ(1 + z/2),
Again, the last generic formula for the distribution-like contains two special cases of interest: the first case is χ0,1 = 0, that gives the distribution-like for TFDE (4.18) and (4.19)
which reduces to (4.4) in the limit β = 1. Meanwhile, the second case, χ0,1 = 1, gives the distribution-like for TFCE (4.29) and (4.30)
for α + β ≤ 1, see Figures 6a and 6b for the graphical representations of the closed-form (4.43). The subordination of the process governed by TFCE to the Gaussian process is verified whenever the distribution-like Q(x, t) is non-negative. In the green curve of Figure 6a the non-negativity condition has been clearly breached due to the choice 1 < α + β < 2 violating the condition α + β < 1 established above. It comes to our attention that the general form of distribution-like (4.41) can be derived by using the Laplace transform of equations (4.23), (4.34) and (4.37) and utilising the useful relations (2.20) and (2.21) in [60]. As applied in previous sections, the Mellin transform can be employed here to check the normalisation of (4.43) and obtain the δ th-order moment of QTFCE(x, t), one can obtain the most interesting result,
(a) Spatial evolution of the distribution-like Q(x, t) of the time-fractional Cattaneo-type equation with two fractional parameters at dimensionless time t = 0.5. (b) Spatial distribution of Q(x, t) of TFCE subordinating to the Gaussian process (non-negative) at different instants of time and fractional parameters α = 0.5, β = 0.4.
which suggests an alternative definition for the distribution of the diffusing substance with alternative governing equations (4.29) and (4.30) for time-fractional finite-speed diffusion, (4.18) and (4.19) for time-fractional diffusion, and (4.7) for the Gaussian diffusion.
Remark 4.3
The relation between the fluxes, (4.36) and (4.39) and the distribution-like (4.43) can be rewritten in the integral forms
where Eα,β (z) is the Mittag-Leffler function defined in (A.22). In addition, the fluxes, (4.35) and (4.38) can be given in terms of the distribution-like (4.42) via
5 Concluding remarks
In this work we addressed four Cattaneo-type models with time fractality based on Riemann-Liouville and Caputo single-order and distributed-order fractional derivatives. Many processes which transit from ballistic or superdiffusive behaviour in the short-time limit to subdiffusive behaviour, or even without transport 〈x2(t)〉 ∼ 1, in the long-time limit are found to be captured by these Cattaneo-like equations. We summarise the MSD of the proposed models in Table 1.
Limiting behaviour of the MSD for the proposed Cattaneo-like models, along with the respective equation numbers.
| Model [resp. equations] | 〈x2(t)〉 | 〈x2(t)〉 |
|---|---|---|
| t → 0+ | t → ∞ | |
| 2-parameter TFCE/GCE-I [(2.3), (2.4), (4.20)] | ∼ tα+β | ∼ tβ |
| DTFCE-I [(2.8), (2.11), (2.12)] | ∼ p1tα+1 | ∼ p2tβ |
| DTFCE-II [(2.9), (2.11), (2.12)] | ∼ tβ+1/p2 | ∼ tα/p1 |
| DTFCE-III [(2.10), (2.11), (2.12)] | ∼ tβ+1/p2 | ∼ t |
Conditional closed-form solutions for the proposed models were derived in terms of the Fox H-function. Alternative formulations for the Green’s function of normal, time-fractional, time-fractional finite-speed diffusion were provided. For the generalised Cattaneo equation of type I with two fractional parameters, this modified Green’s function can be viewed as a gauge for the subordination to the Gaussian process, where the non-negativity of the modified Green’s function is broken in the transition from Gaussian to non-Gaussian. It is worth mentioning that we have disregarded the space-fractality which submits a generic crossover from superballistic/subdiffusion to subdiffusion/superballistic. Such space-fractional kinetic models can be found in the double-order space-fractional diffusion equation in the natural/modified forms [82, 15], the focus of a separate future work.
