A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Fell...

In this paper, superconvergence points are located for the approximation of the Riesz derivative of order α using classical Lobatto-type polynomials when α ∈ (0, 1) and generalized Jacobi functions (GJF) for arbitrary α > 0, respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre-Lobatto expansion. It is ob...

Fractional centred differences and derivatives definitions are proposed generalising to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalisations of the Cauchy integral derivative. To compute this integral, a special two straight line path was use...

Abstract This article deals with the existence, uniqueness and Ulam-Hyers--Rassias stability results for a class of coupled systems implicit fractional differential equations Riesz-Caputo derivative boundary conditions. We will employ Banach’s contraction principle as well Schauder’s fixed point theorem to demonstrate our existence results. provide an example illustrate obtained

In this paper, the Mickens non-standard discretization method which effectively preserves the dynamical behavior of linear differential equations is adapted to solve numerically the fractional order hyperbolic partial differential equations. The fractional derivative is described in the Riesz sense. Special attention is given to study the stability analysis and the convergence of the proposed m...

The aim of this paper is to deduce a discrete version of the fractional Laplacian in matrix form defined on the 1D periodic (cyclically closed) linear chain of finite length. We obtain explicit expressions for this fractional Laplacian matrix and deduce also its periodic continuum limit kernel. The continuum limit kernel gives an exact expression for the fractional Laplacian (Riesz fractional d...

in this article, we survey the asymptotic stability analysis of fractional differential systems with the prabhakar fractional derivatives. we present the stability regions for these types of fractional differential systems. a brief comparison with the stability aspects of fractional differential systems in the sense of riemann-liouville fractional derivatives is also given.

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose Galerkin finite element scheme both the temporal and spatial discretizations. For proposed scheme, derive sharp stability estimates as well optimal priori error estimates. Extensive experiments are conducted to verify promising features of newl...

Abstract In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class fractional differential equations by using generalized Riesz operator. One can view results work as refinement existence theory with Riemann–Liouville, Caputo, classical derivative. Some special cases be derived to obtain corresponding equations. We provide an illustrated example unique...