Fractional derivative Riesz

The space-fractional derivative S /0 x a occurring in the fractional Fokker-Planck equation (22) is called the Riesz fractional derivative. We have already seen that it is implicitly defined by... 

The Riesz fractional derivative is defined explicitly, via the Weyl fractional operator... 

The symmetric Riesz fractional derivative (3.92) is the pseudo-differential operator with symbol - k ". Such a derivative describes a redistribution of particles in the whole space according to the heavy-tailed distribution of the jumps... 

In order to find the process 7(0, we need of the quantities Ah(1 - 7k) that play the role of amplification factor of the Riesz fractional derivatives of the white noise W(t). Even though Wit) is nowhere differentiable, the fractional derivative exists since the kernel of the Riesz integral smooths the singularities (see West et al. 2003) see also Chechkin and Gonchar (2001), Grigoriu (2007), and Ortigueira and Batista (2008)). 

Note that an asymmetric density of jump lengths leads to the Riesz-Feller space-fractional derivative of order a and skewness 9 with the characteristic exponent... 

In order to derive the filter equations, we need of other two relevant definitions the Riesz fractional operator and the Mellin transform. 

The RL fractional operators are related to the evolution offrt) into an assigned interval a -T t). In some cases, like in the probability domain or in power spectral density domain, the functions are defined in the whole range oo oo. In these cases it is important to extend the RL fractional operators to unbounded domain. The extensions of the RL operator in unbounded domain are the Riesz fractional integrals and derivatives. These operators are a combination of left and right RL operators. Riesz fractional integrals and derivatives are in the following denoted as P f) t) and (D f)(f), respectively, and are defined as... 

Figure 8. Time evolution of the Levy flight-PDF in the presence of the superharmonic external potential [Eq. (26)] with c — 4 (quartic Levy oscillator) and Levy index a = 1.2, obtained from the numerical solution of the fractional Fokker-Planck equation, using the Griinwald-Letnikov representation of the fractional Riesz derivative (full line). The initial condition is a 8-function at the origin. The dashed lines indicate the corresponding Boltzmann distribution. The transition from one to two maxima is clearly seen. This picture of the time evolution is typical for 2 < c <4 (see below).

These investigations have been almost entirely based on fractional diffusion and Fokker-Planck equations with a fractional Riesz derivative and have turned out to be a convenient basis for mathematical manipulations, while at the same time being easy to interpret in the context of a dynamical approach. 

In order to find a discrete time and position expression for the fractional Riesz derivative in Eq. (130), we employ the Griinwald-Letnikov scheme [110-112], whence we obtain... 

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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