Appendix A. Mathematical preliminaries
In this appendix we give a short brief for the mathematical preliminaries needed throughout the paper.
A.1 Fractional derivatives and transform techniques
We start with the left-sided Caputo (C) fractional derivative of order α ∈ (0, 1], defined for any well-behaved function f(t) by [66, 35]
∀ a ∈ ℝ, where
The left-sided Riemann-Liouville (RL) fractional derivative of order α ∈ (0, 1] is defined as
The regularised Prabhakar fractional derivative is defined by [67, 22]
where n − 1 < β < n, and
The Laplace transform
where α ∈ (0, 1]. The Fourier transform of any generic function g(x) is defined by
A.2 Fox H-function
The Fox H-function is defined in terms of the Mellin-Barnes integral [60]
where m, n, p, and q are integers satisfying 0 ≤ n ≤ p, 1 ≤ m ≤ q, ai, bj ∈ ℂ, Ai, Bj ∈ ℝ+, i = 1, ⋯, p, j = 1, ⋯, q, and the function Θ(s) is given by
where Γ(⋅) is the Gamma function. The contour Ω in the right side of equation (1) separates the poles of Γ(bj + Bjs), j = 1, ⋯, m from the poles of Γ(1 – ai – Ais), i = 1, ⋯, n.
The Mellin transform, defined for any generic function as 𝓜{f(x)}(z) =
where Θ(z) is given by (A.8).
The inverse Fourier transform of the H-function
where k is the Fourier variable, a, γ, δ ∈ ℝ+, β, n ∈ ℂ, and λ ∈ ℝ+ ∪ {0}.
Remark A.1
The proof of relation (A.10) has been omitted for brevity. For more details about the method, the reader may consult [20, 56, 36].
If the poles of
The following properties are used throughout the paper,
where a, δ ∈ ℝ+, and r ∈ ℕ ∪ {0},
and
A.3 Generalised Mittag-Leffler functions
The Prabhakar generalisation of Mittag-Leffler function (PML) is defined in the series form [24]
where (γ)n is the ascending Pochhammer symbol defined by (γ)0 = 1, (γ)n = γ(γ + 1) ⋯ (γ + n – 1) =
where the short-time behaviour is deduced from the series representation (A.16) and the long-time behaviour can be obtained from the series
The most important properties of PML are its Laplace transform
where s is the Laplace variable, and its relation to the Fox H-function,
The following recurrence relation can be validated by definition (A.16)
When γ = 1, the PML reduces to the generalised Mittag-Leffler function with two parameters,
and when γ = β = 1, the classical Mittag-Leffler function is recovered,
A.4 Unit doublet function
The unit doublet function (differentiator) [64] is defined as the first derivative of the Dirac delta function,
with the operational properties
It can be generalised to the nth differentiator function
where n ∈ ℕ ∪ {0}. When n is negative integers, equation (A.26) defines then the nth integrator function, whereas n = 0 recovers the conventional Dirac delta function (unit impulse function).
Let us introduce the “modified unit doublet” function as a product of the spatial variable x and the unit doublet function,
whose Fourier transform is given by
Similar to the classical definition of the unit impulse function we consider the following forms for the unit doublet and its modification,
A.5 Bernstein functions [78]
Definition A.1
The function f : (0, ∞) → [0, ∞) is a completely monotone function if f ∈ C∞ and f satisfies the Bernstein–Hausdorff–Widder condition; (–1)nf(n)(t) ≥ 0 for all n ∈ ℕ. The family of all completely monotone functions is denoted by 𝓒𝓜𝓕. The functions λα and exp(–aλ), a, λ > 0, α < 0, are typical examples of 𝓒𝓜𝓕. The product of two complete monotone functions and the linear combination of complete monotone functions are also a complete monotone function. The function φ ∈ 𝓒𝓜𝓕 if, and only if, there exists a function f ≥ 0 such that φ(λ) =
Definition A.2
The function f : (0, ∞) → [0, ∞) is a Bernstein function if f ∈ C∞ and f satisfies the condition: (–1)n–1f(n)(t) ≥ 0 for all n ∈ ℕ. The set of all Bernstein functions is denoted by 𝓑𝓕.
