Riesz operator

The numerical solution of both the fractional Fokker-Planck equation in terms of the Griinwald-Letnikov scheme used to find a discretized approximation to the fractional Riesz operator exhibits reliable convergence, as corroborated by direct solution of the corresponding Langevin equation. 

As an example of practical application of the Riesz representation theorem, consider the 2-D gravity problem. The forward gravity operator is given by the formula... 

From the differentiation theorem of Laplace transform, J f /(t) = uP u) —P t = 0), we infer that the left-hand side in (x,t) space corresponds to 0P(x, t)/dt, with initial condition P(x. 0) = 8(x). Similarly in the Gaussian limit a = 2, the right-hand side is Dd2P(x, f)/0x2, so that we recover the standard diffusion equation. For general a, the right-hand side defines a fractional differential operator in the Riesz-Weyl sense (see below) and we find the fractional diffusion equation [52-56]... 

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... 

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 ... 

In this entry some relevant examples on the use of fractional calculus to earthquake ground motion modeled as a stationary normal colored noise are presented. Applications of fractional calculus for the description of mono- and multivariate earthquake accelerations and exact filter equation are obtained in an integral form involving Riesz fractional operator in zero. The latter are related to complex spectral moments by Mellin transform operator. Other relevant application in probability may be found in Cottone et al. (2010) and Di Paola and Pinnola (2012). 

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... 

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