Our purpose is to introduce a notion of weak solution for class abstract fractional differential equations. We point out that the time derivative occurring in equations sense Caputo derivative. prove existence results and strong solutions. To justify theory we develop, apply two examples concrete equations: time-fractional wave Petrovsky systems. Both these are great interest partial

Abstract In this paper, we concentrate on the Lie symmetry structure of a system multi-dimensional time-fractional partial differential equations (PDEs). Specifically, first give an explicit prolongation formula involving Riemann–Liouville derivative for infinitesimal generator in case, and then show that has elegant structure. Furthermore, present two simple conditions to determine generators ...

We develop a mesoscopic modeling framework for diffusion in a crowded environment, particularly with applications in the description of living cells. Through homogenization techniques we effectively coarse-grain a detailed microscopic description into a previously developed internal state diffusive framework. In turn, in a certain deterministic limit this framework connects to macroscopic fract...

Fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. They are useful to model anomalous diffusion, where a plume of particles spreads in a different manner than the classical diffusion equation predicts. An initial value problem involving a space-fractional diffusion equation is an abstract Cauchy problem, whose analyt...

The aim of this work is to describe the qualitative behavior of the solution set of a given system of fractional differential equations and limiting behavior of the dynamical system or flow defined by the system of fractional differential equations. In order to achieve this goal, it is first necessary to develop the local theory for fractional nonlinear systems. This is done by the extension of...

This paper address a new vision for the generalized Mittag-Leffler stability of the fractional differential equations. We mainly focus on a new method, consisting of decomposing a given fractional differential equation into a cascade of many sub-fractional differential equations. And we propose a procedure for analyzing the generalized Mittag-Leffler stability for the given fractional different...

In this paper, a modification of finite integration method (FIM) is combined with the radial basis function (RBF) method to solve a time-fractional convection-diffusion equation with variable coefficients. The FIM transforms partial differential equations into integral equations and this creates some constants of integration. Unlike the usual FIM, the proposed method computes constants of integ...