Definition A.3
The function φ : (0, ∞) → [0, ∞) is said to be a Stieltjes function if there exists f ∈ 𝓒𝓜𝓕 such that φ(λ) =
Definition A.4
The Bernstein function φ(λ) is said to be a complete Bernstein function if, and only if, φ(λ)/λ ∈ 𝓢𝓕. Denoting by 𝓒𝓑𝓕 to the set of all complete Bernstein functions, thus 𝓒𝓑𝓕 ⊂ 𝓑𝓕. The functions λα and λ1–α, where α ∈ [0, 1] and λ > 0, are typical examples of complete Bernstein functions. The linear combination of complete Bernstein functions are also a complete Bernstein function. The set of complete Bernstein functions is not in general closed under multiplication, however, if φ, ψ ∈ 𝓒𝓑𝓕, then [φ(λ)]α [ψ(λ)]β ∈ 𝓒𝓑𝓕 provided that α, β ∈ (0, 1) and α + β ≤ 1. If φ(λ) ∈ 𝓒𝓑𝓕, then exp(–aφ(λ)) ∈ 𝓒𝓜𝓕 for a, λ > 0.
Appendix B. Derivation of the solutions
Here, we consider four different problems of inverting the Laplace-Fourier transform, which are applied in the derivation of our solutions.
B.1 Problem B.1
Let us consider the inversion of the Laplace-Fourier transform of the following equation
where χ0,1 ∈ {0, 1} and γ ≥ β such that γ = β if χ0,1 = 0 and γ > β if χ0,1 = 1. Rearranging the above equation, we have
where
which can be converted to the physical domain by means of (A.10), (A.19) and (A.20), and we obtain
B.2 Problem B.2
If the well-behaved function ϱ(x, t) has the Laplace-Fourier transformed form
where χ0,1 ∈ {0, 1}, γ ≥ α such that γ = α if χ0,1 = 0 and γ > α if χ0,1 = 1, 0 < α < β ≤ 1 and p1 + p2 = 1, then
provided that p1 ≥ p2. The proof of the above claim can be accomplished by firstly rewriting ϱ̂͠(k, s) in the following form
where
Remark B.2
If χ0,1 = 0 and γ = α, then
where p1 ≥ p2, which coincides the solution of the double-order time-fractional diffusion equation of the modified form derived in [36].
B.3 Problem B.3
If the well-behaved function ϱ(x, t) has the form
in Laplace-Fourier space, where 0 < α < β ≤ 1, p1 + p2 = 1, γ1 ≥ α and γ2 ≥ β are given by
then
provided that p1 ≤ p2 for χ0,1 = 0, and p1 < p2 for χ0,1 = 1. Here, we used γ21 = γ2 – γ1, γα = γ1 – α, and γβ = γ2 – β.
In order to validate equation (B.3) we first rehash ϱ̂͠(k, s) to take the form
where
Upon investigating the properties of f2(k, λ; p1, p2), λ = Re(s) > 0, we have two special cases of interest: When χ0,1 = 0, we get the characteristic function for DTFDE-II, given by
Remark B.3
The solution of the double-order time-fractional diffusion equation of the natural form obtained by [72] can be derived from (B.3) by setting χ0,1 = 0, and thus the only non-vanishing term of the second series is the first term, m = 0, namely,
where p1 ≤ p2.
B.4 Problem B.4
If the well-behaved function ϱ(x, t) is given by
in Laplace-Fourier space, where p1 + p2 = 1, γ1 and γ2 are given through
and 0 < α < β ≤ 1, then
provided that p1 ≤ p2 for χ0,1 = 0, and p1 ∈ [0, 0.3], p2 ∈ [0.7, 1], p1 + p2 = 1, and β ∈ [0.4, 1] for χ0,1 = 1.
