It is known that at least ten equivalent definitions of the fractional Laplacian exist in an unbounded domain. Here we derive a further definition based on Mellin transform and it can be used when applied to radial functions. The main finding tested case space-fractional diffusion equation. one-dimensional also considered, such Riesz (namely symmetric Riesz–Feller) derivative established. This ...

In this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz-Caputo derivative. First we prove a generalized Euler-Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional derivative. Next we consider integral dynamic constraints on the problem, for sever...

The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided and two-sided derivatives. An analysis of Riesz potential is done shows that it cannot be considered as a derivative.

In this paper we present in one-dimensional space a numerical solution of a partial differential equation of fractional order. This equation describes a process of anomalous diffusion. The process arises from the interactions within the complex and non-homogeneous background. We presented a numerical method which bases on the finite differences method. We considered pure initial and boundaryini...

The purpose of this paper is to prove some classical estimates for fractional derivatives of functions defined on a Coifman-Weiss space of homogeneous type. In particular the Product Rule and Chain Rule estimates in [KP] and [CW]. The fractional calculus of M. Riesz was extended to these spaces in [GSV]. Our main tools are fractional difference quotients and the square fractional derivative of ...

Continuous limits of discrete systems with long-range interactions are considered. The map of discrete models into continuous medium models is defined. A wide class of long-range interactions that give the fractional equations in the continuous limit is discussed. The one-dimensional systems of coupled oscillators for this type of longrange interactions are considered. The discrete equations of...

in this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the lipschitz type condition. moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

In this paper, we investigate the existence of nontrivial solutions in Bessel Potential space for nonlinearfractional Schrödinger-Poisson system involving distributional Riesz fractional derivative. By using themountain pass theorem combination with perturbation method, prove solutions.

The 1D discrete fractional Laplacian operator on a cyclically closed (periodic) linear chain with finite number N of identical particles is introduced. We suggest a ”fractional elastic harmonic potential”, and obtain the N -periodic fractional Laplacian operator in the form of a power law matrix function for the finite chain (N arbitrary not necessarily large) in explicit form. In the limiting ...