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.

Semistable Lévy motions have stationary independent increments with semistable distributions. They can be realized as scaling limits of simple random walks, extending the familiar Lévy motions. Generators of stable semigroups are fractional derivatives, and the semistable generators provide a new approximation to fractional derivatives. Semistable Lévy motions and semistable generators may be u...

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 the Bagley-Torvik equation as a prototype fractional differential equation with two derivatives is investigated by means of homotopy perturbation method. The results reveal that the present method is very effective and accurate.

Discrete-time fractional derivative filters (1-D and 2-D) are shown to be well approximated from a small set of integer derivatives. A fractional derivative of arbitrary order (and, in 2-D, of arbitrary orientation) can therefore be efficiently computed from a linear combination of integer derivatives of the underlying signal or image.

In this paper we present approximate analytical solution of a time-fractional Zakharov-Kuznetsov equation via the fractional iteration method. The fractional derivatives are described in the Caputo sense. The approximate results show that the fractional iteration method is a very efficient technique to handle fractional partial differential equations.

The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the ...

The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals. This issue is circumvented by deriving fractional-order models from an underlying stochas...

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...