Acknowledgements
RM acknowledges an Alexander von Humboldt Polish Honorary Research Scholarship from the Foundation for Polish Science (Fundacja na rzecz Nauki Polskiej).
References
[1] T.M. Atanackovic, S. Pilipovic, On a constitutive equation of heat conduction with fractional derivatives of complex order. Acta Mechanica229, No 3 (2018), 1111–1121.10.1007/s00707-017-1959-4Search in Google Scholar
[2] E. Awad, On the time-fractional Cattaneo equation of distributed order. Physica A518 (2019), 210–233.10.1016/j.physa.2018.12.005Search in Google Scholar
[3] E. Awad, On the generalized thermal lagging behavior: Refined aspects. J. Thermal Stresses35, No 4 (2012), 293–325.10.1080/01495739.2012.663682Search in Google Scholar
[4] E. Barkai, Y. Garini, R. Metzler, Strange kinetics of single molecules in living cells. Phys. Today65, No 8 (2012), 29–35.10.1063/PT.3.1677Search in Google Scholar
[5] E. Bazhlekova, Subordination in a class of generalized time-fractional diffusion-wave equations. Fract. Calc. Appl. Anal. 21, No 4 (2018), 869–900; 10.1515/fca-2018-0048; https://www.degruyter.com/view/j/fca.2018.21.issue-4/issue-files/fca.2018.21.issue-4.xml.Search in Google Scholar
[6] J.-P. Bouchaud, A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, No 4-5 (1990), 127–293.10.1016/0370-1573(90)90099-NSearch in Google Scholar
[7] J.-P. Bouchaud, A. Comtet, A. Georges, and P. Le Doussal, Classical diffusion of a particle in a one-dimensional random force field. Ann. Phys. (N.Y.)201, No 2 (1990), 285–341.10.1016/0003-4916(90)90043-NSearch in Google Scholar
[8] L. Boyadjiev, Y. Luchko, The neutral-fractional telegraph equation. Math. Modelling Nat. Phenomena12, No 6 (2017), 51–67.10.1051/mmnp/2017064Search in Google Scholar
[9] M. Caputo, Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, No 4 (2001), 421–442.Search in Google Scholar
[10] C. Cattaneo, Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena3, No 3 (1948), 83–101.10.1007/978-3-642-11051-1_5Search in Google Scholar
[11] A. Chechkin, R. Gorenflo, I. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E66, No 4 (2002), Art. 046129.10.1103/PhysRevE.66.046129Search in Google Scholar PubMed
[12] A.V. Chechkin, J. Klafter, I.M. Sokolov, Fractional Fokker-Planck equation for ultraslow kinetics. Europhys. Lett. 63, No 3 (2003), 326–332.10.1209/epl/i2003-00539-0Search in Google Scholar
[13] A. Chechkin, V.Y. Gonchar, R. Gorenflo, N. Korabel, I. Sokolov, Generalized fractional diffusion equations for accelerating subdiffusion and truncated Lévy flights. Phys. Rev. E78, No 2 (2008), Art. 021111.10.1103/PhysRevE.78.021111Search in Google Scholar PubMed
[14] A.V. Chechkin, R. Gorenflo, I.M. Sokolov, V.Y. Gonchar, Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal. 6, No 3 (2003), 259–280.Search in Google Scholar
[15] A. Chechkin, I.M. Sokolov, J. Klafter, Natural and modified forms of distributed-order fractional diffusion equations. In: Fractional Dynamics: Recent Advances, Ed. by J. Klafter, S.C. Lim, R. Metzler, World Scientific, Singapore (2011), 107–127.10.1142/9789814340595_0005Search in Google Scholar
[16] A. Compte, R. Metzler, The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A30, No 22 (1997), 7277–7289.10.1088/0305-4470/30/21/006Search in Google Scholar
[17] A. Dhar, Fractional equation description of an open anomalous heat conduction set-up. J. Stat. Mech. 2019, No 1 (2019), Art. 013205.10.1088/1742-5468/aaf630Search in Google Scholar
