In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with delay Riesz space fractional derivatives. The temporal orders in considered model are taken as 0 <m...

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 ...

Based on recent studies by Guy Jumarie [1] which defines probability density of fractional order and fractional moments by using fractional calculus (fractional derivatives and fractional integration), this study expands the concept of probability density of fractional order by defining the fractional probability measure, which leads to a fractional probability theory parallel to the classical ...

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

In this paper, a new alternating direction implicit Galerkin–Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank–Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived ...

recently, mohiuddine and alghamdi introduced the notion of lacunary statistical convergence in a locally solid riesz space and established some results related to this concept. in this paper, some inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences are established and extensions of a decomposition theorem, a tauberian theorem to the l...

The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generaliz...

Ideas from probability can be very useful to understand and motivate fractional calculus models for anomalous diffusion. Fractional derivatives in space are related to long particle jumps. Fractional time derivatives code particle sticking and trapping. This probabilistic point of view also leads to some interesting extensions, including vector fractional derivatives, and tempered fractional de...

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.