Riemann-Liouville fractional operator

Here, D< )s is the fractional diffusivity defined as (4-2d,) (2d,-4) )(3-dP) j js a constant related to the fractal dimension and R0 is the side length of a square electrode), and dv dtv means the Riemann-Liouville mathematical operator of fractional derivative ... 

Expression (24) reduces to the standard 1 = 0 for q —> n, due to the divergence of the gamma function T(z) for nonpositive integers. The fractional Riemann-Liouville integral operator oDJq fulfills the generalized integration theorem of the Laplace transformation ... 

In the limit a —> 1, the Riemann-Liouville fractional integral oD a reduces to an ordinary integration so that lim i oL>,1- = JjJodr becomes the identity operator that is, Eqs. (15) and (19) simplify to the standard diffusion and Fokker-Planck equations, respectively. 

Recently the new concept of fractional time evolution was introduced [45]. In addition to the usual equilibrium state (96), this concept leads to the possibility of the existence of an equilibrium state with power-law long-time behavior. Here the infinitesimal generator of time evolution is proportional to the Riemann-Liouville fractional differential operator oDvt. By definition of the Riemann-Liouville fractional differentiation operator [231,232] we have... 

Since 0 < a < 1 the exponent in Eq. (137) is 1 — a > 0. The mathematical implication is that M(p) (137) is a multivalued function of the complex variable p. In order to represent this function in the time domain, one should select the schlicht domain using supplementary physical reasons [135]. These computational constraints can be avoided by using the Riemann-Liouville fractional differential operator oDlt a [see definitions (97) and (98)]. Thus, one can easily see that the Laplace transform of... 

Another approach to the problem of anomalous relaxations uses fractal concepts [187-189,200-203], Here the problem is analyzed using the mathematical language of fractional derivatives [194,200-203] based on the previously mentioned Riemann-Liouville fractional differentiation operator... 

Riemann-Liouville fractional derivative expectation operator set of edges of Q... 

The characteristic changes brought about by fractional dynamics in comparison to the Brownian case include the temporal nonlocality of the approach manifest in the convolution character of the fractional Riemann-Liouville operator. Initial conditions relax slowly, and thus they influence the evolution of the system even for long times [62, 116] furthermore, the Mittag-Leffler behavior replaces the exponential relaxation patterns of Brownian systems. Still, the associated fractional equations are linear and thus extensive, and the limit solution equilibrates toward the classical Gibbs-B oltzmann and Maxwell distributions, and thus the processes are close to equilibrium, in contrast to the Levy flight or generalised thermostatistics models under discussion. 

Fractional calculus is referred to derivatives and integrals of order G K or more generally to complex order y = p + it], p K, t] G M. There are many different definitions of fractional operators such as Riemann-Liouville, Riesz, Marchaud, Caputo, etc. (see, e.g., Podlubny 1999 Samko et al. 1993). The various definitions differ with each another by intervals of integration or are simply adaptations of the Riemann-LiouvUle (Mies. In any case all the fractional operators share some common points ... 

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