[18] A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. (Leipzig)322, No 8 (1905), 549–560.10.1002/andp.19053220806Search in Google Scholar
[19] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Tables of Integral Transforms, Vol. 1. McGraw-Hill, New York (1954).Search in Google Scholar
[20] W.G. Glöckle, T.F. Nonnenmacher, Fox function representation of non-Debye relaxation processes. J. Stat. Phys. 71, No 3-4 (1993), 741–757.10.1007/BF01058445Search in Google Scholar
[21] A. Godec, A. V. Chechkin, E. Barkai, H. Kantz, and R. Metzler, Localization and universal fluctuations in ultraslow diffusion processes. J. Phys. A47, No 49 (2014), Art. 492002.10.1088/1751-8113/47/49/492002Search in Google Scholar
[22] R. Garra, R. Gorenflo, F. Polito, Ž. Tomovski, Hilfer–Prabhakar derivatives and some applications. Appl. Math. Comput. 242 (2014), 576–589.10.1016/j.amc.2014.05.129Search in Google Scholar
[23] R. Gorenflo, Y. Luchko, M. Stojanović, Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density. Fract. Calc. Appl. Anal. 16, No 2 (2013), 297-316; 10.2478/s13540-013-0019-6; https://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.Search in Google Scholar
[24] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014).10.1007/978-3-662-43930-2Search in Google Scholar
[25] R.A. Guyer, J.A. Krumhansl, Solution of the linearized phonon Boltzmann equation. Phys. Rev. 148, No 2 (1966), 766–778.10.1103/PhysRev.148.766Search in Google Scholar
[26] S. Havlin and G. H. Weiss, A new class of long-tailed pausing time densities. J. Stat. Phys. 58, No 5-6 (1990), 1267–1273.10.1007/BF01026577Search in Google Scholar
[27] R. Hilfer, Mathematical and physical interpretations of fractional derivatives and integrals. In: Handbook of Fractional Calculus with Applications, Vol. 1, Basic Theory, Ed. by A. Kochubei, Y. Luchko, Berlin, De Gruyter (2019).Search in Google Scholar
[28] R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives. J. Phys. Chem. B104, No 16 (2000), 3914–3917.10.1021/jp9936289Search in Google Scholar
[29] F. Höfling, T. Franosch, Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys. 76, No 4 (2013), Art. 046602.10.1088/0034-4885/76/4/046602Search in Google Scholar
[30] P. De Jagher, A hyperbolic “diffusion equation” taking a finite collision frequency into account. Physica A101, No 2-3 (1980), 629–633.10.1016/0378-4371(80)90200-9Search in Google Scholar
[31] J.-H. Jeon, H.M.-S. Monne, M. Javanainen, R. Metzler, Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. Phys. Rev. Lett. 109, No 18 (2012), Art. 188103.10.1103/PhysRevLett.109.188103Search in Google Scholar PubMed
[32] J.-H. Jeon, M. Javanainen, H. Martinez-Seara, R. Metzler, I. Vattulainen, Protein crowding in lipid bilayers gives rise to non-Gaussian anomalous lateral diffusion of phospholipids and proteins. Phys. Rev. X6, No 2 (2016), Art. 021006.10.1103/PhysRevX.6.021006Search in Google Scholar
[33] D. Jou, G. Lebon, J. Casas-Vázquez, Extended Irreversible Thermodynamics. Springer, Berlin (2014).Search in Google Scholar
[34] N.G. van Kampen, Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam (1981).Search in Google Scholar
[35] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar
[36] T. Langlands, Solution of a modified fractional diffusion equation. Physica A367 (2006), 136–144.10.1016/j.physa.2005.12.012Search in Google Scholar
