Diffusion equation fractional

C. Boundary Value Problems for the Fractional Diffusion Equation HI. The Fractional Fokker-Planck Equation... 

According to Eq. (21), the FFPE (19) involves a slowly decaying, selfsimilar memory so that the present state W (x, t) of the system depends strongly on its history W(x, tr), t1 < t, in contrast to its Brownian counterpart which is local in time. In the force-free case, F(x) = 0, the FFPE (19) reduces to the fractional diffusion equation (15). 

Exemplifying the convenience of the fractional approach, we address the imposition of boundary value problems on the fractional diffusion equation which was demonstrated in Ref. 62. In this force-fiee case for which the kernel, Eq. (27), takes on the homogeneous form K(x,x u) = uw(u) (2( c — x ) — <5(x))/(l - w(u)), one can apply the method of images in order to construct the solution [12]. 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

The solution to the fractional diffusion equation is clearly dependent on fluctuations that have occurred in the remote past note the time lag k in the index on the fluctuations and the fact that it can be arbitrarily large. The extent of the influence of these distant fluctuations on the system response is determined by the relative size of the coefficients in the series. Using Stirling s approximation on the gamma functions determines the size of the coefficients in Eq. (25) as the fluctuations recede into the past, that is, as k — oo we obtain... 

Finally, the fractional calculus was used to construct fractional diffusion equations. One such equation, in particular, models the evolution of the Levy nestable probability density describing Levy diffusion, another mechanism for generating anomalous diffusion. It was shown that this probability density satisfies the scaling relation [Eq. (35)] with the Levy index a such that 8 = 1 /a. The dynamics of a Levy diffusion process, using a Langevin equation, were also considered. The probability density for a simple dissipative process being driven by Levy noise is also Levy but with a change in parameters. This is a possible alternate model of the fluctuations in the interbeat intervals for the human heart shown to be Levy stable over a decade ago [25],... 

This form is useful in the generalization of Eq. (485) to fractional diffusion. The investigation of the diffusion equation (485) began when Louis Bachellier (Jules Poincare s student) wrote his thesis in 1900. It was called The Theory of Speculations and was devoted to the evolution of the stock market. Many of the most famous scientists have contributed to our knowledge of diffusion processes, amongst them Einstein, Langevin, Smoluchowski, Fokker, Planck, Levy, and others. 

Let us now consider the equation for diffusive particle flow with a fractional derivative (570). By Laplace transformation of equation (570), we obtain... 

B. Fractional Diffusion Equation for the Cole-Cole Behavior... 

The second model of Debye or the Debye-Frohlich model may also be generalized to fractional diffusion [8,25] (including inertial effects [26]). Moreover, it has been shown [25] that the Cole-Cole equation arises naturally from the solution of a fractional Fokker-Planck equation in the configuration space of orientations derived from the diffusion limit of a CTRW. The broadening of the dielectric loss curve characteristic of the Cole-Cole spectrum may then be easily explained on a microscopic level by means of the relation [8,24]... 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

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