Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractio...

We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.

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...

In this paper we establish the existence of a square integrable occupation density for two classes of stochastic processes. First we consider a Gaussian process with an absolutely continuous random drift, and secondly we handle the case of a (Skorohod) integral with respect to the fractional Brownian motion with Hurst parameter H > 1 2 . The proof of these results uses a general criterion for t...

Let B H t be a d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1). We study the regularity, in the sense of the Malliavin calculus, of the renormalized self-intersection local time ℓ = T 0 t 0 δ 0 (B H t − B H s)dsdt − E T 0 t 0 δ 0 (B H t − B H s)dsdt , where δ 0 is the Dirac delta function.

Recently, fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering. Especially, the study on stability of fractional differential equations appears to be very important. In this paper, a brief overview on the recent stability results of fractional differential equations and the analytical methods used are provided. These equatio...

The sufficient conditions for existence and partial approximate controllability of fractional stochastic evolution equations with nonlocal initial have been discussed. discussion is based on the variational method, calculus, Schauder's fixed point theorem, analysis. Contrary to results available in literature, non-local function does not need be compact or satisfy Lipschitz's condition. Moreove...

This work identifies the influence of chaos theory on fractional calculus by providing a theorem for existence and stability solution in fractional-order gyrostat model with help fixed-point theorem. We modified an integer order consisting three rotors into attaching rotatory fuel-filled tank provided iterative scheme our proposed as working rule obtained analytical results. Moreover, this is i...

An important class of fractional differential and integral operators is given by the theory calculus with respect to functions, sometimes called ?-fractional calculus. The operational approach has proved useful for understanding extending this topic study. Motivated equations, we present an Laplace transforms functions their relationship functions. This makes generalised much easier analyse app...