[37] P. Langevin, Sur la théorie du mouvement brownien. C. R. Acad. Sci. Paris146, No 10 (1908), 530–533.Search in Google Scholar
[38] Y.B. Lev, D.M. Kennes, C. Klöckner, D.R. Reichman, C. Karrasch, Transport in quasiperiodic interacting systems: From superdiffusion to subdiffusion. Europhys. Lett. 119, No 3 (2017), Art. 37003.10.1209/0295-5075/119/37003Search in Google Scholar
[39] Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374, No 2 (2011), 538–548.10.1016/j.jmaa.2010.08.048Search in Google Scholar
[40] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons & Fractals7, No 9 (1996), 1461–1477.10.1016/0960-0779(95)00125-5Search in Google Scholar
[41] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, No 6 (1996), 23–28.10.1016/0893-9659(96)00089-4Search in Google Scholar
[42] R. Gorenflo, F. Mainardi, Fractional calculus. In: Fractals and Fractional Calculus in Continuum Mechanics, Springer, Berlin (1997), 223-276.10.1007/978-3-7091-2664-6_5Search in Google Scholar
[43] F. Mainardi, G. Pagnini, R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order. Applied Mathematics and Computation187, No 1 (2007), 295–305.10.1016/j.amc.2006.08.126Search in Google Scholar
[44] R.N. Mantegna, H.E. Stanley, Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight. Phys. Rev. Lett. 73, No 22 (1994), 2946–2949.10.1103/PhysRevLett.73.2946Search in Google Scholar
[45] J. Masoliver, Fractional telegrapher’s equation from fractional persistent random walks. Phys. Rev. E93, No 5 (2016), Art. 052107.10.1103/PhysRevE.93.052107Search in Google Scholar
[46] J. Masoliver, G.H. Weiss, Finite-velocity diffusion. Euro. J. Phys. 17, No 4 (1996), 190–196.10.1088/0143-0807/17/4/008Search in Google Scholar
[47] Y. Meroz, I.M. Sokolov, A toolbox for determining subdiffusive mechanisms. Phys. Rep. 573 (2015), 1–29.10.1016/j.physrep.2015.01.002Search in Google Scholar
[48] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, No 1 (2000), 1–77.10.1016/S0370-1573(00)00070-3Search in Google Scholar
[49] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A37, No 31 (2004), R161-R208.10.1088/0305-4470/37/31/R01Search in Google Scholar
[50] R. Metzler, J. Klafter, I.M. Sokolov, Anomalous transport in external fields: Continuous time random walks and fractional diffusion equations extended. Phys. Rev. E58, No 2 (1998), 1621–1633.10.1103/PhysRevE.58.1621Search in Google Scholar
[51] R. Metzler, Generalized Chapman-Kolmogorov equation: A unifying approach to the description of anomalous transport in external fields. Phys. Rev. E62, No 5 (2000), 6233–6245.10.1103/PhysRevE.62.6233Search in Google Scholar
[52] R. Metzler, E. Barkai, and J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82, No 18 (1999), 3563–3567.10.1103/PhysRevLett.82.3563Search in Google Scholar
[53] R. Metzler, E. Barkai, and J. Klafter, Deriving fractional Fokker-Planck equations from a generalised master equation. Europhys. Lett. 46, No 4 (1999), 431–436.10.1209/epl/i1999-00279-7Search in Google Scholar
[54] R. Metzler, J.-H. Jeon, and A. G. Cherstvy, Non-Brownian diffusion in lipid membranes: experiments and simulations. Biochimica et Biophysica Acta (BBA) - Biomembranes1858, No 10 (2016), 2451–2467.10.1016/j.bbamem.2016.01.022Search in Google Scholar
[55] R. Metzler, J.-H. Jeon, A.G. Cherstvy, E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16 (2014), 24128–24164.10.1039/C4CP03465ASearch in Google Scholar
[56] R. Metzler, T.F. Nonnenmacher, Space- and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation. Chem. Phys. 284, No 1-2 (2002), 67-90.10.1016/S0301-0104(02)00537-2Search in Google Scholar
[57] D. Molina-Garcia, T. Sandev, H. Safdari, G. Pagnini, A. Chechkin, R. Metzler, Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers. New J. Phys. 20 (2018), Art. 103027.10.1088/1367-2630/aae4b2Search in Google Scholar
[58] P.M. Morse, H. Feshbach, Methods of Theoretical Physics. McGraw-Hill, New York (1953).Search in Google Scholar
[59] F. Mainardi, A. Mura, G. Pagnini, R. Gorenflo, Time-fractional diffusion of distributed order. J. Vibration and Control14, No 9-10 (2008), 1267–1290.10.1177/1077546307087452Search in Google Scholar
[60] A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-Function: Theory and Applications. Springer, Berlin (2010).10.1007/978-1-4419-0916-9Search in Google Scholar
[61] R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi133, No 1 (1986), 425–430.10.1515/9783112495483-049Search in Google Scholar
[62] T. Nonnenmacher, D. Nonnenmacher, Towards the formulation of a nonlinear fractional extended irreversible thermodynamics. Acta Phys. Hungarica66, No 1-4 (1989), 145–154.10.1007/BF03155787Search in Google Scholar
[63] K. Nørregaard, R. Metzler, C. Ritter, K. Berg-Sørensen, L.B. Oddershede, Manipulation and motion of organelles and single molecules in living cells. Chem. Rev. 117, No 5 (2017), 4342-4375.10.1021/acs.chemrev.6b00638Search in Google Scholar
[64] A.V. Oppenheim, Signals and Systems, MIT OpenCourseWare, Cambridge MA (2011); https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011.Search in Google Scholar
[65] E. Orsingher, L. Beghin, Time-fractional telegraph equations and telegraph processes with Brownian time. Probability Theory and Related Fields128, No 1 (2004), 141–160.10.1007/s00440-003-0309-8Search in Google Scholar
[66] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, Amsterdam (1998).Search in Google Scholar
[67] T.R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Math. J. 19, No 1 (1971), 7–15.Search in Google Scholar
[68] L.F. Richardson, Atmospheric diffusion shown on a distance-neighbour graph. Proc. Roy. Soc. A110, No 756 (1926), 709–737.Search in Google Scholar
[69] H. Qi, X. Jiang, Solutions of the space-time fractional Cattaneo diffusion equation. Physica A390, No 11 (2011), 1876–1883.10.1016/j.physa.2011.02.010Search in Google Scholar
[70] T.Q. Qiu, C.L. Tien, Short-pulse laser heating on metals. Intl. J. Heat and Mass Transfer35, No 3 (1992), 719–726.10.1016/0017-9310(92)90131-BSearch in Google Scholar
[71] T.Q. Qiu, C.L. Tien, Heat transfer mechanisms during short-pulse laser heating of metals. ASME J. Heat Transfer115, No 4 (1993), 835–841.10.1115/1.2911377Search in Google Scholar
[72] T. Sandev, A.V. Chechkin, N. Korabel, H. Kantz, I.M. Sokolov, R. Metzler, Distributed-order diffusion equations and multifractality: Models and solutions. Phys. Rev. E92, No 4 (2015), Art. 042117.10.1103/PhysRevE.92.042117Search in Google Scholar PubMed
[73] T. Sandev, A. Chechkin, H. Kantz, R. Metzler, Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel. Fract. Calc. Appl. Anal. 18, No 4 (2015), 1006–1038; 10.1515/fca-2015-0059; https://www.degruyter.com/view/j/fca.2015.18.issue-4/issue-files/fca.2015.18.issue-4.xml.Search in Google Scholar
[74] T. Sandev, I.M. Sokolov, R. Metzler, A. Chechkin, Beyond monofractional kinetics. Chaos, Solitons & Fractals102 (2017), 210–217.10.1016/j.chaos.2017.05.001Search in Google Scholar
[75] T. Sandev, Z. Tomovski, J.L. Dubbeldam, A. Chechkin, Generalized diffusion-wave equation with memory kernel. J. Phys. A52, No 1 (2019), Art. 015201.10.1088/1751-8121/aaefa3Search in Google Scholar
[76] C. Sándor, A. Libál, C. Reichhardt, C. Olson Reichhardt, Dewetting and spreading transitions for active matter on random pinning substrates. J. Chem. Phys. 146, No 20 (2017), Art. 204903.10.1063/1.4983344Search in Google Scholar PubMed PubMed Central
[77] R.K. Saxena, G. Pagnini, Exact solutions of triple-order time-fractional differential equations for anomalous relaxation and diffusion I: The accelerating case. Physica A390, No 4 (2011), 602–613.10.1016/j.physa.2010.10.012Search in Google Scholar
[78] R.L. Schilling, R. Song, Z. Vondracek, Bernstein Functions: Theory and Applications. De Gruyter, Berlin (2012).10.1515/9783110269338Search in Google Scholar
[79] W. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30, No 1 (1989), 134–144.10.1063/1.528578Search in Google Scholar
[80] Y.G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium. Theory of Probability & Applications27, No 2 (1982), 256-268.10.1007/978-1-4419-6205-8_2Search in Google Scholar
[81] M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. (Leipzig)326, No 14 (1906), 756–780.10.1002/andp.19063261405Search in Google Scholar
[82] I. Sokolov, A. Chechkin, J. Klafter, Distributed-order fractional kinetics. Acta Phys. Polonica B35, No 47 (2004), 1323–1341.Search in Google Scholar
[83] I. Sokolov, Thermodynamics and fractional Fokker-Planck equations. Phys. Rev. E63, No 5 (2001), Art. 056111.10.1103/PhysRevE.63.056111Search in Google Scholar PubMed
[84] P. Tan, Y. Liang, Q. Xu, E. Mamontov, J. Li, X. Xing, L. Hong, Gradual crossover from subdiffusion to normal diffusion: a many-body effect in protein surface water. Phys. Rev. Lett. 120, No 24 (2018), Art. 248101.10.1103/PhysRevLett.120.248101Search in Google Scholar PubMed
[85] A.M. Tawfik, H. Fichtner, R. Schlickeiser, A. Elhanbaly, Analytical solutions of the space-time fractional telegraph and advection-diffusion equations. Physica A491 (2018), 810–819.10.1016/j.physa.2017.09.105Search in Google Scholar
[86] D.Y. Tzou, Macro-to Microscale Heat Transfer: The Lagging Behavior. John Wiley & Sons, New York (2014).10.1002/9781118818275Search in Google Scholar
[87] G.H. Weiss, Some applications of persistent random walks and the telegrapher’s equation. Physica A311, No 3-4 (2002), 381–410.10.1016/S0378-4371(02)00805-1Search in Google Scholar
[88] W. Wyss, The fractional diffusion equation. J. Math. Phys. 27, No 11 (1986), 2782–2785.10.1063/1.527251Search in Google Scholar
[89] M.A. Zaks, A. Nepomnyashchy, Subdiffusive and superdiffusive transport in plane steady viscous flows. Proc. Natl. Acad. Sci. USA116, No 37 (2018), 18245–18250.10.1073/pnas.1717225115Search in Google Scholar PubMed PubMed Central
[90] G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2008).Search in Google Scholar
[91] V. Želi, D. Zorica, Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law. Physica A492 (2018), 2316–2335.10.1016/j.physa.2017.11.150Search in Google Scholar
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